# Locally toroidal polytopes of rank 6 and sporadic groups

Dmitrii V. Pasechnik  Department of Computer Science and Pembroke College,
University of Oxford<EMAIL_ADDRESS>
###### Abstract.

We augment the list of finite universal locally toroidal regular polytopes of
type $\\{3,3,4,3,3\\}$ due to P. McMullen and E. Schulte, adding as well as
removing entries. This disproves a related long-standing conjecture. Our new
universal polytope is related to a well-known $Y$-shaped presentation for the
sporadic simple group $\mathit{Fi}_{22}$, and admits $S_{4}\times
O_{8}^{+}(2){:}S_{3}$ as the automorphism group. We also discuss further
extensions of its quotients in the context of $Y$-shaped presentations. As
well, we note that two known examples of finite universal polytopes of type
$\\{3,3,4,3,3\\}$ are related to $Y$-shaped presentations of orthogonal groups
over $\mathbb{F}_{2}$. Mixing construction is used in a number of places to
describe covers and 2-covers.

##  1\. Introduction and results

Presentations of finite sporadic simple groups as quotients of Coxeter groups
with diagram $Y_{\alpha\beta\gamma}$, cf. Fig. 1(a), were discovered 30 years
ago [6, 7, 23] and remain a subject of considerable interest, cf. e.g. [24, 8,
14, 3, 1, 2].

$\ \ \beta\geq 0$$\alpha\geq 0\ \ $$\gamma\geq 0\mbox{ nodes }\ \
$$Y_{\alpha\beta\gamma}$(a)

$a$$b_{3}$$c_{3}$$b_{2}$$c_{2}$$d_{2}$$b_{1}$$c_{1}$$d_{1}$$\phi$$\phi$$\phi$(b)

Figure 1. $Y$-diagrams; the general case and $Y_{332}$.

For instance, the sporadic simple group $\mathit{Fi}_{22}$ is a quotient of
$Y_{332}$, cf. Fig. 1(b) modulo relations (2) below, cf. [6, p.233]. As
explained in Section 3 below, these relations afford the automorphism $\phi$
of the diagram swapping $b_{1}$, $c_{1}$, and $d_{1}$ with, respectively
$b_{2}$, $c_{2}$, and $d_{2}$. Thus $\mathit{Fi}_{22}$ has a subgroup
isomorphic to a quotient of the Coxeter group with Schläfli symbol
$[3,3,4,3,3]$, i.e. with the string diagram we will also denote by
$[3^{2},4,3^{2}]$, cf. Fig. 2.

$\nu\geq 0$$\mu\geq 0$ Figure 2. $[3^{1+\nu},4,3^{1+\mu}]$-diagram.

Groups of this kind arise in the study of _abstract regular polytopes_ , for
which the book [17] by P. McMullen and E. Schulte is a definite reference.
These objects may be viewed as quotients (satisfying _intersection property_
(1), cf. [17]) of Coxeter complexes of Coxeter groups with $r$-node string
diagram; here $r$ is called the _rank_. Abstract regular polytopes are
classified according to topology of their vertex figures and facets; they are
called _locally $X$_ if the latter have topology of $X$. An up-to-date review
on group-theoretic approaches to the problem of classification of abstract
regular polytopes may be found e.g. in [5].

Following a problem posed by B. Grünbaum [12, p.196], particular attention has
been paid to finite locally toroidal abstract regular polytopes, in this case
the rank is bounded from above by $6$, cf. [17, Lemma 10A.1].

One of the three rank 6 cases is the case corresponding to the group
$[3^{2},4,3^{2}]$, cf. [17, Sect. 12D] (or [16, Sect. 7]), where a
conjecturally complete list of the finite universal examples is given, see
[17, Table 12D1] (which is already in [16, Table V]) and [21, Problem 18]. The
main results of the present paper give one more example, missing in that
table, and remove erroneous infinite series of examples, of which only first
terms actually exist.

In more detail, the facets $\\{3,3,4,3\\}_{\mathbf{s}}$ and vertex figures
$\\{3,4,3,3\\}_{\mathbf{t}}$ here correspond to nontrivial finite quotients
$[3,3,4,3]_{\mathbf{s}}$ and $[3,4,3,3]_{\mathbf{t}}$ of the affine Coxeter
group $\tilde{F}_{4}$, i.e. the groups $[3,3,4,3]$ and $[3,4,3,3]$, with
normal Abelian subgroups either of the form $q^{4}$ or $q^{2}\times(2q)^{2}$,
with $q\geq 2$, where we follow notation from [6] to denote the direct product
$(\mathbb{Z}/s\mathbb{Z})^{k}$ of $k$ copies of the cyclic group of order $s$
by $s^{k}$. The case $q^{4}$ is denoted in [16, 17] by
$\\{3,3,4,3\\}_{(q,0,0,0)}$, and the case $q^{2}\times(2q)^{2}$ by
$\\{3,3,4,3\\}_{(q,q,0,0)}$ (and completely similarly for $\\{3,4,3,3\\}$).

######  Theorem 1.

Let $\Gamma$ be a universal locally toroid rank $6$ abstract regular polytope
with vertex figures of type $\\{3,4,3,3\\}_{(3000)}$ and facets of type
$\\{3,3,4,3\\}_{(2200)}$. Then $\Gamma$ is the 24-fold cover of $\Gamma_{1}$,
where $\Gamma_{1}$ has $v=11200=2^{6}{\cdot}5^{2}{\cdot}7$ vertices and
$f=14175=3^{4}{\cdot}5^{2}{\cdot}7$ facets, and the group
$\Omega:=O_{8}^{+}(2){:}S_{3}$, of order
$g=2^{13}{\cdot}3^{6}{\cdot}5^{2}{\cdot}7$. The group of $\Gamma$ is
isomorphic to $S_{4}\times\Omega$.

Here $\Gamma=\Gamma_{4}$ and $\Gamma_{1}$ are the biggest and the smallest
member of the sequence of covers $\Gamma_{k}$, with $1\leq k\leq 4$. Namely,
$\Gamma_{k}$ is a $k!$-fold cover of $\Gamma_{1}$ and the group of
$\Gamma_{k}$ is isomorphic to $S_{k}\times\Omega$. More details on this are
given in Section 3.

A number of comments on Theorem 1 are in order. The claim of finiteness of
$\Gamma$ depends upon coset enumeration, that was, for robustness purposes,
carried out using two different implementations of the Todd-Coxeter algorithm:
the built-in implementation of GAP system [10], and ACE implementation by G.
Havas and C. Ramsay [13], also available as a GAP package [9]. Both
computations in the case $\mathbf{t}={(3000)}$, $\mathbf{s}=(2200)$ returned
the index of the vertex stabiliser, the subgroup $F:=3^{4}{:}F_{4}$ in the
quotient $\Theta$ of the Coxeter group $[3^{2},4,3^{2}]$ modulo the relations
on Fig. 5 equal to $268800=2^{9}{\cdot}3{\cdot}5^{2}{\cdot}7$. We explain
below that on the other hand the identification of the group of $\Gamma_{1}$
does not need coset enumeration, and can be carried out by hand or with any
software capable of multiplying $24\times 24$ matrices over $\mathbb{F}_{2}$,
the field of 2 elements. It is worth mentioning that both $F$ and the facet
stabiliser $(2^{2}.4^{2}){:}F_{4}$ are maximal subgroups in $\Omega$, although
only $F$ remains maximal upon restricting to the simple subgroup
$O_{8}^{+}(2)$ of $\Omega$.

In Section 3 we sketch how to establish the existence of the group of
$\Gamma_{1}$ as a subgroup in $\mathit{Fi}_{22}$. Apart from this, we discuss
there other connections to the $Y$-shaped presentations already mentioned, and
pose a number of related open problems. In Section 4 we give another proof of
existence of $\Gamma_{1}$ and its covers $\Gamma_{k}$ for $2\leq k\leq 4$ by a
computer-free argument. The latter is an example of an application of
Proposition 3 below.

$\mathit{Fi}_{22}$ is not the only quotient of the Coxeter group $Y_{332}$
related to polytopes of type $\\{3,3,4,3,3\\}$. Namely, one also finds groups
for such polytopes within the 2-modular quotient $G$ of $Y_{332}$, which is
isomorphic to $2^{8}{:}O_{8}^{-}(2){:}2$, as established by R. Griess in [11],
where a general study of $p$-modular quotients of $Y_{\alpha\beta\gamma}$ is
carried out; in case $p=2$ one speaks of subgroups of
$\mathit{GL}_{1+\alpha+\beta+\gamma}(2)$ generated by _transvections_ —images
of the generators of $Y_{\alpha\beta\gamma}$. One such polytope is described
in the following result, and the other one is its 2-quotient. The group $G$
also appears on [6, pp.232-233], as a quotient of $X_{332}$, a re-labelling of
$Y_{332}$. See Section 5 for details.

######  Theorem 2.

Let $\Gamma$ be a locally toroid rank $6$ abstract regular polytope with
vertex figures of type $\\{3,4,3,3\\}$ and facets of type
$\\{3,3,4,3\\}_{(2000)}$. Then its vertex figures are either of type
$\\{3,4,3,3\\}_{(2200)}$, and $\Gamma$ has $128$ facets, $32$ vertices, and
the group of order $2^{18}{\cdot}3^{2}$, or its vertex figures are of type
$\\{3,4,3,3\\}_{(2000)}$, and $\Gamma$ has $32$ facets, $32$ vertices, and the
group of order $2^{15}{\cdot}3^{2}$.

Basically, enumeration of the cosets of a facet stabiliser in the quotient of
the Coxeter group $[3^{2},4,3^{2}]$ by a relation forcing the facets
$\\{3,3,4,3\\}_{(2000)}$ shows that this group is finite, of order
$2^{18}{\cdot}3^{2}$. This _proves_ that the vertex figures are of type
$\\{3,4,3,3\\}_{(2200)}$ (unless further relations are imposed). This
identifies $\Gamma$ with the second entry in [17, Table 12D1] (or [16, Table
V]) for $t=2$, i.e. $\mathbf{t}=(2200)$.

The quotient by a normal subgroup $2^{3}$ produces the other case,
$\mathbf{t}=(2000)$, the first entry in [loc.cit.] for $t=2$. It shows that
the cases with $t>2$ in [loc.cit.] in fact do not arise. We will sketch
alternative computer-free approaches to a proof of this part of the theorem in
Section 5.

No novelty is claimed for the second part of Theorem 2 (e.g. it is already in
[16]), it is added for the sake of completeness.

Currently known (up to taking duals) examples of universal finite polytopes of
type $\\{3,3,4,3,3\\}$ are listed in Table 1. We also give there information
on numbers of vertices $v$, facets $f$, groups, and whether there exists
$\Gamma\in\langle\\{3,3,4,3\\}_{\mathbf{s}},\\{3,4,3,3\\}_{\mathbf{t}}\rangle$
with group embedded as a subgroup in a “$Y$-overgroup“, i.e. in a quotient of
$Y_{332}$ over relations affording $\phi$, as shown on Figure 1(b). The last
example in the table is new, the rest already appear in [16].

