# Hyperspherical Trigonometry and Corresponding Elliptic Functions

Paul Jennings<EMAIL_ADDRESS>Current address withheld  Frank
Nijhoff<EMAIL_ADDRESS>School of Mathematics, Department of Applied
Mathematics, University of Leeds, United Kingdom, LS2 9JT

###### Abstract

We develop the basic formulae of hyperspherical trigonometry in
multidimensional Euclidean space, using multidimensional vector products, and
their conversion to identities for elliptic functions. We show that the basic
addition formulae for functions on the 3-sphere embedded in 4-dimensional
space lead to addition formulae for elliptic functions, associated with
algebraic curves, which have two distinct moduli. We give an application of
these formulae to the cases of a multidimensional Euler top, using them to
provide a link to the Double Elliptic model.

##  I Introduction

Spherical trigonometry, a branch of geometry dealing with the goniometry of
the angles of triangles confined to the surface of a 2- sphere, is an area of
mathematics that has existed since the ancient Greeks, with its foundations
laid by Menelaus and Hipparchus. It is of vital importance for many
calculations in astronomy, navigation and cartography. Further advances made
in the Islamic world, in order to help calculate its Holy Days based on the
phases of the moon, resulted in giving us the basis of spherical trigonometry
in its modern form. In the western world in the 17th century spherical
trigonometry was first considered as a separate mathematical discipline,
independent of astronomy. One of the protagonists of this era, John Napier, in
his work of 1614, treated spherical trigonometry alongside his work
introducing logarithms. Other protagonists include Delambre, Euler, Cagnoli
and l’Huillier.

For obvious reasons the trigonometry of hyperspheres in higher dimensions was
not studied so intently. McMahon produced a number of formulae as
generalisations of some of those from spherical trigonometry.2  More recently,
Sato considered the relationship between the dihedral angles of spherical
simplices and those of their polars.sato  There also exists a significant
amount of work looking into the ‘Law of Sines’, generalisations of the sine
rule from the spherical case. eriksson  Various formulae have also arisen in
the the work of Derevnin, Mednykh and Pashkevich in their work on spherical
volumes.Derevnin ; Derevnin2  Recently, it has been shown by Petrera and Suris
the the cosine rule for spherical triangles and tetrahedra define integrable
systems.Suris

It is well known that elliptic functions are related to spherical trigonometry
through their addition formulae.Lagrange ; Legendre  These functions are
commonly defined through the inversion of integrals, and they consequently
obey differential equations associated with certain algebraic curves. A great
deal of research took place in this area in the 19th century, comprising works
by many of the great mathematicians , including Euler, Jacobi, Legendre and
Frobenius. As for higher dimensional hyperspherical trigonometry no such
connection is known. It may be expected that this link would be through higher
genus elliptic curves, for example Abelian functionsbaker2 . However, this is
not in fact the case. We show that instead the link is through the
‘Generalised Jacobi Elliptic Functions’ explicitly defined by Pawellekpawellek
, and building upon Jacobi’s work in this area,Jacobi  as an elliptic
covering, dependent on two distinct moduli.

In this paper we develop a complete set of formulae for hyperspherical
trigonometry, and explore their link with elliptic functions. We establish a
novel connection between the generalised Jacobi elliptic functions and the
formulae of hyperspherical trigonometry. We show that through this connection
the basic addition formulae of hyperspherical trigonometry lead to addition
formulae for these generalised Jacobi elliptic functions. We apply this
connection to an example, the four-dimensional Euler top. We then show that
this system is equivalent to the one-particle Double Elliptic model.

Our interest in this subject area arises in relation to the tetrahedron
equation, a higher dimensional generalisation of the Yang-Baxter
equation.Zamolodchikov  Zamolodchikov presented an intuitive solution to the
latter dependent on spherical trigonometry, cf. also Baxter . It is therefore
natural for higher generalisations to be solved in terms of higher dimensional
hyperspherical trigonometry.

The outline of this paper is as follows. In section II we review
multidimensional vector products as a higher dimensional analogue of the
standard cross-product of vectors, which we need to obtain the formulae of
hyperspherical trigonometry. In section III we provide a summary of the well-
known formulae of spherical trigonometry. We deduce analogous formulae for the
4 dimensional hyperspherical case, and using the same principles, do the same
for the general $n$-dimensional case. In section IV we review the link between
spherical trigonometry and the Jacobi elliptic functions, and generalise this
to the link between four-dimensional hyperspherical trigonometry and elliptic
functions. Section V is a brief introduction to the generalised Jacobi
elliptic functionspawellek , complete with a link between the formulae of
hyperspherical trigonometry and these functions. We conclude with two
examples, the first being the four-dimensional Euler top, and the second the
double elliptic (DELL) model, providing a connection between the two.

Note: This paper summarises some collaborative results which appeared in the
PhD thesis of one of the authorsJennings-thesis .

##  II Multidimensional Vector Products

###  II.1 Definition and Relations

The formulae involved in spherical trigonometry are dependent on the cross
product between vectors. The vector product $\mathbf{a}\times\mathbf{b}$ in 3D
Euclidean space is a binary operation defined by

$(\mathbf{a}\times\mathbf{b})_{i}=\mathrm{det}(\mathbf{a},\mathbf{b},\mathbf{e}_{i}),\hskip
56.9055pt(i=1,2,3),$ (1)

with $\mathbf{e}_{1}$, $\mathbf{e}_{2}$, $\mathbf{e}_{3}$ the standard unit
vectors in the orthogonal basis. This product obeys the rules:

  * •

Anti-commutative: $\mathbf{a}\times\mathbf{b}=-\mathbf{b}\times\mathbf{a}.$

  * •

Vector Triple Product:
$(\mathbf{a}\times\mathbf{b})\times\mathbf{c}=(\mathbf{a}\cdot\mathbf{c})\mathbf{b}-(\mathbf{b}\cdot\mathbf{c})\mathbf{a}=-\left|\begin{array}[]{cc}\mathbf{a}&\mathbf{b}\\\
\mathbf{a}\cdot\mathbf{c}&\mathbf{b}\cdot\mathbf{c}\\\ \end{array}\right|,$

from which it follows
$(\mathbf{a}\times\mathbf{b})\times\mathbf{c}-(\mathbf{a}\times\mathbf{c})\times\mathbf{b}=\mathbf{a}\times(\mathbf{b}\times\mathbf{c}).$

  * •

Area of a Parallelogram: The modulus of the vector product,
$|\mathbf{a}\times\mathbf{b}|,$ is equivalent to the area of the parallelogram
defined by these vectors, $|\mathbf{a}\times\mathbf{b}|=\sin\theta,$ with
$\theta$ the obtuse angle between them $(0\leq\theta\leq\pi).$

For higher dimensional spherical trigonometry an $n$-ary operation between $n$
vectors is required. A natural vector product in four dimensions will
therefore be a ternary vector product,
$\mathbf{a}\times\mathbf{b}\times\mathbf{c}$, of three vectors in 4D Euclidean
space $E_{4}$, defined in a similar manner, by

$(\mathbf{a}\times\mathbf{b}\times\mathbf{c})\cdot\mathbf{d}=\mbox{det}(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}),\hskip
14.22636pt\mathrm{for\,\,all\,\,vectors}\,\mathbf{d}\in E_{4}.$ (2)

More generally, this could be extended to an $n$-dimensional vector product as
an $n$-ary operation of $n-1$ vectors in $E_{n}$, defined by the
expressionGrassman ; Acosta

$(\mathbf{a}_{1}\times\mathbf{a}_{2}\times\dots\times\mathbf{a}_{n-1})\cdot\mathbf{a}_{n}=\mbox{det}(\mathbf{a}_{1},\mathbf{a}_{2},\dots,\mathbf{a}_{n-1},\mathbf{a}_{n}).$
(3)

It clearly follows from this definition that the multiple vector products are
antisymmetric with respect to the interchangement of their constituent
vectors. Furthermore, these multiple vector products are perpendicular to any
one of their constituent vectors. In fact, the multidimensional vector product
follows directly from the wedge product of the exterior algebra. Janich

The following nested product identity involving five vectors holds for the
triple vector product in $E_{4}=\mathbb{R}^{4}$,

$(\mathbf{a}\times\mathbf{b}\times\mathbf{c})\times\mathbf{d}\times\mathbf{e}=-\left|\begin{array}[]{ccc}\mathbf{a}&\mathbf{b}&\mathbf{c}\\\
(\mathbf{a}\cdot\mathbf{d})&(\mathbf{b}\cdot\mathbf{d})&(\mathbf{c}\cdot\mathbf{d})\\\
(\mathbf{a}\cdot\mathbf{e})&(\mathbf{b}\cdot\mathbf{e})&(\mathbf{c}\cdot\mathbf{e})\\\
\end{array}\right|.$ (4)

This follows from the more general higher-dimensional analogue.

######  Proposition 1.

(Nested Vector Product Identity). For 2n-1 vectors,
$\mathbf{a}_{i}\in\mathbb{R}^{n+1}$, $i=1,\dots,2n-1$, we have the identity

$(\mathbf{a}_{1}\times\mathbf{a}_{2}\times\dots\times\mathbf{a}_{n})\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-1}=-\left|\begin{array}[]{cccc}\mathbf{a}_{1}&\mathbf{a}_{2}&\dots&\mathbf{a}_{n}\\\
(\mathbf{a}_{1}\cdot\mathbf{a}_{n+1})&(\mathbf{a}_{2}\cdot\mathbf{a}_{n+1})&\dots&(\mathbf{a}_{n}\cdot\mathbf{a}_{n+1})\\\
\vdots&\vdots&\ddots&\vdots\\\
(\mathbf{a}_{1}\cdot\mathbf{a}_{2n-1})&(\mathbf{a}_{2}\cdot\mathbf{a}_{2n-1})&\dots&(\mathbf{a}_{n}\cdot\mathbf{a}_{2n-1})\\\
\end{array}\right|.$ (5)

###### Proof.

Consider the determinantal expression

$\mbox{det}(\mathbf{a}_{1},\dots\mathbf{a}_{n+1})=\sum_{i_{1}}\dots\sum_{i_{n+1}}\varepsilon_{i_{1}\dots
i_{n+1}}(\mathbf{a}_{1})_{i_{1}}\dots(\mathbf{a}_{n+1})_{i_{n+1}},$ (6)

where $\varepsilon_{i_{1}\dots i_{n+1}}$ is the $(n+1)$-dimensional Levi-
Civita symbol. From this it follows that

$\begin{split}[(\mathbf{a}_{1}&\times\mathbf{a}_{2}\times\dots\times\mathbf{a}_{n})\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-1}]_{j_{n}}\\\
&=\sum_{i_{1}}\dots\sum_{i_{n+1}}\sum_{j_{1}}\dots\sum_{j_{n}}\epsilon_{j_{1}i_{1}\dots
i_{n}}\epsilon_{j_{1}\dots
j_{n+1}}(\mathbf{a}_{1})_{i_{1}}\dots(\mathbf{a}_{n})_{i_{n}}(\mathbf{a}_{n+1})_{j_{2}}\dots(\mathbf{a}_{2n-1})_{j_{n}}.\end{split}$
(7)

Noting that the Levi-Civita symbol satisfies the following product rule

$\sum_{j_{1}}\epsilon_{j_{1}i_{1}\dots i_{n}}\epsilon_{j_{1}\dots
j_{n+1}}=\left|\begin{array}[]{ccc}\delta_{i_{1}j_{2}}&\dots&\delta_{i_{1}j_{n+1}}\\\
\vdots&&\vdots\\\ \delta_{i_{n}j_{2}}&\dots&\delta_{i_{n}j_{n+1}}\\\
\end{array}\right|,$ (8)

the result follows. ∎

Vectorial addition identities follow from the Plücker relations in projective
geometry. The Plücker relations are identities involving minors of non-square
matrices which are the Plücker coordinates of corresponding Grassmannians.

######  Proposition 2 (Plücker Relations).

For $(2n-2)$ vectors
$\mathbf{a}_{1},\mathbf{a}_{2},\dots,\mathbf{a}_{2n-2}\in\mathbb{R}^{n-1}$,

$\begin{split}(\mathbf{a}_{1},\mathbf{a}_{n+1},\dots,\mathbf{a}_{2n-2})&(\mathbf{a}_{2},\dots,\mathbf{a}_{n})-(\mathbf{a}_{2},\mathbf{a}_{n+1},\dots,\mathbf{a}_{2n-2})(\mathbf{a}_{1},\mathbf{a}_{3},\dots,\mathbf{a}_{n})\\\
&+\dots+(-1)^{n-1}(\mathbf{a}_{n},\mathbf{a}_{n+1},\dots,\mathbf{a}_{2n-2})(\mathbf{a}_{1},\dots,\mathbf{a}_{n-1})=0.\end{split}$
(9)

###### Proof.

Consider $n$ vectors, $\mathbf{a}_{1},\mathbf{a}_{2},\dots,\mathbf{a}_{n}$ in
$\mathbb{R}^{n-1}$. As there are $n$ vectors in $(n-1)$ dimensional space they
must be linearly dependent, hence,

$\left|\begin{array}[]{cccc}(\mathbf{a}_{1})&(\mathbf{a}_{2})&\cdots&(\mathbf{a}_{n})\\\
\mathbf{a}_{1}&\mathbf{a}_{2}&\cdots&\mathbf{a}_{n}\\\ \end{array}\right|=0,$
(10)

where $(\mathbf{a}_{i})$ represents the column vector of the components of
$\mathbf{a}_{i}$, $i\in\\{1,2,\dots,n\\}$. This implies

$\mathbf{a}_{1}(\mathbf{a}_{2},\dots,\mathbf{a}_{n})-\mathbf{a_{2}}(\mathbf{a}_{1},\mathbf{a}_{3},\dots,\mathbf{a}_{n})+\dots+(-1)^{n-1}\mathbf{a}_{n}(\mathbf{a}_{1},\dots,\mathbf{a}_{n-1})=0,$
(11)

which, in turn gives

$\begin{split}(\mathbf{a}_{1},\mathbf{a}_{n+1},\dots,\mathbf{a}_{2n-2})&(\mathbf{a}_{2},\dots,\mathbf{a}_{n})-(\mathbf{a}_{2},\mathbf{a}_{n+1},\dots,\mathbf{a}_{2n-2})(\mathbf{a}_{1},\mathbf{a}_{3},\dots,\mathbf{a}_{n})\\\
&+\dots+(-1)^{n-1}(\mathbf{a}_{n},\mathbf{a}_{n+1},\dots,\mathbf{a}_{2n-2})(\mathbf{a}_{1},\dots,\mathbf{a}_{n-1})=0,\end{split}$
(12)

for some arbitrary
$\mathbf{a}_{n+1},\dots,\mathbf{a}_{2n-2}\in\mathbb{R}^{n-1}$, as required. ∎

######  Corollary 1.

For vectors $\mathbf{a}_{1},\dots,\mathbf{a}_{2n-2}\in\mathbb{R}^{n-1}$,

$\begin{split}&(\mathbf{a}_{1}\times\dots\times\mathbf{a}_{n})\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-1}\\\
&+\mathbf{a}_{n}\times(\mathbf{a}_{1}\times\dots\times\mathbf{a}_{n-1}\times\mathbf{a}_{n+1})\times\mathbf{a}_{n+2}\times\dots\times\mathbf{a}_{2n-1}+\dots\\\
&+\mathbf{a}_{n}\times\dots\times\mathbf{a}_{2n-2}\times(\mathbf{a}_{1}\times\dots\times\mathbf{a}_{n-1}\times\mathbf{a}_{2n-1})\\\
&=\mathbf{a}_{1}\times\dots\times\mathbf{a}_{n-1}\times(\mathbf{a}_{n}\times\dots\times\mathbf{a}_{2n-1}).\end{split}$
(13)

###### Proof.

From the Plüker relations, (9), set $\mathbf{a}_{2n-2}=\mathbf{a}_{n}=e_{i}$
and sum over $i$. This implies

$\begin{split}&(\mathbf{a}_{1}\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-3})\cdot(\mathbf{a}_{2}\times\dots\times\mathbf{a}_{n-1})\\\
&-(\mathbf{a}_{2}\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-3})\cdot(\mathbf{a}_{1}\times\mathbf{a}_{3}\dots\times\mathbf{a}_{n-1})\\\
&+\dots+(-1)^{n}(\mathbf{a}_{n-1}\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-3})\cdot(\mathbf{a}_{1}\times\dots\times\mathbf{a}_{n-2})\\\
&=0,\end{split}$ (14)

which, using
$(\mathbf{a}_{1}\times\mathbf{a}_{2}\times\dots\mathbf{a}_{n-2})\cdot\mathbf{a}_{n-1}=-(\mathbf{a}_{2}\times\dots\times\mathbf{a}_{n-1})\cdot\mathbf{a}_{1}$,
implies

$\begin{split}&\mathbf{a}_{1}\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-4}\times(\mathbf{a_{2}}\times\dots\times\mathbf{a}_{n-1})\\\
&-\mathbf{a}_{2}\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-4}\times(\mathbf{a_{1}}\times\mathbf{a}_{3}\times\dots\times\mathbf{a}_{n-1})\\\
&+\dots+(-1)^{n}\mathbf{a}_{n-1}\times\mathbf{a}_{n+1}\times\dots\times\mathbf{a}_{2n-4}\times(\mathbf{a_{1}}\times\dots\times\mathbf{a}_{n-2})\\\
&=0.\end{split}$ (15)

Summing over $(n-2)$ copies, the result follows. ∎

######  Corollary 2.

More specifically, for
$\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{e}\in\mathbb{R}^{4}$,

$(\mathbf{a}\times\mathbf{b}\times\mathbf{c})\times\mathbf{d}\times\mathbf{e}+(\mathbf{a}\times\mathbf{b}\times\mathbf{d})\times\mathbf{e}\times\mathbf{c}+(\mathbf{a}\times\mathbf{b}\times\mathbf{e})\times\mathbf{c}\times\mathbf{d}=\mathbf{a}\times\mathbf{b}\times(\mathbf{c}\times\mathbf{d}\times\mathbf{e}).$
(16)

This identity will be particularly important in proving the various
hyperspherical identities in the four dimensional case.

##  III Hyperspherical Trigonometry

In this section we review the formulae for spherical trigonometry as a
preparation for developing similar formulae in higher dimensions. The approach
we take is by exploiting the higher dimensional vector product introduced in
section II.

