Comprehensive Catalogue of Polyhedra
Debra Briggs
Supervised by Dr David Yost
Federation University Australia
ABSTRACT
Complete catalogues of polyhedra with up to 8 vertices or 8 faces were
given by Britton & Dunitz (1973) and Federico (1974). The aim of this
project was to produce a more comprehensive catalogue, thus providing a
resource for examples and conjectures. In particular, all polyhedra with
V=F=9 represented as planar graphs are given here, their duals are
calculated and most are classified in terms of Minkowski decomposability.
1
1. Introduction:
Every polyhedron has a graph. It has vertices, it has edges, and therefore it is a graph.
Theorem: Let G be a planar, connected graph. Let V be the number of vertices, E the
number of edges, and F the number of faces in G, including the unbounded face.
Euler's Theorem states that if a 3-dimensional polyhedron has 𝑉 vertices, 𝐸 edges and
𝐹 faces, then 𝑉, 𝐸, 𝐹 are related by the formula 𝑉 − 𝐸 + 𝐹 = 2.
Steinitz’ Theorem: A graph G is the edge graph of a polyhedron iff G is a simple
planar graph which is 3-connected. (Ziegler, 1995, Chapter 4).
Steinitz in 1906 gave a complete characterisation of the triples (𝑉, 𝐸, 𝐹) for which
there exists a polyhedron with 𝑉 vertices, 𝐸 edges and 𝐹 faces.
Steinitz’ Theorem characterises which graphs are graphs of polyhedra. If you have a
polyhedron then its graph is planar and 3-connected. Steinitz’ theorem tells us that the
converse is true. Therefore it reduces the question of characterising polyhedra to graph
theory which simplifies it.
2. Construction of Polyhedra with 9 faces and 9 vertices:
Britton & Dunitz (1973) catalogued all of the Polyhedra up to 8 vertices, and Federico
(1974) used Steinitz’ Theorem to catalogue all of the Polyhedra with up to 8 faces. So
the next step was to catalogue all of the Polyhedra with 9 faces and 9 vertices. As you
can see on the table from Federico (1969) there are 296 of them.
Federico (1969)
Voytekhovsky (2014) gives a historical account of the classification of polyhedra and
mentions that T.P Kirkman initiates the systematic and enumeration of the
combinatorial types of polyhedra without drawing the graphs.
2
Reverend Thomas P. Kirkman, an amateur mathematician in countryside England did a
lot of work on “polyhedra” (Biggs, 1980) in the mid 19th century. Kirkman (1863)
described all polyhedra with up to 16 edges, and some with 17 and 18 edges, but his
notation and terminology is very unique and rather challenging to translate. Some would
even say incomprehensible. Kirkman attempted to construct all “9-acral 9-edra
polyedra” (1878), which was seemingly missed by Voytekhovsky, and using his work I
was able to construct the planar graphs of all 296 polyhedra with 9 faces and 9 vertices.
Since Steinitz’ Theorem was proved in 1906, Kirkman wasn’t aware of it. Therefore he
wasn’t aware that he could make his work simpler by drawing his polyhedra as planar
graphs. So his diagrams are therefore tricky to follow.
While reconstructing his drawings, I was verifying his work and finding the dual pairs,
which no-one has attempted to do for 9 faces and 9-vertices before. A number of
people, e.g. Federico (1969), have used computer programs to verify that there are 296
of them but they’ve never published a catalogue.
Figure 2.1 explains how I preserved Kirkman’s notation for the asymmetrical
polyhedra, the letters 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and 𝐹 are used. Euler’s relation and
if T, Q, P, H, S, and O, denote the number of triangles, quadrilaterals, pentagons,
hexagons, heptagons and octagons,
then 𝑇 + 𝑄 + 𝑃 + 𝐻 + 𝑆 + 𝑂 = 𝐹 = 9
and 3𝑇 + 4𝑄 + 5𝑃 + 6𝐻 + 7𝑆 + 8𝑂 = 2𝐸 = 32
The following solutions to these equations are the only possible combinations of faces
that exist.
Letter Number
Combination
𝐴
10017
𝐵
1107
𝐶
1026
𝐷
216
𝐸
135
𝐹
54
𝐽
100008
Faces
Vertices
1 hexagon, 1 quadrilateral
and 7 triangles.
1 hexagon, 1 pentagon and
7 triangles.
1 hexagon, 2 quadrilaterals
and 6 triangles.
2 pentagons, 1 quadrilateral
and 6 triangles.
1 pentagon, 3 quadrilaterals
and 5 triangles.
5 quadrilaterals and 4
triangles
1 octagon and 8 triangles.
1 vertex of degree 7, 1 of degree
4 and 7 of degree 3.
1 vertex of degree 6, 1 of degree
5 and 7 of degree 3.
1 vertex of degree 6, 2 of degree
4 and 6 of degree 3.
2 vertices of degree 5, 1 of
degree 4 and 6 of degree 3.
1 vertex of degree 5, 3 of degree
4 and 5 of degree 3.
5 vertices of degree 4 and 4 of
degree 3.
1 vertex of degree 8 and 8 of
degree 3.
Figure 2.1
The first letter corresponds to the number of faces and the second letter refers the
degrees of the vertices, for example 𝐸𝐷 represents 1 pentagon, 3 quadrilaterals and 5
triangles, and 2 vertices of degree 5, 1 of degree 4 and 6 of degree 3.
3
I have introduced 𝐽 – 100008 which represents the 𝐽𝐽𝑠, the octagonal pyramid. It may
even be useful for future catalogues.
Within each group there are a certain number of them, so we just number them. The
𝐸𝐸’s have 67 in their group.
Kirkman may have had some logic behind the notation for his symmetrical polyhedra,
but there is no reason to keep symmetrical and asymmetrical separate, so I have
combined them. His polyhedra 𝐾 is now 𝐹𝐷𝑠1 . The 𝑠 meaning it has symmetry.
There are 9 misprints in Kirkman’s (1878) catalogue which I have corrected:
 𝑂9 (now 𝐹𝐸𝑠3 ) was combinatorially equivalent with 𝑂8 ,and 𝐸𝐸33 was
combinatorially equivalent with 𝐸𝐸32 .
 𝑀1−1 (now 𝐹𝐹𝑠8 ) and 𝐸𝐷11 had 10 faces.
 𝐸𝐸22 had 10 vertices.
 𝐸𝐸1 had 8 faces.
