SQU Journal for Science, 2016, 21(2), 150-161 © 2016 Sultan Qaboos University 150 4D Pyritohedral Symmetry Nazife O. Koca*, Amal J.H. Al Qanobi and Mehmet Koca Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Sultanate of Oman,* Email: nazife@squ.edu.om. ABSTRACT: We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W (F4) and W (H4), implying that it is a group relevant to the crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24- cell. It turns out that the relevant lattice is the root lattice of W (D4). The vertices of the dual polytope of the pseudo snub 24-cell consists of the union of three sets: 24-cell, another 24-cell and a new pseudo snub 24-cell. We also derive a new representation for the symmetry group of the pseudo snub 24-cell and the corresponding vertices of the polytopes. Keywords: Pseudoicosahedron; Pyritohedron; Lattice; Coxeter groups and Quaternions. تماثل متعدد األوجه ذو التركيب المشابه لمعدن البايريت في الفضاء االقليدي ذو األبعاد األربعة محمد كوجا نظيفة كوجا، أمل القنوبي و و الذي يمكن صياغته في االبعاد -( pyritohedralوصفنا امتدادا لتماثل تركيب متعددات األوجه من النوع المشابه لبلورة معدن البايرايت ) ستخلص:م باستخدام العدد المركب التخيلي ببلورة معدن البايرايت رباعية االبعادالمتصلة إلى الفضاء االقليدي ذو األبعاد األربعة ثم بنينا عناصر المجموعة -الثالثة ذات الرتبة الرابعة 4H( W (و W )4F (الرباعي )الكواتيرنيونات(، و اتضح أن هذه المجموعة من أكبر المجموعات الفرعية لكل من مجموعات الكوكستر ذوات العشرون 24اشتقاق احداثيات رؤوس تم مما يعني أنها مجموعة ذات صلة بالتركيبات البلورية فضال عن اشباه البلورية في األبعاد األربعة، كما ، و اتضح أن pseudo snub 24-cell يسمىمجسم رباعي األبعاد أهراما مثلثية و التي تمثل أسطح ل 96رباعي األسطح و 24وجها غير المنتظم و هي اتحاد ثالث مجموعات: اربعة و cell-pseudo snub 24رؤوس المزدوج لـ و أن 4D( W (الشبيكة الفراغية ذات الصلة هي جذر الشبيكة الفراغية لـ ، كذلك اشتققنا تمثيال جديدا لمجموعة تماثل جديد pseudo snub 24-cellو ( أخرىcell-24اربعة و عشرون خلية )( و 24cellعشرون خلية ) pseudo snub 24-cell المقابلة.األجسام رباعية األبعاد و رؤوس شبيه ذي العشرين وجها المنتظم، متعدد األوجه ذو التركيب المشابه لمعدن البايرايت، الشبيكة الفراغية، مجموعات كوكستر، : كلمات مفتاحية الرباعيات. 1. Introduction attices in higher dimensions described by the affine Coxeter groups, when projected into lower dimensions, may represent the quasicrystal structures [1-5]. It is known that the 4 A lattice projects into the aperiodic lattice with 5- fold symmetry [1]. There is no doubt that the projections of the higher dimensional lattices may have some implications in physics. The exceptional Coxeter-Weyl group )( 4 FW describes the symmetry of the unique self-dual polytope, the 24-cell, which is the Voronoi cell (Wigner-Seitz cell) of the 4 F lattice. The noncrystallographic Coxeter group )( 4HW is the symmetry of the famous 600-cell and its dual 120-cell [6-7]. In this work we construct the 4D pyritohedral group from 4 D diagram. In technical terms the group  4 3 2 ( ) : W D S C of order 576 [8] can be expressed in terms of quaternions and we will determine its orbits as the pseudo snub 24-cell and L 4D PYRITOHEDRAL SYMMETRY 151 its dual polytope, which are related to the lattice )( 4 DW . All rank 4 Coxeter-Weyl groups can be represented, in compact forms, by quaternion pairs [9]. This paper is organized