###  III.1 Spherical Trigonometry

We review the derivation of the basic formulae of spherical trigonometry, cf.
also 1 . Consider a big spherical triangle on the surface of a $2$-sphere of
unit radius embedded in three dimensional Euclidean space, that is a triangle
bounded by three big circles on $S^{2}$, restricted such that each side is
less than a semicircle, and hence, each angle is less than $\pi$. Take the
centre of the sphere to be the origin in $\mathbf{E}_{3}=\mathbb{R}^{3}$ and
denote the position vectors of the three vertices of the spherical triangle by
$\mathbf{n}_{1},\mathbf{n}_{2}$ and $\mathbf{n}_{3}$, with the angles,
$\theta_{ij}$, between them, corresponding to the edges, being defined by

$\textbf{n}_{i}\cdot\textbf{n}_{j}\equiv\cos\theta_{ij},\hskip
56.9055pti,j=1,2,3.$ (17)

Introduce the vectors

$\textbf{u}_{ij}\equiv\frac{\textbf{n}_{i}\times\textbf{n}_{j}}{\left|\textbf{n}_{i}\times\textbf{n}_{j}\right|},$
(18)

and define the spherical angles, $\alpha_{j}$, between them by

$\textbf{u}_{ij}\cdot\textbf{u}_{jk}\equiv-\cos\alpha_{j}.$ (19)

$\mathbf{n}_{k}$$\mathbf{n}_{j}$$\mathbf{n}_{i}$$\theta_{jk}$$\alpha_{i}$$\theta_{ik}$$\alpha_{j}$$\theta_{ij}$$\alpha_{k}$
Figure 1: A spherical triangle

######  Theorem 1 (Spherical Excess).

The area, $A$, of a spherical triangle is given by its $\mathbf{Spherical}$
$\mathbf{Excess}$,

$A=\alpha_{i}+\alpha_{j}+\alpha_{k}-\pi.$ (20)

###### Proof.

Scläfli’s Differential Volume formula states that the area of a spherical
triangle must satisfy

$\mathrm{d}A=\sum_{i=1}^{3}\mathrm{d}\alpha_{i}.$ (21)

Integrating this gives

$A=\alpha_{i}+\alpha_{j}+\alpha_{k}+c,$ (22)

where $c=-\pi$ is the integration constant determined by considering, say, the
area of a spherical triangle with planar angles
$\alpha_{i}=\alpha_{j}=\alpha_{k}=\frac{\pi}{2}$ and comparing this to the
known area of $1/8$ of a sphere. ∎

This theorem may also be proven by considering the areas of intersecting
lunars1 .

######  Definition 1 (Polar Triangle).

The spherical triangle defined by the vectors $\mathbf{u}_{ij}$,
$\mathbf{u}_{ik}$, $\mathbf{u}_{jk}$ is called the $\mathbf{polar}$ of the
spherical triangle defined by the vectors $\mathbf{n}_{i}$, $\mathbf{n}_{j}$,
$\mathbf{n}_{k}$.

By considering various relations for the scalar and vector products between
the polar vectors $\textbf{u}_{ij}$, we can define a number of important
relations between the angles of a spherical triangle.

######  Proposition 3 (Cosine Rule).

$\cos\alpha_{j}=\frac{\cos\theta_{ik}-\cos\theta_{ij}\cos\theta_{jk}}{\sin\theta_{ij}\sin\theta_{jk}},$
(23)

for all $i,j,k=1,2,3$.

###### Proof.

Consider the scalar product,

$\begin{split}\textbf{u}_{ij}\cdot\textbf{u}_{jk}=&\frac{(\textbf{n}_{i}\times\textbf{n}_{j})\cdot(\textbf{n}_{j}\times\textbf{n}_{k})}{\left|\textbf{n}_{i}\times\textbf{n}_{j}\right|\left|\textbf{n}_{j}\times\textbf{n}_{k}\right|},\\\
=&\frac{\cos\theta_{ij}\cos\theta_{jk}-\cos\theta_{ik}}{\sin\theta_{ij}\sin\theta_{jk}}.\end{split}$
(24)

Hence, the Cosine Rule follows. ∎

######  Definition 2.

The generalised sine function of three variables,
$\sin(\theta_{ij},\theta_{jk},\theta_{ik})$,is defined by

$\begin{split}\sin(\theta_{ij},\theta_{jk},\theta_{ik})&=\sqrt{1-\cos^{2}\theta_{ij}-\cos^{2}\theta_{jk}-\cos^{2}\theta_{ik}+2\cos\theta_{jk}\cos\theta_{ij}\cos\theta_{ik}},\\\
&=\left|\begin{array}[]{ccc}1&\cos\theta_{ij}&\cos\theta_{ik}\\\
\cos\theta_{ij}&1&\cos\theta_{jk}\\\ \cos\theta_{ik}&\cos\theta_{jk}&1\\\
\end{array}\right|^{\frac{1}{2}},\hskip
28.45274pt(0\leq\theta_{ij},\theta_{ik},\theta_{jk}\leq\pi).\end{split}$ (25)

Note that the restrictions for a spherical triangle, that each side is less
than a semi-circle and each angle less than $\pi$, ensure that the generalised
sine function is always real and positive.

This sine function is equivalent to the volume of a parallelepiped with edges
$\mathbf{n}_{i}$, $\mathbf{n}_{j}$ and $\mathbf{n}_{k}$, given by

$\text{Volume}=|\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k})|=\sin(\theta_{ij},\theta_{ik},\theta_{jk}).$
(26)

This identity can be viewed of as a consequence of the triple product when we
embed the 3D vectors $\mathbf{n}_{i}$, $\mathbf{n}_{j}$ and $\mathbf{n}_{k}$
in $E_{4}$ by

$\widetilde{\mathbf{n}}_{i}=\left(\begin{array}[]{c}\mathbf{n}_{i}\\\
0\end{array}\right)\
,\quad\widetilde{\mathbf{n}}_{j}=\left(\begin{array}[]{c}\mathbf{n}_{j}\\\
0\end{array}\right)\
,\quad\widetilde{\mathbf{n}}_{k}=\left(\begin{array}[]{c}\mathbf{n}_{k}\\\
0\end{array}\right)\ ,$ (27)

such that

$\widetilde{\mathbf{n}}_{i}\times\widetilde{\mathbf{n}}_{j}\times\widetilde{\mathbf{n}}_{k}=\left(\begin{array}[]{c}0\\\
0\\\ 0\\\
\det(\mathbf{n}_{i},\mathbf{n}_{j},\mathbf{n}_{k})\end{array}\right)\ ,$ (28)

where the last entry equals
$|\mathbf{n}_{i}\cdot(\mathbf{n}_{j}\times\mathbf{n}_{k})|$.

######  Proposition 4 (Sine Rule).

$\frac{\sin\alpha_{i}}{\sin\theta_{jk}}=\frac{\sin\alpha_{j}}{\sin\theta_{ik}}=\frac{\sin\alpha_{k}}{\sin\theta_{ij}}=\frac{\sin(\theta_{ij},\theta_{jk},\theta_{ik})}{\sin\theta_{ij}\sin\theta_{jk}\sin\theta_{ik}}=k,$
(29)

where $k$ is a constant.

###### Proof.

Consider the ratio

$\frac{|\textbf{u}_{ij}\times\textbf{u}_{jk}|}{|\textbf{n}_{i}\times\textbf{n}_{k}|}=\frac{\sin\alpha_{j}}{\sin\theta_{ik}}.$
(30)

Now, note the vector product

$\textbf{u}_{ij}\times\textbf{u}_{jk}=\frac{(\textbf{n}_{i}\times\textbf{n}_{j})\times(\textbf{n}_{j}\times\textbf{n}_{k})}{|\textbf{n}_{i}\times\textbf{n}_{j}||\textbf{n}_{j}\times\textbf{n}_{k}|}=\frac{\left(\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k})\right)\textbf{n}_{j}}{\sin\theta_{ij}\sin\theta_{jk}}.$
(31)

This implies

$|\textbf{u}_{ij}\times\textbf{u}_{jk}|=\left|\frac{\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k})}{\sin\theta_{ij}\sin\theta_{jk}}\right|=\frac{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}{\sin\theta_{ij}\sin\theta_{jk}},$
(32)

and hence, the sine rule follows. ∎

######  Proposition 5 (Polar Cosine Rule).

$\cos\theta_{jk}=\frac{\cos\alpha_{j}\cos\alpha_{k}+\cos\alpha_{i}}{\sin\alpha_{j}\sin\alpha_{k}},$
(33)

for all $i,j,k=1,2,3$.

###### Proof.

Consider the product

$\frac{(\textbf{u}_{ij}\times\textbf{u}_{jk})\cdot(\textbf{u}_{jk}\times\textbf{u}_{ik})}{|\textbf{u}_{ij}\times\textbf{u}_{jk}||\textbf{u}_{jk}\times\textbf{u}_{ik}|}.$
(34)

This product can be calculated in two ways. First,

$\frac{(\textbf{u}_{ij}\times\textbf{u}_{jk})\cdot(\textbf{u}_{jk}\times\textbf{u}_{ik})}{|\textbf{u}_{ij}\times\textbf{u}_{jk}||\textbf{u}_{jk}\times\textbf{u}_{ik}|}=\frac{-\cos\alpha_{j}\cos\alpha_{k}-\cos\alpha_{i}}{\sin\alpha_{j}\sin\alpha_{k}},$
(35)

and second, using (31),

$\begin{split}\frac{(\textbf{u}_{ij}\times\textbf{u}_{jk})\cdot(\textbf{u}_{jk}\times\textbf{u}_{ik})}{|\textbf{u}_{ij}\times\textbf{u}_{jk}||\textbf{u}_{jk}\times\textbf{u}_{ik}|}&=\frac{1}{\sin\alpha_{j}\sin\alpha_{k}}\left(\frac{\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k})}{\sin\theta_{ij}\sin\theta_{jk}}\textbf{n}_{j}\right)\cdot\left(\frac{\textbf{n}_{i}\cdot(\textbf{n}_{k}\times\textbf{n}_{j})}{\sin\theta_{ik}\sin\theta_{jk}}\textbf{n}_{k}\right),\\\
&=-\frac{\sin^{2}(\theta_{ij},\theta_{ik},\theta_{jk})}{\sin\alpha_{j}\sin\alpha_{k}\sin\theta_{ij}\sin^{2}\theta_{jk}\sin\theta_{ik}}\cos\theta_{jk}.\end{split}$
(36)

Reducing this using the Sine Rule and equating with the previous result gives
the Polar Cosine Rule. ∎

Note that, by now considering

$k^{2}=\frac{\sin^{2}\alpha_{i}}{\sin^{2}\theta_{jk}}=\frac{\sin^{2}\alpha_{i}}{1-\cos^{2}\theta_{jk}},$
(37)

and substituting in the polar cosine rule,

$k^{2}=\frac{\sin^{2}\alpha_{i}}{1-\left(\frac{\cos\alpha_{i}+\cos\alpha_{j}\cos\alpha_{k}}{\sin\alpha_{j}\sin\alpha_{k}}\right)^{2}}=\frac{\sin^{2}\alpha_{i}\sin^{2}\alpha_{j}\sin^{2}\alpha_{k}}{1-\cos^{2}\alpha_{i}-\cos^{2}\alpha_{j}-\cos^{2}\alpha_{k}-2\cos\alpha_{i}\cos\alpha_{j}\cos\alpha_{k}},$
(38)

the constant $k$ may also be written in terms of the spherical angles, so that
the sine rule now becomes

$\frac{\sin\alpha_{i}}{\sin\theta_{jk}}=\frac{\sin\alpha_{j}}{\sin\theta_{ik}}=\frac{\sin\alpha_{k}}{\sin\theta_{ij}}=\frac{\sin\alpha_{i}\sin\alpha_{j}\sin\alpha_{k}}{\sqrt{1-\cos^{2}\alpha_{i}-\cos^{2}\alpha_{j}-\cos^{2}\alpha_{k}-2\cos\alpha_{i}\cos\alpha_{j}\cos\alpha_{k}}}.$
(39)

We will use both forms of the sine rule laterwhen discussing the link between
spherical trigonometry and the Jacobi elliptic functions.

###  III.2 Hyperspherical Trigonometry in 4-Dimensional Euclidean Space

$\theta_{ik}$$\alpha_{k}^{(ijk)}$$\mathbf{n}_{j}$$\phi_{ik}$$\mathbf{n}_{l}$$\theta_{kl}$$\theta_{jk}$$\mathbf{n}_{k}$$\mathbf{n}_{i}$$\theta_{ij}$$\theta_{il}$$\alpha_{i}^{(ijk)}$
Figure 2: A hyperspherical tetrahedron

We now extend these principles to the four dimensional hyperspherical case.
Consider a $3$-sphere embedded in four dimensional Euclidean space,
$\mathbf{E}_{4}=\mathbb{R}^{4}$. In this case, we have four unit vectors,
$\textbf{n}_{i},i=1,\dots,4$ pointing to the four vertices of a hyperspherical
tetrahedron. The angles between these unit vectors, $\theta_{ij}$, are
defined, in the same way as for the spherical case, by

$\textbf{n}_{i}\cdot\textbf{n}_{j}\equiv\cos\theta_{ij},$ (40)

whereas now we must also define orthogonal vectors $\textbf{u}_{ijk}$ to each
hyperplane $[i,j,k]$ (defined by $\mathbf{n}_{i}$, $\mathbf{n}_{j}$,
$\mathbf{n}_{k}$). This is done using the ternary cross product,

$\textbf{u}_{ijk}\equiv\frac{\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{k}}{\left|\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{k}\right|}.$
(41)

The four vectors $\mathbf{u}_{123}$, $\mathbf{u}_{124}$, $\mathbf{u}_{134}$,
$\mathbf{u}_{234}$ define the polar of the hyperspherical tetrahedron, between
which we define dihedral angles $\phi_{jk}$ by

$\textbf{u}_{ijk}\cdot\textbf{u}_{jkl}\equiv-\cos\phi_{jk}.$ (42)

We also have

$|\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{k}|=\sin(\theta_{ij},\theta_{jk},\theta_{ik}),$
(43)

together with

$|\textbf{u}_{ijk}\times\textbf{u}_{jkl}\times\textbf{u}_{kli}|=\sin(\phi_{jk},\phi_{ik},\phi_{kl}).$
(44)

The four faces of the hyperspherical tetrahedron are spherical triangles, with
the various sine and cosine rules for the spherical case still holding true
for these. For ease of notation, we now label the angles in the faces of the
spherical triangles $\alpha_{i}^{(ijk)}$ to indicate which triangle is being
considered.

In addition to the spherical relations, there are now various relations
involving the dihedral angles, $\phi_{ij}$, which connect the various faces of
the hyperspherical tetrahedron. By taking various relations between the scalar
and vector products for the vectors $\textbf{u}_{ijk}$, we can again derive a
number of relations.

######  Proposition 6 (Cosine Rule).

$\cos\phi_{jk}=\frac{-\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{il}\\\
1&\cos\theta_{jk}&\cos\theta_{jl}\\\ \cos\theta_{jk}&1&\cos\theta_{kl}\\\
\end{array}\right|}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{jk},\theta_{jl},\theta_{kl})}.$
(45)

for all $i,j,k,l=1,2,3,4$.

###### Proof.

From the computation

$\begin{split}(\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{k})\cdot(\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l})&=-(\mathbf{n}_{j}\times\mathbf{n}_{k}\times(\mathbf{n}_{j}\times\mathbf{n}_{k}\times\mathbf{n}_{l}))\mathbf{n}_{i},\\\
&=\left|\begin{array}[]{ccc}\textbf{n}_{j}&\textbf{n}_{k}&\textbf{n}_{l}\\\
\textbf{n}_{j}\cdot\textbf{n}_{j}&\textbf{n}_{j}\cdot\textbf{n}_{k}&\textbf{n}_{j}\cdot\textbf{n}_{l}\\\
\textbf{n}_{k}\cdot\textbf{n}_{j}&\textbf{n}_{k}\cdot\textbf{n}_{k}&\textbf{n}_{k}\cdot\textbf{n}_{l}\\\
\end{array}\right|\cdot\mathbf{n}_{i},\\\
&=\left|\begin{array}[]{ccc}\textbf{n}_{i}\cdot\textbf{n}_{j}&\textbf{n}_{i}\cdot\textbf{n}_{k}&\textbf{n}_{i}\cdot\textbf{n}_{l}\\\
\textbf{n}_{j}\cdot\textbf{n}_{j}&\textbf{n}_{j}\cdot\textbf{n}_{k}&\textbf{n}_{j}\cdot\textbf{n}_{l}\\\
\textbf{n}_{k}\cdot\textbf{n}_{j}&\textbf{n}_{k}\cdot\textbf{n}_{k}&\textbf{n}_{k}\cdot\textbf{n}_{l}\\\
\end{array}\right|,\end{split}$ (46)

we find that for the following inner product (for all $i,j,k,l=1,2,3,4$) we
have

$\begin{split}\textbf{u}_{ijk}\cdot\textbf{u}_{jkl}=&\frac{(\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{k})\cdot(\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l})}{\left|\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{k}\right|\left|\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l}\right|},\\\
=&\frac{\left|\begin{array}[]{ccc}\textbf{n}_{i}\cdot\textbf{n}_{j}&\textbf{n}_{i}\cdot\textbf{n}_{k}&\textbf{n}_{i}\cdot\textbf{n}_{l}\\\
\textbf{n}_{j}\cdot\textbf{n}_{j}&\textbf{n}_{j}\cdot\textbf{n}_{k}&\textbf{n}_{j}\cdot\textbf{n}_{l}\\\
\textbf{n}_{k}\cdot\textbf{n}_{j}&\textbf{n}_{k}\cdot\textbf{n}_{k}&\textbf{n}_{k}\cdot\textbf{n}_{l}\\\
\end{array}\right|}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{jk},\theta_{jl},\theta_{kl})}.\\\
\end{split}$ (47)

Hence, the Cosine Rule follows. ∎

We now need to extend to definition of the triple sine function for the four-
dimensional case, to give a further generalisation of the sine function
dependent on six variables.