 𝐸𝐸37 had 8 vertices 𝐸𝐹8 had 8 faces and one of its vertices was of degree 2.
 Each of the above polyhedra were able to be found through their duals.
 𝐶𝐶2 , which was combinatorially equivalent with 𝐶𝐸2 , is self dual. Kirkman
(1878) stated that there are 6 “autopolar” (self-dual) polyhedra within the 𝐶𝐶
polyhedra. I had found 5, so 𝐶𝐶2 must also be self dual.
Ernest Jucovič (1962) catalogued all of the self dual polyhedra with 9 vertices and 9
faces by giving their matrix representation but not diagrams. With his work, I was able
to match his matrices with all of the self-dual polyhedra and draw the correct 𝐶𝐶2 .
a
b
c
d
e
f
g
h
i
A
0
1
1
1
1
1
0
1
0
B
1
1
0
0
0
0
1
0
1
C
1
0
0
0
0
1
1
1
0
D
1
0
0
0
1
1
0
0
0
E
1
0
0
1
1
0
0
0
0
F
1
0
1
1
0
0
0
0
0
G
0
1
1
0
0
0
0
0
1
Jucovič (1961)
H
1
0
1
0
0
0
0
0
1
I
0
1
0
0
0
0
1
1
0
The matrix to the left is one from Jucovič (1961). If
you give each column the letters 𝐴 to 𝐼 and each row
the letters 𝑎 to 𝑖, you get the vertices for each face.
𝑎 (𝐵𝐶𝐷𝐸𝐹𝐻)
𝑏 (𝐴𝐵𝐺𝐼)
𝑐 (𝐴𝐹𝐺𝐻)
𝑑 (𝐴𝐸𝐹)
𝑒 (𝐴𝐷𝐸)
𝑓 (𝐴𝐶𝐷)
𝑔 (𝐵𝐶𝐼)
𝑕 (𝐴𝐶𝐼)
𝑖 (𝐵𝐺𝐻)
Face 𝑎 has 6 vertices so it must be a hexagon.
Conversely, vertex 𝐴 appears on 6 faces so it must be a
vertex of degree 6.
After all of the faces and vertices are organised, the
polyhedra can be drawn as a planar graph as shown
here.
On the other hand, every polyhedron can be represented
as a matrix. It is fairly straightforward to see that if a
polyhedron is self dual, then its vertices and faces can
be numbered in such a way that the incidence matrix is
symmetric.
4
3. Determining Decomposability of Polyhedra of V=F=9
In 1954, Gale defined decomposability and stated its basic results (although he called
it reducibility). Shephard (1963) investigated decomposability more seriously and gave
some useful criteria.
Decomposability is essentially breaking down something to its simplest form. A
polyhedron is decomposable iff it’s the sum of two polyhedra which are not similar. In
two-dimensions, everything is decomposable except the triangle. A square is the sum of
two line segments and a hexagon is the sum of two triangles or the sum of three line
segments (as shown in the pictures below).
or
In three dimensions it’s much harder to determine what’s decomposable and what’s
not. A simple example is a prism. For example, a triangular prism is the sum of a
triangle and the line segments.
+
=
Kallay, in 1982, was working on indecomposable polytopes and came up with an idea
on reducing them to graph theory. He invented a definition of decomposability of
geometric graphs.
He called a geometric graph indecomposable if every local similarity is a similarity or
a constant function (Kallay 1982). A local similarity is a mapping which preserves the
direction of edges. This means if 𝑢𝑣 is an edge and 𝑢 and 𝑣 are adjacent vertices
then 𝑓 𝑢 − 𝑓 𝑣 is either zero or a positive multiple of 𝑢 − 𝑣. A local similarity
between indecomposable geometric graphs is determined by its actions on two points
(and is necessarily connected). Kallay also found some conditions which imply
indecomposablity in terms purely of the graph, which we’ll get to in a moment.
In 1987, Smilansky had some major breakthroughs regarding decomposability. His
simplest and most impressive result is “THEOREM 6.7. If a 3-polytope 𝑃 has more
vertices than facets, then 𝑃 is decomposable.” (1987). He also showed that if 𝐹 is
greater than or equal to 2𝑉 − 6 it implies indecomposability.
For a graph to be combinatorially decomposable, all polyhedra with the same graph
are decomposable.
For a graph to be combinatorially indecomposable, all polyhedra with the same graph
are indecomposable.
For a graph to be conditionally decomposable, there exists 2 polyhedra with the same
graph, one decomposable, one indecomposable (Smilansky, 1987).
5
1
Smilansky’s table shows the polyhedra which satisfy 2 𝑉 + 2 ≤ 𝐹 < 𝑉 are
decomposable. For 2𝑉 − 4 < 𝐹 < 2𝑉 − 7 the polyhedra are all indecomposable. For
𝐹 = 2𝑉 − 7, the polyhedra are either decomposable or indecomposable. In other cases,
there exist examples of decomposable, indecomposable and conditionally
decomposable (ambiguous). Although it doesn’t tell you how many of each type. It just
states that there is at least 1 of each type.
Smilansky (1987)
So it is interesting to classify individually the polyhedra with 𝑉 ≤ 𝐹 ≤ 2𝑉 − 7. This
was essentially known for 𝑉 = 𝐹 = 7, and was completed for 𝑉 = 𝐹 = 8 in Yost
(2007) and for 𝑉 = 8, 𝐹 = 9 in Przeslawski and Yost (2016). In particular, all
polyhedra with 8 or fewer vertices have already been classified, and the next case to
study is 𝑉 = 𝐹 = 9.
Dr David Yost, with my help, has been working through them to determine how many
polyhedra are decomposable, indecomposable and ambiguous with the purpose to see
whether or not an interesting pattern turns up. I have been using certain techniques to
see how many are decomposable, indecomposable, and ambiguous within the 296
polyhedra with 9 faces and 9 vertices.
Most of the polyhedra we study turn out to be indecomposable. In most cases,
indecomposabilty can be seen quickly by applying the following some basic tools.
Kallay showed that if a subgraph G of the graph of a polyhedron 𝑃 is indecomposable
and contains all the vertices of 𝑃, then 𝑃 itself is indecomposable.
6
It was noted in Przeslawski and Yost (2008) that this still holds if 𝐺 just contains at
least one vertex from every face of 𝑃.
The following two are sufficient to prove indecomposability of many polyhedra.