as follows. In Section 2 we introduce the 4D - pyritohedral symmetry derived from D4 diagram. In Section 3 we construct the group  4 3 2 ( ) : W D S C and apply it to a vector to generate the vertices of a polytope which we call “pseudo snub 24-cell”. We find the facets of the pseudo snub 24-cell which consist of the pseudoicosahedra, tetrahedra and triangular pyramids. The vertices of the dual polytope of the pseudo snub 24-cell are constructed. Finally, in Section 4 we present a brief discussion on the physical implications of our technique. 2. 4D crystals with the pyritohedral symmetry derived from D4 diagram In the paper [10] we discussed the pyritohedral group * ],[],[ TTTT  of order 24 which is derived from the Coxeter- Dynkin diagram 3D by the rotation generators and the Dynkin diagram symmetry. The straightforward generalization of this group to 4D is to start with the rotation generators of 4D and impose the Dynkin diagram symmetry. We will see that the generated group from the diagram 4D is nothing other than the group * ],[],[ TTTT  of order 576. It represents the symmetry of the snub 24-cell [6], [8]. The snub 24-cell is a convex uniform polytope in four dimensions consisting of 120 regular tetrahedral and 24 icosahedral cells. It has 96 vertices at each of which five tetrahedra and three icosahedra meet. Snub 24-cell can be constructed from the 24-cell by dividing the edges in the golden ratio and truncating it in a certain way. This truncation transforms the 24 octahedral cells of the 24-cell to the 24 icosahedral cells of the snub 24-cell; the truncated vertices become 24 tetrahedral cells and the gaps in between are filled in by another 96 tetrahedra. 3. Construction of the symmetry group of snub 24-cell 3 2 4 S C DW : )(       The Coxeter-Dynkin diagram 4D is shown in Figure 1 with the quaternionic simple roots. The corresponding weights are determined as 1 2 1 3 1 2 3 4 1 2 3 1 1 1, 1 , (1 ), (1 ). 2 2 e e e e e e e              (1) Note that 2 , 1, 2, 3, 4; ; , j=1,3,4. 2 2 i j T i T T        The set T is given by the group elements 𝑇 = {±1,±𝑒1,±𝑒2,±𝑒3, 1 2 (±1 ± 𝑒1 ± 𝑒2 ± 𝑒3)}, (2) and is called the binary tetrahedral group of order 24 . Another set of 24 quaternions is defined by α 2 =e 1 -e 2 α 1 =1-e 1 α 3 =e 2 -e 3 α 4 =e 2 +e 3 Figure 1. The Coxeter-Dynkin diagram 4D with the simple roots. https://en.wikipedia.org/wiki/Uniform_4-polytope https://en.wikipedia.org/wiki/Cell_(mathematics) NAZIFE O. KOCA ET AL 152 𝑇′ = { 1 √2 (±1 ± 𝑒1), 1 √2 (±1 ± 𝑒2), 1 √2 (±1 ± 𝑒3), 1 √2 (±𝑒1 ± 𝑒2), 1 √2 (±𝑒2 ± 𝑒3), 1 √2 (±𝑒3 ± 𝑒1) }. (3) In terms of quaternionic simple roots, the group generators of )( 4DW can be written as 1 1 1 2 1 2 1 2 3 2 3 2 3 4 2 3 2 3 1 1 [ (1- ), - (1- )] , 2 2 1 1 [ ( ), - ( )] , 2 2 1 1 [ ( ), - ( )] , 2 2 1 1 [ ( ), - ( )] . 