# Angle Spaces and their conformal embeddability in $\mathbb{E}^{n}$

L. Felipe Prieto-Martínez

###### Abstract

In this article we provide a definition of the concept of angle space. Unlike
previous works on this topic, we do it for sets with a notion of betweenness
that are not necessarily metric spaces. After this, we solve two problems
proposed by K. Menger in the Annals of Mathematics in 1931. The first one
consists in characterizing angle spaces that can be embedded in the euclidean
plane $\mathbb{E}^{2}$. We answer this question in a general way
characterizing those that can be embedded in $\mathbb{E}^{n}$. And the second
one concerns a characterization of those angle spaces admitting a distance
function compatible with the angle space structure.

##  1 Introduction

Let $(\mathcal{X},d)$ be a metric or semi-metric space. Menger explored in
various papers (see [8], for example) the notion of _angle_ for these spaces.
Let us denote by $\mathcal{X}^{3*}=\\{(a,b,c)\in\mathcal{X}^{3}:a\neq b,b\neq
c\\}$. K. The author defined _angle functions_ to be maps
$\angle:X^{3*}\to\mathbb{[}0,\pi]$ such that

(1.1) $\angle(A,B,C)=\angle(C,B,A)=\begin{cases}0&\text{if
}d(B,A)+d(A,C)=d(B,C)\text{ or }d(B,C)+d(C,A)=d(B,A)\\\ \pi&\text{if
}d(A,C)=d(A,B)+d(B,C)\\\ >0&\text{in other case}\end{cases}$

For him, an _angle space_ was then a triple $(\mathcal{X},d,\angle)$, where
$X$ is a set, $d$ a distance or semi-distance and $\angle$ an _angle
function_.

Menger already realized that for his triples $(\mathcal{X},d,\angle)$ it is
interesting to discuss if the distance function and the angle function are
compatible, in the sense that they are in the euclidean space, that is, for
every $(B,A,C)\in\mathcal{X}^{3*}$ we have:

(1.2) $d(B,C)^{2}=d(B,C)^{2}+d(A,B)^{2}+2\cdot d(B,C)\cdot
d(A,B)\cdot\cos\angle(B,A,C)$

In this article we use the term _euclidean-compatible_ for a distance and
angle functions satisfying the condition above. _Euclidean-compatiblity_ is
necessary for the triple $(\mathcal{X},d,\angle)$ to be, both isometrically
and conformally, embedded in the euclidean space $\mathbb{E}^{n}$.

After this, other attempts have been done to provide a useful definition of
_angle space_. Always in the setting of metric spaces.

  * •

Some years after the article by Menger W. A. Wilson in [13] studied the
concept of _angle_ for convex complete metric spaces $\mathcal{X}$ satisfying,
what he called, the “four point property”: every set of four points
$\\{A,B,C,D\\}\subset\mathcal{X}$ can be embedded in $\mathbb{E}^{n}$ for some
$n$. In this setting, there is a notion of line, ray and segment, and it is
possible to define angles between them from the euclidean-compatibility
property.

  * •

Of course, the more complicated the structure of the metric space is and the
more axioms that are asked for the angle function, the more properties are
satisfied by $(\mathcal{X},d,\angle)$. This point of view lead to the authors
of [12] to restrict theirselves to studying angles in linear normal spaces.

  * •

On the other hand, the recent paper [9] makes a great effort to offer a
definition of angle between three points for general metric spaces. The
motivation for this approach is, as the author explains, that the definition
suggested for the angle of $p,x,q$ _in the Euclidean plane gives the standard
angle between three points and in a Riemannian manifold coincides with the
usual angle between the geodesics, if $x$ is not in the cut locus of $p$ or
$q$._

In his paper [8], Menger stated the following two problems:

######  Problem 1 (copied literally from page 750 in [8])

It would be desirable to know relations between the angles of an angle-space,
characteristic for those angle-spaces which are conformal to a subset of the
plane and to the plane itself.

######  Problem 2 (copied literally from page 750 in [8])

In a metrical space (…) each three points are congruent to three points of the
plane and, therefore, the definition of distances induces a definition of
angles (viz. the angles in the corresponding plane triangles). A problem
(connected, of course, with the first mentioned) is to find conditions which
are necessary and sufficient in order that in an angle-space, which at the
same time is a metrical space, the angles are identical with the angles
induced by the definition of distance.

In this paper, we provide a simple and general axiomatic definition of _angle
function_ for spaces $\mathcal{X}$ that are not necessary metric or semi-
metric (Definition 7). This eliminates the hassle of discussing the relation
between the distance and angle functions a priori. We defend that the minimum
structure that should be required for $\mathcal{X}$ in order to have a not
artificial notion of angle is a _relation of betweenness_ $\mathcal{B}$. There
exists recent bibliography studying this kind of spaces (see [1, 3]) and it is
reviewed in the following section. Not only seems to us that this context is
more natural to define angles, but it is also more general since every metric
space $(\mathcal{X},d)$ has a natural notion of _betweenness_ :

(1.3) $B\text{ is between }A,C\Longleftrightarrow d(A,B)+d(B,C)=d(A,C)$

This _metric betweenness_ is already captured in the definition by Menger
(1.1). Our definition has, in addition, one more requirement than the obvious
generalization of (1.1) for spaces with a notion of _betweenness_ : the _First
Axiom of Collinearity_. This new axiom seems to us to be basic if we expect to
obtain a reasonable notion of angle. So, for us, _angle spaces_ are triples
$(\mathcal{X},\mathcal{B},\angle)$ where $\mathcal{B}$ is a _relation of
betweenness_ and $\angle$ is an _angle function_ satisfying certain properties
with respect to it. As usual, we say that a function between two angle spaces
is _conformal_ if it preserves the angle.