1. A single edge is indecomposable
2. If 𝐺 is indecomposable and if 𝑣 is not in 𝐺 and if 𝑣 is adjacent to 2 vertices in 𝐺,
then the sum of 𝐺 and 𝑣 and the 2 edges is indecomposable
Clearly 1 and 2 imply that any triangle is indecomposable, and leads for example to
Smilansky (1987. THEOREM 5.1.) If a polyhedron 𝑃 has a strongly connected set of
triangular faces, that touches all the facets of 𝑃, then 𝑃 is indecomposable.
For some more difficult cases, the following two may need to be applied.
3. Non-planar 4-cycle is indecomposable (Przeslawski, K. & Yost, D. 2008,
Proposition 2).
4. If 𝐺 and 𝐻 are two indecomposable graphs, with two vertices in common, and
the edge between those vertices belongs to at most one of 𝐺, 𝐻, then (𝐺 ∪ 𝐻)
minus that edge is an indecomposable geometric graph. (Przeslawski and Yost,
In preparation), E.g. 𝐹𝐷𝑠1 , 𝐷𝐹𝑠1 , 𝐸𝐸23 , 𝐸𝐹𝑠3 , 𝐹𝐸𝑠3 .
Note that 2 is a special case of 4.
There are a few more rules for harder cases, e.g. in Kallay, but they are not necessary
for our purpose.
The literature contains very few conditions which imply decomposability. The
following is practically the only one.
 If there is one face with every vertex of degree 3, and it has at least 2 vertices
outside of that face, then it’s decomposable. (Shephard, 1963)
Theorem. Let 𝑃 be a polyhedron with the same number of vertices and facets (> 4).
Suppose the graph of 𝑃 contains the complete graph on 4 vertices. Then 𝑃 and its dual
are both decomposable.
Proof. If graph of 𝑃 contains 𝐾4 then 𝑃 is the result of stacking a
tetrahedron onto a polyhedron 𝑄 [note, so when you stack it on,
you’re adding one extra vertex and you’re adding three extra faces
and you’re losing one underneath when you “glue” the two
together] So it changes in regards to 𝑉 − 1 vertex and 𝐹 − 2 faces.
Since 𝑉 − 1 > 𝐹 − 2, Smilansky tells us that 𝑄 is decomposable.
Let 𝑣 be the “extra” vertex. Theorem 5.1 from Smilansky (1987) Q is decomposable.
Then there exists a local similarity 𝑓 on 𝑄 which is not a similarity. A function defined
on the vertices of Q but not a similarity. Then if we extend 𝑓, the restriction of 𝑓 to the
𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 and then the three edges connect to the intersection of 𝑄 and the
𝑡𝑒𝑡𝑟𝑎𝑕𝑒𝑑𝑟𝑜𝑛. A local similarity restricted to a triangle which is indecomposable must
be a similarity. And since a similarity is defined as something with a domain of all of
𝑅3 so it must be the restriction of a similarity 𝑔.
We can define a local similarity 𝑕 on 𝑃 by 𝑕 𝑤 = 𝑓 𝑤
for all 𝑤 ∈ 𝑄
𝑕 𝑣 = 𝑔(𝑣)
and 𝑕 is not a similarity.
Hence 𝑃 is decomposable.
7
In polyhedron 𝑅, which is the dual of 𝑃,
𝑣 corresponds to a triangular face 𝐹.
𝑣 belongs to 3 trianglular faces in 𝑃.
Every vertex corresponds to a face in the dual. These three
triangular faces correspond to vertices of degree 3 in 𝑅.
So in 𝑅, 𝐹 is enclosed by 3 vertices of degree 3.
Which implies 𝑅 is decomposable. (Shephard, 1963)
𝑄𝐸𝐷
Interestingly, for every polyhedron 𝑃 we have determined so far, the following
statements are true.
1. 𝑃 is combinatorially decomposable if and only if the dual of 𝑃 is
combinatorially decomposable.
2. 𝑃 is combinatorially indecomposable if and only if the dual of 𝑃 is
combinatorially indecomposable.
3. 𝑃 is conditionally decomposable (ambiguous) if and only if the dual of 𝑃 is
conditionally decomposable.
This is known to be true for all 54 combinatorial types of polyhedra with 𝑉 = 𝐹 up to 8
(Yost, 2007). So far (27th Feb) we have determined the decomposability or otherwise of
291 of the 296 types with 𝑉 = 𝐹 = 9, and no counterexample has appeared. A long
term project would be to see whether or not this is able to be proven.
This motivated the preceding theorem, which is a special case.
Establishing conditional decomposability (ambiguity) of some polyhedra is a little
tricky as you need to find two realisations of polyhedron, one decomposable and one
indecomposable. The following proves this for 𝐹𝐸1 .
Consider the triangle 𝑇 with vertices
𝑂 (0,0,0),
𝐴 (0,1,0) and
𝐵 (0,1,1),
and the quadrilateral 𝑄 with vertices
𝑂 (0,0,0),
𝐶 (2,0,0),
𝐷 (2,0,1) and
𝐸 (1,0, −1).
Then 𝑃 = 𝑇 + 𝑄is the polyhedron with vertices 𝑂, 𝐴, 𝐵, 𝐶, 𝐷, 𝐸,
𝐹 = 𝐴 + 𝐶 (2,1,0),
𝐺 = 𝐴 + 𝐸 (1,1, −1) and
𝐻 = 𝐵 + 𝐷 (2,1,2).
It is not hard to see that 𝐴 + 𝐷 = 𝐵 + 𝐶 (2,1,1) is an interior point of 𝑃, while 𝐻 = 𝐵 +
𝐷 (1,1,0) is midpoint of the edge 𝐴𝐹.
The combinatorial structure of 𝑃 is 𝐹𝐸1 . The quadrilateral faces are 𝑂𝐵𝐻𝐷, 𝐶𝐷𝐻𝐹,
𝐶𝐸𝐺𝐹, 𝑂𝐴𝐸𝐺 and 𝐴𝐵𝐹𝐻. The triangular faces are 𝑂𝐴𝐵, 𝑂𝐶𝐷,𝑂𝐶𝐸 and 𝐴𝐹𝐺.
8
To see an indecomposable realisation, replace A by the nearby point (−0.1,1,0.1). The
new 𝐴 still lies in the planes with equations 𝑥 + 𝑧 = 0 and 𝑦 = 1. This means that
𝑂𝐴𝐸𝐺 is still coplanar, hence a quadrilateral face of the new 𝑃, and the same applies to
𝐴𝐵𝐹𝐻. So the combinatorial structure of 𝑃 has not changed. However the 4-cycle
𝑂𝐴𝐹𝐶 is no longer coplanar (𝑂, 𝐹 and 𝐶 are in the plane 𝑧 = 0), and it touches every
face.