######  Definition 3.

The generalised sine function of six variables,
$\sin(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})$,
is the four dimensional analogue of the triple sine function, and is given by

$\sin(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})=\left|\begin{array}[]{cccc}1&\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{il}\\\
\cos\theta_{ij}&1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{ik}&\cos\theta_{jk}&1&\cos\theta_{kl}\\\
\cos\theta_{il}&\cos\theta_{jl}&\cos\theta_{kl}&1\\\
\end{array}\right|^{\frac{1}{2}},$ (48)
$0\leq\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl}\leq\pi.$
(49)

Note that the restrictions on being a spherical tetrahedron ensure that the
generalised sine function is always real and positive.

This sine function is equivalent to the volume of a 4D parallelotope with
edges $\mathbf{n}_{i}$, $\mathbf{n}_{j}$, $\mathbf{n}_{k}$ and
$\mathbf{n}_{l}$, given by

$\text{Volume}=|\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l})|=\sin(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl}).$
(50)

This relation follows from embedding 4D vectors $\mathbf{n}_{i}$,
$\mathbf{n}_{j}$, $\mathbf{n}_{k}$ and $\mathbf{n}_{l}$ into $E_{5}$ in a
similar manner to the three-dimensional case.

######  Proposition 7 (Sine Rule).

$\begin{split}\frac{\sin(\phi_{ij},\phi_{ik},\phi_{il})}{\sin(\theta_{jk},\theta_{jl},\theta_{kl})}&=\frac{\sin(\phi_{ij},\phi_{jk},\phi_{jl})}{\sin(\theta_{ik},\theta_{il},\theta_{kl})}=\frac{\sin(\phi_{ik},\phi_{jk},\phi_{kl})}{\sin(\theta_{ij},\theta_{il},\theta_{jl})}=\frac{\sin(\phi_{il},\phi_{jl},\phi_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})},\\\
&=\frac{\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{ij},\theta_{il},\theta_{jl})\sin(\theta_{ik},\theta_{il},\theta_{kl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})}=k_{H},\end{split}$
(51)

where $k_{H}$ is constant.

###### Proof.

Consider the triple product

$\begin{split}\textbf{u}_{ijk}\times\textbf{u}_{jkl}\times\textbf{u}_{kli}&=\frac{(\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{k})\times(\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l})\times(\textbf{n}_{k}\times\textbf{n}_{l}\times\textbf{n}_{i})}{|\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{k}||\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l}||\textbf{n}_{k}\times\textbf{n}_{l}\times\textbf{n}_{i}|},\\\
&=\frac{-\left|\begin{array}[]{ccc}\textbf{n}_{i}&\textbf{n}_{j}&\textbf{n}_{k}\\\
\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l})&0&0\\\
0&\textbf{n}_{j}\cdot(\textbf{n}_{k}\times\textbf{n}_{l}\times\textbf{n}_{i})&0\end{array}\right|}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{kl},\theta_{ki},\theta_{li})},\\\
&=\frac{\left(\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l})\right)^{2}}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{kl},\theta_{ki},\theta_{li})}\,\textbf{n}_{k}.\end{split}$
(52)

This implies

$\begin{split}|\textbf{u}_{ijk}\times\textbf{u}_{jkl}\times\textbf{u}_{kli}|&=\frac{\left(\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l})\right)^{2}}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{kl},\theta_{ki},\theta_{li})},\\\
&=\frac{\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{kl},\theta_{ki},\theta_{li})}.\end{split}$
(53)

From this we have that the ratio

$\frac{|\textbf{u}_{ijk}\times\textbf{u}_{jkl}\times\textbf{u}_{kli}|}{|\textbf{n}_{i}\times\textbf{n}_{j}\times\textbf{n}_{l}|}=\frac{\sin(\phi_{ik},\phi_{jk},\phi_{kl})}{\sin(\theta_{ij},\theta_{il},\theta_{jl})}$
(54)

is symmetric with respect to the interchange of the labels $i,j,k,l$. Thus, it
follows that

$\frac{\sin(\phi_{ik},\phi_{jk},\phi_{kl})}{\sin(\theta_{ij},\theta_{il},\theta_{jl})}=\frac{\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{kl},\theta_{ki},\theta_{li})\sin(\theta_{ij},\theta_{il},\theta_{jl})},$
(55)

and hence, the hyperspherical sine rule follows. ∎

There also exists a simpler sine relationship between the standard sine
functions of the central angles and the sine of the dihedral angles.

######  Proposition 8.

$\sin\phi_{kl}=\frac{\sin(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})}{\sin(\theta_{ik},\theta_{il},\theta_{kl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})}\sin\theta_{kl}.$
(56)

In order to prove this proposition, we require a determinantal identity,
attributed to Desnanot for the $n\leq 6$ caseDesnanot , and to Jacobi in the
general case.Jacobi2

######  Theorem 2 (Desnanot-Jacobi Identity).

Let $M$ be a $n\times n$ square matrix, and denote by $M^{p}_{i}$ the matrix
obtained by removing both the $i$-th row and $p$-th column. Similarly, let
$M^{p,q}_{i,j}$ denote the matrix obtained by deleting the $i$-th and $j$-th
rows, and the $p$-th and $q$-th columns, respectively, with $1\leq i,j,p,q\leq
n$. Then,

$\mathrm{det}\left(M\right)\mathrm{det}\left(M^{1,n}_{1,n}\right)=\mathrm{det}\left(M^{1}_{1}\right)\mathrm{det}\left(M^{n}_{n}\right)-\mathrm{det}\left(M^{n}_{1}\right)\mathrm{det}\left(M^{1}_{n}\right).$
(57)

Diagrammatically, this is

$\left|\begin{picture}(49.0,32.0)\end{picture}\right|\times\left|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{picture}(49.0,32.0)\put(3.0,-22.0){\rule{0.56905pt}{51.21504pt}}\put(46.0,-22.0){\rule{0.56905pt}{51.21504pt}}
\put(1.0,-19.0){\rule{48.36958pt}{0.56905pt}}\put(1.0,26.0){\rule{48.36958pt}{0.56905pt}}
\end{picture}\right|=\left|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{picture}(49.0,32.0)\put(3.0,-22.0){\rule{0.56905pt}{51.21504pt}}\put(1.0,26.0){\rule{48.36958pt}{0.56905pt}}
\end{picture}\right|\times\left|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{picture}(49.0,32.0)\put(46.0,-22.0){\rule{0.56905pt}{51.21504pt}}\put(1.0,-19.0){\rule{48.36958pt}{0.56905pt}}
\end{picture}\right|-\left|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{picture}(49.0,32.0)\put(3.0,-22.0){\rule{0.56905pt}{51.21504pt}}\put(1.0,-19.0){\rule{48.36958pt}{0.56905pt}}
\end{picture}\right|\times\left|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\begin{picture}(49.0,32.0)\put(46.0,-22.0){\rule{0.56905pt}{51.21504pt}}\put(1.0,26.0){\rule{48.36958pt}{0.56905pt}}
\end{picture}\right|.$ (58)

###### Proof.

The result follows directly from the Desnanot-Jacobi determinant identity

$\begin{split}&\left|\begin{array}[]{cccc}1&\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{jk}\\\
\cos\theta_{ij}&1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{ik}&\cos\theta_{jk}&1&\cos\theta_{kl}\\\
\cos\theta_{il}&\cos\theta_{jl}&\cos\theta_{kl}&1\\\
\end{array}\right|\left|\begin{array}[]{cc}1&\cos\theta_{kl}\\\
\cos\theta_{kl}&1\\\ \end{array}\right|\\\
=&\left|\begin{array}[]{ccc}1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{jk}&1&\cos\theta_{kl}\\\ \cos\theta_{jl}&\cos\theta_{kl}&1\\\
\end{array}\right|\left|\begin{array}[]{ccc}1&\cos\theta_{ik}&\cos\theta_{jk}\\\
\cos\theta_{ik}&1&\cos\theta_{kl}\\\ \cos\theta_{il}&\cos\theta_{kl}&1\\\
\end{array}\right|\\\
&-\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{ik}&1&\cos\theta_{kl}\\\ \cos\theta_{il}&\cos\theta_{kl}&1\\\
\end{array}\right|\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{jk}\\\
\cos\theta_{jk}&1&\cos\theta_{kl}\\\ \cos\theta_{jl}&\cos\theta_{kl}&1\\\
\end{array}\right|.\end{split}$ (59)

The result follows directly. ∎

From this proposition, it hence follows that the hyperspherical sine rule may
be rewritten as

$\frac{\sin\phi_{ij}}{\sin\theta_{ij}}\frac{\sin\phi_{kl}}{\sin\theta_{kl}}=\frac{\sin\phi_{ik}}{\sin\theta_{ik}}\frac{\sin\phi_{jl}}{\sin\theta_{jl}}=\frac{\sin\phi_{il}}{\sin\theta_{il}}\frac{\sin\phi_{jk}}{\sin\theta_{jk}}=k_{H}.$
(60)

The constant $k_{H}$ may also be neatly expressed in terms of cosines.Derevnin
For this we need the Polar Cosine rule.

######  Proposition 9 (Polar Cosine Rule).

$\cos\theta_{kl}=\frac{\left|\begin{array}[]{ccc}-\cos\phi_{jk}&\cos\phi_{ik}&-\cos\phi_{ij}\\\
1&-\cos\phi_{kl}&\cos\phi_{jl}\\\
-\cos\phi_{kl}&1&-\cos\phi_{il}\end{array}\right|}{\sin(\phi_{ik},\phi_{jk},\phi_{kl})\sin(\phi_{il},\phi_{jl},\phi_{kl})}.$
(61)

for all $i,j,k,l=1,2,3,4$.

###### Proof.

Consider the product

$\begin{split}\frac{(\textbf{u}_{ijk}\times\textbf{u}_{jkl}\times\textbf{u}_{kli})\cdot(\textbf{u}_{jkl}\times\textbf{u}_{kli}\times\textbf{u}_{lij})}{|\textbf{u}_{ijk}\times\textbf{u}_{jkl}\times\textbf{u}_{kli}||\textbf{u}_{jkl}\times\textbf{u}_{kli}\times\textbf{u}_{lij}|}&=\frac{\left|\begin{array}[]{ccc}\textbf{u}_{ijk}\cdot\textbf{u}_{jkl}&\textbf{u}_{ijk}\cdot\textbf{u}_{kli}&\textbf{u}_{ijk}\cdot\textbf{u}_{lij}\\\
\textbf{u}_{jkl}\cdot\textbf{u}_{jkl}&\textbf{u}_{jkl}\cdot\textbf{u}_{kli}&\textbf{u}_{jkl}\cdot\textbf{u}_{lij}\\\
\textbf{u}_{kli}\cdot\textbf{u}_{jkl}&\textbf{u}_{kli}\cdot\textbf{u}_{kli}&\textbf{u}_{kli}\cdot\textbf{u}_{lij}\end{array}\right|}{\sin(\phi_{ik},\phi_{jk},\phi_{kl})\sin(\phi_{il},\phi_{jl},\phi_{kl})},\\\
&=\frac{\left|\begin{array}[]{ccc}-\cos\phi_{jk}&\cos\phi_{ik}&-\cos\phi_{ij}\\\
1&-\cos\phi_{kl}&\cos\phi_{jl}\\\
-\cos\phi_{kl}&1&-\cos\phi_{il}\end{array}\right|}{\sin(\phi_{ik},\phi_{jk},\phi_{kl})\sin(\phi_{il},\phi_{jl},\phi_{kl})}.\end{split}$
(62)

On the other hand, using (52), we have

$\textbf{u}_{ijk}\times\textbf{u}_{jkl}\times\textbf{u}_{kli}=\frac{\left(\textbf{n}_{i}\cdot(\textbf{n}_{j}\times\textbf{n}_{k}\times\textbf{n}_{l})\right)^{2}}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{kl},\theta_{ki},\theta_{li})}\,\textbf{n}_{k}.$
(63)

Reducing this using the Sine Rule and equating the two results gives the
hyperspherical polar cosine rule. ∎

######  Proposition 10.

$\begin{split}\frac{\cos\phi_{ij}\cos\phi_{kl}-\cos\phi_{ik}\cos\phi_{jl}}{\cos\theta_{ij}\cos\theta_{kl}-\cos\theta_{ik}\cos\theta_{jl}}&=\frac{\cos\phi_{ik}\cos\phi_{jl}-\cos\phi_{il}\cos\phi_{jk}}{\cos\theta_{ik}\cos\theta_{jl}-\cos\theta_{il}\cos\theta_{jk}},\\\
&=\frac{\cos\phi_{il}\cos\phi_{jk}-\cos\phi_{ij}\cos\phi_{kl}}{\cos\theta_{il}\cos\theta_{jk}-\cos\theta_{ij}\cos\theta_{kl}},\\\
&=k_{H}.\end{split}$ (64)

###### Proof.

Consider the numerator

$\cos\phi_{ij}\cos\phi_{kl}-\cos\phi_{ik}\cos\phi_{jl},$ (65)

and rewrite this in terms of the central angles using the cosine rule,

$\begin{split}\cos\phi_{ij}\cos\phi_{kl}&-\cos\phi_{ik}\cos\phi_{jl}=\frac{\left|\begin{array}[]{ccc}\cos\theta_{ik}&\cos\theta_{jk}&\cos\theta_{kl}\\\
1&\cos\theta_{ij}&\cos\theta_{il}\\\ \cos\theta_{ij}&1&\cos\theta_{jl}\\\
\end{array}\right|}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{ij},\theta_{il},\theta_{jl})}\frac{\left|\begin{array}[]{ccc}\cos\theta_{ik}&\cos\theta_{il}&\cos\theta_{ij}\\\
1&\cos\theta_{kl}&\cos\theta_{jk}\\\ \cos\theta_{kl}&1&\cos\theta_{jl}\\\
\end{array}\right|}{\sin(\theta_{ik},\theta_{il},\theta_{kl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})}\\\
&-\frac{\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{jk}&\cos\theta_{jl}\\\
1&\cos\theta_{ik}&\cos\theta_{il}\\\ \cos\theta_{ik}&1&\cos\theta_{kl}\\\
\end{array}\right|}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{ik},\theta_{il},\theta_{jk})}\frac{\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{il}&\cos\theta_{ik}\\\
1&\cos\theta_{jl}&\cos\theta_{jk}\\\ \cos\theta_{jl}&1&\cos\theta_{kl}\\\
\end{array}\right|}{\sin(\theta_{ij},\theta_{il},\theta_{jl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})}.\end{split}$
(66)

Multiplying this out and factorising, this reduces to

$\cos\phi_{ij}\cos\phi_{kl}-\cos\phi_{ik}\cos\phi_{jl}=(\cos\theta_{ij}\cos\theta_{kl}-\cos\theta_{ik}\cos\theta_{jl})k_{H},$
(67)

from which the result follows. ∎

Furthermore, in 4D there are also relations involving the spherical angles
$\alpha_{j}^{(j\cdot\cdot)}$ at a common vertex, $\mathbf{n}_{j}$, of the
tetrahedron involving the different spherical triangles.

######  Proposition 11 (Vertex Cosine Rule).

$\cos\phi_{jk}=\frac{\cos\alpha_{j}^{(ijk)}\cos\alpha_{j}^{(jkl)}-\cos\alpha_{j}^{(ijl)}}{\sin\alpha_{j}^{(ijk)}\sin\alpha_{j}^{(jkl)}}.$
(68)

###### Proof.

Expanding out the cosine rule (45) and applying the sine rule (51) to the
denominator gives

$\begin{split}\cos\phi_{jk}=&(\cos\theta_{ij}\cos\theta_{jl}+\cos\theta_{ik}\cos\theta_{kl}-\cos\theta_{ij}\cos\theta_{jk}\cos\theta_{kl}\\\
&-\cos\theta_{ik}\cos\theta_{jk}\cos\theta_{jl}-\cos\theta_{il}+\cos\theta_{il}\cos^{2}\theta_{jk})\\\
&/{\sin\alpha_{j}^{(ijk)}\sin\alpha_{j}^{(jkl)}\sin\theta_{ij}\sin^{2}\theta_{jk}\sin\theta_{jl}}.\end{split}$
(69)

This can be factorised as

$\cos\phi_{jk}=\frac{\left(\frac{\cos\theta_{ik}-\cos\theta_{ij}\cos\theta_{jk}}{\sin\theta_{ij}\sin\theta_{jk}}\right)\left(\frac{\cos\theta_{kl}-\cos\theta_{jk}\cos\theta_{jl}}{\sin\theta_{jk}\sin\theta_{jl}}\right)-\left(\frac{\cos\theta_{il}-\cos\theta_{ij}\cos\theta_{jl}}{\sin\theta_{ij}\sin\theta_{jl}}\right)}{\sin\alpha_{j}^{(ijk)}\sin\alpha_{j}^{(jkl)}}.$
(70)

Applying (23) the result follows. ∎

This set of relations imply, in turn, corresponding vertex polar cosine
relations, expressing the spherical angles at a vertex in terms of the
dihedral angles as follows,

$\cos\alpha_{j}^{(ijl)}=\frac{\cos\phi_{jk}+\cos\phi_{ij}\cos\phi_{jl}}{\sin\phi_{ij}\sin\phi_{jl}},$
(71)

together with a corresponding sine rule of the form

$\frac{\sin\phi_{jk}}{\sin\alpha_{j}^{(ijl)}}=\frac{\sin\phi_{jl}}{\sin\alpha_{j}^{(ijk)}}=\frac{\sin\phi_{ij}}{\sin\alpha_{j}^{(jkl)}}=\frac{\sin(\alpha_{j}^{(ijk)},\alpha_{j}^{(ijl)},\alpha_{j}^{(jkl)})}{\sin\alpha_{j}^{(ijk)}\sin\alpha_{j}^{(ijl)}\sin\alpha_{j}^{(jkl)}},$
(72)

for $i,j,k,l=1,2,3,4$. Note the similarities between these formulae and those
of the spherical case.

We also present a, to our knowledge, new formula which again follows from the
Desnanot-Jacobi identity, which proves particularly useful when providing the
link between hyperspherical trigonometry and elliptic functions given later.