Once we have settled our framework, the rest of the article is devoted to
solving the two problems proposed by Menger. Concerning the second one, since
we are not restricted to the case in which the angle space is a metric space,
we now split the problem in two parts:

###### Problem 3

Characterize metric spaces $(X,d)$ that admits an angle function $\angle$
which is euclidean-compatible to $d$.

###### Problem 4

Characterize angle spaces $(\mathcal{X},\mathcal{B},\angle)$ that admits a
distance function $d$ euclidean-compatible to $\angle$.

The first one is easier, and it is solved in Section 3. The candidate to be
the _angle function_ is obvious ($\angle(A,B,C)$ can be obtained from (1.2)).
This problem was trivial in the setting of Menger, but recall that ours is
slightly more restrictive and we need to check this new _First Axiom of
Collinearity_. The second problem is more complicated. We need to answer the
question first for angle spaces of three and four points (sections 4 and 5)
and then we take care of the general case in Section 6 (see Theorem 17).

After this, in Section 7, we solve the strong version of Problem 1 (Theorem
20):

###### Problem 5

Find relations, for each $n\in\mathbb{N}$, between the angles of an angle-
space, characteristic for those angle-spaces which are conformal to a subset
of $\mathbb{E}^{n}$ or to $\mathbb{E}^{n}$ itself.

To do so, see that if $(\mathcal{X},\mathcal{B},\angle)$ can be conformally
embedded in $\mathbb{E}^{n}$, then it admits a distance function $d$
euclidean-compatible to $\angle$. In this case, checking that
$(\mathcal{X},\mathcal{B},\angle)$ can be conformally embedded into
$\mathbb{E}^{n}$ is equivalent to seeing that $(\mathcal{X},d)$ can be
isometrically embedded in $\mathbb{E}^{n}$. Menger, in his article [6] dated
in 1928, had already provided a characterization of those metric spaces
$\mathcal{X}$ that are isometrically embeddable in $\mathbb{E}^{n}$. We
recommend also the recent article [2] updating the notation and explaining in
detail the results in [6]. This characterization is stated in terms of a
condition that must satisfy all the finite metric subspaces of $n+2$ points of
$\mathcal{X}$. So our solution for Problem 5 is also given in terms of a
condition for all the finite angle subspace of $n+2$ points of $\mathcal{X}$
(but stated only in terms of the angle function, of course).

In Section 8 we provide three open problems and some lines for future work. We
also explain how some angle spaces are related to important old and unsolved
problem in Plane Geometry (_The Inscribed Square Problem_).

##  2 Relation of betweeness. Our definition of angle space

In [4] some axiomatic approaches for the concept of _betweeness_ for geometric
use are discussed. Among all the possibilities, we are interested in the, not
very restrictive, following one:

######  Definition 6 (pseumetric betweenness)

A relation of betweennes in a set $\mathcal{X}$ is a ternary relation
$\mathcal{B}$ satisfying the condition listed below. If
$(A,B,C)\in\mathcal{B}$ we say that $B$ is between $A$ and $C$. And if either
$(B,A,C)\in\mathcal{B}$, $(A,B,C)\in\mathcal{B}$ or $(A,C,B)\in\mathcal{B}$
then we say that $A,B,C$ are collinear.

  1. (B1)

If $(A,B,C)\in\mathcal{B}$, then $A,B,C$ are distinct.

  2. (B2)

If $(A,B,C)\in\mathcal{B}$, then $(C,B,A)\in\mathcal{B}$.

  3. (B3)

If $(A,B,C)\in\mathcal{B}$, then $(B,A,C)\notin\mathcal{B}$.

  4. (B4)

(Strong Transitivity) If
$(P_{1},P_{2},P_{3}),(Q_{1},Q_{2},Q_{3})\in\mathcal{B}$ and these two triples
of points have two points in common, then any possible set of three different
points in $\\{P_{1},P_{2},P_{3}\\}\cup\\{Q_{1},Q_{2},Q_{3}\\}$ is collinear.

This definition provide us a framework for our concept of angle:

######  Definition 7 (angle function)

Let $\mathcal{X}$ be a space with a notion of betweenness $\mathcal{B}$. A
function $\angle:\mathcal{X}^{3*}\to\mathbb{[}0,\pi]$ is an _angle function_
if:

  1. (i)

(required in the original definition by Menger) It is symmetric in the first
and third coordinates $\angle(A,B,C)=\angle(C,B,A)$.

  2. (ii)

(required in the original definition by Menger)

$\angle(A,B,C)=\begin{cases}0&\text{if }A=C,\;(B,A,C)\in\mathcal{B}\text{ or
}(B,C,A)\in\mathcal{B}\\\ \pi&\text{if }(A,B,C)\in\mathcal{B}\\\ >0&\text{in
other case}\end{cases}$

  3. (iii)

(First Axiom of Collinearity) If $(B,C,D)\in\mathcal{B}$, then for all $A\neq
B$, $\angle(A,B,C)=\angle(A,B,D)$.

We have some comments to do about these definitions:

###### Remark 8

In [1] (in relation to some results in [3]) the concept of _pseudometric
betweenness_ is introduced. A relation of pseudometric betweeness is this one
that satisfies $(B_{1})$, $(B_{2})$, $(B_{3})$ (appearing in the previous
definition) and

  1. (B4)’

(Transitivity) If $(A,B,C),(A,C,D)\in\mathcal{B}$, then
$(A,B,D),(B,C,D)\in\mathcal{B}$,

instead of $(B4)$. The reason of the name “pseudometric” is discussed there.
According to this article, it seems that was also Menger in [6] the first one
to prove that every metric space has a natural relation $\mathcal{B}$ of
betweenness (the one given by (1.3)).

We can see that, if a space $\mathcal{X}$ has a notion of _pseudometric
betweenness_ and it is an angle space, then it satisfies the axiom $(B4)$. So
we have preferred to, directly, include it in our definition.

