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ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 13 (2017) 23-30
Classification of convex polyhedra by their rotational orbit Euler characteristic
Jurij Kovic *
IMFM, Jadranska 19, 1000 Ljubljana, Slovenia; UP FAMNIT, Glagoljaska 8, 6000 Koper, Slovenia
Received 2 February 2015, accepted 10 October 2015, published online 22 August 2016
Abstract
Let P be a polyhedron whose boundary consists of flat polygonal faces on some compact surface S(P) (notnecessarily homeomorphic to the sphere S2). Let voR(P),eoR(P), foR(P) be the numbers of rotational orbits of vertices, edges and faces, respectively, determined by the group G = GR (P) of all the rotations of the Euclidean space E3 preserving P. We define the rotational orbit Euler characteristic of P as the number Eor(P) = voR(P) - eoR(P) + foR(P).
Using the Burnside lemma we obtain the lower and the upper bound for EoR(P) in terms of the genus of the surface S(P). We prove that EoR G {2,1,0, -1} for any convex polyhedron P. In the non-convex case EoR may be arbitrarily large or small.
Keywords: Polyhedron, rotational orbit, Euler characteristic. Math. Subj. Class.: 52B05, 52B10
1 Introduction
CONTEXT: Euler (1752) discovered the famous relation v - e + f = 2 between the numbers of vertices v, edges e and faces f of any convex polyhedron. This Euler polyhedron formula was implicitly stated in the formulas of Descartes (1630) p = 2f + 2v - 4,p = 2e, where p is the number of "plane angles" - corners of faces determined by pairs of adjacent edges ([5], p.469).
The number x = v - e + f can be defined for any map (a graph cellularly embedded into a compact surface S) and is called its Euler characteristic. It is related to the genus g of the surface S as follows: x = 2 - 2g ([5], p. 473.) and it may be used for the
*This work is supported in part by the Slovenian Research Agency (research program P1-0294 and research project N1-0032).
E-mail address: jurij.kovic@siol.net (Jurij Kovic)
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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Ars Math. Contemp. 13 (2017) 125-136
classification of surfaces by two parameters: one is x and the other is orientability (or non-orientability) of the surface. In [4] we introduced the concept of Euler orbit characteristic Eo = vo — eo + fo, where vo, eo, fo denote the number of orbits of vertices, edges and faces, respectively, determined by the group of all rotations and reflections of the Euclidean space E3, preserving the polyhedron P, and we used it for the classification of the 92 Johnson solids ([4], p.258). In this paper we introduce a similar concept, called rotational orbit Euler characteristic, and we use it for the classification of convex polyhedra.
Definition 1.1. The rotational Euler orbit characteristic of the polyhedron P is defined as the number EoR = voR — eoR + foR where voR, eoR, foR are the numbers of rotational orbits of the vertices, edges and faces, respectively, of P (these orbits are determined by the group GR(P) of all the rotations of the Euclidean space E3 preserving P).
Proposition 1.2. EoR = 1 for all the Platonic solids and all the n-prisms and n-anti-prisms, while 1 < EoR < 2 for all the Archimedean solids.
Proof. In the Table 1 the number of rotational orbits of Platonic and Archimedean solids are given. These values can be easily found for each solid directly or deduced from the symmetry-type graphs of Platonic and Archimedean solids [3]. The 5 Platonic solids have just one rotational orbit of vertices, edges and faces. The 13 Archimedean solids have at most two rotational orbits of vertices and at most three rotational orbits of edges and faces. The n-prisms and the n-antiprisms have just one rotational orbit of vertices and two rotational orbits of edges and faces.	□
class	solid P	vertex pattern	VOR	eoR	fOR	Eor
I.	tetrahedron	(3.3.3)	1	1	1	1
I.	octahedron	(3.3.3.3)	1	1	1	1
I.	cube	(4.4.4)	1	1	1	1
I.	icosahedron	(3.3.3.3.3)	1	1	1	1
I.	dodecahedron	(5.5.5)	1	1	1	1
II.	cuboctahedron	(3.4.3.4)	1	1	2	2
II.	icosidodecahedron	(3.5.3.5)	1	1	2	2
III.	truncated tetrahedron	(3.6.6)	1	1	2	2
III.	truncated cube	(3.8.8)	1	1	2	2
III.	truncated octahedron	(4.6.6)	1	1	2	2
III.	truncated dodecahedron	(3.10.10)	1	1	2	2
III.	truncated icosahedron	(5.6.6)	1	1	2	2
IV.	rhombicuboctahedron	(3.4.4.4)	1	2	3	2
IV.	rhombicosidodecahedron	(3.4.5.4)	1	2	3	2
V.	truncated cuboctahedron	(4.6.8)	2	3	3	2
V.	truncated icosidodecahedron	(4.6.10)	2	3	3	2
VI.	snub cube	(3.3.3.3.4)	1	3	3	1
VI.	snub dodecahedron	(3.3.3.3.5)	1	3	3	1
VII.	n-prism	(4.4.n)	1	2	2	1
VIII.	n-antiprism	(3.3.n)	1	2	2	1
Table 1: Values of voR, eoR, foR for Platonic and Archimedean solids and for the infinite families of n-prisms and n-antiprisms.
