Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions

Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions Mehmet Koca a), Nazife Ozdes Koca b) and Muna Al-Shu eili c) Department of Physics, College of Science, Sultan Qaboos University P.O. Box 6, Al-Khoud, Muscat, Sultanate of Oman Abstract There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. n this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups W( AAA), W( A), W( B), and W(H ) to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral groupw( B) C so they are not classified in the class of chiral polyhedra. t is noted that vertices of the snub cube and snub dodecahedron can be derived from the vectors, which are linear combinations of the simple roots, by the actions of the proper rotation groupsw( B) C andw( H) C respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by product we obtain the pyritohedral group as the subgroup the Coxeter group W( H ) and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions. a) electronic-mail: kocam@squ.edu.om b) electronic-mail: nazife@squ.edu.om c) electronic-mail: m054946@squ.edu.om

ntroduction t seems that the Coxeter groups and their orbits [] derived from the Coxeter diagrams describe the molecular structures [], viral symmetries [], crystallographic and quasi crystallographic materials [4]. Chirality is a very interesting topic in molecular chemistry and physics. A number of molecules display one type of chirality; they are either leftoriented or right-oriented molecules. n fundamental physics chirality plays very important role. For example a massless Dirac particle has to be either in the left handed state or in the right handed state. No Lorentz transformation exist transforming one state to the other state. The weak interactions which is described by the standard model of high energy physics is invariant under one type of chiral transformations. n three dimensional Euclidean space, which will be the topic of this paper, the chirality is defined as follows: the object which can not be transformed to its mirror image by other than the proper rotations and translations are called chiral objects. For this reason the chiral objects lack the plane and/or central inversion symmetry. n two previous papers we have constructed the vertices of the Platonic Archimedean solids [5] and the dual solids of the Archimedean solids, the Catalan solids [6], using the quaternionic representations of the rank- Coxeter groups. Two of the Archimedean solids, the snub cube and snub dodecahedron are the chiral polyhedra whose symmetries are the proper rotational subgroups of the octahedral group and the icosahedral group respectively. n this paper we use a similar technique of references [5-6] to construct the vertices of the chiral Archimedean solids, snub cube, snub dodecahedron and their duals. They have been constructed by employing several techniques [7-8] but it seems that the method in what follows has not been studied earlier in this context. We follow a systematic method for the construction of the chiral polyhedra. First we begin with the Coxeter diagrams A A A and A which lead to the tetrahedron and icosahedron respectively and prove that they possess larger proper rotational symmetries which transform them to their mirror images so that they are not chiral solids. We organize the paper as follows. n Sec. we construct the Coxeter groupsw( A A A ), W( A ), W( B ), and W( H ) in terms of quaternions. n Sec. we obtain the proper rotation subgroup of the Coxeter groupw( AAA), and determine the vertices of the tetrahedron by imposing some conditions on the general vector expressed in terms of simple roots of the diagram AA A. We prove that the tetrahedron can be transformed to its mirror image by the proper octahedral rotation groupw( B ) C. n Sec.4 we discuss similar problem for the Coxeter-Dynkin diagram A leading to an icosahedron and again prove that it can be transformed by the groupw( B) C to its mirror image which indicates that neither tetrahedron nor icosahedron are chiral solids. Here we also discuss the properties of the pyritohedral group and the constructions of the pyritohedrons. The Sec.5 deals with the construction of the snub Cube and its dual pentagonal icositetrahedron from the proper rotational octahedral symmetryw( B ) C using the same technique employed in Sec. and Sec.4. n Sec.6 we repeat a similar work for the constructions of the snub dodecahedron and its dual pentagonal hexacontahedron from the proper icosahedral group W( H) C A which is isomorphic to the group of even permutations of five 5

letters. n the concluding Sec.7 we point out that our technique can be extended to determine the chiral polyhedra in higher dimensions. Construction of the groups W( A A A), W( A), W( B), and W(H ) in terms of quaternions. Let q q q e 0 i i, ( i,, ) be a real unit quaternion with its conjugate defined by q q0 qe i iand the norm qq qq.the quaternionic imaginary units satisfy the relations ee i j ij ijkek, ( i, j, k,, ) () where ij and ijk are the Kronecker and Levi-Civita symbols and summation over the repeated indices is implicit. The unit quaternions form a group isomorphic to the unitary group SU(). With the definition of the scalar product ( p, q) ( pqqp) ( pq qp), () quaternions generate the four-dimensional Euclidean space. The Coxeter diagram A A A can be represented by its quaternionic roots in Fig. with the normalization. e e e Figure. The Coxeter diagram A A A with quaternionic simple roots. The Cartan matrix and its inverse are given as follows 0 0 C 0 0, 0 0 0 0 C 0 0. () 0 0 For any Coxeter diagram, the simple roots i and their dual vectors i satisfy the scalar product [9] ( i, j ) Cij, ( i, j ) ( C ) ij, ( i, j ) ij ; i, j,,. (4) We note also that they can be expressed in terms of each other: i Cijj, i ( C ) ij j. (5)