2 2 r e e r e e e e r e e e e r e e e e               (4) They generate the Coxeter-Weyl group )( 4DW of order 192 [12]. The subsets of the quaternions  VVVT 0  and 321 VVVT  are defined as follows: These subsets are useful to denote the Coxeter-Weyl group )( 4DW in a compact form: }],[],[],[],[],[],{[)( *** 332211004 VVVVVVVVVVVVDW    . (6) Note that the subset of the Coxeter-Weyl group 4 0 0 2 ( ) {[ , ] [ , ] [ , ]} W D V V V V V V C      (7) represents the proper subgroup and can be directly generated by the rotation generators 𝑟2𝑟1,𝑟2𝑟3,𝑟2𝑟4. Let us impose the Dynkin diagram symmetry which is the permutation group 𝑆3 of the simple roots 𝛼1,𝛼3 and 𝛼4 as shown in Figure 2. },,,{ 3210 1 eeeV  , )( 3211 2 1 eeeV  , even number of (-) sign, )( 3211 2 1 eeeV  , odd number of (-) sign, )}(),({ 3211 2 1 1 2 1 eeeV  , )}(),({ 1322 2 1 1 2 1 eeeV  , )}(),({ 2133 2 1 1 2 1 eeeV  . (5) 4D PYRITOHEDRAL SYMMETRY 153 Generator 1: ],[ qp Generator 2: *],[ 33 ee  Figure 2 . The action of permutation group 𝑺𝟑. The permutation group of order 6 can be generated, for example, by two generators [ , ]p q and * 3 3 [ , ]e e where 1 2 3 1 2 (1 )p e e e    and 1 2 3 1 2 (1 )q e e e    . They are the elements of 3 [ , ] [ , ] with [ , ] [1,1]p q V V p q     and 3 3 0 0 [ , ] [ , ]e e V V     . The group       2 4 C DW )( is invariant under conjugation by the group 𝑆3. We first note that the extension of the group of eq. (7) by the cyclic group of order 3 generated by the generator [𝑝,𝑞] is a group of order 288 which can be denoted by  4 3 2 ( ) : [ , ] W D C T T C  . (8) The extension of the group by full permutation group 3 S is given as the semi-direct product of two groups as:   *4 3 2 ( ) : {[ , ] [ , ] } W D S T T T T C  . (9) As we will see in the next section this is the symmetry group of the snub 24-cell as well as that of any pseudo snub 24- cell. 3.1 Construction of the vertices of the pseudo snub 24-cell The affine Coxeter group 𝑊𝑎(𝐷4) =< 𝑟0,𝑟1,𝑟2,𝑟3,𝑟4 > can be generated by five generators by introducing 𝑟0 as shown in Figure 3. 𝑟0 represents the reflection with respect to the hyperplane bisecting the line from the origin to the highest root 1 2 3 4 1 0 2 1 e             . The action of 𝑟0 on a general vector  is given as 0 2[( , ) 1] ( , ) r           [13]. Figure 3. Affine Coxeter-Dynkin diagram 𝑾𝒂(𝑫𝟒) with simple roots. α α 1 α α 4 α α 1 α 3 α 4 α 2 α 1 α 3 α 4 α 0 NAZIFE O. KOCA ET AL 154 Applying the group Wa (D4) on a simple root we can generate the root lattice. We now derive the vertices of pseudo snub 24- cell in terms of the root lattice vectors of D4. The lattice of D4 is self-dual so that we can express the dual lattice vectors in terms of the weight vectors [14]. Table 1 shows the lattice vectors in terms of the root & weight vectors and quaternions. Table 1. The construction of the lattice from Wa (D4). Wa(D4) generates Lattice Root lattice (Real lattice) Weight lattice (Reciprocal lattice) Vectors 44332211  bbbb  Zb i  44332211  aaaa  Za i  Vectors in terms of quaternions 1 1 2 2 3 3 4 m e m e m e m     1 1 2 2 3 3 4 n e n e n e n     Determine the lattice integereven 2 4 3 0   bm i i 3 0 even integer i i n   We impose the Dynkin-diagram symmetry 𝑆3 on the vector  , that is, 3S    as shown in Figure 2. The group 𝑆3 permutes the weight vectors 𝜔1 ,𝜔3 𝑎𝑛𝑑 𝜔4 but leaves 𝜔2 invariant. Similarly, the generators 𝑟1 ,𝑟3 𝑎𝑛𝑑 𝑟4 are permuted and the generator 𝑟2 is left invariant under the group conjugation of 𝑆3.This implies that the vector  takes the form 2 2 1 1 3 4 ( ), , 1, 2. i a a a i         Z (10) Factorizing by 𝑎2 and defining the rational number 𝑥 = 𝑎1 𝑎2 the vector  reads in terms of quaternions ])()[( 212 121 xeexxa  . (11) Note that the sum of the coefficients of the