MOTIVATION: Similar bounds on EoR exist for the Johnson solids (i.e. convex polyhedra with regular polygonal faces and at least two orbits of vertices [2]). The direct motivation
J. Kovic: Classification of convex polyhedra by their rotational orbit Euler characteristic 25
for writing this paper came from the empirical observation that the values of EoR for the 92 Johnson solids are in a small range between — 1 and 2. This was discovered during the process of constructing a table of 16 parameters of the Johnson solids presented in [4], while the range for Eo for the same solids turned out to be bigger: 0 < Eo < 5. COMPARISON OF Eo AND EoR: The two characteristics behave very differently on the set of all convex polyhedra: the main result of the paper (Theorem 2.1) states that the relation — 1 < EoR < 2 holds for all convex polyhedra, while for Eo there is no fixed upper bound, we can obtain only the following estimate: Eo = vo — eo + fo < vo + fo < voR + foR = < (vor — eoR + foR) + eoR = eor + eoR < 2 + eoR.
Definition 1.3. Let GR(P) = [Ri, R2,..., Rn-i, Rn = Id} be the group of rotational symmetries of the polyhedron P. The poles of the rotation Ri are the points in which the axis of the rotation Ri intersects the surface S(P). Let vp(Ri), ep(Ri), fp(Ri) denote the numbers of poles of Ri in the vertices, edge centers and face centers of P, respectively. The number Ep(Ri) = vp(Ri) — ep(Ri) + fp(Ri) is called the Euler polar characteristics of the rotation Ri.
Lemma 1.4. Let ni denote the number of poles of any non-trivial rotation Ri of the polyhedral map P on the surface S of genus g. Then
ni < 2(g + 1), m e [0, 2, 4,..., 2(g + 1)}, vp(Ri) + ep(Ri) + fp(Ri) = ni, 0 < ep(Ri) < ni, Ep(Ri) = ni — 2ep(Ri), —2(g + 1) < —ni < Ep(Ri) < ni < 2(g + 1).
If the order of the rotation Ri is greater than 2 (i.e. Rn = id and n > 2), then
ep(Ri) = 0 and Ep(Rj) = ni.
Proof. Any line intersecting P has at most 2g intersecting points with S. If P is a convex polyhedron then any nontrivial rotation Ri has exactly two poles, hence ni = 2. If ni > 2 then each segment of the rotational axis ri not lying in the interior of P contributes two poles (hence ni is an even number!) and at least one new handle. Thus it "increases" the genus of S for 1 (since it is well known that the genus counts the numbers of "handles" of a surface), therefore it must be ni < 2(g +1). The poles can be only in vertices, edge centers or face centers, hence vp(Ri) + ep(Ri) + fp(Ri) = ni and 0 < ep(Ri) < ni. Obviously Ep(Ri) = vp(Ri) — ep(Ri) + fp(Ri) = ni — 2ep(Ri). Therefore the upper bound for Ep(Ri) is ni and the lower bound is —ni.	□
Corollary 1.5. If P is a convex polyhedron, then Ep(Ri) = 2 — 2ep (Ri) e [2,0, —2}. If ep =0 then Ep = 2, if ep = 1 then Ep = 1, and if ep = 2 then Ep = —2. If the order of the rotation Ri is greater than 2, then Ep(Ri) =2.
Proof. Every convex polyhedron is homeomorphic (by a radial projection from any point of its interior) to a sphere, which has genus g = 0. Now ni = 2 and the formulas follow from the Lemma 1.4.	□
The next tool we need (in order to prove the main result, Theorem 2.1) is the Burnside lemma, a standard tool for calculating the number of orbits.