𝑄𝐸𝐷
The ambiguity of other polyhedra listed in the table have been proved by similar
arguments.
4. Results
The table explains how to classify each diagram. The meaning of the columns is as
follows:
(1) Order Number. This number refers to the number of the diagram. Simply its
order on the list, no quantitative meaning
(2) Notation. I preserved Kirkman’s (1878) notation for the asymmetrical
polyhedra. The first letter corresponds to the number of faces and the second
letter refers the degrees of the vertices, and since there are 2 or more polyhedra
within most groups, they have been numbered. Kirkman may have had some
logic behind the notation for his symmetrical polyhedra, but there is no reason to
keep symmetrical and asymmetrical separate, so I have combined them. For
example, his polyhedron 𝐾 is now 𝐹𝐷𝑠1 . The 𝑠 meaning it has symmetry
(3) Faces. The digit on the right is the number of triangular faces, the next digit is
the number of quadrilateral faces, the next digit is the number of pentagons, the
next is the number of hexagons, the next is the number of heptagons, and the
next is the number of octagons. For example, if the number is 10017, there’s 1
heptagon, 1 quadrilateral, and 7 triangles.
(4) Vertices. Similar to the face notation, but instead it relates to the degrees of the
vertices. For example, if the number is 216, it has 2 vertices of degree 5, 1 of
degree 4 and 6 of degree 3.
(5) Symmetry. The number in this column represents how many symmetries the
polyhedron has. If there is nothing in the table cell, the polyhedron has no
symmetry and therefore is asymmetrical.
(6) Dual. The Notation in this column refers to the Notation of the dual of the
polyhedron in each particular line. If it says “Self” then the polyhedron is selfdual.
(7) Decomposability. This column states whether the polyhedron is decomposable,
indecomposable, or ambiguous.
(8) Kirkman. The notation in this column corresponds to Kirkman’s (1878) paper
titled “The Enumeration and Construction of the 9-Acral 9-Edra”. The double
asterisk ** indicates the errors found and corrected in Kirkman’s work.
(9) Jucoviĉ. The notation in this column relates to Ernest Jucoviĉ’s (1962) paper
titled “Самосопряженные К–полиэдры” (Roughly translated to Self-Dual KPolyhedra). In his paper, he uses the same notation for different polyhedra so I
added subscripts in order of appearance in his paper.
9
The lines highlighted in orange are the polyhedra which haven’t been classified
regarding their decomposability yet. The rest of the results will be submitted in a
paper co-authored by Dr David Yost once we finish determining decomposability.
5. The Diagrams:
The diagrams are in the same order as the table. Starting with 𝐽𝐽𝑠, the polyhedron with