In [1], using the notion of collinearity, _lines_ and _segments_ are defined.
But we prefer not to introduce more notation in this article. Instead, our
condition of _Strong Transitivity_ allow us to give a definition of
collinearity for sets of three or more points. We say that such a set is
collinear if any subset of of three points in $\mathcal{X}$ is collinear.

###### Remark 9

It is possible to define angle spaces for sets with no notion of betweenness.
In this case, such a relation $\mathcal{B}$ would be induced by
$(A,B,C)\in\mathcal{B}\Longleftrightarrow\angle(A,B,C)=\pi$. But in this case,
axioms $(ii),(iii)$ would be stated in a more artificial way.

###### Remark 10

Menger proposed a definition of angle function removing condition (iii). This
may allow more generality, but we feel that a minimum structure for the space
$X$ is needed. Also we think that a possible reason for this author to do it
this way is that he was mainly interested in angle functions for metric
spaces.

Once we have settled the definition of _betweenness relation_ and of _angle
function_ , an _angle space_ is a triple $(\mathcal{X},\mathcal{B},\angle)$.
We say that two angle spaces are _conformal_ if there exists a one to one
correspondence that preserves the notion of betweeness and the angles.

An angle space is called _trivial_ if all of its triples of points are
collinear. Trivial angle spaces are obviously embeddable in $\mathbb{E}^{1}$
and admit infinitely many distances euclidean-compatible with their angle
function.

See that if an angle space $(\mathcal{X},\mathcal{B},\angle)$ admits an
euclidean-compatible distance function $d$ then the natural definition of
betweenness for $(\mathcal{X},d)$ (see (1.3)) and the one for
$(\mathcal{X},\angle)$ coincide.

##  3 Solution to Problem 3

As we have already discussed, given any metric space $(\mathcal{X},d)$ we have
a betweenness relation in $\mathcal{X}$ and a unique candidate for an angle
function euclidean compatible with $d$. But this unique candidate may fail to
satisfy the axiom of collinearity. For example consider a metric space
$(\mathcal{X},d)$ consisting in four points $\\{A,B,C,D\\}$. Suppose that
$d(A,B)=d(A,C)=d(B,C)=4$, $d(B,D)=d(C,D)=2$ and $d(A,D)=5$.

Figure 1: $D$ is between $C$ and $D$ and $A$ is not collinear to any pair of
points in $\\{B,C,D\\}$.

If we want to define a distance function $\angle$ euclidean-compatible with
$d$ see that

$\cos\angle(D,A,C)=\frac{2^{2}+4^{2}-5^{2}}{2\cdot 2\cdot
4}=-\frac{5}{16},\qquad\cos\angle(B,A,C)=\frac{4^{2}+4^{2}-4^{2}}{2\cdot
4\cdot 4}=\frac{1}{2}$

which contradicts the _First Axiom of Collinearity_.

So we have the following:

###### Proposition 11

A metric space $(\mathcal{X},d)$ admits an angle function $\angle$ euclidean-
compatible with $d$ if and only if for every set of four different points
$\\{A,B,C,D\\}\subset\mathcal{X}$ such that $D$ is between $B$ and $C$ and $A$
is not collinear to any pair of points in $\\{B,C,D\\}$, the relation of
Stewart’s Theorem holds, that is:

(3.1) $d(A,C)^{2}\cdot d(B,D)+d(A,B)^{2}\cdot
d(C,D)=d(B,C)\cdot(d(A,D)^{2}+d(B,D)\cdot d(C,D))$

Proof: The angle function is defined as:

$\angle(B,A,C)=arccos\left(\frac{d(A,B)^{2}+d(A,C)^{2}-d(B,C)^{2}}{2\cdot
d(A,B)\cdot d(A,C)}\right)$

It is well defined since $d$ satisfies the triangular inequality. It satisfies
axioms $(i)$ and $(ii)$ in the definition of angle function. Finally, in order
to satisfy the _First Axiom of Collinearity_ we only need to check that for
every four different points $A,B,C,D$ such that $D$ is between $B,C$,
$\angle(A,B,C)=\angle(A,B,D)$. And this occurs if and only if (3.1) holds.
$\Box$

##  4 Solution to Problem 4 for trigons.

For a space $\mathcal{X}$ with a relation of betweenness, a (non-degenerated)
$n$-gon is a set of $n$ points, such that no set of three of them is
collinear. For $n=3,4,5,6$ we call them _trigons, tetragons, pentagons_ and
_hexagons_. A set of $n$ points where three or more of them are collinear is
called _degenerated $n$-gon_.

Obviously, if $(\mathcal{X},\mathcal{B},\angle)$ admits an euclidean
compatible distance, all of its trigon also do. So the following results shows
a necessary condition for an angle space to admit an euclidean-compatible
distance function:

###### Proposition 12

Consider the angle space $(\mathcal{X},\mathcal{B},\angle)$ and
$\mathcal{X}^{\prime}=\\{A,B,C\\}\subset\mathcal{X}$ to be a non-degenerated
trigon. If $\mathcal{X}$ admits an euclidean-compatible distance function then

$\angle(B,A,C)+\angle(A,B,C)+\angle(A,C,B)=\pi$

Proof: Let us follow the usual convention:

(4.1)
$a=d(B,C),b=d(A,C),c=d(A,B),\qquad\alpha=\angle(C,A,B),\beta=\angle(A,B,C),\gamma=\angle(B,C,A)$

The conditions for euclidean-compatibility are

(4.2) $\begin{cases}a^{2}=b^{2}+c^{2}-2bc\cos\alpha\\\
b^{2}=a^{2}+c^{2}-2ac\cos\beta\\\ c^{2}=a^{2}+b^{2}-2ab\cos\gamma\end{cases}$