$\mathbf{s}$ | $\mathbf{t}$ | $v$ | $f$ | $G$ | “$Y$-overgroup”
---|---|---|---|---|---
(2000) | (2000) | $2^{5}$ | $2^{5}$ | $[2^{15}3^{2}]$ | $O_{8}^{-}(2){:}2$
(2000) | (2200) | $2^{5}$ | $2^{7}$ | $[2^{18}3^{2}]$ | $2^{8}{:}O_{8}^{-}(2){:}2$
(2200) | (2200) | $2^{11}$ | $2^{11}$ | $[2^{24}3^{2}]$ | —
(3000) | (3000) | $2^{2}{\cdot}3^{2}{\cdot}5{\cdot}13$ | $2^{2}{\cdot}3^{2}{\cdot}5{\cdot}13$ | $(3^{2}\times L_{4}(3)).2^{2}$ | —
(2200) | (3000) | $2^{9}{\cdot}3{\cdot}5^{2}{\cdot}7$ | $2^{3}{\cdot}3^{5}{\cdot}5^{2}{\cdot}7$ | $S_{3}\times O_{8}^{+}(2){:}S_{3}$ | $\mathit{Fi}_{22}$
Table 1. The known universal finite polytopes
$\\{\\{3,3,4,3\\}_{\mathbf{s}},\\{3,4,3,3\\}_{\mathbf{t}}\\}$, their groups,
and (whenever they exist) “$Y$-overgroups“.

Note that Lemma 13 gives a construction of a finite polytope (not known to be
universal at present) with $\mathbf{s}=(6600)$, $\mathbf{t}=(3000)$ as the
mix, of the last two entries in Table 1. The mix construction from [17, Sect.
7A] provides a useful technical tool for constructing covers, as well as
2-covers, of abstract polytopes, and, more generally, quotients of Coxeter
complexes.

######  Proposition 3.

(Mix construction [17]). Let $G=\langle g_{1},\dots,g_{r}\rangle$ and
$H=\langle h_{1},\dots,h_{r}\rangle$ be two quotients of an arbitrary Coxeter
group $F$ with $r$ generators $\gamma_{k}$, so that $g_{k}$ and $h_{k}$ are
images of $\gamma_{k}$ under the respective homomorphisms, $1\leq k\leq r$.
Let $G\diamond H:=\langle g_{1}h_{1},g_{2}h_{2},\dots g_{r}h_{r}\rangle\leq
G\times H$. Then $G\diamond H$, called _mix_ of $G$ and $H$, is a quotient of
$F$, with $g_{k}h_{k}$ the image of $\gamma_{k}$, for $1\leq k\leq r$. ∎

Note that in general $G\diamond H$ might fail the intersection property,
despite it holding in $G$ and $H$, cf. [17, Sect. 7A6]. Thus we will need
extra tools to demonstrate the intersection property in $G\diamond H$, in
particular [17, Lemma 12D4], which is Lemma 5 below.

Proposition 3 allows to mix two abstract polytopes with the same diagram
$\mathcal{D}$, or, more generally, with the groups affording the Coxeter
relations from $\mathcal{D}$ (in the latter case the actual Coxeter diagram
might be a “quotient”, in the same sense as Weyl group $A_{2}\times A_{2}$
admits the relations of $F_{4}$). The corresponding algebraic system is a
meet-semilattice, corresponding to the partially ordered set of certain normal
subgroups in the Coxeter group with diagram $\mathcal{D}$. One application of
this is to construct covers of abstract polytopes, used in Section 4 to
analyse the group of $\Gamma$ in Theorem 1. For the sake of completeness, in
Section 6, cf. Proposition 11, we show how to use it to identify the group of
the universal polytope $\\{\\{3,3,4,3\\}_{(3000)},\\{3,4,3,3\\}_{(3000)}\\}$
with $(3^{2}\times L_{4}(3)).2^{2}$, the example studied in a great detail in
[18, Sect.4.2].

Another application of Proposition 3 is to construct certain 2-covers of
abstract polytopes. For instance, one constructs a finite abstract polytope of
type
$\\{\\{3,3,4,3\\}_{\mathbf{s}},\\{3,4,3,3\\}_{\mathbf{t}}\\}$ for a new tuple
$(\mathbf{s},\mathbf{t})$ of parameters, mixing a pair of examples from Table
1. More details on this are given in Section 6.

##  2\. Notation and preliminaries

Our notation for Coxeter groups and diagrams is standard. A Coxeter group $G$
is generated by generators $g_{1},\dots,g_{r}$, with the relations
$g_{i}^{2}=(g_{i}g_{j})^{k_{ij}}=1$, for $1\leq i<j\leq r$ and $2\leq
k_{ij}<\infty$ (more generally, one may assume $k_{ij}=\infty$, but we will
not use this here). Graphically this is drawn as an $r$-node graph (diagram)
$\mathcal{D}(G)$ with $k_{ij}-2$ edges joining vertices $i$ and $j$ (or, more
generally, with labels $k_{ij}$ on the edges for which $k_{ij}\geq 5$, but we
will not use this here). Groups $G$ for which $\mathcal{D}(G)$ is connected
are called _irreducible_ ; it is a necessary (although not sufficient)
condition for the irreducibility of the natural reflection representation of
$G$ with generators $g_{i}$ being reflections in $\mathbb{R}^{r}$ with respect
to hyperplanes $H_{i}$, with angles between $H_{i}$ and $H_{j}$ determined by
$k_{ij}$. Subgroups $P_{I}:=\langle g_{i}\mid i\in I\rangle\leq G$ for
$I\subseteq\\{1,\dots,r\\}$ are called _parabolic_ (called _special
parabolics_ in [17]).

For instance, the diagrams $Y_{\alpha,\beta,0}$, $Y_{\alpha,1,1}$, $Y_{221}$,
$Y_{321}$, $Y_{421}$ (cf. Fig. 1) correspond to irreducible finite Coxeter
groups $A_{\alpha+\beta}$, $D_{\alpha+2}$, $E_{6}$, $E_{7}$, and $E_{8}$,
which are also the Weyl groups of the respective root systems so named. Note
that the group $A_{n}$ is isomorphic to the symmetric group $S_{n+1}$. If
$\mathcal{D}(G)$ is an $r$-path, maybe with multiple edges, one talks about a
_string_ diagram, and encodes it as _Schläfli symbol_
$[k_{12},k_{23},\dots,k_{r-1,r}]$ (assuming the $g_{i}$ are ordered
consecutively on the path $\mathcal{D}(G))$. We have already mentioned an
example of a string diagram $A_{n}$; its Schläfli symbol is $[3^{n-1}]$, where
we abbreviated $n-1$ consecutive $3$’s as $3^{n-1}$. Other examples we need
here of irreducible finite Coxeter groups with string diagram are $F_{4}$,
with Schläfli symbol $[3,4,3]$, cf. Fig. 2, and $B_{n}$, with Schläfli symbol
$[4,3^{n-2}]$.

Further, we are interested in quotients of $G$ that satisfy _intersection
property_ , that is

(1) $G_{I}\cap G_{J}=G_{I\cap J}\qquad\text{for all
}I,J\subseteq\\{1,\dots,r\\}.$

For instance, finite quotients $p^{6}{:}E_{6}$, of the affine Coxeter group
$\tilde{E}_{6}$, i.e. the Coxeter group with diagram $Y_{222}$, and finite
quotients $p^{4}{:}F_{4}$ of the affine Coxeter group $\tilde{F}_{4}$, (the
group with Schläfli symbol $[3,3,4,3]$) satisfy the intersection property
whenever $p\geq 2$.

Quotients of $G$ with string diagram that satisfy (1) are called _string
$C$-groups_ in [16, 17], and they correspond to _abstract $r$-polytopes_. The
order complexes of these polytopes are quotients of the _Coxeter complex_ [25,
Chapter 2], [20, 4] of $G$, a simplicial complex built from the cosets of the
parabolic subgroups $G_{I}\leq G$, $I\subseteq\\{1,\dots,r\\}$, with respect
to the respective normal subgroups (the latter might be trivial), cf. [17,
Sect. 2C, 3D]. In particular for finite irreducible $G$ they are classical
polytopes: $n$-simplices for $A_{n}$, $n$-cubes for $B_{n}$, the _$24$ -cell_
$\\{3,4,3\\}$ for $F_{4}$, etc. Abstract regular polytopes are also known as
satisfying the intersection property thin diagram geometries with string
diagram, see e.g. [20, 4]. By $\langle\mathcal{P}_{f},\mathcal{P}_{v}\rangle$
we denote the class of all polytopes with facets isomorphic to
$\mathcal{P}_{f}$ and vertex figures isomorphic to $\mathcal{P}_{v}$. This
class contains unique polytope, called the _universal polytope_
$\\{\mathcal{P}_{f},\mathcal{P}_{v}\\}$, which covers each member of the
class, cf. [17, Thm.4A2].

In the sequel we will need a classification of abstract polytopes of type
$\\{3,3,4,3\\}$ and, dually, $\\{3,4,3,3\\}$. Due to duality, it suffices to
deal with the former, corresponding to the quotients of the affine Coxeter
group $\tilde{F}_{4}$, see Fig. 3. (In our notation for _conjugation_ ,
$x^{y}=y^{-1}xy$.)

$a$$b$$c$$d$$e$

$\sigma:=d^{cb}$, $\tau:=c^{de}$

Figure 3. Affine Coxeter group $\tilde{F}_{4}$, nodes labelled by generators.

$F_{4}$ acts on the lattice
$\mathbb{Z}^{4}\cup((\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})+\mathbb{Z}^{4}$
of $\mathbb{R}^{4}$-vectors with coordinates being integer or half-integer,
cf. e.g. [16, p.23], and one needs to classify the invariant sublattices of
finite index. The latter may be generated (as a submodule) by a single vector,
of the shape $\mathbf{s}:=(s,0,0,0)$, or $\mathbf{s^{\prime}}:=(s,s,0,0)$,
with $2\leq s\in\mathbb{Z}$. Combining [16, Theorems 3.4 and 3.5], one obtains
the following.

######  Theorem 4.

[16] Let $\Pi$ be a finite abstract regular polytope
$\\{3,3,4,3\\}_{\mathbf{t}}$. Then for $\mathbf{t}=\mathbf{s}$ its group is
isomorphic to $[3,3,4,3]$ subject to the extra relation
$(a\sigma\tau\sigma)^{s}=1$, with $\tau$ and $\sigma$ as on Fig. 3, whereas
for $\mathbf{t}=\mathbf{s^{\prime}}$ the extra relation is
$(a\sigma\tau)^{2s}=1$. ∎

In what follows we denote the groups arising in Theorem 4 by
$[3,3,4,3]_{\mathbf{t}}$ (or, if appropriate, by $[3,4,3,3]_{\mathbf{t}}$).

The mix construction in Proposition 3 needs extra tools to show that
intersection property holds in our mixes. One of these is [17, Lemma 12D4] and
some of its corollaries proved immediately after [loc.cit.] which we state
here in our context.

######  Lemma 5.

Let $H=[3,3,4,3]_{(s,0,0,0)}$ with $s$ an odd prime. Then the
$[3,4,3]$-parabolic is maximal in $H$. Consequently, let $G$ be a finite
quotient of $[3^{2},4,3^{2}]$ such that the following holds for its
parabolics: $G_{\\{0,\dots,4\\}}\cong[3,3,4,3]_{(s,0,0,0)}$ and
$G_{\\{1,\dots,5\\}}\cong[3,4,3,3]_{(t,0,0,0)}$ (or $[3,4,3,3]_{(t,t,0,0)}$),
$t\geq 2$, and so that $G_{\\{0,\dots,4\\}}$ and $G_{\\{1,\dots,5\\}}$ are
proper subgroups of $G$. Then (1) holds in $G$. ∎

##  3\. Sporadic $Y$-presentations and their twists

Our Theorem 1 will follow at once from the following.