one octagon and eight triangles, e.g. the octagonal pyramid.
Apart from the background face, the remaining faces are colour coded for easy
identification. The pentagons are blue, the quadrilaterals are green and the triangles
are maroon.
10
Number Notation Faces
Vertices Symmetry
One octagon& eight triangles
1
100008 100008 16
𝐽𝐽𝑠
One heptagon, one quadrilateral & seven triangles
2
10017
10017
𝐴𝐴
3
10017
1026
𝐴𝐶
4
10017
216
2
𝐴𝐷𝑠
One hexagon, one pentagon & seven triangles
5
1107
1107
𝐵𝐵
6
1107
1026
2
𝐵𝐶𝑠
7
1107
216
𝐵𝐷
8
1107
135
𝐵𝐸1
9
1107
135
𝐵𝐸2
10
1107
54
2
𝐵𝐹𝑠
One hexagon, two quadrilaterals& six triangles
11
1026
10017
𝐶𝐴
12
1026
1107
2
𝐶𝐵𝑠
13
1026
1026
2
𝐶𝐶𝑠1
14
1026
1026
2
𝐶𝐶𝑠2
15
1026
1026
2
𝐶𝐶𝑠3
16
1026
1026
2
𝐶𝐶𝑠4
17
1026
1026
𝐶𝐶1
18
1026
1026
𝐶𝐶2
19
1026
1026
𝐶𝐶3
20
1026
1026
𝐶𝐶4
21
1026
1026
𝐶𝐶5
22
1026
1026
𝐶𝐶6
23
1026
1026
𝐶𝐶7
24
1026
1026
𝐶𝐶8
25
1026
216
2
𝐶𝐷𝑠
26
1026
216
𝐶𝐷1
27
1026
216
𝐶𝐷2
28
1026
216
𝐶𝐷3
29
1026
135
2
𝐶𝐸𝑠1
30
1026
135
2
𝐶𝐸𝑠2
31
1026
135
2
𝐶𝐸𝑠3
32
1026
135
2
𝐶𝐸𝑠4
33
1026
135
𝐶𝐸1
34
1026
135
𝐶𝐸2
35
1026
135
𝐶𝐸3
36
1026
135
𝐶𝐸4
37
1026
135
𝐶𝐸5
11
Dual
Decomposability
Kirkman
Jucovič
Self
Indecomposable
𝐶
(319; 1)
Self
𝐶𝐴
𝐷𝐴𝑠
Indecomposable
Indecomposable
Indecomposable
𝐴𝐴
𝐴𝐶
𝐿1
(325; 42)1
Self
𝐶𝐵𝑠
𝐷𝐵
𝐸𝐵2
𝐸𝐵1
𝐹𝐵𝑠
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
𝐵𝐵
𝐿2
𝐵𝐷
𝐵𝐸1
𝐵𝐸2
(329; 87)
𝐴𝐶
𝐵𝐶𝑠
Self
Self
𝐶𝐶𝑠4
𝐶𝐶𝑠3
𝐶𝐶5
Self
Self
Self
𝐶𝐶1
Self
Self
Self
𝐷𝐶𝑠
𝐷𝐶3
𝐷𝐶2
𝐷𝐶1
𝐸𝐶𝑠1
𝐸𝐶𝑠2
𝐸𝐶𝑠3
𝐸𝐶𝑠4
𝐸𝐶1
𝐸𝐶2
𝐸𝐶3
𝐸𝐶4
𝐸𝐶5
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
𝐶𝐴
𝐿−1
2
𝐷
𝑁2
𝐿4
𝐿−1
4
𝐶𝐶1
𝐶𝐶2 **
𝐶𝐶3
𝐶𝐶4
𝐶𝐶5
𝐶𝐶6
𝐶𝐶7
𝐶𝐶8
𝑂11
𝐶𝐷1
𝐶𝐷2
𝐶𝐷3
𝐿3
𝐿−1
5
𝑁1
𝑁3
𝑂7
𝐶𝐸1
𝐶𝐸2
𝐶𝐸3
𝐶𝐸4
𝐶𝐸5
(325; 42)2
(330; 35)2
(319; 2)
(330; 33)2
(325; 34)
(325; 31)
(330; 35)2
(325; 29)
38
1026
135
𝐶𝐸6
39
1026
135
𝐶𝐸7
40
1026
135
𝐶𝐸8
41
1026
135
𝐶𝐸9
42
1026
135
𝐶𝐸10
43
1026
135
𝐶𝐸11
44
1026
135
𝐶𝐸12
45
1026
135
𝐶𝐸13
46
1026
54
2
𝐶𝐹𝑠1
47
1026
54
2
𝐶𝐹𝑠2
48
1026
54
𝐶𝐹
Two pentagons, one quadrilateral & six triangles
49
216
10017
2
𝐷𝐴𝑠
50
216
1107
𝐷𝐵
51
216
1026
2
𝐷𝐶𝑠
52
216
1026
𝐷𝐶1
53
216
1026
𝐷𝐶2
54
216
1026
𝐷𝐶3
55
216
216
2
𝐷𝐷𝑠
56
216
216
𝐷𝐷1
57
216
216
𝐷𝐷2
58
216
216
𝐷𝐷3
59
216
216
𝐷𝐷4
60
216
216
𝐷𝐷5
61
216
216
𝐷𝐷6
62
216
216
𝐷𝐷7
63
216
216
𝐷𝐷8
64
216
216
𝐷𝐷9
65
216
216
𝐷𝐷10
66
216
135
2
𝐷𝐸𝑠
67
216
135
𝐷𝐸1
68
216
135
𝐷𝐸2
69
216
135
𝐷𝐸3
70
216
135
𝐷𝐸4
71
216
135
𝐷𝐸5
72
216
135
𝐷𝐸6
73
216
135
𝐷𝐸7
74
216
135
𝐷𝐸8
75
216
135
𝐷𝐸9
76
216
135
𝐷𝐸10
77
216
135
𝐷𝐸11
78
216
135
𝐷𝐸12
12
EC_6
EC_7
EC_8
EC_9
EC_10
EC_11
EC_12
EC_13
FCs_1
FCs_2
FC
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
𝐶𝐸6
𝐶𝐸7
𝐶𝐸8
𝐶𝐸9
𝐶𝐸10
𝐶𝐸11
𝐶𝐸12
𝐶𝐸13
𝐴𝐷𝑠
𝐵𝐷
𝐶𝐷𝑠
𝐶𝐷3
𝐶𝐷2
𝐶𝐷1
Self
Self
𝐷𝐷10
𝐷𝐷8
Self
Self
Self
Self
𝐷𝐷3
Self
𝐷𝐷2
𝐸𝐷𝑠
𝐸𝐷1
𝐸𝐷2
𝐸𝐷3
𝐸𝐷4
𝐸𝐷5
𝐸𝐷6
𝐸𝐷7
𝐸𝐷8
𝐸𝐷9
𝐸𝐷10
𝐸𝐷11
𝐸𝐷12
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
𝐿−1
1
𝐷𝐵