######  Proposition 12.

The following identity holds between the various tetrahedral angles:

$\begin{split}&\frac{\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})\,\cos\alpha_{l}^{(jkl)}\,\sin\theta_{jl}\,\sin\theta_{kl}}{\sin^{2}(\theta_{jk},\theta_{jl},\theta_{kl})}\\\
&=\sin(\theta_{ij},\theta_{il},\theta_{jl})\,\sin(\theta_{ik},\theta_{il},\theta_{kl})\,\left(\cos\phi_{jl}\,\cos\phi_{kl}-\cos\phi_{il}\right)\end{split}$
(73)

###### Proof.

Consider the Desnanot-Jacobi determinantal identity applied to
$\sin^{4}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})$.
This yields:

$\begin{split}\left|\begin{array}[]{cc}\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{kl}&1\end{array}\right|&\left|\begin{array}[]{cccc}1&\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{il}\\\
\cos\theta_{ij}&1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{ik}&\cos\theta_{jk}&1&\cos\theta_{kl}\\\
\cos\theta_{il}&\cos\theta_{jl}&\cos\theta_{kl}&1\end{array}\right|\\\
&=\left|\begin{array}[]{ccc}1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{jk}&1&\cos\theta_{kl}\\\
\cos\theta_{jl}&\cos\theta_{kl}&1\end{array}\right|\,\left|\begin{array}[]{ccc}1&\cos\theta_{ik}&\cos\theta_{il}\\\
\cos\theta_{ij}&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{il}&\cos\theta_{kl}&1\end{array}\right|\\\
&-\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{il}\\\
1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{jl}&\cos\theta_{kl}&1\end{array}\right|\,\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{ik}&1&\cos\theta_{kl}\\\
\cos\theta_{il}&\cos\theta_{kl}&1\end{array}\right|\end{split}$ (74)

from which the result follows. ∎

In Appendix A, we consider the expansion of one of the radial vectors,
$\mathbf{n}_{i}$, in terms of the other radial vectors. We compare this with
the expansion in terms of an orthogonal frame, obtained by Gram-Schmidt
orthonormalisation to obtain the four-parts formula. We follow a similar
procedure for the orthogonal vectors, $\mathbf{u}_{ijk}$, to obtain the five-
parts formula. We believe these expansions will be of particular use in
considering a model associated with quadrilateral nets introduced by Sergeev
in Sergeev .

###  III.3 m-Dimensional Hyperspherical Trigonometry

Some of the results in this subsection were derived in sato  from a different
perspective. Consider an $(m-1)$-dimensional hyperspherical simplex on the
surface of an $m$-dimensional hypersphere, an $(m-1)$-sphere. Let the $m$
vectors, $\mathbf{n}_{1},\mathbf{n}_{2},\dots,\mathbf{n}_{m}$ be the position
vectors of the $m$ vertices of this simplex, such that the edges of this
simplex are given by $\theta^{[1]}_{i_{j}i_{k}}$, with

$\mathbf{n}_{i_{j}}\cdot\mathbf{n}_{i_{k}}\equiv\cos\theta^{[1]}_{i_{j}i_{k}},$
(75)

for $i_{j},i_{k}\in\\{1,2,\dots,m\\}$. Now define orthogonal vectors
$\mathbf{u}_{i_{1}i_{2}\dots i_{m-1}}$ using the $(m-1)$-ary vector product,

$\mathbf{u}_{i_{1}i_{2}\dots
i_{m-1}}\equiv\frac{\mathbf{n}_{i_{1}}\times\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m-1}}}{|\mathbf{n}_{i_{1}}\times\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m-1}}|}.$
(76)

Define the angles between these orthogonal vectors by

$\mathbf{u}_{i_{1}i_{2}\dots i_{m-1}}\cdot\mathbf{u}_{i_{2}i_{3}\dots
i_{m}}\equiv-\cos\theta^{[m-1](i_{1}\dots i_{m})}_{i_{2}\dots i_{m-1}}.$ (77)

Note that the superscript $(i_{1}\dots i_{m})$ denotes the simplex that is
being considered. In the highest dimensional case this may be omitted. We also
have

$|\mathbf{n}_{i_{1}}\times\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m-1}}|=\sin\left(\Theta^{[1]}_{i_{1}\dotsi_{m-1}}\right),$
(78)

where $\sin\left(\Theta^{[1]}_{i_{1}\dotsi_{m-1}}\right)$ is the generalised
sine function of $m(m-1)/2$ variables,

$\sin\left(\Theta^{[1]}_{i_{1}\dotsi_{m-1}}\right)=\sin\left(\left\\{\theta^{[1]}_{i_{j}i_{k}}\bigg{|}i_{j}<i_{k},\,i_{j},i_{k}=1,\dots,m-1\right\\}\right),$
(79)

together with

$|\mathbf{u}_{i_{1}\dots i_{m-1}}\times\mathbf{u}_{i_{2}\dots
i_{m}}\times\mathbf{u}_{i_{3}\dots
i_{m}i_{1}}\times\dots\times\mathbf{u}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}|=\sin\left(\Theta^{[m-1]}_{i_{1}\dots i_{m}}\right),$ (80)

where similarly $\sin\left(\Theta^{[m-1]}_{i_{1}\dots i_{m}}\right)$ is the
multiple sine of all of the angles between each pair of orthogonal vectors.
There follow a number of identities, similar to those for the lower
dimensional cases.

######  Proposition 13 (Cosine Rule).

$\cos\theta^{[m-1]}_{i_{2}\dots
i_{m-1}}=-\frac{\left|\begin{array}[]{ccc}\cos\theta^{[1]}_{i_{1}i_{2}}&\dots&\cos\theta^{[1]}_{i_{1}i_{m}}\\\
\vdots&\ddots&\vdots\\\
\cos\theta^{[1]}_{i_{m-1}i_{2}}&\dots&\cos\theta^{[1]}_{i_{m-1}i_{m}}\\\
\end{array}\right|}{\sin\left(\Theta^{[1]}_{i_{1}\dots
i_{m-1}}\right)\sin\left(\Theta^{[1]}_{i_{2}\dots i_{m}}\right)}.$ (81)

###### Proof.

Consider the scalar product

$\begin{split}\mathbf{u}_{i_{1}\dots i_{m-1}}\cdot\mathbf{u}_{i_{2}\dots
i_{m}}&=\frac{(\mathbf{n}_{i_{1}}\times\dots\times\mathbf{n}_{i_{m-1}})\cdot(\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m}})}{|\mathbf{n}_{i_{1}}\times\dots\times\mathbf{n}_{i_{m-1}}||\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m}}|},\\\
&=\frac{\left|\begin{array}[]{ccc}\mathbf{n}_{i_{1}}\cdot\mathbf{n}_{i_{2}}&\dots&\mathbf{n}_{i_{1}}\cdot\mathbf{n}_{i_{m}}\\\
\vdots&\ddots&\vdots\\\
\mathbf{n}_{i_{m-1}}\cdot\mathbf{n}_{i_{2}}&\dots&\mathbf{n}_{i_{m-1}}\cdot\mathbf{n}_{i_{m}}\\\
\end{array}\right|}{\sin\left(\Theta^{[1]}_{i_{1}\dots
i_{m-1}}\right)\sin\left(\Theta^{[1]}_{i_{2}\dots i_{m}}\right)},\\\
&=\frac{\left|\begin{array}[]{ccc}\cos\theta^{[1]}_{i_{1}i_{2}}&\dots&\cos\theta^{[1]}_{i_{1}i_{m}}\\\
\vdots&\ddots&\vdots\\\
\cos\theta^{[1]}_{i_{m-1}i_{2}}&\dots&\cos\theta^{[1]}_{i_{m-1}i_{m}}\\\
\end{array}\right|}{\sin\left(\Theta^{[1]}_{i_{1}\dots
i_{m-1}}\right)\sin\left(\Theta^{[1]}_{i_{2}\dots i_{m}}\right)}.\end{split}$
(82)

Hence, the cosine rule follows. ∎

######  Proposition 14 (Sine Rule).

$\frac{\sin\left(\Theta^{[1](i_{m-1})}_{i_{1}\dots
i_{m}}\right)}{\sin\left(\Theta^{[m-1]}_{i_{m}i_{1}\dots
i_{m-2}}\right)}==\frac{\sin^{m-2}\left(\Theta^{[1]}_{i_{1}\dots
i_{m}}\right)}{\sin\left(\Theta^{[m-1]}_{i_{1}\dots
i_{m-1}}\right)\dots\sin\left(\Theta^{[m-1]}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\right)}=k,\;\textrm{a\;constant},$ (83)

where

$\sin\left(\Theta^{[1](i_{m-1})}_{i_{1}\dots
i_{m}}\right)=\sin\left(\left\\{\theta^{[1]}_{j_{1}\dots
j_{m-1}}\bigg{|}i_{m-1}=j_{k}\,\mathrm{for\,some}\,k\in\\{1,\dots,m-1\\}\right\\}\right).$
(84)

###### Proof.

Consider the ratio

$\frac{|\mathbf{u}_{i_{1}\dots i_{m-1}}\times\mathbf{u}_{i_{2}\dots
i_{m}}\times\dots\times\mathbf{u}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}|}{|\mathbf{u}_{i_{m}i_{1}\dots
i_{m-2}}|}=\frac{\sin\left(\Theta^{[1](i_{m-1})}_{i_{1}\dots
i_{m}}\right)}{\sin\left(\Theta^{[m-1]}_{i_{m}i_{1}\dots i_{m-2}}\right)}.$
(85)

However,

$\begin{split}&(\mathbf{u}_{i_{1}\dots i_{m-1}}\times\mathbf{u}_{i_{2}\dots
i_{m}}\times\dots\times\mathbf{u}_{i_{m-1}i_{m}i_{1}\dots i_{m-3}})\\\
&=\frac{(\mathbf{n}_{i_{1}}\times\dots\times\mathbf{n}_{i_{m-1}})\times\dots\times(\mathbf{n}_{i_{m-1}}\times\mathbf{n}_{i_{m}}\times\mathbf{n}_{i_{m-1}}\dots\times\mathbf{n}_{i_{m-3}})}{|\mathbf{n}_{i_{1}}\times\dots\times\mathbf{n}_{i_{m-1}}|\dots|\mathbf{n}_{i_{m-1}}\times\mathbf{n}_{i_{m}}\times\mathbf{n}_{i_{m-1}}\dots\times\mathbf{n}_{i_{m-3}}|},\\\
&=\frac{-\left|\begin{array}[]{ccc}\mathbf{n}_{i_{1}}&\dots&\mathbf{n}_{i_{m-1}}\\\
\mathbf{n}_{i_{1}}\cdot(\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m}})&\dots&\mathbf{n}_{i_{m-1}}\cdot(\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m}})\\\
\vdots&\ddots&\vdots\\\
\mathbf{n}_{i_{1}}\cdot(\mathbf{n}_{i_{m-1}}\times\mathbf{n}_{i_{m}}\times\mathbf{n}_{i_{1}}\times\dots\times\mathbf{n}_{i_{m-3}})&\dots&\mathbf{n}_{i_{m-1}}\cdot(\mathbf{n}_{i_{m-1}}\times\mathbf{n}_{i_{m}}\times\mathbf{n}_{i_{1}}\times\dots\times\mathbf{n}_{i_{m-3}})\end{array}\right|}{\sin\left(\Theta^{[m-1]}_{i_{1}\dots
i_{m-1}}\right)\dots\sin\left(\Theta^{[m-1]}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\right)},\\\
&=\frac{(\mathbf{n}_{i_{1}}\cdot(\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m}}))^{m-2}\mathbf{n}_{i_{m-1}}}{\sin\left(\Theta^{[m-1]}_{i_{1}\dots
i_{m-1}}\right)\dots\sin\left(\Theta^{[m-1]}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\right)}.\\\ \end{split}$ (86)

Hence,

$\begin{split}|\mathbf{u}_{i_{1}\dots i_{m-1}}\times\mathbf{u}_{i_{2}\dots
i_{m}}\times\dots\times\mathbf{u}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}|&=\frac{(\mathbf{n}_{i_{1}}\cdot(\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m}}))^{m-2}}{\sin(\Theta^{[m-1]}_{i_{1}\dots
i_{m-1}})\dots\sin\left(\Theta^{[m-1]}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\right)},\\\ &=\frac{\sin^{m-2}\left(\Theta^{[1]}_{i_{1}\dots
i_{m}}\right)}{\sin\left(\Theta^{[m-1]}_{i_{1}\dots
i_{m-1}}\right)\dots\sin\left(\Theta^{[m-1]}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\right)},\end{split}$ (87)

and so, the sine rule follows. ∎

######  Proposition 15 (Polar Cosine Rule).

$\cos\theta^{[1]}_{i_{m-1}i_{m}}=\mathrm{det}(X),$ (88)

where

$(X)_{jk}=\left\\{\begin{array}[]{cc}-1;&j=k-1,\\\
-\cos\theta^{[m-1]}_{i_{1}\dots i_{j-1}i_{j+1}\dots i_{k-2}i_{k}\dots
i_{m}};&j\neq k-1,\;m\;\mathrm{odd},\\\
(-1)^{j+k+1}\cos\theta^{[m-1]}_{i_{1}\dots i_{j-1}i_{j+1}\dots
i_{k-2}i_{k}\dots i_{m}};&j\neq-1,\;m\;\mathrm{even}.\end{array}\right.$ (89)

###### Proof.

Consider the product

$\begin{split}&(\mathbf{u}_{i_{1}\dots i_{m-1}}\times\mathbf{u}_{i_{2}\dots
i_{m}}\times\mathbf{u}_{i_{3}\dots
i_{m}i_{1}}\times\dots\times\mathbf{u}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}})\cdot(\mathbf{u}_{i_{2}\dots
i_{m}}\times\dots\times\mathbf{u}_{i_{m}i_{1}\dots i_{m-2}})\\\
&=\left|\begin{array}[]{ccc}\mathbf{u}_{i_{1}\dots
i_{m-1}}\cdot\mathbf{u}_{i_{2}\dots i_{m}}&\dots&\mathbf{u}_{i_{1}\dots
i_{m-1}}\cdot\mathbf{u}_{i_{m}i_{1}\dots i_{m-2}}\\\ \vdots&\ddots&\vdots\\\
\mathbf{u}_{i_{m-1}i_{m}i_{1}\dots i_{m-3}}\cdot\mathbf{u}_{i_{2}\dots
i_{m}}&\dots&\mathbf{u}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\cdot\mathbf{u}_{i_{m}i_{1}\dots i_{m-2}}\end{array}\right|,\\\
&=\mathrm{det}(X),\end{split}$ (90)

where $X$ is as defined previously. Alternatively, recalling (86) gives

$\begin{split}&(\mathbf{u}_{i_{1}\dots i_{m-1}}\times\mathbf{u}_{i_{2}\dots
i_{m}}\times\mathbf{u}_{i_{3}\dots
i_{m}i_{1}}\times\dots\times\mathbf{u}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}})\cdot(\mathbf{u}_{i_{2}\dots
i_{m}}\times\dots\times\mathbf{u}_{i_{m}i_{1}\dots i_{m-2}}),\\\
&=\left(\frac{(\mathbf{n}_{i_{1}}\cdot(\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m}}))^{m-2}\mathbf{n}_{i_{m-1}}}{\sin\left(\Theta^{[m-1]}_{i_{1}\dots
i_{m-1}}\right)\dots\sin\left(\Theta^{[m-1]}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\right)}\right)\cdot\left(\frac{(\mathbf{n}_{i_{2}}\cdot(\mathbf{n}_{i_{3}}\times\dots\times\mathbf{n}_{i_{m}}\times\mathbf{n}_{i_{m1}}))^{m-2}\mathbf{n}_{i_{m}}}{\sin\left(\Theta^{[m-1]}_{i_{2}\dots
i_{m}}\right)\dots\sin\left(\Theta^{[m-1]}_{i_{m}i_{1}\dots
i_{m-2}}\right)}\right),\\\
&=\frac{(\mathbf{n}_{i_{1}}\cdot(\mathbf{n}_{i_{2}}\times\dots\times\mathbf{n}_{i_{m}}))^{2m-4}\cos\theta^{[1]}_{i_{m-1}i_{m}}}{\sin(\Theta^{[m-1]}_{i_{1}\dots
i_{m-1}})\sin^{2}\left(\Theta^{[m-1]}_{i_{2}\dots
i_{m}}\right)\dots\sin^{2}\left(\Theta^{[m-1]}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\right)\sin\left(\Theta^{[m-1]}_{i_{m}i_{1}\dots i_{m-2}}\right)},\\\
&=\frac{\sin^{2m-4}(\Theta^{[1]}_{i_{1}\dotsi_{m}})\cos\theta^{[1]}_{i_{m-1}i_{m}}}{\sin(\Theta^{[m-1]}_{i_{1}\dots
i_{m-1}})\sin^{2}\left(\Theta^{[m-1]}_{i_{2}\dots
i_{m}}\right)\dots\sin^{2}\left(\Theta^{[m-1]}_{i_{m-1}i_{m}i_{1}\dots
i_{m-3}}\right)\sin\left(\Theta^{[m-1]}_{i_{m}i_{1}\dots i_{m-2}}\right)}.\\\
\end{split}$ (91)

By applying the sine rule, the Polar Cosine rule then follows. ∎

Note that the facets of the $(m-1)$-dimensional simplex are
$(m-2)$-dimensional simplices with the various cosine and sine rules still
holding true in these facets. The same applies to these $(m-2)$-dimensional
facets, and so on, forming a complete hierarchy of cosine and sine rules
between the angles for the $j$-dimensional facets, and those for the
$(j-1)$-dimensional facets. So, in terms of the cosine rule we have

$\cos\theta^{[j]}_{i_{2}\dots
i_{j-1}}=\frac{-\left|\begin{array}[]{cc}\cos\theta^{[j-1](i_{1}\dots
i_{j-1})}_{i_{2}\dots i_{j-1}}&\cos\theta^{[j-1](i_{1}\dots
i_{j-2}i_{j})}_{i_{2}\dots i_{j-1}}\\\ 1&\cos\theta^{[j-1](i_{2}\dots
i_{j})}_{i_{2}\dots i_{j-1}}\\\
\end{array}\right|}{\sin\theta^{[j-1](i_{1}\dots i_{j-1})}_{i_{2}\dots
i_{j-1}}\sin\theta^{[j-1](i_{2}\dots i_{j})}_{i_{2}\dots i_{j-1}}}.$ (92)

This can be extended so that the angles of any facet can be expressed in terms
of those for any other dimensional facets to give a cosine rule of the form

$\cos\theta^{[j]}_{i_{2}\dots
i_{j-1}}=\frac{-\left|\begin{array}[]{cccc}\cos\theta^{[k](i_{1}\dots
i_{k}i_{k+1})}_{i_{2}\dots i_{k}}&\cos\theta^{[k](i_{1}\dots
i_{k}i_{k+2})}_{i_{2}\dots i_{k}}&\dots&\cos\theta^{[k](i_{1}\dots
i_{k}i_{j})}_{i_{2}\dots i_{k}}\\\ \cos\theta^{[k](i_{2}\dots
i_{k}i_{k+1})}_{i_{2}\dots i_{k+1}i_{k+1}}&\cos\theta^{[k](i_{2}\dots
i_{k+2})}_{i_{2}\dots i_{k}}&\dots&\cos\theta^{[k](i_{2}\dots
i_{k+1}i_{j})}_{i_{2}\dots i_{k}}\\\ \vdots&\vdots&\ddots&\vdots\\\
\cos\theta^{[k](i_{j-1}i_{2}\dots i_{k+1})}_{i_{2}\dots
i_{k}}&\cos\theta^{[k](i_{j-1}i_{2}\dots i_{k}i_{k+2})}_{i_{2}\dots
i_{k}}&\dots&\cos\theta^{[k](i_{j-1}i_{2}\dots i_{k}i_{j})}_{i_{2}\dots
i_{k}}\\\ \end{array}\right|}{\sin\left(\Theta^{[k](i_{1}\dots
i_{k})}_{i_{2}\dots i_{k}}\right)\sin\left(\Theta^{[k](i_{2}\dots
i_{k+1})}_{i_{2}\dots i_{k}}\right)}.$ (93)

From these cosine rules, there arise corresponding sine rules. Specifically,
note that

$\sin\theta^{[j]}_{i_{2}\dots i_{j}}=k_{j-1}\sin\theta^{[j-1]}_{i_{2}\dots
i_{j-1}},$ (94)

where

$k_{j-1}=\frac{\sin\left(\theta^{[j-1](i_{1}\dots i_{j})}_{i_{2}\dots
i_{j-1}},\theta^{[j-1](i_{2}\dots i_{j+1})}_{i_{2}\dots
i_{j-1}},\theta^{[j-1](i_{1}\dots i_{j-1}i_{j+1})}_{i_{2}\dots
i_{j-1}}\right)}{\sin\left(\theta^{[j-1](i_{1}\dots i_{j})}_{i_{2}\dots
i_{j-1}}\right)\sin\left(\theta^{[j-1](i_{2}\dots i_{j+1})}_{i_{2}\dots
i_{j-1}}\right)\sin\left(\theta^{[j-1](i_{1}\dots i_{j-1}i_{j+1})}_{i_{2}\dots
i_{j-1}}\right)},$ (95)

with $k_{j-1}$ being constant. Hence, from this, it follows that a full
hierarchy of intertwined sine rules exists between the angles of any two
different dimensional facets.