######  Theorem 6.

Let $\Gamma$ be a locally toroid rank $6$ abstract regular polytope with
vertex figures of type $\\{3,4,3,3\\}_{\mathbf{s}}$ and facets of type
$\\{3,3,4,3\\}_{\mathbf{t}}$. Let $\mathbf{s}={(3000)}$ and
$\mathbf{t}=(2200)$. Then $\Gamma$ is finite and isomorphic to $\Gamma_{k}$,
for some $1\leq k\leq 4$, defined in and after Theorem 1. In particular,
$\Gamma$ is a quotient of $\Gamma_{4}$, and the group of $\Gamma_{k}$ is
isomorphic to $S_{k}\times\Omega$.

According to [6], the sporadic simple group $\mathit{Fi}_{22}$ has a
presentation $Y_{332}$ on Fig. 1(b) subject to extra relations
$S=f_{12}=f_{21}=1$, where

(2) $S=(ab_{1}c_{1}ab_{2}c_{2}ab_{3}c_{3})^{10},\quad
f_{ij}=(ab_{i}b_{j}b_{k}c_{i}c_{j}d_{i})^{9},\text{ where
}\\{i,j,k\\}=\\{1,2,3\\}.$

The following observation was the starting point of this project.

######  Lemma 7.

Let $\phi$ be the automorphism of the diagram $Y_{332}$ from Fig. 1(b). Then
$S^{\phi}$ and $S$ generate the same normal subgroup, and
$f_{12}^{\phi}=f_{21}$, $f_{21}^{\phi}=f_{12}$.

###### Proof.

Note that $S,S^{\phi}\in Y_{222}=\tilde{E}_{6}$, and it is straightforward to
check that they generate the same $E_{6}$-invariant sublattice in the
6-dimensional $E_{6}$-invariant lattice. The rest of the statement is obvious.
∎

In view of Lemma 7, $\mathit{Fi}_{22}$ contains a quotient $\Theta=\langle
c_{3},b_{3},a,b_{1}b_{2},c_{1}c_{2},d_{1}d_{2}\rangle$ of a Coxeter group with
diagram $[3^{2},4,3^{2}]$.

$d_{1}d_{2}$$c_{1}c_{2}$$b_{1}b_{2}$$a$$b_{3}$$c_{3}$ Figure 4. Coxeter group
with diagram $[3^{2},4,3^{2}]$ from $\phi$.

One immediately identifies $\Theta_{f}=\langle
b_{3},a,b_{1}b_{2},c_{1}c_{2},d_{1}d_{2}\rangle<\Theta$ with a subgroup in
$2^{6}{:}Sp_{6}(2)$ presented by $Y_{331}$ with extra relations
$f_{12}=f_{21}=1$, and $\Theta_{v}=\langle
c_{3},b_{3},a,b_{1}b_{2},c_{1}c_{2}\rangle<\Theta$ with a subgroup in
$3^{5}{:}E_{6}$ presented by $Y_{222}$ with an extra relation $S=1$, see [6,
p.232-233].

Direct computations either by coset enumeration, or in appropriate $E_{6}$\-
and $E_{7}$-invariant lattices allow one to identify $\Theta_{v}$ and
$\Theta_{f}$ as groups classified by Theorem 4. By Lemma 5, we have the
following.

######  Lemma 8.

The following isomorphisms hold: $\Theta_{f}=[3,4,3,3]_{(2200)}$, and
$\Theta_{v}=[3,3,4,3]_{(3000)}$. The intersection property holds in $\Theta$.
∎

At this point we can already conclude that the example of
$\\{3,3,4,3,3\\}$-polytope provided by $\Theta$ does not appear in [16, Table
V], as the pair of vectors $\mathbf{s}=(3000)$ and $\mathbf{t}=(2200)$
determining the vertex figure and the facet is not present there.

As $\phi$ on Fig. 1(b) induces an outer automorphism of $\mathit{Fi}_{22}$ and
centralises $\Theta$, one concludes by inspecting the list of maximal
subgroups of $\mathit{Fi}_{22}$ that $\Theta\leq O_{8}^{+}(2){:}S_{3}$, and
enumeration of cosets of $\Theta$ in $\mathit{Fi}_{22}$, or analysis of
maximal subgroups of $O_{8}^{+}(2){:}S_{3}$ confirm that in fact
$\Theta=O_{8}^{+}(2){:}S_{3}$, identifying it with the group of the example
$\Gamma_{1}$ in our Theorem 6.

######  Remark 1.

The problem of recognising groups presented by $Y_{332}$ with similar extra
relations, generalising ones on [6, pp.232-233], has been brought to the
author’s attention by Alexander A. Ivanov. Namely, one is interested in the
general setting where subgroups $Y_{331}=\tilde{E}_{7}$ and
$Y_{222}=\tilde{E}_{6}$ are made finite by factoring out the corresponding
finite index invariant sublattices.

It is strikingly parallel to the problem of classifying the finite quotients
of $[3,3,4,3,3]$ we are concerned with in this text. The simplest case, where
$Y_{222}$ becomes $2^{6}{:}E_{6}$, has been essentially settled in [11] (one
gets orthogonal groups over $\mathbb{F}_{2}$, in some cases acting on its
natural $\mathbb{F}_{2}$-module), but in general it is open. We remark that
Proposition 3 allows to construct more examples, e.g. we can get a subgroup of
$\mathit{Fi}_{22}\times 2^{8}{:}O_{8}^{-}(2)$ by applying it to the two known
examples.

We hope to explore this topic further in another publication.

######  Problem 1.

The question of understanding similarly constructed subgroups of the Monster-
related groups with $Y$-presentations is largely open, although they ought to
exist by arguments similar to the $Y_{332}$-case considered here.

Preliminary experiments with enumerating cosets of one of our finite
$[3^{2},4,3^{2}]$-quotients, namely the group of $\Gamma_{1}$, isomorphic to
$O_{8}^{+}(2){:}S_{3}$, in $[3^{2},4,3^{3}]$, indicates that this gives the
group $F_{4}(2)$, which is a subgroup of a quotient of $Y_{333}$ isomorphic to
${}^{2}\,E_{6}(2)$. More precisely, the enumeration returns index 12673024,
which is 4 times the index of $O_{8}^{+}(2){:}S_{3}$ in $F_{4}(2)$.

We should also mention that the $[3,4,3^{3}]$-subgroup in this group, which
corresponds to a locally toroidal polytope of the type considered in [17,
Sect. 12C], is isomorphic to $L_{4}(3){:}2$, shown to be a subgroup in
$F_{4}(2)$ by L. Soicher [24] by an argument involving a diagram automorphism
similar to $\phi$.

##  4\. Another existence proof of $\Gamma_{1}$, and existence of its covers

Here we present another, direct, existence proof of $\Gamma_{1}$, and complete
the proof of Theorem 6 by constructing its covers.

We begin by writing down a presentation $\mathcal{P}$ for $\Theta$, the group
of the _universal cover_ of the examples from Theorem 6. In our context it
simply means that in $\mathcal{P}$ we do not impose any relations that involve
all the generators. As $\Theta$ is the group of the (unique) universal
polytope, we can conclude without loss in generality that $\mathcal{P}$ is

$a$$b$$c$$d$$e$$f$

$\sigma:=d^{cb}$, $\tau:=c^{de}$, $1=(a\sigma\tau)^{4}=(f\tau\sigma\tau)^{3}$.

Figure 5. The presentation $\mathcal{P}$ for $\Theta$, nodes labelled by
generators.

######  Lemma 9.

There is a surjective homomorphism $\hat{}:\Theta\to O_{8}^{+}(2){:}S_{3}$,
with its image, the group of $\Gamma_{1}$, satisfying the intersection
property.

###### Proof.

The embedding will be given by matrices in $\mathit{GL}_{24}(2)$, with all the
elements, except images $\hat{a}$ and $\hat{f}$ of $a$ and $f$, permutation
matrices. Namely, $\langle b,c,d,e\rangle=F_{4}$ embeds as the permutation
group in its natural action on the vertices of the 24-cell. The latter action
has an imprimitivity system with three blocks of size 8, which will correspond
to the three natural 8-dimensional submodules for $O_{8}^{+}(2){:}2$. We
choose the generators so that the blocks are $\\{1..8\\}$, $\\{9..16\\}$, and
$\\{17..24\\}$, and so that the centre of $F_{4}$ is the permutation
$(1,2)(3,4)\dots(23,24)$. Let us slightly abuse notation and view the
following as specifications of $24\times 24$-permutation matrices over
$\mathbb{F}_{2}$.

$\displaystyle\hat{b}=$
$\displaystyle\prod_{i\in\\{0,8,16\\}}(i+3,i+8)(i+4,i+7),$
$\displaystyle\hat{c}=$ $\displaystyle(3,6)(4,5)(11,14)(12,13)(17,24)(18,23),$
$\displaystyle\hat{d}=$ $\displaystyle(5,6)\prod_{i=9}^{16}(i,i+8),$
$\displaystyle\hat{e}=$ $\displaystyle(21,22)\prod_{i=1}^{8}(i,i+8).$

In terms of the 24-cell, the size 8 imprimitivity block of a vertex $v$
consists of the antipodal to $v$ vertex of the 24-cell, and the vertices
antipodal to $v$ in each of the 6 octahedra (facets of the 24-cell) on $v$.
The stabiliser of the block in $F_{4}$ is
$\langle\hat{b},\hat{c},\hat{d},\hat{\tau}\rangle$.

Define matrices over $\mathbb{F}_{2}$, using $I_{k}$ to denote $k\times k$
identity matrix, and $E_{ij}$ to denote the matrix with $1$ in the position
$ij$ and $0$ elsewhere.

$\displaystyle A$ $\displaystyle:=\begin{pmatrix}1&0&0&0&0&0&0&0\\\
0&1&1&1&1&1&0&0\\\ 1&0&1&0&0&0&0&1\\\ 1&0&0&1&0&0&0&1\\\ 1&0&0&0&1&0&0&1\\\
1&0&0&0&0&1&0&1\\\ 0&0&1&1&1&1&1&0\\\ 0&0&0&0&0&0&0&1\end{pmatrix},$
$\displaystyle F$ $\displaystyle:=\begin{pmatrix}0&0&0&0&1&1&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&1&1\\\ 0&0&0&0&0&0&1&0\\\ 0&1&0&0&0&0&0&0\\\
1&1&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&1&1&0&0&0&0\end{pmatrix},$
$\displaystyle\hat{a}$ $\displaystyle:=I_{3}\otimes A,$ $\displaystyle\hat{f}$
$\displaystyle:=\begin{pmatrix}I_{8}+E_{21}&0&0\\\ 0&0&F\\\
0&F&0\end{pmatrix}.$

Now it is routine to check that the relations in $\mathcal{P}$ hold for just
defined $\hat{a}$,…, $\hat{f}$. As well, the elements $\hat{a}$,…, $\hat{f}$
leave invariant the quadratic form
$\Phi:=\sum_{j=1}^{24}x_{j}^{2}+\sum_{i=1}^{12}x_{2i-1}x_{2i}$; this is
immediate for the permutation matrices $\hat{b}$,…, $\hat{e}$, as they commute
with the centre of $\langle\hat{b},\hat{c},\hat{d},\hat{e}\rangle$, and a
direct computation for $\hat{a}$ and $\hat{f}$.