quaternionic units is an even integer as we mentioned earlier. For 0 and 2 1 1  ,x the vector  belongs to the set of quaternions, T  which is known to be the 24-cell. For these particular values of x, the group   *4 3 2 ( ) : {[ , ] [ , ] } W D S T T T T C  generates 24 vertices of 𝑇′. The 24-cell has 24 vertices and 24 octahedral cells where 6 octahedra meet at one vertex. The polytope 24-cell constitutes the unit cell of the 𝐷4 lattice. In terms of the quaternionic sets when T represents the unit cell of the root lattice, then the set 𝑇′ √2 represents its Voronoi cell. Applying the group elements represented by eq. (9) on the vector in (11) we obtain 96 vertices of pseudo snub 24-cell as the orbit of the group  4 3 2 ( ) : W D S C as expected for 576 6 96 . We list 96 vertices of the pseudo snub 24-cell omitting the overall factor 𝑎2 as follows: }.)()(,)()(,)()( ,)()(,)()(,)()( ,)()(,)()(,)()( ,)()(,)()(,)()({ 321321321 313131 323232 212121 211 211 121 211 211 121 211 121 121 211 121 121 )( xeexexexexxeexxeex xeexxexexxexxex exxexexexxxeexx exxexexexxxeexxxS     (12) dual dual 4D PYRITOHEDRAL SYMMETRY 155 The mirror image of the pseudo snub 24-cell can be obtained by applying any reflection generator of 𝐷4. For example, applying 𝑟2 on 𝑆(𝑥) in (12) interchanges 1 2e e and leaves the other quaternionic units unchanged. Then under the action of the mirror operator 𝑟2 one can obtain the mirror image of the pseudo snub 24-cell in (12). Both the pseudo snub 24-cell and its mirror image lie in the 𝐷4 lattice. It is clear that for 0 and 2 1 1  ,x the vertices of (12) reduces to the set 𝑇′. Excluding these values of x, the set of vertices represent a pseudo snub 24-cell. Only in the limit 2 51 x ( ,x ) the vertices given in (13) represent the snub 24-cell [8]. Since in this case x is not a rational number, the vertices do not belong to the lattice 𝐷4. 𝑆 = { 1 2 (±𝜏 ± 𝑒1 ± 𝜎𝑒3), 1 2 (±𝜏 ± 𝑒2 ± 𝜎𝑒1), 1 2 (±𝜏 ± 𝑒3 ± 𝜎𝑒2), 1 2 (±𝜎 ± 𝑒1 ± 𝜏𝑒3), 1 2 (±𝜎 ± 𝑒2 ± 𝜏𝑒1), 1 2 (±𝜎 ± 𝑒3 ± 𝜏𝑒2), 1 2 (±1 ± 𝜏𝑒1 ± 𝜎𝑒2), 1 2 (±1 ± 𝜏𝑒2 ± 𝜎𝑒3), 1 2 (±1 ± 𝜏𝑒3 ± 𝜎𝑒1), 1 2 (±𝜎𝑒1 ± 𝜏𝑒2 ± 𝑒3), 1 2 (±𝜎𝑒2 ± 𝜏𝑒3 ± 𝑒1), 1 2 (±𝜎𝑒3 ± 𝜏𝑒1 ± 𝑒2).} . (13) Table 2 summarizes the action of the group 3 2 4 S C DW : )(       on  given in (11) for certain x values. Table 2. Action of the group  4 3 2 ( ) : W D S C on  . x Vertices Polytope generated #vertices #cells Type of cell 0 and 2 1 1  ,x 𝑇′ in (3) 24-cell 24 24 24 octahedra 𝑥 = 𝜎 S in (13) Snub 24-cell 96 144 24 icosahedra, 24 tetrahedra, 96 tetrahedra 𝑥 = 𝜏 Mirror image of S Snub 24-cell 96 144 24 icosahedra, 24 tetrahedra, 96 tetrahedra Any other 𝑥 S (x) in (12) Pseudo snub 24-cell 96 144 24 pseudoicosahedra, 24 tetrahedra, 96 triangular pyramid 3.2 Determination of facets of pseudo 24-cell It is known that every vertex of the snub 24-cell is surrounded by three icosahedra and five tetrahedra. We shall prove that the facets of the pseudo snub 24-cell consist of pseudo icosahedra, tetrahedra and triangular pyramids. In the pseudo snub 24-cell three pseudoicosahedra, one tetrahedron and four triangular pyramids meet at the same vertex. Now we discuss the details of this structure. It is evident from the 𝐷4 diagram that each of the following sets of rotation generators (𝑟1𝑟2,𝑟2𝑟3),(𝑟3𝑟2,𝑟2𝑟4) and (𝑟4𝑟2,𝑟2𝑟1) generate