##  IV Link between Hyperspherical Trigonometry and Elliptic Functions

###  IV.1 Link between Spherical Trigonometry and Jacobi Elliptic Functions

There exists a well-known link between the formulae of spherical trigonometry
and the Jacobi elliptic functions through their addition formulaeLagrange ;
Legendre . Here, we follow a derivation given by Irwinirwin , but with a
slight modification, taking as a starting point the sine rule for spherical
trigonometry,

$\frac{\sin\alpha_{i}}{\sin\theta_{jk}}=\frac{\sin\alpha_{j}}{\sin\theta_{ik}}=\frac{\sin\alpha_{k}}{\sin\theta_{ij}}=k,$
(96)

where

$k=\frac{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}{\sin\theta_{ij}\sin\theta_{ik}\sin\theta_{jk}},$
(97)

and considering the expression

$W=k^{2}\sin^{2}\theta_{ij}\sin^{2}\theta_{ik}\sin^{2}\theta_{jk}-\sin^{2}(\theta_{ij},\theta_{ik},\theta_{jk}).$
(98)

It follows that the derivative of $W$ with respect to one of the angles,
$\theta_{ij}$, is

$\begin{split}\frac{\partial
W}{\partial\theta_{ij}}&=-2\sin\theta_{ij}(\cos\theta_{ij}-\cos\theta_{ik}\cos\theta_{jk}-k^{2}\cos\theta_{ij}\sin^{2}\theta_{ik}\sin^{2}\theta_{jk}),\\\
&=2\sin\theta_{ij}\sin\theta_{ik}\sin\theta_{jk}\cos\alpha_{i}\cos\alpha_{j}.\end{split}$
(99)

If the radial angles $\theta_{ij}$, $\theta_{ik}$ and $\theta_{jk}$ all vary
in such a way as to keep $k$ constant, then since this is equivalent to $W=0$,
this implies

$\cos\alpha_{i}\cos\alpha_{j}\mathrm{d}\theta_{ij}+\cos\alpha_{i}\cos\alpha_{k}\mathrm{d}\theta_{ik}+\cos\alpha_{j}\cos\alpha_{k}\mathrm{d}\theta_{jk}=0,$
(100)

which is of course equivalent to

$\frac{\mathrm{d\theta_{ij}}}{\cos\alpha_{k}}+\frac{\mathrm{d\theta_{jk}}}{\cos\alpha_{i}}+\frac{\mathrm{d\theta_{ik}}}{\cos\alpha_{j}}=0.$
(101)

However, the sine rule gives

$\cos\alpha_{i}=\pm\sqrt{1-k^{2}\sin^{2}\theta_{jk}},\,\mathrm{etc.},$ (102)

and so, as the signs of $\theta_{ij}$, $\theta_{ik}$ and $\theta_{jk}$ can be
chosen arbitrarily, without loss of generality they can all be chosen to be
positive, giving

$\frac{\mathrm{d\theta_{ij}}}{\sqrt{1-k^{2}\sin^{2}\theta_{ij}}}+\frac{\mathrm{d\theta_{ik}}}{\sqrt{1-k^{2}\sin^{2}\theta_{ik}}}+\frac{\mathrm{d\theta_{jk}}}{\sqrt{1-k^{2}\sin^{2}\theta_{jk}}}=0$
(103)

Writing $\theta_{ij}=\mathrm{am}(a_{k})$, with

$a_{k}=\int_{0}^{\mathrm{am}(a_{k})}\frac{dt}{\sqrt{1-k^{2}\sin^{2}t}},$ (104)

and similarly, $\theta_{ik}=\mathrm{am}(a_{j})$ and
$\theta_{jk}=\mathrm{am}(a_{i})$, the integral of the differential relation,
(103), implies that

$a_{i}+a_{j}+a_{k}=\delta,\,\mathrm{constant},$ (105)

together with

$\begin{split}&\sin\theta_{ij}=\sin(\mathrm{am}(a_{k}))=\mathrm{sn}(a_{k}),\\\
&\sin\theta_{ik}=\sin(\mathrm{am}(a_{j}))=\mathrm{sn}(a_{j}),\\\
&\sin\theta_{jk}=\sin(\mathrm{am}(a_{i}))=\mathrm{sn}(a_{i}).\end{split}$
(106)

Therefore, having introduced uniformising variables $a_{i}$, $i=1,2,3,$
associated with the three spherical angles $\theta_{jk}$, the various
spherical trigonometric functions can be identified with the Jacobi elliptic
functions by using the identifications

$\sin(\theta_{jk})=\text{sn}(a_{i};k)\iff\sin\alpha_{i}\equiv
k\,\text{sn}(a_{i};k),$ (107)

in which $k$ is the modulus of the elliptic function given by (97). These
identifications, through the usual relations between the three Jacobi elliptic
functions $\mathrm{sn}$, $\mathrm{cn}$ and $\mathrm{dn}$,

$\begin{array}[]{cc}\mathrm{cn}^{2}(u;k)+\mathrm{sn}^{2}(u;k)=1,\hskip
28.45274pt\mathrm{dn}^{2}(u;k)+k^{2}\mathrm{sn}^{2}(u;k)=1,\\\ \end{array}$
(108)

lead to

$\cos\theta_{jk}=\text{cn}(a_{i};k),\,\mathrm{and}\,\cos\alpha_{i}=\text{dn}(a_{i};k),$
(109)

for $i,j,k$ cyclic. Addition formulae follow readily from the various
spherical trigonometric relations. In particular, the cosine and polar cosine
rules yield the relations

$\begin{array}[]{c}\mathrm{cn}(a_{i})=\mathrm{cn}(a_{j})\mathrm{cn}(a_{k})+\mathrm{sn}(a_{j})\mathrm{sn}(a_{k})\mathrm{dn}(a_{i}),\\\
\mathrm{dn}(a_{i})=-\mathrm{dn}(a_{j})\mathrm{dn}(a_{k})+k^{2}\mathrm{sn}(a_{j})\mathrm{sn}(a_{k})\mathrm{cn}(a_{i}),\end{array}$
(110)

respectively. Solving these relations as functions of $a_{i}$ gives

$\mathrm{cn}(a_{i})=\frac{\mathrm{dn}(a_{j})\mathrm{dn}(a_{k})\mathrm{sn}(a_{j})\mathrm{sn}(a_{k})-\mathrm{cn}(a_{j})\mathrm{cn}(a_{k})}{1-k^{2}\mathrm{sn}^{2}(a_{j})\mathrm{sn}^{2}(a_{k})},$
(111)
$\mathrm{dn}(a_{i})=\frac{\mathrm{dn}(a_{j})\mathrm{dn}(a_{k})-k^{2}\mathrm{cn}(a_{j})\mathrm{cn}(a_{k})\mathrm{sn}(a_{j})\mathrm{sn}(a_{k})}{1-k^{2}\mathrm{sn}^{2}(a_{j})\mathrm{sn}^{2}(a_{k})},$
(112)

the addition formulae for the Jacobi elliptic functions with

$\begin{array}[]{c}\mathrm{cn}(a_{i})=-\mathrm{cn}(a_{j}+a_{k}),\\\
\mathrm{dn}(a_{i})=\mathrm{dn}(a_{j}+a_{k}).\\\ \end{array}$ (113)

The Jacobi Elliptic functions are periodic in $K(k)$ and $K^{\prime}(k)$ as

$\begin{array}[]{c}\mathrm{cn}(a_{j}+2mK+2niK^{\prime};k)=(-1)^{m}\mathrm{cn}(a_{j};k),\\\
\mathrm{dn}(a_{j}+2mK+2niK^{\prime};k)=(-1)^{n}\mathrm{dn}(a_{j};k),\\\
\end{array}$ (114)

for all $a_{j}$, and so we must restrict $\delta$ such that
$a_{i}+a_{j}+a_{k}=2K+2iK^{\prime}$. The addition formula for $\mathrm{sn}$
follows as a consequence. From these addition formulae a number of intertwined
addition relations follow

$\text{cn}(a_{j})\text{sn}(a_{i}+a_{j})=\text{sn}(a_{i})\text{dn}(a_{j})+\text{dn}(a_{i})\text{sn}(a_{j})\text{cn}(a_{i}+a_{j}),$
(115)
$\text{dn}(a_{i})\text{sn}(a_{i}+a_{j})=\text{cn}(a_{i})\text{sn}(a_{j})+\text{sn}(a_{i})\text{cn}(a_{j})\text{dn}(a_{i}+a_{j}),$
(116)
$\text{sn}(a_{i})\text{cn}(a_{i}+a_{j})+\text{sn}(a_{j})\text{dn}(a_{i}+a_{j})=\text{cn}(a_{i})\text{dn}(a_{j})\text{sn}(a_{i}+a_{j}).$
(117)

Denoting

$w_{1}(a_{j})=\frac{\rho}{\text{sn}(a_{j})},\,w_{2}(a_{j})=\frac{\rho\,\text{cn}(a_{j})}{\text{sn}(a_{j})},\,w_{3}(a_{j})=\frac{\rho\,\text{dn}(a_{j})}{\text{sn}(a_{j})},$
(118)

these addition formulae may be rewritten as

$w_{i}(a_{j})w_{j}(a_{i})+w_{j}(a_{k})w_{k}(a_{j})+w_{k}(a_{i})w_{i}(a_{k})=0,\,\,i,j,k=1,2,3,$
(119)

with $a_{1}+a_{2}+a_{3}=2K+2iK^{\prime}$. Note that this relation is the
functional Yang-Baxter relationSklyanin .

Now, by considering the reciprocal of k,

$\frac{\sin\theta_{jk}}{\sin\alpha_{i}}=\frac{\sin\theta_{ik}}{\sin\alpha_{j}}=\frac{\sin\theta_{ij}}{\sin\alpha_{k}}=\frac{1}{k},$
(120)

following Irwin’s method, we can derive a similar, yet slightly simpler,
relationship. First, recall

$k=\frac{\sin\alpha_{k}}{\sin\theta_{ij}}=\frac{\sin\alpha_{i}\sin\alpha_{j}\sin\alpha_{k}}{\sqrt{1-\cos^{2}\alpha_{i}-\cos^{2}\alpha_{j}-\cos^{2}\alpha_{k}-2\cos\alpha_{i}\cos\alpha_{j}\cos\alpha_{k}}},$
(121)

and let

$\overline{W}=\frac{1}{k^{2}}\sin^{2}\alpha_{i}\sin^{2}\alpha_{j}\sin^{2}\alpha_{k}-(1-\cos^{2}\alpha_{i}-\cos^{2}\alpha_{j}-\cos^{2}\alpha_{k}-2\cos\alpha_{i}\cos\alpha_{j}\cos\alpha_{k}).$
(122)

Now, consider the derivative $\partial\overline{W}/\partial\alpha_{i}$,

$\partial\overline{W}/\partial\alpha_{i}=-2\cos\theta_{ij}\cos\theta_{ik}\sin\alpha_{i}\sin\alpha_{j}\sin\alpha_{k}.$
(123)

Varying $\alpha_{i}$, $\alpha_{j}$ and $\alpha_{k}$ as to keep k constant, it
follows that

$\frac{\mathrm{d}\alpha_{i}}{\sqrt{1-\frac{1}{k^{2}}\sin^{2}\alpha_{i}}}+\frac{\mathrm{d}\alpha_{j}}{\sqrt{1-\frac{1}{k^{2}}\sin^{2}\alpha_{i}}}+\frac{\mathrm{d}\alpha_{k}}{\sqrt{1-\frac{1}{k^{2}}\sin^{2}\alpha_{i}}}=0.$
(124)

Writing $\alpha_{i}=\mathrm{am}(b_{i})$, with

$b_{i}=\int_{0}^{am(b_{i})}\frac{\mathrm{d}t}{\sqrt{1-\frac{1}{k^{2}}\sin^{2}t}},$
(125)

and similarly, $\alpha_{j}=\mathrm{am}(b_{j})$ and
$\alpha_{k}=\mathrm{am}(b_{k})$, the integral of (124) implies

$b_{i}+b_{j}+b_{k}=\gamma,\,\,\mathrm{constant},$ (126)

together with

$\begin{split}&\sin\alpha_{i}=\sin(\mathrm{am}(b_{i}))=\mathrm{sn}(b_{i}),\\\
&\sin\alpha_{j}=\sin(\mathrm{am}(b_{j}))=\mathrm{sn}(b_{j}),\\\
&\sin\alpha_{k}=\sin(\mathrm{am}(b_{k}))=\mathrm{sn}(b_{k}).\\\ \end{split}$
(127)

Therefore, having introduced spherical angles $b_{i}$, $i=1,2,3$, this time
associated with the three spherical angles $\alpha_{i}$, the various spherical
trigonometric functions can be identified with the Jacobi elliptic functions
via the identifications

$\sin\alpha_{i}=\mathrm{sn}\left(b_{i};\frac{1}{k}\right)\iff\sin\theta_{jk}=\frac{1}{k}\mathrm{sn}\left(b_{i};\frac{1}{k}\right),$
(128)

in which $1/k$ is the modulus of the elliptic function. These identifications
lead to

$\cos\alpha_{i}=\mathrm{cn}\left(b_{i};\frac{1}{k}\right),\,\,\mathrm{and}\,\,\cos\theta_{jk}=\mathrm{dn}\left(b_{i};\frac{1}{k}\right).$
(129)

In this case, the Jacobi elliptic function addition formulae are satisfied
providing we have

$\begin{split}&\mathrm{cn}(b_{i})=\mathrm{cn}(b_{j}+b_{k}),\\\
&\mathrm{dn}(b_{i})=-\mathrm{dn}(b_{j}+b_{k}).\end{split}$ (130)

The periodicity ensures $\gamma=2K$,

$b_{i}+b_{j}+b_{k}=2K.$ (131)

Note that making the identification in this way results in a slightly simpler
restriction, dependent only on $K$, and not also $K^{\prime}$.