Suppose that $\alpha$ is the biggest angle between $\alpha,\beta,\gamma$. From
the last equation we obtain, since $c>0$, that
$c=\sqrt{a^{2}+b^{2}-2ab\cos\gamma}$. Let us substitute this value in the
other two equation to obtain:

$\begin{cases}b^{2}-ab\cos\gamma=b(\sqrt{a^{2}+b^{2}-2ab\cos\gamma})\cos\alpha\\\
a^{2}-ab\cos\gamma=a(\sqrt{a^{2}+b^{2}-2ab\cos\gamma})\cos\beta\end{cases}$

Since $A,B,C$ are different, $a,b\neq 0$. Also, since the points are not
collinear, $sin\alpha,sin\beta\neq 0$. So we can divide both equations by $ab$
to obtain:

(4.3)
$\begin{cases}\frac{b}{a}-cos\gamma=(\sqrt{(\frac{b}{a}-cos\gamma)^{2}+sin^{2}\gamma})cos\alpha\\\
\frac{a}{b}-cos\gamma=(\sqrt{(\frac{a}{b}-cos\gamma)^{2}+sin^{2}\beta})cos\beta\end{cases}\longrightarrow\begin{cases}\frac{b}{a}=\frac{sin(\alpha\pm_{1}\gamma)}{sin\alpha}\\\
\frac{a}{b}=\frac{sin(\beta\pm_{2}\gamma)}{sin\beta}\end{cases}$

In each of these last two equations we have introduced an incorrect solution.
Since $\alpha$ is the biggest angle, then $cos\alpha,cos\beta>0$. And the sign
is $+$ in both equations. So we have that:

$sin(\alpha+\gamma)sin(\beta+\gamma)=sin\alpha sin\beta\longrightarrow-
cos(\alpha+\beta+2\gamma)+cos(\alpha-\beta)=-cos(\alpha+\beta)+cos(\alpha-\beta)\longrightarrow$
$\longrightarrow cos(\alpha+\beta+2\gamma)=cos(\alpha+\beta)$

which, taking into account that $\alpha,\beta,\gamma\in[0,\pi]$, implies
$\pi=\alpha+\beta+\gamma$. $\Box$

If an angle space satisfies that for every trigon (degenerated or not)
$\\{A,B,C\\}$ we have that $\angle(ABC)+\angle(BCA)+\angle(CAB)=\pi$, then we
say that it is an _euclidean_ angle space. See that if $\\{A,B,C\\}$ is a
degenerated trigon, then automatically
$\angle(ABC)+\angle(BCA)+\angle(CAB)=\pi$.

As a consequence of the proposition above, we have the following result for
trigons:

######  Theorem 13 (Law of Sines for Euclidean Angle Spaces)

Let $(\mathcal{X},\mathcal{B},\angle)$ be an euclidean angle space where
$\mathcal{X}=\\{A,B,C\\}$ is a trigon. Then there is a unique distance
function $d$, up to multiplication by a constant, euclidean-compatible with
$\angle$. And we have that:

(4.4)
$\frac{d(B,C)}{\sin\alpha}=\frac{d(A,C)}{\sin\beta}=\frac{d(A,B)}{\sin\gamma}$

Proof: We have to show that (4.2) has a unique solution in the indeterminates
$a,b,c$, up to multiplication by a constant, with all its elements greater
than 0. We continue from (4.3). Doing the same as in the previous proof for
the indeterminate $c$ we reach the following:

$\begin{cases}a=a\\\ \frac{b}{a}=\frac{sin(\alpha+\gamma)}{sin\alpha}\\\
\frac{c}{a}=\frac{sin(\alpha+\beta)}{sin\alpha}\end{cases}\longrightarrow\begin{cases}a=a\\\
\frac{b}{a}=\frac{sin(\beta)}{sin\alpha}\\\
\frac{c}{a}=\frac{sin(\gamma)}{sin\alpha}\end{cases}$

(4.4) is a direct consequence of this expression. $\Box$

See also that if $(X,\mathcal{B},\angle)$ is an angle space, $\mathcal{X}$ is
a trigon, and $\angle(A,B,C)+\angle(B,C,A)+\angle(C,A,B)=\pi$, then it can be
conformally embedded in $\mathbb{E}^{2}$.

##  5 Solution to Problem 4 for tetragons.

The condition of being euclidean is not sufficient to ensure that an angle
space $(\mathcal{X},\mathcal{B},\angle)$ admits an euclidean compatible
distance function. See for instante the tetragon in the following image, for
which we have drawn also a “plane diagram” (analogous to the net of a
polyhedron, used for papercraft models).

None of the possible sets of three points are collinear. All the angles in
grey equal to $\pi/3$. Suppose that $d(C,D)=1$. It is easy to see (Law of
Sines) that the, the distances $d(A,C)=d(A,D)=d(B,C)=d(B,D)=1$. Finally see
that if $\angle(CAB)=\angle(ABD)=\pi/3$, $\angle(ACB)=\pi/6$,
$\angle(ADB)=\pi/5$ the the side $AB$ would have a different value according
to the Law of Sines in the trigons $\\{A,B,C\\}$ and $\\{A,B,D\\}$.

So to characterize angle spaces that admit euclidean-compatible distance
functions, we are going to begin with tetragons. We have already discussed, as
trivial, the case in which all the points in the space, four in this case, are
collinear. So two more cases are left: the one in which three points are
collinear but the four of them are not, and the one in which no set of three
points is collinear.