a proper subgroup of the tetrahedral group of order 12 as shown in Figure 4. NAZIFE O. KOCA ET AL 156 1 2 3 Figure 4. Three proper subgroups of the tetrahedral group from the 𝑫𝟒 diagram. Let us discuss how one of these groups acts on the vertex  . Let us take the group ],[, 443221  TTrrrr  (case 1 in Figure 4) which implies that the group consists of 12 elements leaving the weight vector T 4  invariant. The group generators transform quaternionic units as follows: 1 2 1 2 3 3 2 3 1 2 3 1 :1 1, :1 1, . r r e e e e r r e e e e         (14) The 12 vertices generated from 1 2 [(1 2 ) (1 ) ]x x e xe      (here we dropped the overall scale factor) the group  3221 rrrr , are determined as: 1 2 1 2 1 2 2 3 2 3 2 3 1 3 1 3 1 3 1 2 3 (1 2 ) (1 ) , (1 2 ) (1 ) , (1 ) (1 2 ) , (1 2 ) (1 ) , (1 2 ) (1 ) , (1 ) (1 2 ) , (1 2 ) (1 ) , (1 ) (1 2 ) , (1 ) (1 2 ) , (1 2 ) (1 ) , x x e xe x x e x e x xe x e x x e xe x x e x e x xe x e x xe x e x x e x e x x e xe x e xe x e x                                         1 2 3 1 2 3 (1 ) (1 2 ) , (1 ) (1 2 ) .e x e x e x e x e xe        (15) Since the vector 4  is left invariant by the generators in (14) the center of the polyhedron of (15) can be taken as 4  up to a scale factor. One may check that the set in (15) is also left invariant under the group element * ],[ 44  . Therefore, the set (15) is invariant under the larger group * ],[],[ 4444  TTTTT h  of order 24 isomorphic to the pyritohedral group. If we define a new set of unit quaternions 03302201140 peppeppepp  ,,, and express the vectors in (15) in terms of the new set of quaternions, then the set of vertices in (15) represent a pseudoicosahedron as shown in Figure 5(a). When the set of generators  4223 rrrr , (case 2 in Figure 4) and  1224 rrrr , (case 3 in Figure 4) respectively are applied to the vector  , one generates two more pseudo icosahedra with the centers represented by 1  and 3  respectively. The groups generating the vertices of the second and the third pseudo icosahedra can be written respectively as TTTTT  11111  ,],[],[ * and TTTTT  33333  ,],[],[ * .These groups are isomorphic to the pyritohedral group ]},[],{[ TTTTTh   . This is obtained if 1 1  is substituted above in the basis vectors 1 2 3 , and e e e in 3D. Note that [1,−1] ∈ [𝑇,𝑇] is also an element of the group [𝑇,𝑇] which commutes with all elements and sends a quaternion to its negative 𝑞 → −𝑞. With the generator [1,−1] one obtains a larger group TttTtTtTtTCTh  2 ,],[],[ * of order 48 leaving the vector ±𝑡 invariant so that the group can be embedded in the group 3 2 4 S C DW : )(       12 different ways. The group [𝑇,𝑇] ∪ [𝑇,𝑇]∗ also leaves the set of quaternions 𝑇′ invariant. The largest subgroup which leaves t T   invariant can be written as TttTtTtTtTCT d  2 ,],[],[ * . Now we continue to discuss the polyhedral facets having  as a vertex. Let us consider the following five sets of rotational generators obtained from the generators of the Coxeter-Weyl group 𝑊(𝐷4): 4D PYRITOHEDRAL SYMMETRY 157 1 3 3 4 4 1 ( , , )r r r r r r , 2 1 2 3 2 4 ( , , )r r r r r r , 1 2 1 3 1 4( , , )r r r r r r , 3 1 3 2 3 4( , , )r r r r r r , 4 1 4 2 4 3( , , )r r r r r r . (16) The first two sets of generators are invariant under the conjugation of the permutation group 𝑆3 