###  IV.2 Link between Hyperspherical Trigonometry and Elliptic Functions

We now look to provide a similar link for the 4 dimensional hyperspherical
case and elliptic functions following a similar procedure as before. We take
as a starting point the hyperspherical sine rule relation,

$\frac{\sin(\phi_{ik},\phi_{jk},\phi_{kl})}{\sin(\theta_{ij},\theta_{il},\theta_{jl})}=\frac{\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\sin(\theta_{ij},\theta_{il},\theta_{jl})\sin(\theta_{ik},\theta_{il},\theta_{kl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})}=k,$
(132)

from which we introduce

$W=\sin^{4}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})-k^{2}\sin^{2}(\theta_{ij},\theta_{ik},\theta_{jk})\sin^{2}(\theta_{ij},\theta_{il},\theta_{jl})\sin^{2}(\theta_{ik},\theta_{il},\theta_{kl})\sin^{2}(\theta_{jk},\theta_{jl},\theta_{kl}).$
(133)

Taking $W$’s derivative with respect to $\theta_{ij}$ gives

$\begin{split}\frac{\partial
W}{\partial\theta_{ij}}=&2\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})\left(\frac{\partial\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})}{\partial\theta_{ij}}\right)\\\
&-k^{2}\left(\frac{\partial\sin^{2}(\theta_{ij},\theta_{ik},\theta_{jk})}{\partial\theta_{ij}}\right)\sin^{2}(\theta_{ij},\theta_{il},\theta_{jl})\sin^{2}(\theta_{ik},\theta_{il},\theta_{kl})\sin^{2}(\theta_{jk},\theta_{jl},\theta_{kl})\\\
&-k^{2}\sin^{2}(\theta_{ij},\theta_{ik},\theta_{jk})\left(\frac{\partial\sin^{2}(\theta_{ij},\theta_{il},\theta_{jl})}{\partial\theta_{ij}}\right)\sin^{2}(\theta_{ik},\theta_{il},\theta_{kl})\sin^{2}(\theta_{jk},\theta_{jl},\theta_{kl}).\end{split}$
(134)

Considering the derivatives of the squares of the triple sine functions, we
have

$\begin{split}\frac{\partial\sin^{2}(\theta_{ij},\theta_{ik},\theta_{jk})}{\partial\theta_{ij}}&=2\cos\theta_{ij}\sin\theta_{ij}-2\sin\theta_{ij}\cos\theta_{ik}\cos\theta_{jk}\\\
&=2\cos\alpha_{k}^{(ijk)}\sin\theta_{ij}\sin\theta_{ik}\sin\theta_{jk},\end{split}$
(135)

and similarly,

$\frac{\partial\sin^{2}(\theta_{ij},\theta_{il},\theta_{jl})}{\partial\theta_{ij}}=2\cos\alpha_{l}^{(ijl)}\sin\theta_{ij}\sin\theta_{il}\sin\theta_{jl}.$
(136)

We also have, for the six term sine function,

$\frac{\partial\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})}{\partial\theta_{ij}}=-2\sin\theta_{ij}\cos\phi_{kl}\sin(\theta_{ik},\theta_{il},\theta_{kl})\sin(\theta_{jk},\theta_{jl},\theta_{kl}).$
(137)

Substituting these back into (134) gives

$\begin{split}\frac{\partial
W}{\theta_{ij}}=&-2\sin\theta_{ij}\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})\\\
&\times\Bigg{(}2\cos\phi_{kl}\sin(\theta_{ik},\theta_{il},\theta_{kl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\\\
&+\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})\left(\frac{\cos\alpha_{k}^{(ijk)}\sin\theta_{ik}\sin\theta_{jk}}{\sin^{2}(\theta_{ij},\theta_{ik},\theta_{jk})}+\frac{\cos\alpha_{l}^{(ijl)}\sin\theta_{il}\sin\theta_{jl}}{\sin^{2}(\theta_{ij},\theta_{il},\theta_{jl})}\right)\Bigg{)}.\end{split}$
(138)

In order to simplify this derivative, we now consider the Desnanot-Jacobi
determinantal identity,

$\begin{split}&\left|\begin{array}[]{cc}\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{kl}&1\\\
\end{array}\right|\left|\begin{array}[]{cccc}1&\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{il}\\\
\cos\theta_{ij}&1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{ik}&\cos\theta_{jk}&1&\cos\theta_{kl}\\\
\cos\theta_{il}&\cos\theta_{jl}&\cos\theta_{kl}&1\\\ \end{array}\right|\\\
&=\left|\begin{array}[]{ccc}1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{jk}&1&\cos\theta_{kl}\\\ \cos\theta_{jl}&\cos\theta_{kl}&1\\\
\end{array}\right|\left|\begin{array}[]{ccc}1&\cos\theta_{ik}&\cos\theta_{il}\\\
\cos\theta_{ij}&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{il}&\cos\theta_{kl}&1\\\
\end{array}\right|-\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{il}\\\
1&\cos\theta_{jk}&\cos\theta_{jl}\\\ \cos\theta_{jl}&\cos\theta_{kl}&1\\\
\end{array}\right|\left|\begin{array}[]{ccc}\cos\theta_{ij}&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{ik}&1&\cos\theta_{kl}\\\ \cos\theta_{il}&\cos\theta_{kl}&1\\\
\end{array}\right|,\end{split}$ (139)

which, reduces neatly to

$\frac{\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})\cos\alpha_{l}^{(jkl)}\sin\theta_{jl}\sin\theta_{kl}}{\sin^{2}(\theta_{jk},\theta_{jl},\theta_{kl})}=\sin(\theta_{ij},\theta_{il},\theta_{jl})\sin(\theta_{ik},\theta_{il},\theta_{kl})\left(\cos\phi_{jl}\cos\phi_{kl}-\cos\phi_{il}\right).$
(140)

Substituting this expression into (138), the derivative reduces to

$\frac{\partial
W}{\partial\theta_{ij}}=-2\sin\theta_{ij}\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{ik},\theta_{il},\theta_{kl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\left(\cos\phi_{il}\cos\phi_{jl}+\cos\phi_{ik}\cos\phi_{jk}\right),$
(141)

which, using (56), may be rewritten as

$\begin{split}\frac{\partial
W}{\partial\theta_{ij}}=-2\sin(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})&\sin(\theta_{ik},\theta_{il},\theta_{kl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{ij},\theta_{il},\theta_{jl})\sin(\theta_{ij},\theta_{ik},\theta_{jk})\\\
&\cdot\sin\phi_{ij}\left(\cos\phi_{il}\cos\phi_{jl}+\cos\phi_{ik}\cos\phi_{jk}\right).\end{split}$
(142)

If we vary the six ‘$\theta$’s such that k remains constant, then since this
is equivalent to $W=0$, setting $\mathrm{d}W=0$ implies

$\sum_{perm}\sin\phi_{ij}\left(\cos\phi_{il}\cos\phi_{jl}+\cos\phi_{ik}\cos\phi_{jk}\right)\mathrm{d}\theta_{ij}=0,$
(143)

where $perm$ denotes all six index pairs $ij,ik,il,jk,jl,kl$. In spite of the
similarity between (143) and the analogous formula (100) in the spherical
case, the former cannot be simplified along the same line as in the spherical
case. A way out, albeit not quite satisfactory, is to focus on a special
symmetric case, in which the opposite dihedral angles are equivalent, i.e.

$\phi_{ij}=\phi_{kl},\quad\phi_{ik}=\phi_{jl},\quad\phi_{il}=\phi_{jk}\ ,$

which we will refer to as the symmetric hyperspherical tetrahedron.

Restricting ourselves to this symmetric case, the differential relation (143)
reduces to

$\sum_{perm}\sin\phi_{ij}\,\cos\phi_{ik}\,\cos\phi_{jk}\,\mathrm{d}\theta_{ij}=0,$
(144)

or equivalently

$\sum_{perm}\frac{\sin\phi_{ij}}{\cos\phi_{ij}}\,\mathrm{d}\theta_{ij}=0.$
(145)

The restrictions on the tetrahedron being symmetric also simplify the
hyperspherical sine rule (51), reducing it to

$\sin^{2}\phi_{ij}=k_{H}\sin^{2}\theta_{ij}\ ,$ (146)

and substituting this into (145) we obtain

$\sum_{perm}\frac{\sqrt{k_{H}}\,\sin\theta_{ij}}{\sqrt{1-k_{H}\sin^{2}\theta_{ij}}}\,\mathrm{d}\theta_{ij}=0,$
(147)

which yields a sum of elliptic integrals of the form

$\int\frac{\sqrt{k_{H}}\,u}{\sqrt{(1-k_{H}u^{2})(1-u^{2})}}\,\mathrm{d}u\ .$

Thus, for the case of a symmetric hyperspherical tetrahedron, the
hyperspherical trigonometric functions which in a sense uniformise the
hyperspherical sine rule (132) are again associated with elliptic functions,
suggesting that for the general case this remains true. However, in that
general case the connection becomes a bit more entangled and require the
introduction of a generalised class of elliptic functions, introduced by
Pawellekpawellek .

##  V Generalised Jacobi Elliptic Functions and their link to Hyperspherical
Trigonometry

In this section we review a class of ’generalised’ Jacobi elliptic functions
as introduced by Pawellekpawellek , and provide a link between these functions
and the formulae of hyperspherical trigonometry.

###  V.1 Generalised Jacobi Elliptic Functions

In pawellek , Pawellek introduced the generalised Jacobi elliptic functions
$s(u,k_{1},k_{2})$, $c(u,k_{1},k_{2})$, $d_{1}(u,k_{1},k_{2})$ and
$d_{2}(u,k_{1},k_{2})$. These functions are algebraic curves with two distinct
moduli, and are based upon Jacobi’s elliptic functionsJacobi . After assuming
without loss of generality that $1>k_{1}>k_{2}>0$ as moduli parameters, they
are defined as the inversion of the hyperelliptic integrals

$u(x,k_{1},k_{2})=\int_{0}^{x=s(u)}\frac{dt}{\sqrt{(1-t^{2})(1-k_{1}^{2}t^{2})(1-k_{2}^{2}t^{2})}},$
(148a)
$u(x,k_{1},k_{2})=\int_{x=c(u)}^{1}\frac{dt}{\sqrt{(1-t^{2})(k_{1}^{\prime
2}+k_{1}^{2}t^{2})(k_{2}^{\prime 2}+k_{2}^{2}t^{2})}},$ (148b)
$u(x,k_{1},k_{2})=k_{1}\int_{x=d_{1}(u)}^{1}\frac{dt}{\sqrt{(1-t^{2})(t^{2}-k_{1}^{\prime
2})(k_{1}^{2}-k_{2}^{2}+k_{2}^{2}t^{2})}},$ (148c)
$u(x,k_{1},k_{2})=k_{2}\int_{x=d_{2}(u)}^{1}\frac{dt}{\sqrt{(1-t^{2})(t^{2}-k_{2}^{\prime
2})(k_{2}^{2}-k_{1}^{2}+k_{1}^{2}t^{2})}},$ (148d)

respectively, with $k_{i}^{\prime}=\sqrt{1-k_{i}^{2}}$. These generalised
Jacobi functions are associated with an algebraic curve of the form

$\mathcal{C}:y^{2}=(1-x^{2})(1-k_{1}^{2}x^{2})(1-k_{2}^{2}x^{2}),$ (149)

which can be modeled as a Riemann surface of genus 2, but is, in fact, a
double cover of an elliptic curve $\mathcal{E}$,

$\mathcal{E}:w^{2}=z(1-z)(1-k_{1}^{2}z)(1-k_{2}^{2}z),$ (150)

via the cover map $\mathcal{C}\overset{\pi}{\rightarrow}\mathcal{E}$,

$(w,z)=\pi(y,x)=(xy,x^{2}).$ (151)

The functions satisfy a number of identities,

$\begin{split}&c^{2}(u)=1-s^{2}(u),\hskip
28.45274ptd_{1}^{2}(u)=1-k_{1}^{2}s^{2}(u),\hskip
28.45274ptd_{2}^{2}(u)=1-k_{2}^{2}s^{2}(u),\\\
&d_{i}^{2}(u)-k_{i}^{2}c^{2}(u)=1-k_{i}^{2},\hskip 14.22636pti=1,2;\hskip
28.45274ptk_{1}^{2}d_{2}^{2}(u)-k_{2}^{2}d_{1}^{2}(u)=k_{1}^{2}-k_{2}^{2},\\\
\end{split}$ (152)

and are related to the Jacobi elliptic functions by

$\displaystyle s(u,k_{1},k_{2})$
$\displaystyle=\frac{\mathrm{sn}(k_{2}^{\prime}u,\kappa)}{\sqrt{k_{2}^{\prime
2}+k_{2}^{2}sn^{2}(k_{2}^{\prime}u,\kappa)}},$ $\displaystyle
c(u,k_{1},k_{2})$
$\displaystyle=\frac{k_{2}^{\prime}\mathrm{cn}(k_{2}^{\prime}u,\kappa)}{\sqrt{1-k_{2}^{2}cn^{2}(k_{2}^{\prime}u,\kappa)}},$
(153a) $\displaystyle d_{1}(u,k_{1},k_{2})$
$\displaystyle=\frac{\sqrt{k_{1}^{2}-k_{2}^{2}}\mathrm{dn}(k_{2}^{\prime}u,\kappa)}{\sqrt{k_{1}^{2}-k_{2}^{2}\mathrm{dn}^{2}(k_{2}^{\prime}u,\kappa)}},$
$\displaystyle d_{2}(u,k_{1},k_{2})$
$\displaystyle=\frac{\sqrt{k_{1}^{2}-k_{2}^{2}}}{\sqrt{k_{1}^{2}-k_{2}^{2}\mathrm{dn}^{2}(k_{2}^{\prime}u,\kappa)}},$
(153b)

with

$\kappa=\frac{k_{1}^{2}-k_{2}^{2}}{1-k_{2}^{2}}.$ (154)

The first derivatives of these functions are given by

$\displaystyle s^{\prime}(u)$ $\displaystyle=c(u)d_{1}(u)d_{2}(u),$
$\displaystyle c^{\prime}(u)$ $\displaystyle=-s(u)d_{1}(u)d_{2}(u),$ (155)
$\displaystyle d_{1}^{\prime}(u)$ $\displaystyle=-k_{1}^{2}s(u)c(u)d_{2}(u),$
$\displaystyle d_{2}^{\prime}(u)$ $\displaystyle=-k_{2}^{2}s(u)c(u)d_{1}(u).$
(156)

These generalised Jacobi functions with moduli $k_{1}$ and $k_{2}$ satisfy the
following addition formulae:

$s(u\pm v)=\frac{s(u)d_{2}(u)c(v)d_{1}(v)\pm
s(v)d_{2}(v)c(u)d_{1}(u)}{\sqrt{\left[d_{2}^{2}(u)d_{2}^{2}(v)-\kappa^{2}k_{2}^{{}^{\prime}4}s^{2}(u)s^{2}(v)\right]^{2}+k^{2}_{2}\left[s(u)d_{2}(u)c(v)d_{1}(v)\pm
s(v)d_{2}(v)c(u)d_{1}(u)\right]}},$ (157a) $c(u\pm
v)=\frac{c(u)d_{2}(u)c(v)d_{2}(v)\mp
k_{2}^{{}^{\prime}2}s(u)d_{1}(u)s(v)d_{1}(v)}{\sqrt{\left[d_{2}^{2}(u)d_{2}^{2}(v)-\kappa^{2}k_{2}^{{}^{\prime}4}s^{2}(u)s^{2}(v)\right]^{2}+k^{2}_{2}\left[s(u)d_{2}(u)c(v)d_{1}(v)\pm
s(v)d_{2}(v)c(u)d_{1}(u)\right]}},$ (157b) $d_{1}(u\pm
v)=\frac{d_{1}(u)d_{2}(u)d_{1}(v)d_{2}(v)\mp\kappa^{2}k_{2}^{{}^{\prime}2}s(u)c(u)s(v)c(v)}{\sqrt{\left[d_{2}^{2}(u)d_{2}^{2}(v)-\kappa^{2}k_{2}^{{}^{\prime}4}s^{2}(u)s^{2}(v)\right]^{2}+k^{2}_{2}\left[s(u)d_{2}(u)c(v)d_{1}(v)\pm
s(v)d_{2}(v)c(u)d_{1}(u)\right]}},$ (157c) $d_{2}(u\pm
v)=\frac{d_{2}^{2}(u)d_{2}^{2}(v)-\kappa^{2}k_{2}^{{}^{\prime}4}s^{2}(u)s^{2}(v)}{\sqrt{\left[d_{2}^{2}(u)d_{2}^{2}(v)-\kappa^{2}k_{2}^{{}^{\prime}4}s^{2}(u)s^{2}(v)\right]^{2}+k^{2}_{2}\left[s(u)d_{2}(u)c(v)d_{1}(v)\pm
s(v)d_{2}(v)c(u)d_{1}(u)\right]}}.$ (157d)

These formulae follow from the addition formula for the standard Jacobi
elliptic functions.

###  V.2 Link between Hyperspherical Trigonometry and the Generalised Jacobi
Elliptic Functions

We are also able to show a connection between the generalised Jacobi elliptic
functions and hyperspherical trigonometry. As the generalised Jacobi elliptic
functions are dependent on two moduli, then so must the hyperspherical
trigonometry be. It is the interplay of these moduli that govern the
connection. Recall, that for a hyperspherical tetrahedron, we have

$\sin\alpha_{i}^{(ijk)}=k_{1}\sin\theta_{jk},$ (158)

and

$\sin\phi_{il}=k_{2}\sin\alpha_{i}^{(ijk)},$ (159)

for all $i,j,k,l=1,2,3,4$, where

$k_{1}=\frac{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}{\sin\theta_{ij}\sin\theta_{ik}\sin\theta_{jk}},$
(160)

and

$k_{2}=\frac{\sin(\alpha_{i}^{(ijk)},\alpha_{i}^{(ijl)},\alpha_{i}^{(ikl)})}{\sin\alpha_{i}^{(ijk)}\sin\alpha_{i}^{(ijl)}\sin\alpha_{i}^{(ikl)}},$
(161)

respectively. Separately, these each have the same link to the Jacobi elliptic
functions as the functions of a spherical triangle. They imply that

$\sin\phi_{il}=k_{1}k_{2}\sin\theta_{jk}.$ (162)

Introducing uniformising variables $a_{jk}$, $j,k=1,2,3,4$ with $k>j$,
associated with the six $\theta_{jk}$, such that

$a_{jk}=\int_{0}^{\theta_{jk}}\frac{\mathrm{d}t}{\sqrt{(1-k_{1}^{2}\sin^{2}t)(1-k_{1}^{2}k_{2}^{2}\sin^{2}t)}},$
(163)

with $\theta_{jk}=\mathrm{am}(a_{jk})$, the various hyperspherical
trigonometric functions can be identified with Pallewek’s generalised Jacobi
elliptic functions via the identifications

$s(a_{jk})\equiv\sin\theta_{jk}\iff k_{1}s(a_{jk})=\sin\alpha_{i}^{(ijk)}\iff
k_{1}k_{2}s(a_{jk})=\sin\phi_{il},$ (164)

in which, $k_{1}$ and $k_{1}k_{2}$ are the two moduli of the functions, and
are as given earlier. From these and the identities listed previously it
follows that

$c(a_{jk})=\cos\theta_{jk},\;d_{1}(a_{jk})=\cos\alpha_{i}^{(ijk)},\;d_{2}(a_{jk})=\cos\phi_{il}.$
(165)

Note that under these identifications the identities (152) for the generalised
Jacobi functions are still obeyed.

The modulus $k_{1}$ governs the relations between the spherical trigonometry
of each of the faces, and the elliptic functions. The second modulus,
$k_{1}k_{2}$, acts as an overall modulus for the spherical tetrahedron. Note
its dependence on the first modulus, $k_{1}$. These moduli remain constant, so
that if one of the vertices of the spherical tetrahedron were moved, the
others must be adjusted to compensate. These movements result in a change to
the faces of the tetrahedron, but not to $k_{1}$, the facial modulus.

##  VI Examples

As an example of where these hyperspherical formulae and their connection with
the generalised Jacobi elliptic functions may be used, we consider a
multidimensional generalisation of the Euler top in relation to Nambu
mechanicsNambu . We provide a link between this example and the double
elliptic model.