𝑂10
𝐷𝐶1
𝐷𝐶2
𝐷𝐶3
𝐼
𝑃1−1
𝐶𝐹
𝐻
𝐷𝐷1
𝐷𝐷2
𝐷𝐷3
𝐷𝐷4
𝐷𝐷5
𝐷𝐷6
𝐷𝐷7
𝐷𝐷8
𝐷𝐷9
𝐷𝐷10
𝑁11
𝐷𝐸1
𝐷𝐸2
𝐷𝐸3
𝐷𝐸4
𝐷𝐸5
𝐷𝐸6
𝐷𝐸7
𝐷𝐸8
𝐷𝐸9
𝐷𝐸10
𝐷𝐸11
𝐷𝐸12
(330; 33)1
(324; 78)
(327; 77)
(330; 43)1
(330; 35)4
(330; 35)1
(327; 75)
79
216
135
𝐷𝐸13
80
216
135
𝐷𝐸14
81
216
135
𝐷𝐸15
82
216
135
𝐷𝐸16
83
216
135
𝐷𝐸17
84
216
135
𝐷𝐸18
85
216
135
𝐷𝐸19
86
216
135
𝐷𝐸20
87
216
135
𝐷𝐸21
88
216
54
2
𝐷𝐹𝑠1
89
216
54
2
𝐷𝐹𝑠2
90
216
54
2
𝐷𝐹𝑠3
91
216
54
𝐷𝐹1
92
216
54
𝐷𝐹2
93
216
54
𝐷𝐹3
94
216
54
𝐷𝐹4
95
216
54
𝐷𝐹5
96
216
54
𝐷𝐹6
97
216
54
𝐷𝐹7
98
216
54
𝐷𝐹8
One pentagon, three quadrilaterals & five triangles
99
135
1107
𝐸𝐵1
100
135
1107
𝐸𝐵2
101
135
1026
2
𝐸𝐶𝑠1
102
135
1026
2
𝐸𝐶𝑠2
103
135
1026
2
𝐸𝐶𝑠3
104
135
1026
2
𝐸𝐶𝑠4
105
135
1026
𝐸𝐶1
106
135
1026
𝐸𝐶2
107
135
1026
𝐸𝐶3
108
135
1026
𝐸𝐶4
109
135
1026
𝐸𝐶5
110
135
1026
𝐸𝐶6
111
135
1026
𝐸𝐶7
112
135
1026
𝐸𝐶8
113
135
1026
𝐸𝐶9
114
135
1026
𝐸𝐶10
115
135
1026
𝐸𝐶11
116
135
1026
𝐸𝐶12
117
135
1026
𝐸𝐶13
118
135
216
2
𝐸𝐷𝑠
13
𝐸𝐷13
𝐸𝐷14
𝐸𝐷15
𝐸𝐷16
𝐸𝐷17
𝐸𝐷18
𝐸𝐷19
𝐸𝐷20
𝐸𝐷21
𝐹𝐷𝑠1
𝐹𝐷𝑠2
𝐹𝐷𝑠3
𝐹𝐷1
𝐹𝐷2
𝐹𝐷3
𝐹𝐷4
𝐹𝐷6
𝐹𝐷5
𝐹𝐷7
𝐹𝐷8
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
𝐷𝐸13
𝐷𝐸14
𝐷𝐸15
𝐷𝐸16
𝐷𝐸17
𝐷𝐸18
𝐷𝐸19
𝐷𝐸20
𝐷𝐸21
𝐵𝐸2
𝐵𝐸1
𝐶𝐸𝑠1
𝐶𝐸𝑠2
𝐶𝐸𝑠3
𝐶𝐸𝑠4
𝐶𝐸1
𝐶𝐸2
𝐶𝐸3
𝐶𝐸4
𝐶𝐸5
𝐶𝐸6
𝐶𝐸7
𝐶𝐸8
𝐶𝐸9
𝐶𝐸10
𝐶𝐸11
𝐶𝐸12
𝐶𝐸13
𝐷𝐸𝑠
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Decomposable
𝐸𝐵1
𝐸𝐵2
𝐺
𝐿6
𝑀2−1
𝐷𝐹1
𝐷𝐹2
𝐷𝐹3
𝐷𝐹4
𝐷𝐹5
𝐷𝐹6
𝐷𝐹7
𝐷𝐹8
𝐿5
𝑁4
𝑁8
𝑂5
𝐸𝐶1
𝐸𝐶2
𝐸𝐶3
𝐸𝐶4
𝐸𝐶5
𝐸𝐶6
𝐸𝐶7
𝐸𝐶8
𝐸𝐶9
𝐸𝐶10
𝐸𝐶11
𝐸𝐶12
𝐸𝐶13
𝑁6
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
𝐸𝐷1
𝐸𝐷2
𝐸𝐷3
𝐸𝐷4
𝐸𝐷5
𝐸𝐷6
𝐸𝐷7
𝐸𝐷8
𝐸𝐷9
𝐸𝐷10
𝐸𝐷11
𝐸𝐷12
𝐸𝐷13
𝐸𝐷14
𝐸𝐷15
𝐸𝐷16
𝐸𝐷17
𝐸𝐷18
𝐸𝐷19
𝐸𝐷20
𝐸𝐷21
𝐸𝐸𝑠1
𝐸𝐸𝑠2
𝐸𝐸𝑠3
𝐸𝐸𝑠4
𝐸𝐸𝑠5
𝐸𝐸𝑠6
𝐸𝐸𝑠7
𝐸𝐸𝑠8
𝐸𝐸𝑠9
𝐸𝐸1
𝐸𝐸2
𝐸𝐸3
𝐸𝐸4
𝐸𝐸5
𝐸𝐸6
𝐸𝐸7
𝐸𝐸8
𝐸𝐸9
𝐸𝐸10
𝐸𝐸11
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
216
216
216
216
216
216
216
216
216
216
216
216
216
216
216
216
216
216
216
216
216
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
𝐷𝐸1
𝐷𝐸2
𝐷𝐸3
𝐷𝐸4
𝐷𝐸5
𝐷𝐸6
𝐷𝐸7
𝐷𝐸8
𝐷𝐸9
𝐷𝐸10
𝐷𝐸11
𝐷𝐸12
𝐷𝐸13
𝐷𝐸14
𝐷𝐸15
𝐷𝐸16
𝐷𝐸17
𝐷𝐸18
𝐷𝐸19
𝐷𝐸20
𝐷𝐸21
Self
Self
𝐸𝐸𝑠6
Self
𝐸𝐸𝑠9
𝐸𝐸𝑠3
𝐸𝐸𝑠8
𝐸𝐸𝑠7
𝐸𝐸𝑠5
𝐸𝐸21
𝐸𝐸24
Self
𝐸𝐸27
𝐸𝐸12
Self
𝐸𝐸15
𝐸𝐸14
𝐸𝐸30
Self
Self
2
2
2
2
2
2
2
2
2
14
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Ambiguous
Ambiguous
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
𝐸𝐷1
𝐸𝐷2
𝐸𝐷3
𝐸𝐷4
𝐸𝐷5
𝐸𝐷6
𝐸𝐷7
𝐸𝐷8
𝐸𝐷9
𝐸𝐷10
𝐸𝐷11 **
𝐸𝐷12
𝐸𝐷13
𝐸𝐷14
𝐸𝐷15
𝐸𝐷16
𝐸𝐷17
𝐸𝐷18
𝐸𝐷19
𝐸𝐷20
𝐸𝐷21
𝑁5
𝑁9
𝑂1
𝑂2
𝑂3
𝑂4
𝑂6
𝑂14
𝑂15
𝐸𝐸1 **
𝐸𝐸2
𝐸𝐸3
𝐸𝐸4
𝐸𝐸5
𝐸𝐸6
𝐸𝐸7
𝐸𝐸8
𝐸𝐸9
𝐸𝐸10
𝐸𝐸11
(332; 7)
(331; 63)
(330; 41)
(332; 19)
(329; 81)
(332; 13)1
(330; 29)
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
𝐸𝐸12
𝐸𝐸13
𝐸𝐸14
𝐸𝐸15
𝐸𝐸16
𝐸𝐸17
𝐸𝐸18
𝐸𝐸19
𝐸𝐸20
𝐸𝐸21
𝐸𝐸22
𝐸𝐸23
𝐸𝐸24
𝐸𝐸25
𝐸𝐸26
𝐸𝐸27
𝐸𝐸28
𝐸𝐸29
𝐸𝐸30
𝐸𝐸31
𝐸𝐸32
𝐸𝐸33
𝐸𝐸34
𝐸𝐸35
𝐸𝐸36
𝐸𝐸37
𝐸𝐸38
𝐸𝐸39
𝐸𝐸40
𝐸𝐸41
𝐸𝐸42
𝐸𝐸43
𝐸𝐸44
𝐸𝐸45
𝐸𝐸46
𝐸𝐸47
𝐸𝐸48
𝐸𝐸49
𝐸𝐸50
𝐸𝐸51
𝐸𝐸52
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
𝐸𝐸5
𝐸𝐸28
𝐸𝐸8
𝐸𝐸7
𝐸𝐸31
Self
Self
𝐸𝐸46