As this point we have $\hat{\Theta}\leq O_{8}^{+}(2){:}S_{3}$. To see that the
latter is in fact equality, one can either inspect the list of the maximal
subgroups of $O_{8}^{+}(2){:}S_{3}$, or investigate the action of the
stabiliser
$H:=\langle\hat{a},\hat{b},\hat{c},\hat{d},\hat{\tau},\hat{f}\rangle$ of the
subspace spanned by the first 8 coordinates in $\hat{\Theta}$, and see that it
acts transitively on the 135 nonzero $\Phi$-isotropic vectors there, by
computing the orbit of the all-1 vector, and further identifying $H$ with a
rank 3 permutation group $O_{8}^{+}(2){:}2$ in its natural action on isotropic
vectors.

Finally, the intersection property holds, by [16]. ∎

Let $\psi:\Theta\to S_{4}$ be a map defined on generators of $\Theta$, with
$\psi(a):=(1,2)$, $\psi(b):=(2,3)$, $\psi(c):=(3,4)$, and
$\psi(d)=\psi(e)=\psi(f)=()$. It turns out that it extends to a homomorphism
onto. (Note that there is no symmetry here: if we try to send $a$, $b$, $c$ to
the identity instead, we get a trivial group).

######  Lemma 10.

The map $\psi$ is a homomorphism onto. Also, there is a homomorphism from
$\Theta$ onto $S_{4}\times O_{8}^{+}(2){:}S_{3}$, with its image the group of
$\Gamma_{4}$.

###### Proof.

For the first part, observe that the Coxeter relations of $\Theta$ hold in
$\psi(\Theta)$. Also, $\psi(\sigma)=()$ and $\psi(\tau)=\psi(c)=(3,4)$. It
follows that the rest of the relations hold, as well, and we are done.

For the second part, we construct an embedding of $\Theta$ into
$G:=S_{4}\times O_{8}^{+}(2){:}S_{3}=\psi(\Theta)\times\hat{\Theta}$, as
prescribed by Proposition 3. Consider $H:=\psi(\Theta)\diamond\Theta\leq G$;
recall that it is generated by $\psi(a)\hat{a}$,…, $\psi(f)\hat{f}$.
Trivially, by checking the relations, $H$ is a homomorphic image of $\Theta$.
Moreover, the intersection property holds in $H$ by Lemma 5.

It remains to show $H=G$. Consider an element of $O_{8}^{+}(2)$ of order 5, as
a word $\hat{w}:=w(\hat{a},\dots,\hat{f})$ in our generators, and the
corresponding word $w:=w(\psi(a)\hat{a},\dots,\psi(f)\hat{f})\in H$. Then
$w=w(\psi(a),\dots,\psi(f))\hat{w}$. Thus $w^{12}=\hat{w}^{2}\in\hat{\Theta}$
is of order 5. As $O:=O_{8}^{+}(2)$ is generated by such elements, and as
$\hat{\Theta}$ is generated by $O$ and $\hat{d}$, $\hat{e}$, $\hat{f}$, we
have $\hat{\Theta}\leq H$. Therefore $H=G$, as claimed. ∎

##  5\. The case $\mathbf{s}={(2000)}$

First, we prove Theorem 2. Let
$\Gamma=\\{\\{3,3,4,3\\}_{(2000)},\\{3,4,3,3\\}\\}$. Then its group $\Theta$
has presentation as follows.

$a$$b$$c$$d$$e$$f$

$\sigma:=d^{cb}$, $\tau:=c^{de}$, $1=(a\sigma\tau\sigma)^{2}$.

Enumeration of cosets of the facet stabiliser $\langle a,b,c,d,e\rangle$ in
$\Theta$ shows that $|\Theta|=2^{18}{\cdot}3^{2}$. Working in the resulting
permutation representation one computes the orders of $f\tau\sigma\tau$ and
$f\tau\sigma\tau\sigma$ to be 4, thus identifying the stabiliser of the vertex
with the one corresponding to the type $\\{3,4,3,3\\}_{(2200)}$, as claimed.

The polytope $\Gamma$ corresponds to the second entry in [17, Table 12D1] (or
[16, Table V]) for $t=2$. The quotient by a normal subgroup $2^{3}$ in
$\Theta$ produces the other case, $\mathbf{t}=(2000)$, the first entry in
[loc.cit.] for $t=2$. It shows that the cases with $t>2$ in [loc.cit.] in fact
do not arise. ∎

More conceptually, one can notice that it is proved on [17, page 463] that the
number of vertices in $\Gamma$ equals 32. In particular, it means that the
index of the subgroup $H:=\langle b,c,d,e,f\rangle$ in $\Theta$ is 32. Let
$\pi:\Theta\to\hat{\Theta}$ be the permutation representation of $\Theta$ on
the cosets of $H$, and $K$ its kernel. As $H$ is a quotient of
$\tilde{F}_{4}$, and $K$ is a normal subgroup there, we see that $K$ is very
small. It cannot contain a faithful 4-dimensional module for $F_{4}$, as the
latter would admit an action of $a$ commuting with the action of $\langle
d,e,f\rangle$, but still inducing a faithful action of $\langle a,b\rangle$
(as the action of $\langle b,c\rangle$ would be faithful, and the latter is
conjugate to $\langle a,b\rangle$). This implies that $|K|\leq 8$.

Another possibility is to carry out the computation of a presentation for $H$
applying the Reidemeister–Schreier algorithm (which produces a presentation of
a finite index subgroup of a finitely presented group) cf. e.g. [15], and find
that $H=[3,4,3^{2}]_{(2200)}$.

###  5.1. Embedding of the group of $\Gamma$ in $Y$-group.

Consider the 2-modular quotient $G$ of $Y_{332}$, which is isomorphic to
$G:=2^{8}{:}O_{8}^{-}(2){:}2$, as established by R. Griess in [11,
pp.277-278]. Then $\Theta$ can be identified with its subgroup, as shown on
Figure 4, under the automorphism $\phi$ of the $Y$-diagram on Figure 1(b).
Specifically, the relation forcing the $\\{3,3,4,3\\}_{(2000)}$-facet turns
the subgroup $Y_{222}$ into $2^{6}{:}U_{4}(2).2$, and, respectively, $Y_{331}$
into $2^{7}{:}Sp_{6}(2)$.

Adding further relation forcing the $\\{3,4,3,3\\}_{(2000)}$-vertex figure
kills $O_{2}(G)$, and one obtains a quotient of
$\\{\\{3,3,4,3\\}_{(2000)},\\{3,4,3,3\\}_{(2000)}\\}$ by a subgroup of order
4.

The group $G$ also features on [6, pp.232-233] as a quotient of $X_{332}$, a
re-labelling of $Y_{332}$, with the node $c_{3}$ of the latter denoted by
$a_{3}$, by an explicitly given relation $W$. In complete analogy with the
$\mathit{Fi}_{22}$ situation, the relation $W$ affords the automorphism
$\phi$, and allows one to establish the existence of $\Gamma$ directly.

######  Remark 2.

In his PhD thesis L. Soicher [22, Thm.A.2] proved that the group
$[3^{m},4,3^{n}]$, subject to a relation killing the centre of the
$[3,4,3]$-parabolic, is isomorphic to $2^{mn}{:}(S_{m+1}\times S_{n+1})$. Note
that for $m=n=2$ this example has the cover
$\\{\\{3,3,4,3\\}_{(2000)},\\{3,4,3,3\\}_{(2000)}\\}$ from the last part of
Theorem 2 (see also the first entry in [16, Table V]).

Another “tower” of finite $[3^{m},4,3^{n}]$-examples with unbounded $m$ and
$n$ may be constructed from 2-modular quotients of the group
$Y_{\alpha\alpha\gamma}$ by twisting with $\phi$, generalising the observation
in the beginning of Section 5.1. There the $[3^{2},4,3^{2}]$-subgroups will
correspond to $\\{\\{3,3,4,3\\}_{(2000)},\\{3,4,3,3\\}_{(2200)}\\}$ from
Theorem 2.

##  6\. The mix construction

Our first application of the mix construction in Proposition 3 was Lemma 10.
Here we present more applications of the mix construction. The following
constitutes another proof of a result in [18, Sect.4.2].

######  Lemma 11.

The group of $\Gamma=\\{\\{3,3,4,3\\}_{(3000)},\\{3,4,3,3\\}_{(3000)}\\}$ is
isomorphic to $(3^{2}\times L_{4}(3)).2^{2}$.

###### Proof.

(Sketch) From coset enumeration, we know the order of the group in question.
The group $\hat{\Theta}:=L_{4}(3){:}2_{2}\cong\mathit{PGO}_{6}^{+}(3)$ has a
maximal subgroup isomorphic to $[3,3,4,3]_{(3000)}$, according to [6]. As
well, its central extension by the group of order 2, which is in fact split,
is constructed in [18, Sect.4.2] as a quotient of the group $\Theta$ of
$\Gamma$ (more precisely, as the reduction modulo 3 of the natural
6-dimensional representation of the Coxeter group $[3,3,4,3,3]$).

Let $\Omega$ be the quotient of $\Theta$ obtained by imposed the extra
relation that the generators (in the Coxeter diagram) of the “middle” dihedral
subgroup of order 8 commute. It is easy to check that $\Omega=S_{3}\times
S_{3}$. Consider the mix $P:=\hat{\Theta}\diamond\Omega$. One sees that the
$P$ contains a copy of the simple group $L_{4}(3)$, by considering words in
$P$ corresponding to elements of $\hat{\Theta}$ of order 13. Thus it also
contains $O_{3}(\Omega)=3^{2}$. It remains to observe that the Abelian
invariants of $\Omega$ and of $[3^{2},4,3^{2}]$ are equal to $(2,2)$, implying
that the $2^{2}$ acting on $P$ is generated by $P$. Finally, $P$ satisfies the
intersection property by Lemma 5. ∎

###  6.1. Constructing 2-covers

The following will be needed for rank 6 cases.

######  Proposition 12.

Let $\mathbf{s}=(s,s^{\prime},0,0)\in\\{(s,s,0,0),(s,0,0,0)\\}$ and
$\mathbf{t}=(t,t^{\prime},0,0)\in\\{(t,t,0,0),(t,0,0,0)\\}$. Denote
$\ell:=\operatorname{lcm}(s,t)$. The mix $G\diamond H$ of
$G=[3,3,4,3]_{\mathbf{s}}$ and $H=[3,3,4,3]_{\mathbf{t}}$ is
$[3,3,4,3]_{(\ell,\ell,0,0)}$ in the following cases

  1. (1)

$s^{\prime}=s$, $t^{\prime}=t$;

  2. (2)

$s^{\prime}=s$, $t^{\prime}=0$, $2\ell=\operatorname{lcm}(2s,t)$.

  3. (3)

$s^{\prime}=0$, $t^{\prime}=t$, $2\ell=\operatorname{lcm}(s,2t)$.

Otherwise $G\diamond H$ is $[3,3,4,3]_{(\ell,0,0,0)}$.

###### Proof.

To establish the intersection property (1), we embed $G\diamond H$
“diagonally” into $G\times H$ considered as an affine matrix group over the
suitable finite ring. This shows that $G\diamond H$ is isomorphic to some
$[3,3,4,3]_{\mathbf{u}}$ and so (1) holds. It remains to compute the relation
$\mathbf{u}$ implied by $\mathbf{s}$ and $\mathbf{t}$.