###  VI.1 Multidimensional Euler Top

The Euler top describes a free top, moving in the absence of any external
torque. In their paper on Nambu Quantum mechanicsMinic , Minic and Tze
reformulated the mechanics of this top in terms of a Nambu system of order
three, which they then solved in terms of the Jacobi elliptic functions. We
consider a higher dimensional analogue of their example, formulating the top’s
mechanics in terms of a Nambu system of order four, and then solving this
system in terms of the Generalised Jacobi elliptic functions.

The Euler top admits two conserved quantities, the total energy,

$H_{1}=\frac{1}{2}\sum_{i=1}^{3}\frac{1}{I_{i}}M_{i}^{2},$ (166)

and the square of the angular momentum,

$H_{2}=\frac{1}{2}\sum_{i=1}^{3}M_{i}^{2},$ (167)

where $I_{i}$ denote the principal moments of inertia. Minic and Tze take
these both to be Hamiltonians for the the system, with the action of the top
given by

$S=\int M_{1}dM_{2}\wedge dM_{3}-H_{1}dH_{2}\wedge dt.$ (168)

The equations of motion,

$\frac{dM_{i}}{dt}=\\{H_{1},H_{2},M_{i}\\},\,i=1,2,3,$ (169)

follow from $\delta S=0$, where $\\{H_{1},H_{2},M_{i}\\}$ is the Nambu
3-bracket, a ternary bracket, defined by NambuNambu , satisfying the axioms
for Nambu brackets given by Takhtajantakhtajan :

  * •

Skew-symmetry

$\\{A_{1},A_{2},A_{3}\\}=(-1)^{\epsilon(p)}\\{A_{p(1},A_{p(2)},A_{p(3)}\\},$
(170)

where $p(i)$ is the permutation of indices and $\epsilon(p)$ is the parity of
the permutation.

  * •

Derivation (the Leibniz rule)

$\\{A_{1}A_{2},A_{3},A_{4}\\}=A_{1}\\{A_{2},A_{3},A_{4}\\}+\\{A_{1},A_{3},A_{4}\\}A_{2}.$
(171)

  * •

Fundamental Identity (analogue of Jacobi identity)

$\begin{split}\left\\{\left\\{A_{1},A_{2},A_{3}\right\\},A_{4},A_{5}\right\\}&+\left\\{A_{3},\left\\{A_{1},A_{2},A_{4}\right\\},A_{5}\right\\}+\left\\{A_{3},A_{4},\left\\{A_{1},A_{2},A_{5}\right\\}\right\\}\\\
&=\left\\{A_{1},A_{2},\left\\{A_{3},A_{4},A_{5}\right\\}\right\\}.\end{split}$
(172)

Note the similarity between the Fundamental Identity and the vectorial
identity (16). Without loss of generality, taking $I_{3}>I_{2}>I_{1}$, this
gives the equations of motions as

$\frac{dM_{1}}{dt}=\left(\frac{1}{I_{3}}-\frac{1}{I_{2}}\right)M_{2}M_{3},$
(173a)
$\frac{dM_{2}}{dt}=\left(\frac{1}{I_{1}}-\frac{1}{I_{3}}\right)M_{1}M_{3},$
(173b)
$\frac{dM_{3}}{dt}=\left(\frac{1}{I_{2}}-\frac{1}{I_{1}}\right)M_{1}M_{2},$
(173c)

which have solutions of the form

$M_{1}(t)=K_{1}\mathrm{sn}(c(t-t_{0});k),$ (174a)
$M_{2}(t)=K_{2}\mathrm{cn}(c(t-t_{0});k),$ (174b)
$M_{3}(t)=K_{3}\mathrm{dn}(c(t-t_{0});k),$ (174c)

with

$A_{1}^{2}=\frac{-K^{2}k^{2}I_{1}^{2}I_{2}I_{3}}{(I_{2}-I_{1})(I_{3}-I_{1})}\
,\quad
A_{2}^{2}=\frac{K^{2}k^{2}I_{1}I_{2}^{2}I_{3}}{(I_{2}-I_{1})(I_{3}-I_{2})}\
,\quad
A_{3}^{2}=\frac{K^{2}k^{2}I_{1}I_{2}I_{3}^{2}}{(I_{3}-I_{1})(I_{3}-I_{2})}\ ,$

with $K$ constant, and where the modulus $k$ of the Jacobi elliptic functions
is given by

$k=\frac{I_{2}-I_{1}}{I_{3}-I_{2}}\,\frac{2H_{1}I_{3}-H_{2}}{H_{2}-2I_{1}H_{1}}.$
(175)

Using Irwin’s correspondenceirwin , these solutions may be reparameterised in
terms of spherical trigonometry. Note that in this case
$\theta_{ij}=\alpha_{k}$ for all $i,j,k=1,2,3$.

A generalisation of this method can be made in higher dimensions, and in the
case of a four dimensional space, we formulate the mechanics of the top in
terms of a Nambu mechanical system of order four, which we then solve using
Pawellek’s generalised Jacobi elliptic functions discussed in the previous
section. Consider a spinning top given by the action

$S=\int M_{1}dM_{2}\wedge dM_{3}\wedge dM_{4}-H_{1}dH_{2}\wedge dH_{3}\wedge
dt,$ (176)

where the cross products are the triple cross products defined earlier, and
$H_{1},$ $H_{2}$ and $H_{3}$ are Hamiltonians, given by

$H_{1}=\frac{1}{2}\sum_{i=1}^{4}M_{i}^{2},\quad
H_{2}=\frac{1}{2}\sum_{i=1}^{4}\alpha_{i}M_{i}^{2},\quad
H_{3}=\frac{1}{2}\sum_{i=1}^{4}\beta_{i}M_{i}^{2},$ (177)

with $\alpha_{i}$ and $\beta_{i}$ constants. The top’s equations of motion are
given by the Nambu-Poisson brackets

$\frac{dF}{dt}=\\{F,H_{1},H_{2},H_{3}\\},$ (178)

where

$\\{F,G,H,I\\}=\epsilon^{ijkl}\partial_{M_{i}}F\partial_{M_{j}}G\partial_{M_{k}}H\partial_{M_{l}}I.$
(179)

This gives the intertwined differential system of four variables

$\dot{M}_{1}=\left(\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}+\alpha_{3}\beta_{4}-\alpha_{4}\beta_{3}+\alpha_{4}\beta_{2}-\alpha_{2}\beta_{4}\right)M_{2}M_{3}M_{4},$
(180a)
$\dot{M}_{2}=\left(\alpha_{1}\beta_{4}-\alpha_{4}\beta_{1}+\alpha_{3}\beta_{1}-\alpha_{1}\beta_{3}+\alpha_{4}\beta_{3}-\alpha_{3}\beta_{4}\right)M_{1}M_{3}M_{4},$
(180b)
$\dot{M}_{3}=\left(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}+\alpha_{2}\beta_{4}-\alpha_{4}\beta_{2}+\alpha_{4}\beta_{1}-\alpha_{1}\beta_{4}\right)M_{1}M_{2}M_{4},$
(180c)
$\dot{M}_{4}=\left(\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}+\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}+\alpha_{3}\beta_{2}-\alpha_{2}\beta_{3}\right)M_{1}M_{2}M_{3}.$
(180d)

These equations can be integrated in closed form in terms of the generalised
Jacobi elliptic functions $s,$ $c,$ $d_{1}$ and $d_{2},$ giving

$\begin{array}[]{c}M_{1}(t)=A_{1}s(K(t-t_{0});k_{1},k_{2}),\\\
M_{2}(t)=A_{2}c(K(t-t_{0});k_{1},k_{2}),\\\
M_{3}(t)=A_{3}d_{1}(K(t-t_{0});k_{1},k_{2}),\\\
M_{4}(t)=A_{4}d_{2}(K(t-t_{0});k_{1},k_{2}),\\\ \end{array}$ (181)

with $A_{i},$ $K$ and $t_{0}$ all constants satisfying

$\displaystyle\frac{A_{1}K}{A_{2}A_{3}A_{4}}$ $\displaystyle=$
$\displaystyle\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}+\alpha_{3}\beta_{4}-\alpha_{4}\beta_{3}+\alpha_{4}\beta_{2}-\alpha_{2}\beta_{4},$
$\displaystyle-\frac{A_{2}K}{A_{1}A_{3}A_{4}}$ $\displaystyle=$
$\displaystyle\alpha_{1}\beta_{4}-\alpha_{4}\beta_{1}+\alpha_{3}\beta_{1}-\alpha_{1}\beta_{3}+\alpha_{4}\beta_{3}-\alpha_{3}\beta_{4},$
$\displaystyle-\frac{k_{1}^{2}A_{3}K}{A_{1}A_{2}A_{4}}$ $\displaystyle=$
$\displaystyle\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}+\alpha_{2}\beta_{4}-\alpha_{4}\beta_{2}+\alpha_{4}\beta_{1}-\alpha_{1}\beta_{4},$
$\displaystyle-\frac{k_{2}^{2}A_{4}K}{A_{1}A_{2}A_{3}}$ $\displaystyle=$
$\displaystyle\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}+\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}+\alpha_{3}\beta_{2}-\alpha_{2}\beta_{3}.$

Hence, the motion of the top may be entirely parameterised in terms of the
generalised Jacobi elliptic functions, and therefore, through the link with
hyperspherical trigonometry, the angles of hyperspherical tetrahedra. Note
again that this identification ensures that
$\theta_{ij}=\alpha_{k}=\phi_{kl}$, for all $i,j,k,l=1,2,3,4$.

###  VI.2 Double Elliptic Systems (DELL)

The so-called DELL, or double-elliptic, model is a conjectured generalisation
of the Calogero-Moser and Ruijsenaars-Schneider models (integrable many-body
systems), which is elliptic in both the momentum and position variables. So
far, only the 2-particle model (reducing to one degree of freedom) has been
explicitly constructedbraden ; braden2 . A possible Hamiltonian for the three-
particle model was later suggested in terms of Riemann theta functionsAminov ;
MirMor , supported by numerical evidence in MirMor00  although it remained to
be proven.

The Dell Hamiltonian for this 2-particle model,

$H(P,Q)=\alpha(Q)\mathrm{cn}\left(P\sqrt{k^{\prime
2}+k^{2}\alpha^{2}(Q)};\frac{k\alpha(Q)}{\sqrt{k^{\prime
2}+k^{2}\alpha^{2}(Q)}}\right),$ (182)

with

$\alpha^{2}(Q)=\alpha^{2}_{rat}(Q)=1-\frac{2g^{2}}{Q^{2}},$ (183)

can easily be parameterised in terms of Pawellek’s generalised Jacobi elliptic
functions using the identity

$\mathrm{cn}(k^{\prime}_{2}u,\kappa)=\frac{c(u;k_{1},k_{2})}{d_{2}(u;k_{1},k_{2})},\quad\kappa^{2}=\frac{k_{1}^{2}-k_{2}^{2}}{{k^{\prime}}_{2}^{2}}\
,$ (184)

given by Pawellek. From this, it follows that the Hamiltonian takes the much
neater form

$H(P,Q)=\alpha(Q)\frac{c(P;k_{1},k_{2})}{d_{2}(P;k_{1},k_{2})},$ (185)

with

$\begin{array}[]{c}k_{1}=k,\\\ k_{2}=k\sqrt{(1-\alpha^{2}(Q))}.\end{array}$
(186)

The Hamilton equations of motion read

$\displaystyle\dot{P}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial
Q}=-\alpha^{\prime}_{ell}(Q;\widetilde{k})\frac{c(P;k_{1},k_{2})}{d_{2}(P;k_{1},k_{2})}\
,$ (187a) $\displaystyle\dot{Q}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial
P}=-{k^{\prime}}_{2}^{2}\alpha^{\prime}_{ell}(Q;\widetilde{k})\frac{s(P;k_{1},k_{2})d_{1}(P;k_{1},k_{2})}{d_{2}^{2}(P;k_{1},k_{2})}\
,$ (187b)

where

$\alpha^{\prime}_{ell}(Q;\widetilde{k})=\frac{d\alpha_{ell}(Q;\widetilde{k})}{dQ}=4g^{2}{\rm
sn}(Q;\widetilde{k})\,{\rm cn}(Q;\widetilde{k})\,{\rm dn}(Q;\widetilde{k})\ .$

Therefore, from this it makes sense to take a closer look at the model in
terms of the generalised Jacobi elliptic functions.

The following system of four quadrics in $\mathbb{C}^{6}$ provides a phase
space for this two-body double elliptic system:

$\begin{array}[]{lc}Q_{1}:&x_{1}^{2}-x_{2}^{2}=1,\\\
Q_{2}:&x_{1}^{3}-x_{3}^{2}=k^{2},\\\
Q_{3}:&-g^{2}x_{1}^{2}+x_{4}^{2}+x_{5}^{2}=1,\\\
Q_{4}:&-g^{2}x_{1}^{2}+x_{4}^{2}+\widetilde{k}^{-2}x_{6}^{2}=\widetilde{k}^{-2},\end{array}$
(188)

where $g$ is a coupling constant, and $x_{i}$, $i=1,2,3,4$ are affine
coordinates. The first pair of equations provides the embedding of an elliptic
curve, while the second pair provides a second elliptic curve which is locally
fibred over the first. Note that when $g=0$, the system simply becomes two
copies of elliptic curves embedded in $\mathbb{C}^{3}\times\mathbb{C}^{3}$.
The Poisson brackets are given by

$\\{x_{i},x_{j}\\}=\varepsilon_{ijk_{1}\cdots k_{4}}\frac{\partial
Q_{1}}{\partial x_{k_{1}}}\frac{\partial Q_{2}}{\partial
x_{k_{2}}}\frac{\partial Q_{3}}{\partial x_{k_{3}}}\frac{\partial
Q_{4}}{\partial x_{k_{4}}}\ ,$ (189)

with the polynomials $Q_{i}$ yielding the Casimirs of the algebra. The
relevant Poisson brackets for this system of quadraics are thenbraden2

$\begin{aligned} \dot{x}_{1}&=\\{x_{1},x_{5}\\}=x_{2}x_{3}x_{4}x_{6},\\\
\dot{x}_{3}&=\\{x_{3},x_{5}\\}=x_{1}x_{2}x_{4}x_{6},\\\
\dot{x}_{5}&=\\{x_{5},x_{5}\\}=0,\end{aligned}\qquad\begin{aligned}
\dot{x}_{2}&=\\{x_{2},x_{5}\\}=x_{1}x_{3}x_{4}x_{6},\\\
\dot{x}_{4}&=\\{x_{4},x_{5}\\}=x_{1}x_{2}x_{3}x_{6},\\\
\dot{x}_{6}&=\\{x_{6},x_{5}\\}=0.\end{aligned}$ (190)

These equations of motion can solved in terms of Jacobi elliptic functions,
where the $x_{4},x_{5},x_{6}$ depend on complicated expressions for the moduli
(as in (182)), cf. braden2 , (we omit the formulae), but they are more readily
solved in terms of the generalised Jacobi elliptic functions, namely by the
following expressions:

$\begin{aligned} x_{1}&=A_{1}s(K(t-t_{0});k_{1},k_{2}),\\\
x_{3}&=A_{3}d_{1}(K(t-t_{0});k_{1},k_{2}),\\\
x_{5}&=E,\,\mathrm{the\,\,energy},\end{aligned}\qquad\begin{aligned}
x_{2}&=A_{2}c(K(t-t_{0});k_{1},k_{2}),\\\
x_{4}&=A_{4}d_{2}(K(t-t_{0});k_{1},k_{2}),\\\
x_{6}&=K,\,\mathrm{constant},\end{aligned}$ (191)

with

$\begin{aligned} A_{1}^{2}&=\frac{k_{1}k_{2}}{\sqrt{-g^{2}}},\\\
A_{3}^{2}&=\frac{-k_{2}}{k_{1}\sqrt{-g^{2}}},\end{aligned}\qquad\begin{aligned}
A_{2}^{2}&=\frac{-k_{1}k_{2}}{\sqrt{-g^{2}}},\\\
A_{4}^{2}&=\frac{k_{1}\sqrt{-g^{2}}}{k_{2}},\end{aligned}$ (192)

respectively, identifying $k_{1}=1/\widetilde{k}$ and
$k_{2}=\widetilde{k}\sqrt{-2g^{2}}$. Note that $K$ is related to the energy,
$E$, through the final two quadrics

$\tilde{k}^{2}(1-E^{2})=1-K^{2}.$ (193)

This connection through the quadrics suggests that the generalised Jacobi
functions form the natural parametrisation for the 2-particle DELL model. Note
that these equations of motion also coincide with those for the four-
dimensional Euler top example of the previous subsection. Hence, these two
systems must be equivalent up to scaling.

##  VII Conclusion

We have investigated a novel connection between the formulae governing
4-dimensional hyperspherical geometry and elliptic functions. This connection
is via the generalised Jacobi elliptic functions, defined through Abelian
integrals associated with a doubly elliptic cover of an elliptic curve. These
generalised Jacobi elliptic functions have two distinct moduli, $k_{1}$ and
$k_{2}$, and are expressible in terms of the usual Jacobi elliptic functions,
. As such these functions may be of interest for the study of certain elliptic
integrable models where solutions in terms of elliptic functions with
different moduli appear, for example the Q4 lattice equation of the ABS
classification, and the DELL model. The link between addition formulae and
hyperspherical geometry suggests that discrete integrable models, such as
discretisation of the DELL model, may be derived exploiting this connection.
This is a link that has been exploited in the spherical case by Petrera and
SurisSuris  in producing an integrable map, in the sense of multidimensional
consistency, based upon the cosine and polar cosine rules. They have shown
this map to be related to the Hirota-Kimura discretisation of the Euler top.
Whilst this touches slightly on our work, we are more concerned with the
higher dimensional hyperspherical trigonometry, and its links to elliptic
functions. The Dell model, which arises in our case, although equivalent to a
higher dimensional Euler top, seems to be different from Petrera and Suris’
model.

We have also generalised the nested structure of hyperspherical trigonometry
here to any dimension using higher-dimensional vector products, although the
inter-relations between the formulae become increasingly complicated.