𝐸𝐸22
𝐸𝐸1
𝐸𝐸20
Self
𝐸𝐸2
Self
Self
𝐸𝐸4
𝐸𝐸13
Self
𝐸𝐸9
𝐸𝐸16
𝐸𝐸45
𝐸𝐸63
Self
Self
𝐸𝐸38
𝐸𝐸42
𝐸𝐸36
𝐸𝐸65
𝐸𝐸48
Self
𝐸𝐸37
𝐸𝐸44
𝐸𝐸43
𝐸𝐸32
𝐸𝐸19
Self
𝐸𝐸40
𝐸𝐸62
Self
𝐸𝐸54
𝐸𝐸60
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
15
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Ambiguous
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
𝐸𝐸12
𝐸𝐸13
𝐸𝐸14
𝐸𝐸15
𝐸𝐸16
𝐸𝐸17
𝐸𝐸18
𝐸𝐸19
𝐸𝐸20
𝐸𝐸21
𝐸𝐸22 **
𝐸𝐸23
𝐸𝐸24
𝐸𝐸25
𝐸𝐸26
𝐸𝐸27
𝐸𝐸28
𝐸𝐸29
𝐸𝐸30
𝐸𝐸31
𝐸𝐸32
𝐸𝐸33 **
𝐸𝐸34
𝐸𝐸35
𝐸𝐸36
𝐸𝐸37 **
𝐸𝐸38
𝐸𝐸39
𝐸𝐸40
𝐸𝐸41
𝐸𝐸42
𝐸𝐸43
𝐸𝐸44
𝐸𝐸45
𝐸𝐸46
𝐸𝐸47
𝐸𝐸48
𝐸𝐸49
𝐸𝐸50
𝐸𝐸51
𝐸𝐸52
(331; 63)
(324; 1)
(332; 11)1
(331; 54)
(331; 55)2
(331; 55)1
(326; 72)1
(326; 72)2
(329; 16)
(330; 24)
(325; 33)
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
𝐸𝐸53
𝐸𝐸54
𝐸𝐸55
𝐸𝐸56
𝐸𝐸57
𝐸𝐸58
𝐸𝐸59
𝐸𝐸60
𝐸𝐸61
𝐸𝐸62
𝐸𝐸63
𝐸𝐸64
𝐸𝐸65
𝐸𝐸66
𝐸𝐸67
𝐸𝐹𝑠1
𝐸𝐹𝑠2
𝐸𝐹𝑠3
𝐸𝐹𝑠4
𝐸𝐹1
𝐸𝐹2
𝐸𝐹3
𝐸𝐹4
𝐸𝐹5
𝐸𝐹6
𝐸𝐹7
𝐸𝐹8
𝐸𝐹9
𝐸𝐹10
𝐸𝐹11
𝐸𝐹12
𝐸𝐹13
𝐸𝐹14
𝐸𝐹15
𝐸𝐹16
𝐸𝐹17
𝐸𝐹18
𝐸𝐹19
𝐸𝐹20
𝐸𝐹21
𝐸𝐹22
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
𝐸𝐸61
𝐸𝐸51
Self
Self
𝐸𝐸58
𝐸𝐸57
Self
𝐸𝐸52
𝐸𝐸53
𝐸𝐸49
𝐸𝐸33
Self
𝐸𝐸39
Self
Self
𝐹𝐸𝑠1
𝐹𝐸𝑠2
𝐹𝐸𝑠3
𝐹𝐸𝑠4
𝐹𝐸1
𝐹𝐸2
𝐹𝐸3
𝐹𝐸4
𝐹𝐸5
𝐹𝐸6
𝐹𝐸7
𝐹𝐸8
𝐹𝐸9
𝐹𝐸10
𝐹𝐸11
𝐹𝐸12
𝐹𝐸13
𝐹𝐸15
𝐹𝐸16
𝐹𝐸17
𝐹𝐸18
𝐹𝐸20
𝐹𝐸21
𝐹𝐸22
𝐹𝐸19
𝐹𝐸14
2
2
2
2
16
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
Ambiguous
Indecomposable
Indecomposable
Indecomposable
Ambiguous
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
𝐸𝐸53
𝐸𝐸54
𝐸𝐸55
𝐸𝐸56
𝐸𝐸57
𝐸𝐸58
𝐸𝐸59
𝐸𝐸60
𝐸𝐸61
𝐸𝐸62
𝐸𝐸63
𝐸𝐸64
𝐸𝐸65
𝐸𝐸66
𝐸𝐸67
𝑁7
𝑂12
𝑂13
𝐿7
𝐸𝐹1
𝐸𝐹2
𝐸𝐹3
𝐸𝐹4
𝐸𝐹5
𝐸𝐹6
𝐸𝐹7
𝐸𝐹8 **
𝐸𝐹9
𝐸𝐹10
𝐸𝐹11
𝐸𝐹12
𝐸𝐹13
𝐸𝐹14
𝐸𝐹15
𝐸𝐹16
𝐸𝐹17
𝐸𝐹18
𝐸𝐹19
𝐸𝐹20
𝐸𝐹21
𝐸𝐹22
(329; 19)
(330; 21)
(330; 43)2
(330; 31)
(332; 14)
(329; 20)
Five quadrilaterals & four triangles
242
54
1107
𝐹𝐵𝑠
243
54
1026
𝐹𝐶𝑠1
244
54
1026
𝐹𝐶𝑠2
245
54
1026
𝐹𝐶
246
54
216
𝐹𝐷𝑠1
247
54
216
𝐹𝐷𝑠2
248
54
216
𝐹𝐷𝑠3
249
54
216
𝐹𝐷1
250
54
216
𝐹𝐷2
251
54
216
𝐹𝐷3
252
54
216
𝐹𝐷4
253
54
216
𝐹𝐷5
254
54
216
𝐹𝐷6
255
54
216
𝐹𝐷7
256
54
216
𝐹𝐷8
257
54
135
𝐹𝐸𝑠1
258
54
135
𝐹𝐸𝑠2
259
54
135
𝐹𝐸𝑠3
260
54
135
𝐹𝐸𝑠4
261
54
135
𝐹𝐸1
262
54
135
𝐹𝐸2
263
54
135
𝐹𝐸3
264
54
135
𝐹𝐸4
265
54
135
𝐹𝐸5
266
54
135
𝐹𝐸6
267
54
135
𝐹𝐸7
268
54
135
𝐹𝐸8
269
54
135
𝐹𝐸9
270
54
135
𝐹𝐸10
271
54
135
𝐹𝐸11
272
54
135
𝐹𝐸12
273
54
135
𝐹𝐸13
274
54
135
𝐹𝐸14
275
54
135
𝐹𝐸15
276
54
135
𝐹𝐸16
277
54
135
𝐹𝐸17
278
54
135
𝐹𝐸18
279
54
135
𝐹𝐸19
280
54
135
𝐹𝐸20
281
54
135
𝐹𝐸21
𝐵𝐹𝑠
𝐶𝐹𝑠1
𝐶𝐹𝑠2
𝐶𝐹
𝐷𝐹𝑠1
𝐷𝐹𝑠2
𝐷𝐹𝑠3
𝐷𝐹1
𝐷𝐹2
𝐷𝐹3
𝐷𝐹4
𝐷𝐹6
𝐷𝐹5
𝐷𝐹7
𝐷𝐹8
𝐸𝐹𝑠1
𝐸𝐹𝑠2
𝐸𝐹𝑠3
𝐸𝐹𝑠4
𝐸𝐹1
𝐸𝐹2
𝐸𝐹3
𝐸𝐹4
𝐸𝐹5
𝐸𝐹6
𝐸𝐹7
𝐸𝐹8
𝐸𝐹9
𝐸𝐹10
𝐸𝐹11
𝐸𝐹12
𝐸𝐹13
𝐸𝐹22
𝐸𝐹14
𝐸𝐹15
𝐸𝐹16
𝐸𝐹17
𝐸𝐹21
𝐸𝐹18
𝐸𝐹19
2
2
2
2
2
2
2
2
2
2
17
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
𝐿−1
3
𝐸
𝑃1
𝐹𝐶
𝐾
𝐿−1
6
𝑀2
𝐹𝐷1
𝐹𝐷2
𝐹𝐷3
𝐹𝐷4
𝐹𝐷5
𝐹𝐷6
𝐹𝐷7
𝐹𝐷8
Indecomposable
Indecomposable
Indecomposable
Ambiguous
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Decomposable
Decomposable
Indecomposable
Indecomposable
𝑁10
𝑂8
𝑂9 **
𝐿−1