but the next three sets of generators are permuted among each other. Let us determine the vertices of the polyhedra under the action of five sets. 1. The set of vertices 1 2 1 3 1 3 3 4 1 2 4 1 1 3 (1 2 ) (1 ) , (1 ) (1 2 ) , (1 2 ) (1 ) , (1 ) (1 2 ) , x x e xe r r x x e xe r r x x e xe r r x x e xe                         (17) represents a tetrahedron of edge length 2x as shown in Figure 5(b). Its center can be represented by 𝑝(1) = 𝜔2 = 1 + 𝑒1up to some scale factor. Since the Dynkin diagram symmetry 𝑆3 also leaves 𝜔2 = 1 + 𝑒1 invariant, the group 𝐶2×𝐶2 generated by the generators (𝑟1𝑟3,𝑟3𝑟4,𝑟4𝑟1) can be extended by the 𝑆3 symmetry to a group * ],[],[ 2222  TTTT  of order 24 isomorphic to the tetrahedral group [10]. Since 3S , the group TTTTTT d  22222  ,],[],[ * leaves the vertices of the tetrahedron in (17) invariant (Figure 5b). Note that this not a pyritohedral symmetry. This tells us that the number of tetrahedra generated by the conjugate tetrahedral groups is also 24. Extension of the group by the generator [1,−1] leads to the group TTTTT  22222  ,],[],[ * that is isomorphic to the octahedral group [10]. 2. The set of vertices .)()( ,)()( ,)()( ,)()( 3242 3232 2112 21 121 121 211 121 exexxrr exexxrr exxexrr xeexx     (18) determines a triangular pyramid with a base of equilateral triangle with sides 2x and the other edges of length √2(𝑥2 + 𝑥 + 1). The triangular pyramid is depicted in Figure 5 (c). The hyperplane determined by these four vertices in (18) is orthogonal to the vector 21 122 exexp )()()(  . The group generated by these generators extended by the group 𝑆3 is the full group of symmetry 3 2 4 S C DW : )(       . 3. The set of vertices .)()( ,)()( ,)()( ,)()( 3141 3131 2121 21 211 211 1 21 121 exexxrr exexxrr exexxrr xeexx     (19) defines another triangular pyramid with the same edge lengths as above. The vector orthogonal to the hyperplane determined by the vertices in (19) is 21 213 eexxp  )()()( . 4. The set of vertices .)()( ,)()( ,)()( ,)()( 2143 3123 3113 21 121 121 211 121 exexxrr exxexrr xeexxrr xeexx     (20) also determines a triangular pyramid as above. The vector which is orthogonal to the hyperplane determined by the vectors of (20) can be computed as 31 124 eexxp  )()()( . 5. The set of vertices NAZIFE O. KOCA ET AL 158 .)()( ,)()( ,)()( ,)()( 2134 3124 3114 21 121 121 211 121 exexxrr exxexrr xeexxrr xeexx     (21) defines another triangular pyramid with the same edge lengths as above. The vector orthogonal to the hyperplane of the vectors in (21) is 31 125 eexxp  )()()( . (a) (b) (c ) Figure 5. Facets of pseudo snub 24-cell: (a) Pseudoicosahedron for 𝒙 = 𝝉, (b) Tetrahedron with edges for 1x  , (c) Triangular pyramid. Since the group is the full symmetry group, all vectors 𝑝(𝑖), 𝑖 = 2,3,4,5 lie in the same orbit of the group 3 2 4 S C DW : )(       . Below we list these five vectors up to a some scale factors representing the centers of the above polyhedra expressed in terms of the weight vectors 4 3 2 1 ,,,, i i  : .)