Let $\mathbf{s}$ and $\mathbf{t}$ have the same “format”, i.e. either
$s^{\prime}=t^{\prime}=0$ or $s^{\prime}=s$, $t^{\prime}=t$. Then the only
difference between the presentations of $G$ and $H$ is the exponent of the
extra non-Coxeter relation in Theorem 4, and the claim follows.

It remains to deal with the case of different “formats”. Let $G=\langle
a,b,c,d\rangle$, as in Theorem 4. Respectively, let $H=\langle
a^{\prime},b^{\prime},c^{\prime},d^{\prime}\rangle$, and denote
$\sigma^{\prime}:=d^{\prime c^{\prime}b^{\prime}}$, $\tau^{\prime}:=e^{\prime
c^{\prime}d^{\prime}}$.

Let $s^{\prime}=s$ and $t^{\prime}=0$. Thus the orders of $a\sigma\tau$ and
$a\sigma\tau\sigma$ equal to $2s$, while the orders of
$a^{\prime}\sigma^{\prime}\tau^{\prime}$ and
$a^{\prime}\sigma^{\prime}\tau^{\prime}\sigma^{\prime}$ equal $2t$ and $t$,
respectively. Then the order of
$aa^{\prime}\sigma\sigma^{\prime}\tau\tau^{\prime}$ equals $2\ell$, while the
order of
$aa^{\prime}\sigma\sigma^{\prime}\tau\tau^{\prime}\sigma\sigma^{\prime}$
equals $\operatorname{lcm}(2s,t)$. We get the case
$[3,3,4,3]_{(\ell,\ell,0,0)}$ if and only if the two latter orders are equal.
The case of $s^{\prime}=0$, $t^{\prime}=t$ is dealt with in the same way. ∎

This immediately implies

######  Lemma 13.

There exist $\\{\\{3343\\}_{\mathbf{s}},\\{3433\\}_{\mathbf{t}}\\}$ for
$(\mathbf{s},\mathbf{t})=(3000,6600)$.

###### Proof.

Apply the mix construction to $\\{\\{3343\\}_{(3000)},\\{3433\\}_{(2200)}\\}$
and
$\\{\\{3343\\}_{(3000)},\\{3433\\}_{(3000)}\\}$, compute $\mathbf{s}$ and
$\mathbf{t}$ using Proposition 12, and note that the intersection property
holds by Lemma 5. ∎

######  Remark 3.

One can also consider other mixes of examples in Table 1 to obtain, e.g.
$(\mathbf{s},\mathbf{t})=(6000,2200)$; however, we do not know whether the
intersection property holds in this group and other similar examples, as Lemma
5 would not apply.

######  Remark 4.

Analogously one can try to construct new examples of
$\\{\\{3,4,3,3\\}_{\mathbf{s}},\\{4,3,3,4\\}_{\mathbf{t}}\\}$. Here one needs
to be able to compute in the meet-semilattice generated by the parameters
$\mathbf{t}$ of the groups $[4,3,3,4]_{\mathbf{t}}$, for $\mathbf{t}$ in the
corresponding column of [16, Table VI] and in the examples from [19]. We leave
this to another publication.

### Acknowledgements

The author thanks Alexander A. Ivanov for bringing the topic of
$Y_{332}$-presentations, cf. Remark 1, to his attention, Daniel Allcock,
Leonard Soicher for useful suggestions and remarks, and an anonymous referee
for catching errors—in particular misuse of the mix construction—in a previous
version of this text. The author is supported in part by the EU Horizon 2020
research and innovation programme, grant agreement OpenDreamKit No 676541.

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the model’s sensitivity to the specific galaxies randomly selected for
training. Nevertheless, these variations do not alter the overall observable
trends for the different questions.

When comparing the various questions, the ‘disk-edge-on’ task not only has the
lowest mean deviation (as discussed in Sect. 5.2) when trained with the
complete set, but it also achieves a deviation below $10\%$ after training
with just 100 galaxies. This is even more impressive as only $6.1\,\%$ of the
galaxies are relevant (see Table 1), although Zoobot learns from all galaxies.
This further indicates that identifying disk galaxies is easier to learn than
other tasks of the decision tree. Similarly for the ‘bulge-size’ question, the
model achieves a deviation below $12\%$ after training with only 100 images.
Since these tasks were included in all GZ decision trees, this outcome can be
interpreted as a demonstration of the effectiveness of fine-tuning.
Furthermore, training on only 100 random galaxies leads for the ‘smooth-or-
featured’ question to deviations below $12\,\%$. This question was included in
all GZ decision trees as the first question and was thus answered by all
volunteers, therefore requiring fewer new examples compared to other tasks.

In contrast, for the ‘has-spiral-arms’ question, 60 000 galaxies are required
to achieve deviations below $12\,\%$. Despite the inclusion in all GZ decision
trees, a substantial number of examples are still necessary to accurately
predict the corresponding vote fractions. This observation suggests that
detecting spiral arms might pose a greater challenge for Euclid images
compared to the galaxies in the pretraining datasets. Additionally, questions
related to clumps in galaxies exhibit similar patterns, requiring a range of
10 000 to 60 000 random galaxies to achieve a deviation below $12\%$. From the
campaigns involved in the pretraining of Zoobot, these clump-related questions
were exclusively included in the GZC campaign. Consequently, the impact of
this pretraining is likely less effective for these tasks. Moreover, given
that spiral arms and clumps involve finer structures, the associated tasks are
inherently more complex and need a larger number of training examples.

###  5.2 Analysis of the best performing model

In this section, we analyse the performance of Zoobot for emulated Euclid VIS
images with the lowest averaged vote fraction mean deviation $\bar{\delta}$,
i.e., the best performing model, as derived in Sect. 5.1. We show examples of
Zoobot’s output, then investigate the performance with standard classification
metrics after discretizing the vote fractions (Sect. 5.2.1) and demonstrate
how our model can be used to find spiral galaxies in a given dataset (Sect.
5.2.2). Next, we analyse the predicted vote fractions directly by looking at
the mean (Sect. 5.2.3) and the histograms (Sect. 5.2.4) of the deviations from
their respective volunteer vote fractions, and by investigating their redshift
and magnitude dependence (Sect. 5.2.5). Finally, we compare the model
performance between HST and Euclid images (Sect. 5.2.6).

Figure 6: Four examples of the predictions of Zoobot following the structure
of the GZH decision tree (see Table 1 and Fig. 19) for galaxies (inverted
greyscale, image ID given above each image) from the complete test set. For
every question, the answer with the highest predicted vote fraction (denoted
in the parenthesis) is selected. The black bars represent a length of $None$.
Figure 7: Examples of galaxies with the highest predicted vote fractions of
Zoobot for (a) spiral, (b) completely round, (c) disk, (d) barred and (e)
clumpy galaxies from the complete test set. Above each galaxy image, the
corresponding image ID and the predicted vote fraction in percent is given.
The black bars represent a length of $None$.

To verify the quality of the predictions, four examples of Zoobot’s output on
different galaxies from the complete test set are shown in Fig. 6. The
selected answer for every question is the one with the highest predicted vote
fraction, while the asked questions follow the structure of the GZH decision
tree (see Table 1 and Fig. 19). In Fig. 7, we show four galaxies from the
complete test set with the highest predicted vote fractions for five example
answers, i.e., (a) spiral, (b) completely rounded, (c) disk, (d) bar and (e)
clumpy, to demonstrate the quality of Zoobot’s predictions.

####  5.2.1 Discrete classifications

To get an intuitive sense of Zoobot’s performance for the different morphology
tasks, we convert the predicted vote fractions into discrete values by binning
them to the class with the highest predicted vote fraction. However, it is
important to note that these metrics only provide a basic indication of
Zoobot’s performance and do not fully capture its ability to predict
morphology, as the information is simplified and reduced.

We evaluate the discretized predictions with standard classification metrics
for the different classes. Accuracy $A$ is the fraction of correct predictions
for both the positive and negative class among the total number of galaxy
images $N_{\textrm{total}}$. It is calculated as

$A=\frac{N_{\textrm{TP}}+N_{\textrm{TN}}}{N_{\textrm{total}}},$ (4)

where $N_{\textrm{TP}}$ is the number of true positives and $N_{\textrm{TN}}$
the number of true negatives.

Precision $P$ is the fraction of correct classifications among the galaxies
predicted to belong to a particular class. It is calculated as

$P=\frac{N_{\textrm{TP}}}{N_{\textrm{TP}}+N_{\textrm{FP}}},$ (5)

where $N_{\textrm{FP}}$ is the number of false positives.

Recall $R$ is defined as the fraction of correct classifications among the
galaxies of a certain class and calculated as

$R=\frac{N_{\textrm{TP}}}{N_{\textrm{TP}}+N_{\textrm{FN}}},$ (6)

where $N_{\textrm{FN}}$ is the number of false negatives.

The $F_{1}$-score combines precision and recall by taking their harmonic mean.
Thus, it is a more general measure for evaluating model performance. It is
calculated as

$F_{1}=2\,\frac{P\,R}{P+R}.$ (7)

All of these metrics have values between $0$ and $1$. Some classification
tasks have an imbalanced number of galaxies for the different classes.
Moreover, there are some morphology tasks with more than two answers (see
Table 1). Therefore, we calculate the above metrics by treating each class as
the positive class and averaging over the results. We also provide the
$F_{1}$-score weighted by the number of galaxies for the different classes,
$F_{1}^{\star}$, similar to Walmsley et al. (2022a).

The performance of the model for a particular classification task can be
summarized by a confusion matrix. The rows of this two-dimensional matrix
correspond to the predicted classes, while the columns correspond to the
ground truth classes. The diagonal elements are the fraction of correct
classifications, while the other elements correspond to false classifications.