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Ars Math. Contemp. 13 (2017) 125-136
Lemma 1.6. (Burnside lemma) Let a group G act on some set Q. Let |G| = n denote the number of elements of G and let lFix(g)l denote the number of elements a of the set Q, preserved by the given element g of the group: g(a) = a. Then the number of orbits Qo of the set Q is given by the formula
Qo = G E IFix(g)l-
|G| geo
If the group of rotational symmetries of a convex polyhedron is the cyclical group Cn then the exact value for the Euler orbit characteristic EoR (P) can be obtained by a straightforward application of the Burnside lemma. This is a generalization of the similar result for the spherical polyhedra ([4], p.253).
Proposition 1.7. Let P be a convex polyhedron. If GR(P) = Cn where Cn is generated by a rotation R1 (thus Rn = I), then EoR(P) G {0,1,2}.
Proof. The identity transformation fixes all vertices, edges and faces while the other n — 1 rotations fix only the poles. Hence we get by the Burnside lemma and using the Euler formula v — e + f = 2 (valid for any convex polyhedron) the following formulas for the numbers of rotational orbits:
vor = ~(v + (n — 1)vp), n
eoR = —(e + (n — 1)e), n
foR = —(f + (n — 1)fp),
n
Eor =1(2+(n — 1)Ep(Ri)), n
and using Ep(R\) = 2 — 2ep(R\) we see: if ep(R\) = 0 then Ep(R\) = 2 and EoR = 2; if ep(Ri) = 1 then n = 2, Ep(Rx) = 0 and EoR = 1; if ep(R1) = 2 then n = 2, Ep(Rl) = —2 and EoR = 0. Thus, if n> 2 then EoR = 2.	□
2 The main result
Theorem 2.1. Let P be a polyhedron with faces on the surface S of genus g. Then
1	n— 1
Eor(P) = n(x(P) + E Ep(Ri)),
i=i
and we get the following bounds on EoR(P):
n—1	n—1
1(x(P) — E 2(g +1)) < eor(p) < 1(x(P) + E 2(g +1)).
n	z—'	n	z—'
i=1 i=1
If P is a convex polyhedron, then
— 1 < EoR(P) < 2.
J. Kovic: Classification of convex polyhedra by their rotational orbit Euler characteristic 27
Proof. Let n be the number of elements in the group GR(P). The identity transformation fixes each vertex, edge or face. Every rotation Ri fixes vp(Ri) vertices, ep(Ri) edges and fp(Ri) faces. Therefore, by the Burnside lemma:
VOR = 1 (v + Vp(R 1) +----Vp(Rn-1)),
eon = —(e + ep(R 1) +----ep(Rn-1)),
n
fOR = 1(f + fp(R1) + ••• fp (Rn — 1)),
n
Eon = —(X + Ep(R1) + ■ ■ ■ Ep(Rn—1)). Using -2(g +1) < Ep(Ri) < 2(g + 1) (proved in Lemma 1.4) we get
n—1	n—1
— (x - V 2(g + 1)) < Eon < — (x + V 2(g + 1)). nn
i=1 i=1
If P is a convex polyhedron, then g = 0, x = 2, hence
Eor < 1 (2 + (n - 1)2) = 2, n
114
Eor > 1(2 + (n - 1)(-2)) = 1(4 + n(-2)) > -2 + - > -1, nnn
because n > 0 and EoR must be an integer.	□
Thus there are 4 classes C2, C1, Co, C—1 of convex polyhedra, whose Eor are 2,1, 0, -1, respectively.
Is the lower bound EoR = -1 actually obtained, and (if it is so) for which convex polyhedra? And is there any simple description of these four classes?
Proposition 2.2. Let a, b, c be the numbers of rotations R in the group GR(P) of a convex polyhedron for which Ep(Ri) equals 2, 0 and -2, respectively, and let n be the number of elements in GR(P). Then
12
Eor(P) = 1(2 + a ■ 2 + c(-2)) = -(1 + a - c). nn
Thus the number 1 + a - c is an integer multiple of n. The numbers a and c can be (for each of the 4 possible values of EoR) expressed by b, n and EoR.
Proof. This formula follows immediately from the Burnside lemma. Also, it is clear that
a + b + c +1 = n,
hence
a + c = n - b - 1.