We say that an angle space satisfies the _Second Axiom of Collinearity_ if for
every tetragon $\\{A,B,C,D\\}$ such that $(A,B,C)\in\mathcal{B}$ and $D$ is
not collinear to any of them, we have that:

(5.1)
$\angle(ADC)=\angle(ADB)+\angle(BDC),\qquad\text{and}\qquad\angle(ABD)+\angle(CBD)=\pi$

We have the following:

###### Lemma 14

Let an euclidean angle space $(\mathcal{X},\mathcal{B},\angle)$ where
$\mathcal{X}=\\{A,B,C,D\\}$ is a tetragon such that $(A,B,C)\in\mathcal{B}$
and $D$ not collinear to any of them. It admits an euclidean-compatible
distance function $d$ if and only if it satisfies the _Second Axiom of
Collinearity_.

Proof: We are in the following situation:

Denote by $x=d(A,B)$, $x^{\prime}=d(B,C)$, $a=d(A,D)$, $a^{\prime}=d(C,D)$,
$h=d(B,D)$ and $\alpha=\angle(ABD)$. $d(A,C)=x+x^{\prime}$, since
$(A,B,C)\in\mathcal{B}$. Denote also by $\alpha^{\prime}=\angle(C,B,D)$,
$\beta=\angle(BAD)=\angle(CAD)$, $\beta^{\prime}=\angle(BCD)=\angle(ACD)$,
$\gamma=\angle(ADB)$, $\gamma^{\prime}=\angle(BDC)$, $\delta=\angle(A,D,C)$.
As a consequence of the Law of Sines, we have:

(5.2)
$\begin{cases}\frac{a}{\sin\alpha}=\frac{h}{\sin\beta}=\frac{x}{\sin\gamma}\\\
\frac{a^{\prime}}{\sin\alpha^{\prime}}=\frac{h}{\sin\beta^{\prime}}=\frac{x^{\prime}}{\sin\gamma^{\prime}}\\\
\frac{x+x^{\prime}}{\sin\delta}=\frac{a}{\sin\beta^{\prime}}=\frac{a^{\prime}}{\sin\beta}\end{cases}$

From this system, we obtain that:

$\begin{cases}\frac{\sin\gamma}{\sin\alpha}a+\frac{\sin\gamma^{\prime}}{\sin\alpha^{\prime}}a^{\prime}=\frac{\sin\delta}{\sin\beta}a^{\prime}\\\
\frac{\sin\gamma}{\sin\alpha}a+\frac{\sin\gamma^{\prime}}{\sin\alpha^{\prime}}a^{\prime}=\frac{\sin\delta}{\sin\beta^{\prime}}a\end{cases}$

This system must have a non-zero solution. So the determinant of the matrix of
coefficients (once the equations are suitably fixed) must be zero thus we
obtain that:

$\frac{\sin\gamma^{\prime}}{\sin\alpha^{\prime}\sin\beta^{\prime}}+\frac{\sin\gamma}{\sin\alpha\sin\beta}=\frac{\sin\delta}{\sin\beta\sin\beta^{\prime}}$

and then, since $\sin\gamma=\sin(\alpha+\beta)$,
$\sin\gamma^{\prime}=\sin(\alpha^{\prime}+\beta^{\prime})$ this yields:

$\sin\beta\sin\beta^{\prime}(cotan\alpha+cotan\alpha^{\prime})+\sin(\beta+\beta^{\prime})=\sin\delta$

but, since $\delta+\beta+\beta^{\prime}=\pi$, we have that
$sin(\beta+\beta^{\prime})=sin\delta$ and so
$cotan\alpha=-cota\alpha^{\prime}$ and $\alpha=\pi-\alpha^{\prime}$.

Now since:

$\begin{cases}\alpha+\beta+\gamma=\pi\\\
\alpha^{\prime}+\beta^{\prime}+\gamma^{\prime}=\pi\\\
\beta+\beta^{\prime}+\delta=\pi\end{cases}$

we can conclude that $\delta=\beta+\beta^{\prime}$. Finally, assuming that
(5.1) holds we have that (5.2) is equivalent to:

$\begin{cases}\frac{a}{\sin\alpha}=\frac{h}{\sin\beta}=\frac{x}{\sin\gamma}\\\
\frac{a^{\prime}}{\sin\alpha^{\prime}}=\frac{h}{\sin\beta^{\prime}}=\frac{x^{\prime}}{\sin\gamma^{\prime}}\\\
\frac{a}{\sin\beta^{\prime}}=\frac{a^{\prime}}{\sin\beta}\end{cases}$

which has a unique solution up to multiples in the variables
$a,a^{\prime},x,x^{\prime},h$.$\Box$

######  Proposition 15 (metrizability condition for tetragons)

Consider an euclidean angle space $(\mathcal{X},\mathcal{B},\angle)$ where
$\mathcal{X}=\\{A,B,C,D\\}$ is a tetragon. There is a unique distance function
$d$ in $X$ which is euclidean-compatible with $\angle$ if and only if it
satisfies the _metrizability conditions_ :

(5.3)
$\begin{cases}\sin\angle(DBA)\cdot\sin\angle(BCA)\cdot\sin\angle(CDA)=\sin\angle(CBA)\cdot\sin\angle(DCA)\cdot\sin\angle(BDA)\\\
\sin\angle(CAB)\cdot\sin\angle(DCB)\cdot\sin\angle(ADB)=\sin\angle(DAB)\cdot\sin\angle(ACB)\cdot\sin\angle(CDB)\\\
\sin\angle(BAC)\cdot\sin\angle(DBC)\cdot\sin\angle(ADC)=\sin\angle(DAC)\cdot\sin\angle(ABC)\cdot\sin\angle(BDC)\\\
\sin\angle(BAD)\cdot\sin\angle(CBD)\cdot\sin\angle(ACD)=\sin\angle(CAD)\cdot\sin\angle(ABD)\cdot\sin\angle(BCD)\end{cases}$

Proof: Consider the point $D$ and the following convention:

The distances $x_{a},x_{b},x_{c}$ are determined, up to multiplication by a
constant, by the angles in the triangles $\\{A,B,D\\}$, $\\{A,C,D\\}$ and
$\\{B,C,D\\}$ according to:

$\begin{cases}\frac{x_{a}}{\sin\angle(ABD)}=\frac{x_{b}}{\sin\angle(BAD)}\\\
\frac{x_{a}}{\sin\angle(ACD)}=\frac{x_{c}}{\sin\angle(DAC)}\\\
\frac{x_{b}}{\sin\angle(BCD)}=\frac{x_{c}}{\sin\angle(CBD)}\end{cases}$

if and only if
$\sin\angle(ACD)\cdot\sin\angle(BAD)\cdot\sin\angle(CBD)=\sin\angle(BCD)\cdot\sin\angle(CAD)\cdot\sin\angle(ABD)$.