Table 3: Classification metrics of the model on the complete test set for all galaxies corresponding to Fig. 8a. Precision $P$, recall $R$ and $F_{1}$-score are calculated using the unweighted average of all classes. We also show the weighted $F_{1}$-score in the $F_{1}^{\star}$ column. For ‘has-spiral-arms’, we also provide the metrics for finding spiral galaxies in the complete test set (printed in italic), corresponding to the confusion matrix in Fig. 9a. Question | $N_{\textrm{total}}$ | $A$ | $P$ | $R$ | $F_{1}$ | $F_{1}^{\star}$
---|---|---|---|---|---|---
smooth-or-featured | 15 236 | 0.885 | 0.835 | 0.811 | 0.822 | 0.884
disk-edge-on | 986 | 0.982 | 0.963 | 0.957 | 0.960 | 0.982
has-spiral-arms | 764 | 0.916 | 0.584 | 0.725 | 0.614 | 0.935
| 10 746 | 0.965 | 0.864 | 0.936 | 0.896 | 0.966
bar | 974 | 0.878 | 0.822 | 0.744 | 0.774 | 0.869
bulge-size | 975 | 0.822 | 0.542 | 0.563 | 0.549 | 0.823
how-rounded | 9915 | 0.874 | 0.872 | 0.868 | 0.869 | 0.874
bulge-shape | 84 | 0.893 | 0.866 | 0.875 | 0.870 | 0.894
spiral-winding | 746 | 0.709 | 0.683 | 0.672 | 0.677 | 0.709
spiral-arm-count | 745 | 0.678 | 0.450 | 0.353 | 0.375 | 0.653
clumpy-appearance | 2265 | 0.874 | 0.867 | 0.850 | 0.857 | 0.873
clump-count | 328 | 0.546 | 0.516 | 0.413 | 0.390 | 0.539
galaxy-symmetrical | 225 | 0.880 | 0.884 | 0.690 | 0.737 | 0.860
clumps-embedded | 226 | 0.850 | 0.791 | 0.819 | 0.803 | 0.853
Table 4: Same classification metrics as in Table 3, but for galaxies with confident volunteer responses (i.e., one answer has a vote fraction above 0.8) corresponding to Fig. 8b. For ‘has-spiral-arms’, we also provide the metrics for finding spiral galaxies in the complete test set (printed in italics), corresponding to the confusion matrix in Fig. 9b. Question | $N_{\textrm{total}}$ | $A$ | $P$ | $R$ | $F_{1}$ | $F_{1}^{\star}$
---|---|---|---|---|---|---
smooth-or-featured | 1963 | 0.995 | 0.995 | 0.993 | 0.994 | 0.995
disk-edge-on | 907 | 0.998 | 0.994 | 0.994 | 0.994 | 0.998
has-spiral-arms | 666 | 0.950 | 0.542 | 0.975 | 0.564 | 0.971
| 10 553 | 0.970 | 0.859 | 0.952 | 0.899 | 0.971
bar | 511 | 0.977 | 0.968 | 0.907 | 0.935 | 0.976
bulge-size | 85 | 0.976 | 0.905 | 0.991 | 0.940 | 0.978
how-rounded | 5119 | 0.979 | 0.979 | 0.977 | 0.978 | 0.979
bulge-shape | 36 | 0.917 | 0.864 | 0.946 | 0.893 | 0.921
spiral-winding | 46 | 0.978 | 0.933 | 0.982 | 0.954 | 0.979
spiral-arm-count | 202 | 0.941 | 0.396 | 0.534 | 0.402 | 0.943
clumpy-appearance | 1307 | 0.970 | 0.967 | 0.959 | 0.963 | 0.970
clump-count | 64 | 0.828 | 0.540 | 0.513 | 0.525 | 0.868
galaxy-symmetrical | 115 | 0.974 | 0.986 | 0.850 | 0.905 | 0.972
clumps-embedded | 85 | 0.941 | 0.844 | 0.966 | 0.890 | 0.946
Figure 8: Confusion matrices for five selected morphology questions after
binning to the class with the highest predicted vote fraction. The confusion
matrices for the other questions are shown in the Appendix. The colour map
corresponds to the fraction of the ground truth values for the different
classes (also denoted in the confusion matrices). Figure 9: Confusion matrices
for the task of finding spiral galaxies in the complete test set by applying
the selection cuts suggested in Willett et al. (2017).

The resulting metrics are listed in Table 3. For five selected morphology
tasks, we show the corresponding confusion matrices in Fig. 8a. We calculate
the same metrics for galaxies from the complete test set where the volunteers
are confident, e.g., one answer has a vote fraction higher than $0.8$. Through
this procedure, we can analyse the model performance against confident labels
(Domínguez Sánchez et al. 2019; Walmsley et al. 2022a). The results are shown
in Table 4. The corresponding confusion matrices for selected questions are
shown in Fig. 8b. We present all confusion matrices for the remaining tasks in
Appendix C.

For the majority of the morphology questions, the accuracy is higher than
$97\%$. For all other questions the accuracy is above $91\%$ except for the
question of the ‘clump-count’ where it is only $82.8\%$. The $F_{1}$-scores
are all above $89\,\%$ except for the ‘has-spiral-arms’, ‘spiral-arm-count’
and ‘clump-count’ questions.

The accuracy for all galaxies, as shown in Table 3, is generally lower
compared to confidently classified galaxies, ranging from $54.6\%$ (‘clump-
count’) to $98.2\%$ (‘disk-edge-on’). This outcome is expected, since the
ground truth labels themselves carry inherent uncertainty. Considering that
volunteers may not reach a consensus in these cases, it can be inferred that
answering morphology questions for such galaxies could be challenging.
Particularly for complex morphologies such as the number and winding of spiral
arms, the size of the bulge and the number of clumps, the performance of the
model is lower than for other questions that are less complex, such as
determining whether a galaxy is a disk viewed edge-on. This can be attributed
to several factors: the limited number of examples for these classes included
in the training dataset, making them less represented, and the inherent
difficulty associated with accurately identifying these morphological
features.

Furthermore, counting spiral arms and clumps are especially difficult
classification tasks, as there are in both cases six classes that can be
selected and some arms or clumps might be difficult to identify. Moreover, the
distributions of the answers are imbalanced, with classes containing only one
(‘5-plus’ spiral arms) or no examples (one clump) that contribute equally to
the averaged metrics. Thus, the $F_{1}$-scores for confident volunteer
responses are substantially lower than compared to other questions. In
numerous instances, the predicted count for spiral arms and clumps is off by
just one number from the ground truth count. Consequently, the discrete
metrics provided do not fully capture the capabilities of Zoobot. Instead, the
predicted vote fractions are preferable for assessing the number of spiral
arms or clumps.

For the ‘has-spiral-arms’ question, there are only three confident ‘no’
examples, while there are 663 galaxies confidently classified as spiral in the
test set (see Fig. 8b). Thus, the test set in this binary case is extremely
unbalanced and the derived metrics are therefore not reflecting Zoobot’s
overall ability of finding spiral galaxies in a given dataset. We demonstrate
this in the following section by not only utilizing the ‘has-spiral-arms’
question, but the whole decision tree to utilize the full ability of Zoobot.

####  5.2.2 Finding spiral galaxies in the test set

We demonstrate how Zoobot can be used to find spiral galaxies in a given
dataset. Similar to the approach used for volunteer vote fractions $f$, we
applied the suggested criteria from Willett et al. (2017) for selecting spiral
galaxies in the complete test set. These criteria were: $f_{\textrm{edge-
on,no}}>0.25$, $f_{\textrm{clumpy,no}}>0.3$, and $f_{\textrm{features}}>0.23$.
Additionally, we excluded galaxies where the conditions mentioned above apply,
but the number of volunteers was insufficient (as Zoobot is only predicting
vote fractions), using the suggested cutoff of $N_{\textrm{spiral}}\geq 20$.
For the final catalogue, we chose a vote fraction of $f_{\textrm{spiral}}>0.5$
to identify spiral galaxies. Thus, all galaxies for which the conditions were
fulfilled were classified as spiral, while all the others were classified as
not spiral. For the predicted vote fractions, we applied the same cuts. Once
more, we measure the performance for confident labels (volunteer vote fraction
for the final answer greater than $0.8$ or smaller than $0.2$).

Zoobot achieves an accuracy of $96.5\,\%$ for finding spiral galaxies in the
complete test set, with an $F_{1}$-score of $89.6\,\%$. The corresponding
confusion matrix is shown in Fig. 9a and the corresponding metrics listed in
Table 3. On confident labels, Zoobot achieves an accuracy of $97.0\,\%$ with
an $F_{1}$-score of $89.9\,\%$ as shown in Fig. 9b and in Table 4. These
values demonstrate that Zoobot is indeed well suited for identifying spiral
galaxies in a given dataset.

####  5.2.3 Vote fraction mean deviations

Figure 10: Vote fraction mean deviations $\delta_{i}$ of the model predictions
and the volunteer labels for the different morphology answers $i$ (see Eq. 3).
The model was trained with all galaxies from the complete set. The deviations
are displayed for all galaxies of the test set and for galaxies within a
magnitude interval with $m=m_{I814W}$. Lower $\delta_{i}$ indicates better
performance. The black dashed line marks $12\,\%$ vote fraction mean
deviation. Figure 11: Histograms of the vote fraction deviations
$f_{\textrm{pred}}-f_{\textrm{gt}}$ between the predicted $f_{\textrm{pred}}$
and volunteer vote fractions $f_{\textrm{gt}}$ for five selected questions.
For each answer, the number of galaxies $n$, the standard deviation $\sigma$
and the kurtosis $w$ is given. Figure 12: Vote fraction mean deviations
$\delta_{i}$ for all corresponding answers $i$ (different colours, denoted in
legend) of the different GZH questions (see Table 1) as a function of redshift
$z$ for the relevant galaxies of the complete test set (where at least half of
the volunteers voted). Figure 13: Similar as Fig. 12, but with the vote
fraction mean deviations $\delta_{i}$ depending on the magnitude $m$. Figure
14: Histograms of the predicted and volunteer vote fractions for the ‘has-
spiral-arms’ ‘yes’ answer for faint and high redshift galaxies. Figure 15:
Redshift and magnitude dependence of the ‘has-spiral-arms’ vote fraction mean
deviations $\delta_{i}$ for the model trained on the original HST COSMOS
images. Compared to the Euclid images (Figs. 12 and 13), the deviations for
high redshift and faint galaxies are substantially smaller.

Next, we evaluate the model performance by analysing the predicted vote
fractions directly. We show the vote fraction mean deviations $\delta_{i}$ for
all answers $i$ corresponding to different morphology types in Fig. 10.
Moreover, we display how the performance varies with magnitude by selecting
only galaxies from different magnitude intervals.

For almost all answers (36 of 40 answers), the vote fraction mean deviation is
below $12\,\%$, while the performance varies between different answers. As
before, the question with the lowest deviation for all answers is ‘disk-edge-
on’. This can be attributed to the fact that it represents a less intricate
feature, making it relatively easy to discern and learn. Conversely, questions
related to spiral arms and clumps consistently yield the highest deviations.
Once again, this can be explained with the inherent complexity of these
questions, as they involve finer and more intricate structural details. We
expect the quality of the morphological predictions to be better if more
relevant labels for these morphology types were available, as indicated in
Fig. 5.

The dependence of the vote fraction mean deviation on the magnitude differs
between answers. The mean deviation shows no substantial dependence on the
magnitude for the ‘smooth-or-features’, ‘disk-edge-on’ and ‘how-rounded’
questions. For the ‘has-spiral-arms’ question, on the other hand, the
differences between the deviations are the largest. While the model performs
better for brighter galaxies ($m<20.5$) with a mean deviation below $10\,\%$,
for faint galaxies ($m>22$) the deviations are the largest overall
($\sim\,27\,\%$). This indicates that identifying spiral arms in faint
galaxies is a relatively difficult task. This is not surprising, as spiral
arms are a finer structure. Once spiral arms are identified, the other tasks
related to spiral arms, i.e., determining the winding of the spiral arms and
counting them, do not show such a strong magnitude dependence. Finally,
although clumps appear more frequently for faint galaxies, the model performs
better in the case of bright galaxies.

####  5.2.4 Histograms of the vote fraction deviations

While we already investigated the mean of the (absolute) vote fraction
deviations $f_{\textrm{pred}}-f_{\textrm{gt}}$, we now analyse the
corresponding histograms for five selected questions in Fig. 11. Positive
values indicate that Zoobot predicts a higher vote fraction than the
volunteers, and negative that the volunteer vote fraction is higher.

For most answers, the distributions are centred at 0, indicating that for most
galaxies the vote fraction deviations are relatively small. The distributions
are symmetrical around the centre, indicating that the model does not have a
substantial bias. The widths of the distributions correspond to the mean vote
fraction deviations (see Fig. 10), as expected.