7
𝐹𝐸1
𝐹𝐸2
𝐹𝐸3
𝐹𝐸4
𝐹𝐸5
𝐹𝐸6
𝐹𝐸7
𝐹𝐸8
𝐹𝐸9
𝐹𝐸10
𝐹𝐸11
𝐹𝐸12
𝐹𝐸13
𝐹𝐸14
𝐹𝐸15
𝐹𝐸16
𝐹𝐸17
𝐹𝐸18
𝐹𝐸19
𝐹𝐸20
𝐹𝐸21
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
𝐹𝐸22
𝐹𝐹𝑠1
𝐹𝐹𝑠2
𝐹𝐹𝑠3
𝐹𝐹𝑠4
𝐹𝐹𝑠5
𝐹𝐹𝑠6
𝐹𝐹𝑠7
𝐹𝐹𝑠8
𝐹𝐹1
𝐹𝐹2
𝐹𝐹3
𝐹𝐹4
𝐹𝐹5
𝐹𝐹6
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
135
54
54
54
54
54
54
54
54
54
54
54
54
54
54
𝐸𝐹20
Self
Self
Self
𝐹𝐹𝑠8
Self
𝐹𝐹𝑠7
𝐹𝐹𝑠6
𝐹𝐹𝑠4
Self
𝐹𝐹3
𝐹𝐹2
Self
Self
Self
8
8
2
2
2
2
2
2
18
Indecomposable
Decomposable
Ambiguous
Indecomposable
Ambiguous
Ambiguous
Indecomposable
Indecomposable
Indecomposable
Indecomposable
Indecomposable
𝐹𝐸22
𝐴
𝐵
𝐽
𝑀1
𝐹
𝑃2
𝑃2−1
𝑀1−1 **
𝐹𝐹1
𝐹𝐹2
𝐹𝐹3
𝐹𝐹4
𝐹𝐹5
𝐹𝐹6
(332; 1)
(332; 6)
(332; 10)
(332; 13)2
(329; 2)
(331; 60)
(331; 57)
(332; 20)
19
20
21
22
23
24
25
26
27
Reference List
Biggs, N. (1981). T. P. Kirkman, Mathematician. Bulletin of the London
Mathematical Society, 13(2), pp.97-120.
Briggs, D & Yost, D. In preparation.
Britton, D & Dunitz, J.D. (1973). A Complete Catalogue of Polyhedra with Eight or
Fewer Vertices. Acta Crystallographica, series A, 29(4), 367-371.
Federico, P.J. (1969). Enumeration of polyhedra: The number of 9-hedra. Journal of
Combinatorial Theory, 7(2), pp.155-161.
Federico, P.J. (1974). Polyhedra with 4 to 8 Faces. Geometriae Dedicata, 3(4), 469481.
Gale, D. (1954). Irreducible convex sets. Proceedings of the International Congress
of Mathematicians, Amsterdam, 2, 217-218.
Jucovič, E. (1962). Самосопряженные К–полиэдры. Matematicko-fyzikálny
časopis, 12(1), 1-22.
Kallay, M. (1982). Indecomposable Polytopes. Israel Journal of
Mathematics, 41(3), 235-243.
Kirkman, T.P. (1863). Applications of the Theory of the Polyedra to the
Enumeration and Registration of Results. Philosophical Transactions of the Royal
Society of London, 12, 341-380.
Kirkman, T.P. (1878). The Enumeration and Construction of the 9-Acral 9-Edra,
and on the Construction of Polyedra. Proceedings of the Literary & Philosophical
Society of Liverpool, 32, 177-215.
Przeslawski, K. and Yost, D. (2008). Decomposability of polytopes. Discrete &
Computational Geometry,39, 460-468,
Przeslawski, K. and Yost, D. More indecomposable polyhedra, in preparation.
Shephard, G.C. (1963). Decomposable Convex Polyhedra. Mathematika: A Journal of
Pure and Applied Mathematic, 10(2), 89-95.
Smilansky, Z (1987). Decomposability of Polytopes and Polyhedra. Geometriae
Dedicata, 24(1), 29-49.
Voytekhovsky, Y. (2014). E. S. Fedorov’s algorithm of the generation of the
combinatorial diversity of convex polyhedra: The latest results and applications.
Journal of Structural Chemistry, 55(7), pp.1293-1307.
Yost, D. (2007). Some Indecomposable Polyhedra. Optimization, 56(5), 715-724.
Ziegler, G.M. (1995). Lectures on Polytopes, Graduate Texts in Mathematics, Vol.
152, Springer-Verlag, New York, 1995.
28