On one hand, looking at these triangles we have that:

$\begin{cases}\frac{x_{a}\sin\angle(ABD)}{\sin\angle(ABD)}=\frac{x_{b}\sin\angle(ABD)}{\sin\angle(BAD)}=c\\\
\frac{x_{a}\sin\angle(ADC)}{\sin\angle(ACD)}=\frac{x_{c}\sin\angle(ADC)}{\sin\angle(DAC)}=b\\\
\frac{x_{b}\sin\angle(BDC)}{\sin\angle(BCD)}=\frac{x_{c}\sin\angle(BDC)}{\sin\angle(CBD)}=a\end{cases}$

On the other hand and according to this, the distance is well defined if and
only if:

$\frac{a}{\sin\angle(BAC)}=\frac{b}{\sin\angle(ABC)}=\frac{c}{\sin\angle(ACB)}$

which happens as a consequence of (5.3). Repeat the same argument for all the
vertices. $\Box$

##  6 Solution to Problem 4

Let $(\mathcal{X},\mathcal{B},\angle)$ be an euclidean angle space that
satisfies the _Second Axiom of Collinearity_ and the tetragon metrizability
condition. Let $A,B,C,D$ such that not all of them are collinear. Define:

(6.1) $d_{A,B,\lambda}(C,D)=\begin{cases}\lambda&\text{if
}\\{A,B\\}=\\{C,D\\}\\\
\frac{\sin\angle(ABD)}{\sin\angle(ADB)}\cdot\lambda&\text{if }A=C\\\
\frac{\sin\angle(ABC)}{\sin\angle(ACB)}\cdot\lambda&\text{if }A=D\\\
\frac{\sin\angle(BAD)}{\sin\angle(ADB)}\cdot\lambda&\text{if }B=C\\\
\frac{\sin\angle(BAC)}{\sin\angle(ACB)}\cdot\lambda&\text{if }B=D\\\
\frac{\sin\angle(CAB)}{\sin\angle(ADC)}\cdot
d(A,C)=\frac{\sin\angle(CBD)}{\sin\angle(BCD)}\cdot d(B,D)=&\text{if
}\\{A,B\\}\cap\\{C,D\\}\neq\emptyset\\\
\qquad=\frac{\sin\angle(CAB)}{\sin\angle(ACD)}\cdot
d(A,D)=\frac{\sin\angle(CBD)}{\sin\angle(BDC)}\cdot d(B,C)\end{cases}$

Figure 2: All the possibilities listed in the definition of $d_{A,B,\lambda}$
(except the first one).

See that if this angle space admits an euclidean-compatible distance $d$ such
that $d(A,B)=\lambda$, then $d(C,D)=d_{A,B,\lambda}(C,D)$ for every $C,D$ such
that $A,B,C,D$ are not all of them collinear.

###### Lemma 16

Let $(\mathcal{X},\mathcal{B},d)$ be an euclidean angle space that satisfies
the _Second Axiom of Collinearity_ and the tetragon metrizability condition.
The following two statements are equivalent, and both will be referred as the
_Global Compatiblity Condition_ :

  1. (i)

For every $A,B,A^{\prime},B^{\prime},C,D$ such that $\\{A,B,C,D\\}$,
$\\{A,B,A^{\prime},B^{\prime}\\}$, $\\{A^{\prime},B^{\prime},C,D\\}$ have at
least three different points that are not collinear. Then

$d_{A,B,\lambda}(A^{\prime},B^{\prime})=\lambda^{\prime}\Longrightarrow
d_{A^{\prime},B^{\prime},\lambda^{\prime}}(C,D)=d_{A,B,\lambda^{\prime}}(C,D)$

  2. (ii)

For every non-degenerated pentagon $\\{A,B,C,D,E\\}\subset\mathcal{X}$

(6.2)
$\sin\angle(DAE)\cdot\sin\angle(ABD)\cdot\sin\angle(ACB)\cdot\sin\angle(ADC)\cdot\sin\angle(CED)=$
$=\sin\angle(CAD)\cdot\sin\angle(ABC)\cdot\sin\angle(DCE)\cdot\sin\angle(ADB)\cdot\sin\angle(AED)$

and for every non-degenerated hexagon $\\{A,B,C,D,E,F\\}\subset\mathcal{X}$

(6.3) $sin\angle(EAF)\cdot sin\angle(ABE)\cdot sin\angle(ACB)\cdot
sin\angle(CDA)\cdot sin\angle(CED)\cdot sin\angle(EFC)=$ $=sin\angle(CAD)\cdot
sin\angle(ABC)\cdot sin\angle(ECF)\cdot sin\angle(CDE)\cdot
sin\angle(AEB)\cdot sin\angle(EFA)$

Proof: Condition $(i)$ is equivalent to saying that any hexagon in
$\mathcal{X}$, degenerated or not, admits a well defined euclidean-compatible
distance function. The angle space being euclidean, the _Second Axiom of
Collinearity_ and the tetragon metrizability conditions, ensure that trigons
and tetragons admit an euclidean-compatible distance function. So we only need
to care about non-degenerated pentagons and hexagons. The argument is very
similar to the one in the proof of Proposition 15.