In contrast, for the ‘has-spiral-arms’ answers, the distributions are not
symmetrical. While the maximum of the distribution is at 0, indicating that
most deviations are small, the volunteers’ vote fractions for a galaxy to be
spiral are higher than predicted from Zoobot. This can be explained by the
high imbalance (mean vote fraction for ‘yes’ is $90.7\,\%$) of the relevant
‘has-spiral-arms’ answers (see Sect. 5.2.1) in combination with the intrinsic
difficulty of this task.

The ‘disk-edge-on’ question also has an imbalance that is slightly less
extreme (mean vote fraction for ‘no’ is $83.9\,\%$). As it is easier to learn,
the vote fraction mean deviation is much smaller ($\sim\,4\,\%$) than for
‘has-spiral-arms’ ($\sim\,11\,\%$). Thus, the imbalance does not lead to a
substantial asymmetry of the distribution.

####  5.2.5 Magnitude and redshift dependence

Next, we investigate the magnitude and redshift dependence of the mean
deviations $\delta_{i}$ between the predicted and the volunteer vote
fractions. These are shown in Figs. 12 and 13 for the different questions. For
some galaxies, there was no redshift information available. Thus, these
galaxies were excluded.

In general, the vote fraction mean deviation shows no substantial dependence
on magnitude and redshift. For relatively easier morphology tasks (e.g.,
‘disk-edge-on‘ and ‘smooth-or-featured’) the deviations are smaller than for
more complex ones such as tasks related to spiral arms, bars and clumps.

For the ‘has-spiral-arms’ question, the vote fraction mean deviation shows a
strong increase for $z>1$ up to almost $50\,\%$. The same effect can be
observed for fainter galaxies ($m_{I814W}>21.5$ in Fig. 13), although the
deviation is smaller. This indicates again that the difficulty of identifying
spiral arms for high redshift and faint galaxies is substantially higher than
other morphology tasks.

In Fig. 14, we investigate the volunteer and model vote fraction for the ‘yes’
answer of the ‘has-spiral-arms’ question in a histogram for high-redshift and
faint galaxies. For the majority of the galaxies, the volunteer vote fraction
is above $90\,\%$, meaning that the volunteers are confidently classifying
most galaxies to be spiral. Zoobot, on the other hand, shows a wider range of
predicted vote fractions, with most being above $70\,\%$. While there are no
galaxies (for high redshifts) or only one (for faint galaxies) galaxy to be
confidently classified to have no spiral arms (vote fraction below $20\,\%$),
Zoobot is confidently predicting that only two galaxies have no spiral arms.
Therefore, Zoobot is not misclassifying galaxies, it is just not as confident
as the volunteers. This could be explained with the lower resolution and the
additional noise for the emulated Euclid images compared to the original HST
images.

To check this, we looked at the redshift and magnitude dependence for Zoobot
trained on the original HST images in Fig. 15. The vote fraction mean
deviations are, although still present, substantially smaller than for Euclid
images supporting our interpretation. For a practical use, to identify spiral
galaxies in high-redshift and high $m$ ranges, the selection cuts can be
lowered when applying Zoobot.

####  5.2.6 Comparing performance to original HST images

Figure 16: Vote fraction mean deviations of the model predictions and the
volunteer labels for the different answers of the decision tree for the
emulated Euclid images and original HST images for the models trained on 60
940 galaxies from the complete set.

While we already investigated the influence of the lower resolution and the
additional noise of the Euclid images for identifying spiral arms, we show in
Fig. 16 the performance of the models trained on all galaxies of the complete
dataset on the same Euclid and original HST images (see Sect. 2) for all
answers. The results are very similar for HST and Euclid images, indicating
that in general, the lower resolution and noise in the Euclid images do not
lead to a substantial change in performance.

For many questions, such as ‘smooth-or-features’, ‘disk-edge-on’, or ‘how-
rounded’, the vote fraction mean deviation for HST and Euclid images is almost
the same, supporting the previous discussion that these features depend less
on the resolution. On the other hand, for the questions related to spiral
arms, bars and clumps, the performance of Zoobot trained on HST images is
better than for Zoobot trained on Euclid images. This is in agreement with the
previous discussion that detecting spiral arms, bars, and clumps are finer
features and their detection depends on resolution (which is approximately two
times poorer for Euclid compared to HST).

##  6 Adapting Zoobot to a new morphology type

We trained Zoobot to emulated Euclid images with the GZH decision tree (see
Table 1). Therefore, our model could be directly used for real Euclid images,
but is restricted to answering only the questions of the GZH decision tree.
However, for Euclid, there might be additional or other galaxy morphology
tasks that are currently not included in our Zoobot model. We show that Zoobot
can be easily adapted to a new morphology task that is not included in the GZH
tasks with the example of peculiar galaxies.

###  6.1 Adaption procedure

Figure 17: Examples of galaxies and their expert labels to be a peculiar
galaxy or not. The black bars represent a length of $None$. There are no
distinct morphological features that characterize peculiar galaxies, making
this classification task rather difficult.

Peculiar galaxies are a type of irregular galaxy, with disorganised structure,
often at high redshifts that do not typically fall into smooth, spheroid, or
disk classes. The class of peculiar galaxies was not included in the GZH
questions. Instead, we used labels for the same emulated Euclid VIS images
(see Sect. 2) from a different source, namely expert classification from the
Euclid Zoo project111https://www.zooniverse.org/projects/sandorkruk/euclid-
zoo/. Euclid Zoo was an internal classification project in the Euclid
Consortium, with astronomers as classifiers. In total, 2006 galaxies were
classified with $N=1$ to $N=3$ expert classifications per galaxy image. We
selected a galaxy from the dataset to be classified as peculiar if the vote
fraction for the peculiar class is larger than $50\,\%$, resulting in $231$
galaxy images for the peculiar class. In Fig. 17, we show examples of galaxy
images and their corresponding labels. We then applied a 70/10/20 percent
train/validation/test split, leading to 1404 images for training, 200 for
validation and 402 for testing. Next, we balanced our train and validation set
by randomly dropping galaxy images that are not peculiar, leading to the same
number of ‘not peculiar’ and ‘peculiar’ galaxy images. In total, 308 galaxy
images were used for training and 54 for validation.

Similar to Walmsley et al. (2022b), we used our best performing model (trained
on all images from the complete set) and replaced the final output layer by a
new model ‘head’, simply consisting of a final dense layer with a sigmoid
activation function. We utilized the Adam optimizer (Kingma & Ba 2015). We
then trained the new model with the dataset of peculiar galaxies while
applying the same augmentation as before, i.e., random flips and rotations. To
avoid overfitting, we stopped training as soon as the validation loss was not
decreasing for 20 consecutive epochs. After training, Zoobot calculated
predictions for the 402 images of the test set.

###  6.2 Performance of the adapted Zoobot

Table 5: The classification metrics accuracy $A$, precision $P$, recall $R$ and the unweighted and weighted $F_{1}$-scores $F_{1}$ and $F_{1}^{\star}$ for identifying peculiar galaxies in the test set with different confidence thresholds $c_{\textrm{th}}$. The corresponding confusion matrices are shown in Fig. 18. $c_{\textrm{th}}$ | $N_{\textrm{total}}$ | $A$ | $P$ | $R$ | $F_{1}$ | $F_{1}^{\star}$
---|---|---|---|---|---|---
$0.5$ | 402 | $0.791$ | $0.670$ | $0.829$ | $0.689$ | $0.823$
$0.75$ | 402 | $0.915$ | $0.803$ | $0.823$ | $0.812$ | $0.917$

The performance of the model is listed in Table 5 when evaluated with the
classification metrics introduced in Sect. 5.2.1. The corresponding confusion
matrices are displayed in Fig. 18. The model achieves an accuracy of
$79.1\,\%$ for a model confidence threshold of $c_{\textrm{th}}=0.5$ for
selecting a galaxy to be peculiar. By applying $c_{\textrm{th}}=0.75$ for our
model predictions, we obtain a higher accuracy of $91.5\,\%$ and a higher
$F_{1}$-score of $81.2\,\%$ due to significantly higher precision. The values
indicate that Zoobot performs well at the task of finding peculiar galaxies.

The accurate identification of peculiar galaxies is particularly impressive,
considering that it is a relatively challenging task even for an expert, due
to the lack of clear morphological features. In addition to the inherent
difficulty, there were only 231 examples of peculiar galaxies, of which 20 %
were not included in the training dataset. This underscores our earlier
discussion regarding the effectiveness of fine-tuning. Our Zoobot model was
initially trained on all major GZ campaigns (as described in Sect. 3) and
subsequently on GZH using emulated Euclid images, making it well-suited for
adaptation to a new Euclid morphology task.

This shows that Zoobot can easily be adapted to new problems, even if these
are difficult and do not have many examples. For the application to Euclid,
our trained model can be used as a first step to predict detailed morphology
for Euclid with the GZH questions and can then be adapted to a new task in an
effective way without requiring large labelled sets of galaxy images. Thus, in
practice, if an astronomer is interested in finding all examples of a
particular galaxy morphological type that is not included in the GZH questions
for a given set of real Euclid images, the following steps can be applied.
First, a dataset needs to be labelled that is then used to fine-tune the
trained Zoobot model to the new galaxy morphology task. Once the model is
fine-tuned, it can be used to classify all images of a given set of Euclid
images.

Figure 18: Confusion matrices for finding peculiar galaxies for Zoobot
pretrained on the emulated Euclid images and the GZH tasks at (a) $50\,\%$ and
(b) $75\,\%$ cutoff thresholds $c_{\textrm{th}}$ for selecting peculiar
galaxies.

##  7 Summary and conclusions

This paper introduces automated and detailed predictions of galaxy morphology
for emulated Euclid images. These emulated images were generated by converting
HST COSMOS images to Euclid VIS images, considering the Euclid PSF and
adjusting them to match the expected noise level of Euclid. The automated
predictions were created using Zoobot, a Python package for creating deep
learning models that classify galaxy morphology and adapting (‘fine-tuning’)
those models to new surveys and tasks. We fine-tuned a pre-existing Zoobot
model (trained on 450 000 non-Euclid galaxies from Galaxy Zoo) using emulated
Euclid images and labels derived from the Galaxy Zoo: Hubble (GZH) volunteer
responses.

The model is able to accurately predict the detailed morphologies for emulated
Euclid galaxy images. It predicts various aspects, including the presence and
quantity of clumps, detection, and counting of spiral arms, measurement of
their winding, identification of disk galaxies, detection of bars, and
determination of the presence, shape, and size of the central bulge, as well
as measurement of the shape of featureless galaxies (refer to Table 1).

The Zoobot model fine-tuned on 60 000 available emulated Euclid images with
GZH labels achieves a mean deviation of the predicted vote fraction from the
volunteer classifications averaged over all answers of $9.5\,\%$ and below
$12\,\%$ for nearly all answers individually (36 out of 40, as depicted in
Fig. 10). Additionally, it achieves an accuracy above $91\%$ for 12 of 13
questions when considering confident volunteer responses (refer to Table 4).
However, the model’s performance varies across different morphology classes.