()( )()( ,)()( ,))(()( ,)( 2431 2431 2431 2431 2 15 14 13 12 1      xp xp xp xxp p      (22) Now their symmetries are more transparent under the permutation group: 𝑆3 permutes 𝜔1,𝜔3 𝑎𝑛𝑑 𝜔4 but leaves 𝜔2invariant; )(1p and )(2p are invariant under the group 𝑆3 but the others are permuted to each other. Pseudo snub 24-cell consists of 𝑁0 = 96 vertices, 𝑁3 = 24 + 24 + 96 = 144 cells consisting of pseudo icosahedra, tetrahedra and triangular pyramids respectively. It has 𝑁1 = 432 edges and 𝑁2 = 480 faces. These numbers satisfy the Euler characteristic formula 𝑁0 − 𝑁1+𝑁2 − 𝑁3 = 0. Table 3 summarizes the facets of pseudo snub 24-cell. Table 3. Facets of pseudo snub 24-cell. Group generators Group which generates the vertices Vertices Center of polyhedron Polyhedron (facet) generated < 𝑟1𝑟2,𝑟2𝑟3 > *],[],[ 4444  TTTT  in (15) 𝜔4 pseudoicosahedron < 𝑟3𝑟2,𝑟2𝑟4 > * ],[],[ 1111  TTTT  𝜔1 pseudoicosahedron < 𝑟4𝑟2,𝑟2𝑟1 > * ],[],[ 3333  TTTT  𝜔3 pseudoicosahedron 1 3 3 4 4 1 ( , , )r r r r r r *],[],[ 2222  TTTT   in (17) 𝑃(1) = 𝜔2 tetrahedron 4D PYRITOHEDRAL SYMMETRY 159 2 1 2 3 2 4 ( , , )r r r r r r 3 2 4 S C DW : )(       in (18) 𝑃(2) = (1 + 𝑥)(𝜔1 + 𝜔3 + 𝜔4) − 𝑥𝜔2 triangular pyramid 1 2 1 3 1 4 ( , , )r r r r r r in (19) 𝑃(3) = −𝜔1 + 𝜔3 + 𝜔4 + (1 + 𝑥)𝜔2 3 1 3 2 3 4 ( , , )r r r r r r , in (20) 𝑃(4) = 𝜔1 − 𝜔3 + 𝜔4 + (1 + 𝑥)𝜔2 4 1 4 2 4 3 ( , , )r r r r r r in (21) 𝑃(5) = 𝜔1 + 𝜔3 − 𝜔4 + (1 + 𝑥)𝜔2 Table 3. Contd. 3.3 Construction of the vertices of the dual polytope of the pseudo snub-24 cell To construct the dual of the pseudo snub 24-cell we need to determine the centers (orthogonal vectors to the hyperplane) of the pseudoicosahedra, the tetrahedron and the four pyramids up to some scale factors. The centers of the first three pseudo icosahedra can be taken as the weight vectors 𝜔1,𝜔3 𝑎𝑛𝑑 𝜔4 as shown in Table 3. The other vectors in (22) can be taken as the center (orthogonal vector to the hyperplane) of the tetrahedron and the centers of the four pyramids up to some scale vectors. Let us denote by 𝑐(𝑖), 𝑖 = 1,2,…,5, the centers of the respective tetrahedron and the pyramids and define (23) So we have eight vertices including 𝜔1,𝜔3 𝑎𝑛𝑑 𝜔4 . To determine the actual centers of these polyhedra the hyperplane defined by the eight vectors must be orthogonal to the vector  .This will determine the scale factors and the centers of the cells which can be written as: 1 3 1 2 3 4 1 2 3 1 2 1 1 1, (1 ), (1 ), 2 2 1 2 (1) (1 ), 2 3 1 2 ( ) ( ), 2, 3, 4, 5. 3 7 3 e e e e e e x c e x x c i p i i x x                     (24) These eight vectors now determine the vertices of one facet of the dual polytope of the pseudo snub 24-cell. The center of this facet is the vector  . This is a convex solid with 8 vertices 15 edges and 9 faces possessing 𝑆3 symmetry. One can generate the vertices of the dual polytope by applying the group  4 3 2 ( ) : W D S C on the vertices (24) representing one of the facets of the dual polytope. One can display them as the union of three sets 2 1 2 1 2 ( ) 2 ( ) ( ) 2 3 3 7 3 x x T T R x x x x       (25) where TT  and are given in (2-3) and 𝑅(𝑥) can be written as follows 1 2 1 2 1 2 2 3 2 3 2 3 3 1 3 1 3 1 1 2 3 1 ( ) { (2 ) (1 ) , (1 ) (2 ) , 1 (1 ) (2 ) , (2 ) (1 ) , (1 ) (2 ) , 1 (1 ) (2 ) , (2 ) (1 ) , (1 ) (2 ) , 1 (1 ) (2 ) , (2 ) (1 ) , (1 ) R x x e x e x x e e x e x e x e x e x x e e x e x e x e x e x x e e x e x e x e e x e e x                                                       2 3 1 2 3 (2 ) , (1 ) (2 ) }.e x e x e x e e       (26) (1) (1), ( ) ( ), 2, 3, 4, 5c p c i p i i    