Let us begin with hexagons. Every non-degenerated hexagon admits an euclidean-
compatible distance function if and only if for every labeling of its points
$\\{A,B,C,D,E,F\\}$ we have

$d_{A,B,\lambda}(C,D)=\lambda^{\prime}\Longrightarrow
d_{A,B,\lambda}(E,F)=d_{C,D,\lambda^{\prime}}(E,F)$

and, according to (6.1), we can see that this is equivalent to (6.3).

We can proceed in a similar way to show that every pentagon admits a well
defined euclidean-compatible distance function $d$ if and only if for every
labeling of its points $\\{A,B,C,D,E\\}$ we have (6.3). $\Box$

######  Theorem 17 (angle spaces that admit an euclidean-compatible distance)

For $(\mathcal{X},\mathcal{B},\angle)$ be an angle space with 4 our more
points. Then it admits an euclidean-compatible distance function $d$ if and
only if it satisfies the following conditions:

  1. (A)

it is euclidean,

  2. (B)

it satisfies the Second Axiom of Collinearity,

  3. (C)

it satisfies the tetragon metrizability condition and

  4. (D)

it satisfies the Global Compatibility Condition.

In this case, the distance function $d$ is unique up to multiplication by a
constant.

Proof: Fix two points $A,B\in\mathcal{X}$. We are going to construct the
euclidean-compatible distance function $d$. Impose, that $d(P,P)=0$ and that
$d(A,B)=\lambda$. Let $d_{A,B,\lambda}$ be the function defined in (6.1).

Consider any pair of distinct points $P,Q\in\mathcal{X}$. If $A,B,P,Q$ are not
all of them collinear, then we can define $d(P,Q)$ from (6.1). See that, in
this case, $d(P,Q)=d(Q,P)$ and that $d(P,Q)>0$.

If we have a pair of points $P,Q$ simultaneously collinear to $A,B$ then, by
hypothesis, we can choose a fifth point $R$ which is not collinear to $A,B$,
neither to $C,D$. Define $\mu=d_{A,B\lambda}(A,R)$ and then define
$d(P,Q)=d_{A,R,\mu}(P,Q)$. $d(P,Q)$ is, then, greater than 0. And it is well
defined (it does not depend on the choice of $R$) according to the _Global
Compatibility Condition_. The symmetry condition for the distance function is
also guaranteed.

Suppose that $d$ does not satisfy the condition of euclidean-compatibility,
that is, there exists a non-degenerated trigon $\\{A^{\prime},B^{\prime},D\\}$
such that:

$\frac{d(A^{\prime},B^{\prime})}{sin\angle(A^{\prime}DB^{\prime})}\neq\frac{d(A^{\prime},D)}{sin\angle(A^{\prime}B^{\prime}D)}$

Then this would contradict the _Global Compatibility Condition_ in the
pentagon $\\{A,B,A^{\prime},B^{\prime},D\\}$.

The fact that $d$ must satisfy the triangular inequality is a consequence of
the euclidean-compatibility.$\Box$

##  7 Solution to Problems 1 and 5

C. L. Morgan provides in [10] a characterization of the metric spaces
$(\mathcal{X},d)$ that can be embedded in $\mathbb{E}^{n}$ which is, to our
pourposes, more suitable than the one given by Menger in [7]. To explain his
result we need to introduce some notation.

Let $(\mathcal{X},d)$ be a metric space. Define, for every
$A,B,C\in\mathcal{X}$ the quantity

$\langle
A,B,C\rangle=\frac{1}{2}\left(d(A,C)^{2}+d(B,C)^{2}-d(A,B)^{2}\right)$

In the context of this article, our $(n+1)$-gons are called $n$-simplices. For
each $(n+1)$-gon or $n$-simplex $\\{X_{0},\ldots,X_{n}\\}\subset\mathcal{X}$
define:

$\text{Vol}_{n}(\\{X_{0},\ldots,X_{n}\\})=\frac{1}{n!}\sqrt{D_{n}(X_{0},\ldots,X_{n})}$

where $D_{n}(X_{0},\ldots,X_{n})$ is the determinant of the $n\times n$ matrix
which $i,j$ entry equals $\langle X_{i},X_{j},X_{0}\rangle$. This function
$\text{Vol}_{n}$ is well defined and is the so called simplicial $n$-volume.
For general metric spaces, it can be a complex number.

We say that a space $(\mathcal{X},d)$ is $n$-_flat_ if the simplicial
$n$-volume of every $(n+1)$-gon is a real number. In this case, its dimension
is the largest natural number (if it exists) for which there is an $(n+1)$-gon
with positive simplicial volume. See that if, for some $n$, there is no
$(n+1)$-gon with positive simplicial volume then there is no $(k+1)$-gon with
positive simplicial volume for $k\geq n$. We have that:

######  Theorem 18 (proved by Morgan in [10])

A metric space $(\mathcal{X},d)$ can be embedded in $\mathbb{E}^{n}$ if and
only if it is $k$-flat for every $1\leq k\geq n$, and of dimension less than
or equal to $n$.