For the top questions of the decision tree (global morphology type – ‘smooth-
or-features’, disc orientation – ‘disk-edge-on’ or ‘bulge-size’) the model is
able to predict within $10\,\%$ of the volunteers’ vote fraction, after being
trained with only 1000 randomly selected galaxies. For other questions, such
as ‘how-rounded’, ‘spiral-arm-count’ or ‘bulge-shape’, 10 000 training
galaxies are needed, while questions related to the more complex morphologies
such as ‘has-spiral-arms’, ‘bar’, ‘spiral-winding’ or ‘clumpy-appearance’ the
full training set of 60 000 galaxies is required to reach $12\,\%$ deviation
from the volunteer classifications. This suggests that more examples for
complex morphology classes improves the performance of the model. Finally, our
investigations of using the complete sample of available galaxies for training
($m_{I814W}<23.5$), or a subset of the brightest galaxies ($m_{I814W}<22.5$),
suggests that the difference in performance is minimal; the number of galaxies
with complex morphologies used for training has a higher impact.

What do our results imply for Euclid?

  * •

Zoobot, trained on emulated Euclid galaxies using volunteer labels from GZH,
shows accurate predictions (within $10\%$ of human classifications) for global
morphology (smooth versus featured), disk orientation (edge-on versus face-
on), and bulge size.

  * •

To enhance the model’s performance in predicting more complex detailed
morphologies, such as bars, spiral arms, and clumps, additional labels are
required. Based on Fig. 5, approximately 60 000 randomly selected galaxies
would be needed to achieve a global vote fraction deviation below $10\%$ and
maintain deviations below $12\%$ for all labels. These additional labels could
be obtained by initiating a Galaxy Zoo project for Euclid, utilizing Euclid Q1
data.

  * •

Our experiments indicate minimal performance differences when selecting
galaxies with $m_{I814W}<23.5$ or brighter galaxies with $m_{I814W}<22.5$.
Therefore, we suggest expanding the pool of explored galaxies for Euclid, with
a restriction of $I_{\scriptscriptstyle\rm E}<23.5$ (assuming VIS magnitudes
are reasonably similar to I814W magnitudes). Fainter galaxies were not tested
as no morphological labels were available for these galaxies. We expect the
fraction of galaxies with features to decrease at higher magnitudes and
smaller sizes, as observational effects cause these galaxies to appear
smoother.

  * •

Zoobot can be adapted to new Euclid morphology tasks using a few new labels.
We demonstrated this adaptability by successfully training Zoobot for a new
class of peculiar galaxies, consisting of only 261 examples, achieving an
accuracy of $91.5\%$ (Fig. 18). Consequently, for new classes, it is feasible
to set up a dedicated Galaxy Zoo-style workflow where volunteers are asked
simple binary questions related to the morphology of the specific class of
interest. The exact number of required labelled galaxies depends on the
specific morphology class (Fig. 5).

  * •

The proposed morphology classification scheme for Euclid is outlined in a
companion paper (Euclid Collaboration: OU-MER, in prep.).

Currently, the generation of structural parameters describing galaxy
morphology with a morphology fitting code is included in the Euclid data
pipeline (Euclid Collaboration: Bretonnière et al. 2023). The algorithm
generates morphological parameters for single and double Sérsic components,
and the measurements are reliable up to $I_{\scriptscriptstyle\rm E}=23$ for
one component and $I_{\scriptscriptstyle\rm E}<21$ for two components. Euclid
Collaboration: Bretonnière et al. (2023) concludes that for at least 400
million galaxies, robust structural parameters will be delivered by the Euclid
Data Releases.

We estimate the number of galaxies for which detailed morphologies could be
measured with our deep learning model. With $m_{I814W}<23.5$, there are
approximately $70$ 000 galaxies in an area of $(1.2\times 1.2)$
deg${}^{2}=1.44\,$deg2 of the HST COSMOS survey. Scaling up to the total sky
area measured by the Euclid Wide Survey, 15 000 deg2, and assuming that VIS
magnitudes are similar to the $I$814W magnitudes of HST, there would be
approximately $800$ million galaxies with reliably measured morphologies up to
$I_{\scriptscriptstyle\rm E}=23.5$ (focusing on the brighter magnitudes,
$I_{\scriptscriptstyle\rm E}<22.5$, the estimated count would be approximately
$300$ million galaxies). This is close to the 400 million galaxies estimated
by Euclid Collaboration: Bretonnière et al. (2023) for
$I_{\scriptscriptstyle\rm E}<23$. Accounting for an average vote fraction of
$29\,\%$ for galaxies to display features, we conclude that of the 800 million
measured galaxies from the Euclid Wide Survey approximately 230 million
galaxies will display complex morphology. This closely matches the 250 million
galaxies estimated to have complex structures by Euclid Collaboration:
Bretonnière et al. (2022) for the Euclid Wide Survey and the Euclid Deep
Survey.

In conclusion, we have successfully showcased the feasibility of generating
high-quality and detailed morphology predictions for future Euclid images. Our
trained Zoobot model is readily deployable in the Euclid pipeline to produce
morphological catalogues for Euclid images with Q1 data. As additional labels
for more complex morphologies are obtained, the performance of Zoobot will
improve for the upcoming Data Release 1 (DR1). Moreover, the model can be
easily adapted to new morphology classes that are of interest to astronomers
as new labels are gathered through crowd-sourcing projects.

###### Acknowledgements.

The Euclid Consortium acknowledges the European Space Agency and a number of
agencies and institutes that have supported the development of Euclid, in
particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian
Science Policy, the Canadian Euclid Consortium, the French Centre National
d’Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish
Space Research Institute, the Fundação para a Ciência e a Tecnologia, the
Ministerio de Ciencia, Innovación y Universidades, the National Aeronautics
and Space Administration, the National Astronomical Observatory of Japan, the
Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the
Romanian Space Agency, the State Secretariat for Education, Research and
Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom
Space Agency. A complete and detailed list is available on the Euclid web site
(http://www.euclid-ec.org).

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##  Appendix A The GZH decision tree

We show the original GZH decision tree (Willett et al. 2017) in Fig. 19.

Figure 19: GZH decision tree (Willett et al. 2017). The questions T06, T08,
T13, T14 and T15 were not included in this study (see Sect. 2).

##  Appendix B Experimenting with different initial weights

We also conducted the experiment described in the main paper with different
initial weights, i.e., Zoobot pretrained on different data. The weights
described in the main text are denoted as weights B. Additionally, we tested
with Zoobot pretrained on only GZD-5 (Walmsley et al. 2022a, weights A) and
Zoobot without pretraining (random weights, weights C). We show the dependence
of the averaged vote fraction mean deviation $\bar{\delta}$ on the number of
galaxies $N_{\textrm{train}}$ in Fig. 20. To compare the models, we use in all
cases the predictions on the same complete test set of 15 236 images (see
Table 2). Additionally, we show for all answers the deviations for all models
trained on 100, 1000 and 10 000 galaxies in Figs. 21, 22 and 23.

Figure 20: The vote fraction mean deviation averaged over all answers
depending on the number of galaxies used for training for initial weights A
(pretraining on GZD-5), B (pretraining on all major GZ campaigns except GZH)
and C (no pretraining) and training on the bright and complete training set.
In all cases, the predictions were done on the complete test set. Lower values
indicate better performance.

With increasing number of training galaxies, the average mean deviation
$\bar{\delta}$ is decreasing: The more galaxy examples (of different types)
are used for training, the better the model predictions get for all answers.
In the regime with $N_{\textrm{train}}<20$, the models perform similarly. With
more training galaxies, the performance of the model is for all numbers of
training galaxies best with initial weights B, followed by weights A and then
weights C. The model pretrained on all Galaxy Zoo campaigns except GZH leads,
for the same number of galaxies, to a better performance than for a
pretraining with only GZD labels. This is due to the better generalization of
the model in the pretraining. The model without pretraining (weights C) shows
the worst performance of the three, as expected.

Especially, for 100 to 10 000 galaxies, the difference between pretrained
models and models used from scratch is most evident. Thus, for a limited
number of labelled galaxies, transfer learning is substantially more effective
for training to a new problem than training from scratch. In comparison to
weights A and B, for weights C, there is a difference between training with
bright and with random (complete) galaxies, i.e., bright galaxies lead to a
better model performance especially between 100 and 10 000 galaxies. This
difference could be explained with the pretraining for the models with weights
A and B. While these have seen many types of galaxies with different
magnitudes, they are more reliable and the magnitude cut does not have such a
big impact. For weights C, the model was not trained before and thus, learns
the galaxies morphologies for the first time. Bright galaxies with morphology
that is easier accessible in general seem to be more effective when training
from scratch.

More details of the differences between models are shown in Figs. 21, 22 and
23. The impact of pretraining can be best seen for the ‘disk-edge-on’
question. For the pretrained models (weights A and B), 100 galaxy images are
enough in training to get below a deviation of $10\,\%$. This deviation is
reached for the model from scratch (weights C) at 10 000 galaxies. In
contrast, for the questions related to clumps, the differences between the
models for 100, 1000, and 10 000 training galaxies are relatively similar,
indicating that the influence of pretraining is smaller for these questions.
Only for weights B questions regarding clumps were included in the
pretraining, supporting this interpretation.

Figure 21: Vote fraction mean deviations of the model predictions and the
volunteer labels for the different answers of the decision tree for the models
trained on 100 galaxies of the different datasets (bright and complete) and
with different initial weights (weights A, B and C). Lower $\delta_{i}$
indicates better performance. Figure 22: Similar to Fig. 21 with
$N_{\textrm{train}}=1000$ galaxies. Figure 23: Similar to Fig. 21 with
$N_{\textrm{train}}=10\,000$ galaxies.

##  Appendix C Further confusion matrices

We show all confusion matrices for the different GZH questions that were not
included in the main paper on all galaxies and on galaxies with confident
volunteer responses in Figs. 24 and 25, respectively corresponding to the
metrics in Tables 3 and 4.

Figure 24: Confusion matrices continued from Fig. 8 after binning to the class
with the highest predicted vote fraction. The colour map corresponds to the
fraction of the ground truth values for the different classes. To improve the
readability, for the tasks with more than three answers, only the percentage
is stated. Figure 25: Fig. 24 continued.

##  Appendix D Volunteer uncertainty

We show in Fig. 26 the mean vote fraction deviation $\delta_{i}$ for the
‘features’ answer of the ‘smooth-or-features’ question, depending on the
volunteer vote fraction $f_{gt}$. In general, the deviations are smaller for
confident volunteer responses (vote fraction lower than $0.2$ or greater than
$0.8$) compared to more uncertain volunteer responses (vote fraction between
$0.2$ and $0.8$). As expected, the model performs better for confident
volunteer responses and with increasing uncertainty in the volunteer responses
the deviations also increase. Moreover, an asymmetry of the deviations can be
observed, as deviations are substantially smaller for vote fractions below
$0.2$ compared to vote fractions above $0.8$. This could be explained with the
fact that most galaxies of the dataset do not display features. Additionally,
the deviations for the lowest volunteer vote fractions are higher than for
vote fractions of $\sim 0.1$. This is due to the characteristic of Zoobot to
predict for the most extreme volunteer vote fractions (close to 0 or 1) less
extreme vote fractions (Walmsley et al. 2022a), which should not affect
practical use.

Figure 26: Vote fraction mean deviations $\delta_{i}$ of the model predictions
and the volunteer labels for the ‘smooth-or-features_features’ answer
depending on the volunteer vote fraction $f_{gt}$.

##  Appendix E Reproducibility

The Zoobot CNN is publicly available at https://github.com/mwalmsley/zoobot.
The code for the creation of the images, the training of the Zoobot CNN and
for the analysis of the results is publicly available at
https://github.com/baussel/ZoobotEuclid.