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Ars Math. Contemp. 13 (2017) 125-136
The equation n(1 + a - c) = EoR implies 2(1 + a - c) = EoR ■ n and
EOr ■ n
c — a = 1 —
2
Then (a + c) + (c - a) = 2c = n - b - 1 + 1 - ^^p = (2-E°°R)n - b and
_ (2 - EoR)n - 2b
c = 4 •
Similarly, 2a = n - b - 1 - 1 + ^^, hence
n ■ (2 + Eor) - (2b + 4)
a =-4-•
For example, if EoR = -1 and b = 0 then a = n - 1 and c = ^n -In that case n must be divisible by 4.	□
Example 2.3. To find such a solid with 4 symmetries we have to look for one having three rotations with poles in edge centers! The lower bound EoR = -1 is really obtained for the Johnson solid J84 (Snub Disphenoid, see Figure 1), where voR = 2, eoR = 6, foR = 3, hence EoR = 2 - 6 + 3 = -1. Here the number of symmetries is 4 (the identity transformation and 3 rotations of order two with axes going through edge centers), b = 0, a = 0 and c = 3. Thus this lower bound -1 is sharp.
Figure 1: The Johnson solid J84, also known as the Snub Disphenoid.
Remark 2.4. A rotational axis of a non-convex polyhedron may have more than 2 "poles". As a consequence, there is no upper or lower bound for EoR in the non-convex case.
3 Classification of convex polyhedra
As an immediate consequence of the formulas in Proposition 3 we get:
J. Kovic: Classification of convex polyhedra by their rotational orbit Euler characteristic 29
Corollary 3.1. The four classes C2,C1,C0,C-1 of convex polyhedra (whose EoR are 2,1,0, — 1, respectively) can be characterized as follows: C2: a — c = n — 1 C1: a — c = —1 + n Co: a — c = —1
C-1: a — c = —1 — n, all poles in edge centers, a = 0. Corollary 3.2. If a > n/2 then EoR e {1,2}.
Proof. The relation a > n/2 is a sufficient condition for a — c > 0 (since there is also the identity transformation in the group GR (P)) that holds only for polyhedra from C2 and
Ci.	□
Corollary 3.3. Let q be the number of all rotations R e GR(P) with the property that R has the same rotational axis as some k-fold rotation for any k > 2. If q > n/2, where n is the order of the group GR, then EoR(P) e {1, 2}.
Proof. No rotation with such an axis can have any of its two poles in an edge center, hence
a > q > n/2, therefore EoR(P) e {1, 2}.	□
Now we can classify convex polyhedra with respect to their rotational symmetry groups and their rotational orbit Euler characteristic.
The only possible rotational groups of the Euclidean space E3 are the rotational groups of 1) the n-gonal pyramid, 2) the n-gonal dipyramid or prism, 3) the regular tetrahedron, 4) the cube or the regular octahedron, 5) the regular dodecahedron or the regular icosahedron ([1], p.34).
Theorem 3.4. Convex polyhedra with at least one rotational symmetry can be classified by their GR and by their EoR into 13 classes (in Table 2 the impossible cases are marked with $):
	C2	Ci	Co	C_i
Cn cyclical group				0
Dn dihedral group				
T tetrahedron group			0	0
O octahedron group			0	0
D dodecahedron group			0	0
Table 2: Classification of convex polyhedra by GR and EoR.
Proof. If Gr = Cn then EoR = -1 (by Proposition 2). If GR G {T, O, D} then q > n/2 (Table 3) hence P cannot be in C0 or C_i (by Corollary 3.3).	□
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Ars Math. Contemp. 13 (2017) 125-136
	2	3	4	5	n	q
T tetrahedron	3	4			3 + 4.2 + 1 = 12	9
O cube	6	4	3		6 + 4.2 + 3.3 + 1 = 24	17
D dodecahedron	15	10		6	15 + 10.2+12.4 + 1= 60	44
Table 3: Numbers of the 2-,3-,4-,5-fold axes in the solids P with the rotational groups T, O or D, orders n of the groups T, O, D and the numbers q of all rotations R G GR(P) with the property that R has the same rotational axis as some k-fold rotation for any k > 2.
References
[1]	H. S. M. Coxeter, W.O.J. Moser, Generators and Relations for Discrete Groups, Springer Verlag, Berlin, Gottingen, Heidelberg, 1957.
[2]	N. W. Johnson, Convex polyhedra with regular faces, Canad. J. Math. 18 (1966), 169-200.
[3]	J. Kovic, Symmetry-type graphs of Platonic and Archimedean solids, Math. Commun. 16 (No 2) (2011), 491-507.
[4]	J. Kovic, Characterization of convex polyhedra with regular polygonal faces by minimal number of parameters, J. Combin. Math. Combin. Comput. 89 (2014), 249-263.
[5]	J. Stillwell, Mathematics and Its History, Springer, New York, 2010.