If the metric space $(\mathcal{X},d)$ has also an angle function euclidean
compatible with $d$, then

$\langle A,B,C\rangle=d(A,C)\cdot d(B,C)\cdot cos(\angle(A,C,B))$

and so

$D_{n}(x_{0},\ldots,x_{n})=(\Pi_{j=1}^{n}d(x_{0},x_{j})^{2})\begin{array}[]{|
c c c |}cos\angle(x_{1},x_{0},x_{1})&\ldots&cos\angle(x_{1},x_{0},x_{n})\\\
\vdots&&\vdots\\\
cos\angle(x_{n},x_{0},x_{1})&\ldots&cos\angle(x_{n},x_{0},x_{n})\rangle\end{array}$

Let $(\mathcal{X},\mathcal{B},\angle)$ be an angle space that admits an
euclidean compatible distance function $d$. It can be conformally embedded in
$\mathbb{E}^{n}$ if and only if $(\mathcal{X},d)$ can be isometrically
embedded in $\mathbb{E}^{n}$. So everything together proves that:

######  Corollary 19 (isometrical embeddabililty in terms of angles)

Let $(\mathcal{X},\mathcal{B},\angle)$ be an angle space that admits an
euclidean compatible distance function. This angle space can be conformally
embedded in $\mathbb{E}^{n}$ if and only if the two following properties hold:

  1. (1)

Forall $1\leq k\leq n$, $\\{X_{0},\ldots,X_{k}\\}\subset\mathcal{X}$,
$\begin{array}[]{| c c c
|}cos\angle(X_{1},X_{0},X_{1})&\ldots&cos\angle(X_{1},X_{0},X_{k})\\\
\vdots&&\vdots\\\
cos\angle(X_{k},X_{0},X_{1})&\ldots&cos\angle(X_{k},X_{0},X_{n})\rangle\end{array}\geq
0$.

  2. (2)

Forall subset $\\{X_{0},\ldots,X_{n+1}\\}\subset\mathcal{X}$,
$\begin{array}[]{| c c c
|}cos\angle(X_{1},X_{0},X_{1})&\ldots&cos\angle(X_{1},X_{0},X_{n+1})\\\
\vdots&&\vdots\\\
cos\angle(X_{n+1},X_{0},X_{1})&\ldots&cos\angle(X_{n+1},X_{0},X_{n+1})\rangle\end{array}=0$.

As a consequence of Theorem 17 and of the previous corollary we finally
obtain:

######  Theorem 20 (solution for Problem 5)

Let $(\mathcal{X},\mathcal{B},\angle)$ be an angle space. It can be
conformally embedded in $\mathbb{E}^{2}$ if and only if it can be endowed with
an euclidean-compatible distance function, that is:

  1. (A)

it is euclidean,

  2. (B)

it satisfies the tetragon compatibility condition,

  3. (C)

it satisfies the Second Axiom of Collinearity,

  4. (D)

it satisfies the Global Compatibility Condition,

and the resultant metric space can be isometrically embedded in
$\mathbb{E}^{n}$, that is:

  1. (1)

Forall $1\leq k\leq n$, $\\{X_{0},\ldots,X_{k}\\}\subset\mathcal{X}$,
$\begin{array}[]{| c c c
|}cos\angle(X_{1},X_{0},X_{1})&\ldots&cos\angle(X_{1},X_{0},X_{k})\\\
\vdots&&\vdots\\\
cos\angle(X_{k},X_{0},X_{1})&\ldots&cos\angle(X_{k},X_{0},X_{n})\rangle\end{array}\geq
0$.

  2. (2)

Forall subset $\\{X_{0},\ldots,X_{n+1}\\}\subset\mathcal{X}$,
$\begin{array}[]{| c c c
|}cos\angle(X_{1},X_{0},X_{1})&\ldots&cos\angle(X_{1},X_{0},X_{n+1})\\\
\vdots&&\vdots\\\
cos\angle(X_{n+1},X_{0},X_{1})&\ldots&cos\angle(X_{n+1},X_{0},X_{n+1})\rangle\end{array}=0$.

##  8 Final Comments

Our result of conformal embeddability of angle spaces
$(\mathcal{X},\mathcal{B},\angle)$, as the one by Menger for isometrical
embedabbility of metric spaces $(\mathcal{X},d)$, is more useful when
$\mathcal{X}$ is a finite set. If $\mathcal{X}$ has an infinite number of
points, conditions (A), (B), (C), (D), (1) and (2) in Theorem 20 may not be
easily checked. So some work is still left to be done. It would be desirable
to find easily verifiable criterions for conformal embeddability for angle
spaces with an infinite set of points.

Consider the class $\mathcal{T}$ of angle spaces
$(\mathcal{X},\mathcal{B},\angle)$ such that $\mathcal{X}$ is also a
topological space homeomorphic to the unit circle $S^{1}$. There is a
correspondence between Jordan curves an elements in $\mathcal{T}$ and between
knots and elements in $\mathcal{T}$. This new viewpoints for these objects may
be of interest. See for example that a well known unsolved and old conjecture,
the _Inscribed Square Problem_ (see [5, 11]), can be stated in this terms, via
this correspondence:

###### Conjecture 21

Let an angle space $(\mathcal{X},\mathcal{B},\angle)$ which is conformal to a
Jordan curve. Then there exists a tetragon $\\{A,B,C,D\\}$ such that:

$\angle{A,B,C}=\pi/2,\angle{B,C,D}=\pi/2,\angle{C,D,A}=\pi/2,\angle{D,A,B}=\pi/2$

and the rest of agles in the tetragon equal $\pi/4$.

This lead to the following:

###### Problem 22

Find necessary and sufficient conditions for an angle space
$(\mathcal{X},\mathcal{B},\angle)$ to be conformal to a Jordan curve (in terms
of easily verifiable properties of the angle function $\angle$).

Finally, it is also possible to consider problems of conformal embeddability
of angle spaces in other angle spaces. Also for non-euclidean geometries. For
example:

###### Problem 23

Characterize those angle spaces $(\mathcal{X},\mathcal{B},\angle)$ that can be
conformally embedded in the unit sphere $\mathcal{S}^{2}$.

###### Problem 24

Characterize those angle spaces $(\mathcal{X},\mathcal{B},\angle)$ that can be
conformally embedded in the Poincaré Disk.

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