In recent years there has been substantial progress in obtaining formulas for
hyperspherical tetrahedra in 4D in terms of arc lengths and dihedral angles,
cf. Murakami ; Murakami2 ; Murakami3 . These generalise results for symmetric
tetrahedra obtained earlier Derevnin ; Derevnin2 , refining the formula for
the general case found inChoKim . The expressions involve Schäfli type
symbols, and can be expressed in terms of dilogarithms (Lobachevsky functions
in the hyperbolic case). The occurrence of such functions in Lagrangians of
certain discrete integrable systemsLobbNij , seems to point to suggest a
connection between the two types objects from a geometrical perspective.

Finally, the occurrence of the Pawellek functions, which involve elliptic
functions of two distinct moduli, is reminiscent of the bi-elliptic addition
formulaeAHN  that occur naturally in the seed and soliton solutions of the
important Q4 lattice equation, discovered by AdlerAdler .

###### Acknowledgements.

Paul Jennings was supported by the UK Engineering and Physical Sciences
Research Council (ESPRC) when this work was conducted. Part of the work was
done while the second author was supported by a Royal Society/Leverhlme Trust
Senior Research Fellowship (2011-12). He is currently supported by the EPSRC
grant EP/W007290/1.

##  Appendix A Four and Five Parts Formulae

We write a general position vector on the surface of a sphere in terms of its
basis vectors, and derive the four and five parts formulae.

Given 2 vectors in three dimensions, we can express any other vector on the
sphere in terms of these by

$\mathbf{n}_{i}=A\mathbf{n}_{j}+B\mathbf{n}_{k}+C\mathbf{n}_{j}\times\mathbf{n}_{k},$
(194)

with

$\left(\begin{array}[]{c}A\\\ B\\\
\end{array}\right)=\left(\begin{array}[]{cc}1&\cos\theta_{jk}\\\
\cos\theta_{jk}&1\\\
\end{array}\right)^{-1}\left(\begin{array}[]{c}\cos\theta_{ij}\\\
\cos\theta_{ik}\\\
\end{array}\right)=\frac{1}{\sin^{2}\theta_{jk}}\left(\begin{array}[]{c}\sin\theta_{ik}\sin\theta_{jk}\cos\alpha_{k}\\\
\sin\theta_{ij}\sin\theta_{jk}\cos\alpha_{j}\\\ \end{array}\right),$ (195)

and

$C=\frac{\mathbf{n}_{i}\cdot(\mathbf{n}_{j}\times\mathbf{n}_{k})}{\sin^{2}\theta_{jk}}=\frac{\sin\alpha_{i}\sin\theta_{ik}\sin\theta_{ij}}{\sin^{2}\theta_{jk}},$
(196)

giving

$\mathbf{n}_{i}=\frac{\sin\theta_{ik}\cos\alpha_{k}}{\sin\theta_{jk}}\mathbf{n}_{j}+\frac{\sin\theta_{ij}\cos\alpha_{j}}{\sin\theta_{jk}}\mathbf{n}_{k}+\frac{\sin\alpha_{i}\sin\theta_{ik}\sin\theta_{ij}}{\sin^{2}\theta_{jk}}\mathbf{n}_{j}\times\mathbf{n}_{k}.$
(197)

Using the spherical sine rule this reduces to

$\mathbf{n}_{i}=\frac{\sin\alpha_{j}\cos\alpha_{k}}{\sin\alpha_{i}}\mathbf{n}_{j}+\frac{\sin\alpha_{k}\cos\alpha_{j}}{\sin\alpha_{i}}\mathbf{n}_{k}+\frac{\sin\alpha_{i}\sin\sin\alpha_{k}}{\sin\alpha_{i}}\mathbf{n}_{j}\times\mathbf{n}_{k}.$
(198)

Using the Gram-Schmidt Process $\mathbf{n}_{i}$ may be written in terms of an
orthonormal basis,

$\mathbf{n}_{i}=A^{\prime}\mathbf{n}_{j}+B^{\prime}\mathbf{n}_{k}^{\prime}+C^{\prime}\mathbf{n}_{j}\times\mathbf{n}_{k}^{\prime},$
(199)

where $\mathbf{n}_{j}$ is as is before, and $\mathbf{n}_{k}^{\prime}$ is given
by

$\mathbf{n}_{k}^{\prime}=\frac{\mathbf{n}_{k}-(\mathbf{n}_{j}\cdot\mathbf{n}_{k})\mathbf{n}_{j}}{|\mathbf{n}_{k}-(\mathbf{n}_{j}\cdot\mathbf{n}_{k})\mathbf{n}_{j}|}=\frac{\mathbf{n}_{k}-(\mathbf{n}_{j}\cdot\mathbf{n}_{k})\mathbf{n}_{j}}{\sin\theta_{jk}}.$
(200)

This implies

$C^{\prime}=(\sin\theta_{jk})C,\hskip
42.67912ptB^{\prime}=(\sin\theta_{jk})B,\hskip
42.67912ptA^{\prime}=A+\cot\theta_{jk}B^{\prime},$ (201)

from which it follows that

$\mathbf{n}_{i}=\frac{\sin\theta_{ik}\cos\alpha_{k}+\cos\theta_{jk}\sin\theta_{ij}\cos\alpha_{j}}{\sin\theta_{jk}}\mathbf{n}_{j}+\sin\theta_{ij}\cos\alpha_{j}\mathbf{n}_{k}^{\prime}+\sin\theta_{ij}\sin\alpha_{j}\mathbf{n}_{j}\times\mathbf{n}_{k}^{\prime}.$
(202)

Equating the two expression we have for $\mathbf{n_{i}}$ gives the four-parts
formula

$\cot\theta_{ij}\sin\theta_{jk}=\cot\alpha_{k}\sin\alpha_{j}+\cos\theta_{jk}\cos\alpha_{j}.$
(203)

Similarly, for

$\mathbf{u}_{ij}=P\mathbf{u}_{ik}+Q\mathbf{u}_{kj}+R\mathbf{u}_{ik}\times\mathbf{u}_{kj}$
(204)

it follows that

$\mathbf{u}_{ij}=\frac{\sin\alpha_{j}}{\sin\alpha_{k}}\cos\theta_{jk}\mathbf{u}_{ik}+\frac{\sin\alpha_{i}}{\sin\alpha_{k}}\cos\theta_{ik}\mathbf{u}_{kj}-\frac{\sin\theta_{ik}\sin\theta_{kj}}{\sin\theta_{ij}}\mathbf{u}_{ik}\times\mathbf{u}_{kj},$
(205)

or alternatively, using the spherical sine rule,

$\mathbf{u}_{ij}=\frac{\sin\theta_{ik}\cos\theta_{jk}}{\sin\theta_{ij}}\mathbf{u}_{ik}+\frac{\sin\theta_{jk}\cos\theta_{ik}}{\sin\theta_{ij}}\mathbf{u}_{kj}-\frac{\sin\theta_{ik}\sin\theta_{kj}}{\sin\theta_{ij}}\mathbf{u}_{ik}\times\mathbf{u}_{kj}.$
(206)

Similarly, using the Gram-Schmidt Process,

$\mathbf{u}_{ij}=\frac{\sin\alpha_{j}\cos\theta_{jk}-\cos\alpha_{k}\sin\alpha_{i}\cos\theta_{ik}}{\sin\alpha_{k}}\mathbf{u}_{ik}+\sin\alpha_{i}\cos\theta_{ik}\mathbf{u}_{kj}^{\prime}+\sin\theta_{kj}\sin\alpha_{j}\mathbf{u}_{ij}\times\mathbf{u}_{kj}^{\prime}.$
(207)

Equating the two expression we have for $\mathbf{u}_{ij}$ gives the five-parts
formula

$\cot\theta_{jk}\sin\alpha_{j}=\cos\alpha_{i}\sin\alpha_{k}+\cos\theta_{ik}\sin\alpha_{i}\cos\alpha_{k}.$
(208)

In 4-dimensional Euclidean space, for the vectors
$\mathbf{n}_{i},\mathbf{n}_{j},\mathbf{n}_{k},\mathbf{n}_{l}\in E_{4}$ of a
hyperspherical tetrahedron we have the expansions:

$\displaystyle\mathbf{n}_{i}$ $\displaystyle=$
$\displaystyle\frac{\sin(\theta_{ik},\theta_{il},\theta_{kl})\,\cos\phi_{kl}}{\sin(\theta_{jk},\theta_{jl},\theta_{kl})}\mathbf{n}_{j}+\frac{\sin(\theta_{ij},\theta_{il},\theta_{jl})\,\cos\phi_{jl}}{\sin(\theta_{jk},\theta_{jl},\theta_{kl})}\mathbf{n}_{k}+\frac{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\,\cos\phi_{jk}}{\sin(\theta_{jk},\theta_{jl},\theta_{kl})}\mathbf{n}_{l}+$
(210)
$\displaystyle+\frac{\sqrt{\sin(\theta_{ij},\theta_{ik},\theta_{jk})\,\sin(\theta_{ij},\theta_{il},\theta_{jl})\,\sin(\theta_{ik},\theta_{il},\theta_{kl})\,\sin(\phi_{ij},\phi_{ik},\phi_{il})}}{\left(\sin(\theta_{jk},\theta_{jl},\theta_{kl})\right)^{2}}\,\mathbf{n}_{j}\times\mathbf{n}_{k}\times\mathbf{n}_{l}\
$

as well as

$\displaystyle\mathbf{n}_{i}$ $\displaystyle=$
$\displaystyle\cos\theta_{ij}\,\mathbf{n}_{j}+\sin\theta_{ij}\cos\alpha^{(ijk)}_{j}\,\mathbf{n}^{\prime}_{k}+\sin\theta_{ij}\sin\alpha^{(ijk)}_{j}\cos\phi_{jk}\,\mathbf{n}_{l}^{\prime}$
(212)
$\displaystyle~{}~{}~{}~{}~{}+\sin\theta_{ij}\sin\alpha^{(ijk)}_{j}\sin\phi_{jk}\,\mathbf{n}_{j}\times\mathbf{n}^{\prime}_{k}\times\mathbf{n}^{\prime}_{l}\
,$

in a (Gram-Schmidt) orthogonal basis
$\mathbf{n}_{j},\mathbf{n}^{\prime}_{k},\mathbf{n}^{\prime}_{l}$. Comparing
the results provides the following set of equations:

$\begin{split}\frac{\sin(\theta_{jk},\theta_{jl},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}\cos\phi_{jk}=&\cos\theta_{il}\Bigg{(}1-\tan\theta_{il}\cot\theta_{ik}\cos\phi_{ij}\frac{\sin\alpha_{i}^{(ijl)}}{\sin\alpha_{i}^{(ijk)}}\\\
&\
-\tan\theta_{il}\cot\theta_{ij}\cos\alpha_{i}^{(ijl)}\left(1-\tan\alpha_{i}^{(ijl)}\cot\alpha_{i}^{(ijk)}\cos\phi_{ij}\right)\Bigg{)}\
,\\\
\frac{\sin(\theta_{ik},\theta_{il},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}\cos\phi_{ik}=&\frac{\sin\theta_{il}\cos\alpha_{i}^{(ijl)}}{\sin\theta_{ij}}\left(1-\tan\alpha_{i}^{(ijl)}\cot\alpha_{i}^{(ijk)}\cos\phi_{ij}\right)\
,\end{split}$ (213)

which form the hyperspherical analogue of the four-parts formula.

Similarly, for the expansions of the polar vectors
$\mathbf{u}_{ijk},\mathbf{u}_{ijl},\mathbf{u}_{ikl},\mathbf{u}_{jkl}$ we have

$\begin{split}\mathbf{u}_{ijk}=&\frac{\sin(\theta_{ij},\theta_{il},\theta_{jl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}\cos\theta_{kl}\mathbf{u}_{ijl}-\frac{\sin(\theta_{jk},\theta_{jl},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}\cos\theta_{il}\mathbf{u}_{jlk}+\frac{\sin(\theta_{ik},\theta_{il},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}\cos\theta_{jl}\mathbf{u}_{lki}\\\
&+\frac{\sin(\theta_{ij},\theta_{il},\theta_{jl})\sin(\theta_{jk},\theta_{jl},\theta_{kl})\sin(\theta_{ik},\theta_{il},\theta_{kl})}{\sin(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})}\mathbf{u}_{ijl}\times\mathbf{u}_{jlk}\times\mathbf{u}_{kli}\end{split}$
(214)

and in terms of an orthonormal basis,

$\begin{split}\mathbf{u}_{ijk}=&\Bigg{(}\frac{\sin(\theta_{ij},\theta_{il},\theta_{jl})\cos\theta_{kl}}{\sin(\theta_{ij},\theta_{ik},\theta_{jk})}-\left(\frac{\cos\phi_{jk}-\cos\phi_{jl}\cos\phi_{ij}}{\sin\phi_{jl}}\right)\cos\phi_{jl}\\\
&\hskip
72.26999pt+\frac{\cos\theta_{j}l\cos\phi_{il}\sin\phi_{jl}\sin(\phi_{ij},\phi_{jk},\phi_{jl})}{\sin(\phi_{il},\phi_{jl},\phi_{kl})}\Bigg{)}\mathbf{u}_{ijl}\\\
&+\frac{\cos\phi_{jk}-\cos\phi_{ij}\cos\phi_{jl}}{\sin\phi_{jl}}\mathbf{u}_{jlk}^{\prime}+\frac{\cos\theta_{jl}\sin(\phi_{ij},\phi_{jk},\phi_{jl})}{\sin\phi_{ij}}\mathbf{u}_{lki}^{\prime}\\\
&+\sin(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})\mathbf{u}_{ijl}\times\mathbf{u}_{jlk}^{\prime}\times\mathbf{u}_{kli}^{\prime}.\end{split}$
(215)

Comparing the results provides

$\begin{split}\cos\phi_{jk}\cos\phi_{jl}\sin(\theta_{ij},\theta_{ik},\theta_{jk})=&\cos\phi_{ij}\sin\phi_{jl}\sin(\theta_{ij},\theta_{ik},\theta_{jk})\\\
&+\cos\theta_{kl}\sin\phi_{jl}\sin(\theta_{ij},\theta_{il},\theta_{jl})\\\
&+\cos^{2}\phi_{jl}\cos\phi_{ij}\sin(\theta_{ij},\theta_{ik},\theta_{jk})\\\
&+\cos\theta_{jl}\cos\phi_{il}\sin^{2}\phi_{jl}\sin(\theta_{ik},\theta_{il},\theta_{kl}),\end{split}$
(216)

the hyperspherical analogue of the five-parts formula.

##  Appendix B Angle Addition Formulas

We derive the cosine addition formulae by collapsing a spherical triangle. We
then extend this to see what happens when we collapse a hyperspherical
tetrahedron. Recall that the generalised sine function of three variables
gives the volume of a three dimensional parallelepiped defined by vectors
$\mathbf{n}_{i},\mathbf{n}_{j}$ and $\mathbf{n}_{k}$ embedded in four
dimensional Euclidean space,

$\sin(\theta_{ij},\theta_{ik},\theta_{jk})=\left|\begin{array}[]{ccc}1&\cos\theta_{ij}&\cos\theta_{ik}\\\
\cos\theta_{ij}&1&\cos\theta_{jk}\\\ \cos\theta_{ik}&\cos\theta_{jk}&1\\\
\end{array}\right|^{\frac{1}{2}}.$ (217)

When the three vectors $\mathbf{n}_{i},\mathbf{n}_{j}$ and $\mathbf{n}_{k}$
become coplanar, the volume of the parallelepiped collapses to zero, and
hence, so does the generalised sine function. When this occurs

$\sin^{2}(\theta_{ij},\theta_{ik},\theta_{jk})=0.$ (218)

By expanding this out and completing the square in terms of $\cos\theta_{ij}$,
this reduces to

$(\cos\theta_{ij}-\cos\theta_{ik}\cos\theta_{jk})^{2}=\sin^{2}\theta_{ik}\sin^{2}\theta_{jk}.$
(219)

Solving this gives

$\cos\theta_{ij}=\cos\theta_{ik}\cos\theta_{jk}\pm\sin\theta_{ik}\sin\theta_{jk}.$
(220)

This occurs when $\theta_{ij}=\theta{ik}\mp\theta{jk}$, hence giving us the
standard addition formula for cosine,

$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B.$ (221)

Similarly, the generalised sine function of six variables gives the volume of
a four dimensional parallelotope defined by vectors
$\mathbf{n}_{i},\mathbf{n}_{j},\mathbf{n}_{k}$ and $\mathbf{n}_{l}$ embedded
in five dimensional Euclidean space,

$\sin(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})=\left|\begin{array}[]{cccc}1&\cos\theta_{ij}&\cos\theta_{ik}&\cos\theta_{il}\\\
\cos\theta_{ij}&1&\cos\theta_{jk}&\cos\theta_{jl}\\\
\cos\theta_{ik}&\cos\theta_{jk}&1&\cos\theta_{kl}\\\
\cos\theta_{il}&\cos\theta_{jl}&\cos\theta_{kl}&1\\\
\end{array}\right|^{\frac{1}{2}}.$ (222)

When the four vectors $\mathbf{n}_{i},\mathbf{n}_{j},\mathbf{n}_{k}$ and
$\mathbf{n}_{l}$ become linearly dependent, the volume of the 4-parallelepiped
collapses to zero, and hence, so does the generalised sine function. When this
occurs, recalling the hyperspherical sine rule,

$\sin^{2}(\theta_{ij},\theta_{ik},\theta_{il},\theta_{jk},\theta_{jl},\theta_{kl})=0\iff
k_{H}=0.$ (223)

This gives

$\left.\begin{array}[]{c}\sin(\phi_{ij},\phi_{ik},\phi_{il})=0\\\
\sin(\phi_{ij},\phi_{jk},\phi_{jl})=0\\\
\sin(\phi_{ik},\phi_{jk},\phi_{kl})=0\\\
\sin(\phi_{il},\phi_{jl},\phi_{kl})=0\\\ \end{array}\right\\},$ (224)

which, in turn from the spherical case implies

$\left.\begin{array}[]{c}\phi_{ij}+\phi_{ik}+\phi_{il}=0\\\
\phi_{ij}+\phi_{jk}+\phi_{jl}=0\\\ \phi_{ik}+\phi_{jk}+\phi_{kl}=0\\\
\phi_{il}+\phi_{jl}+\phi_{jl}=0\\\ \end{array}\right\\},$ (225)

and hence,

$\phi_{ij}+\phi_{ik}+\phi_{il}+\phi_{jk}+\phi_{jl}+\phi_{kl}=0,$ (226)

the sum of the dihedral angles is zero.

## References

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