NAZIFE O. KOCA ET AL 160 As we substitute 𝑥 = − 1 2 in (25) what we obtain is set T, as expected because it is the dual of the set T  . If 𝑥 = − 2 3 , the dual polytope does not exist. The roots of the quadratic equation 2 3 7 3x x  =0 are irrational numbers which are already excluded since the vertices of the pseudo snub 24-cell and its dual must remain in the 𝐷4 lattice for x to be rational number. One wonders whether the vertices in (26) represent any familiar polytope. If we replace x by 𝑥 = − 1 𝑦+1 , the set 𝑅(𝑥 → − 1 𝑦+1 ), in (26) takes exactly the same form of 𝑆(𝑦) in (13) apart from a scale factor. This implies that 𝑅(𝑥) represents another pseudo snub 24-cell. This proves that every dual of a pseudo snub 24-cell includes another pseudo snub 24-cell in addition to the two sets of 24-cells T and T  . The pseudo snub 24-cell 𝑆(𝑥) turns out to be snub 24-cell whose cells are regular icosahedra and tetrahedra when or x x   . However, ( ) and ( )R R  represent two pseudo snub 24-cells, the mirror images of each other. In brief, when ( )S x represents a snub 24-cell in its dual, there exists an orbit with 96 vertices representing another pseudo snub 24-cell, albeit the coefficients of the unit quaternions are irrational numbers [8]. Table 4 summarizes the way to obtain the dual of the snub 24-cell in terms of its vertices. Table 4. Dual polytope of the pseudo snub 24-cell. One facet (polyhedron) of pseudo snub 24-cell vertices of one facet of the dual of the pseudo snub 24-cell = Centers of polyhedron Apply  4 3 2 ( ) : W D S C on the vertices The dual of the pseudo snub 24-cell Pseudoicosahedron (1) 𝝎𝟒 Elements belong to T 24-cells (24 vertices) Pseudoicosahedron (2) 𝝎𝟏 Pseudoicosahedron (3) 𝝎𝟑 Tetrahedron (1) (1)c p Elements belong to ( 𝟏 + 𝟐𝒙 𝟐 + 𝟑𝒙 )√𝟐𝑻′ 24-cells (24 vertices) Triangular pyramid (1) (2) (2)c p Elements belong to ( )R x Pseudo snub 24-cell (96 vertices) Triangular pyramid (2) (3) (3)c p Triangular pyramid (3) (4) (4)c p Triangular pyramid (4) (5) (5)c p 4. Conclusion We have studied the extension of the pyritohedral group  3 2 2 ( ) : W D C C  in 3D to the group  4 3 2 ( ) : W D S C acting in 4D Euclidean space. We have constructed the 4D polytope with 96 vertices (pseudo snub 24-cell) with the facets as 24 pseudo icosahedron, 24 tetrahedron and 96 triangular pyramids. We also derived its dual polytope with 144 vertices forming three orbits under the group. The explicit construction of the group in terms of quaternions has been worked out. The relevance of the group and the polytopes to the root lattice of the affine Coxeter group 𝑊𝑎(𝐷4) has been pointed out. We have pointed out that the dual of the pseudo snub 24-cell is the union of the sets 𝑇 𝑎𝑛𝑑 𝑇′ representing two 24-cells dual to each other and the set 𝑆(𝑥) with 96 vertices. For rational values of x the vertices belong to the lattice 𝐷4. 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Physics A: Mathematical and General, 2001, 34(5), 11201-11213. 13. Humphreys, J.E. Reflection groups and Coxeter groups, Cambridge University Press, 1990. 14. Conway, J.H. and Sloene, N.J.A. Sphere packings, lattices and groups. Springer Science and Business Media, 2013, 290. Received 6 June 2016 Accepted 22 September 2016 http://arxiv.org/abs/1204.4567v1