Let be an arbitrary quaternionic simple root. Then the reflection of an arbitrary vector with respect to the plane orthogonal to the simple root is given by [0] r [, ]. (6) Our notations for the rotary reflections and the proper rotations will be[ p, q] and [ p, q ] respectively where p and q are arbitrary quaternions. The Coxeter group W( A A A) is generated by three commutative group elements r [ e, e ], r [ e, e ], r [ e, e ]. (7) They generate an elementary Abelian group W( A AA) CCCof order 8. ts proper rotation subgroup elements are given by [,], rr [ e, e ], rr [ e, e ], rr [ e, e ]. (8) The next Coxeter group which will be used is the tetrahedral groupw( A ) Td diagram with its quaternionic roots is shown in Fig.. A S 4. ts e e e e e e Figure. The Coxeter diagram with quaternionic simple roots. A The Cartan matrix of the Coxeter diagram and its inverse matrix are given respectively by the matrices 0 C, C 4 4. (9) 0 The generators of the Coxeter group W( A ) are given by A r [ ( ee), ( e e )], r [ ( ee), ( e e )], r [ ( e e), ( e e )] (0) 4

The group elements of the Coxeter group which is isomorphic to the tetrahedral group of order 4 can be written compactly by the set [] W( A ) {[ p, p] [ t, t ] }, p T, t T. () Here T and T represent respectively the binary tetrahedral group of order 4 and the coset representative T O T where O is the binary octahedral group of quaternions of order 48 []. The Coxeter diagram B leading to the octahedral group W( B ) Oh is shown in Fig.. e e e e e 4 Figure. The Coxeter diagram B with quaternionic simple roots. The Cartan matrix of the Coxeter diagram B and its inverse matrix are given by 0 C, C. () 0 The generators, r [ ( ee), ( e e)], r [ ( e e ), ( e e )], r [ e, e ] generate the octahedral group which can be written as () W( B ) Aut( A ) S C {[ p, p] [ p, p] [ t, t ] [ t, t ] }, p Tt, T. (4) 4 A shorthand notation could bewb ( ) {[ TT, ] [ TT, ] [ T, T] [ T, T] }. Note that we have three maximal subgroups of the octahedral group W( B ), namely, the tetrahedral group W( A ), the chiral octahedral group consisting of the elementsw( B) C {[ T, T] [ T, T]}, and the pyritohedral group consisting of the elementsth A4C {[ T, T] [ T, T] }. The pyritohedral symmetry represents the symmetry of the pyritohedrons, an irregular dodecahedron, with irregular pentagonal faces which occurs in iron pyrites. The Coxeter diagram H leading to the icosahedral group is shown in Fig. 4. 5

e Figure 4. The Coxeter diagram of The Cartan matrix of the diagram 5 e H e e e with quaternionic simple roots.(t is assumed that the simple roots are multiplied by ) and its inverse are given as follows: H 0 C 0, C 4. (5) The generators, r [ e, e)], r [ ( e e e ), ( e e e )], r [ e, e] (6) generate the icosahedral group W( H ) {[ p, p] [ p, p] } A C, ( p, p ), (7) h 5 5 5 or shortly, W( H) {[, ] [, ] }. Here, and is the set of 0 quaternionic elements of the binary icosahedral group [0]. The chiral icosahedral group is represented by the set W( H)/ C A5 {[, ]} which is isomorphic to the even permutations of five letters. Note also that the pyritohedral group is a maximal subgroup of the Coxeter groupw( H ). All finite subgroups of the groups O() an d O(4) in terms of quaternions can be found in reference []. A general vector in the dual space is represented by the vector aa a ( aa a). We will use the notation O( ) W( G) O( aa a) for the orbit of the Coxeter groupw( G ) generated from the vector where the letter G represents the Coxeter diagram. We follow the Dynkin notation to represent an arbitrary vector ( aaa ) in the dual space and drop the basis vectors i, i=,,. n the Lie algebraic representation theory the components (aaa ) of the vector are called the Dynkin indices [] which are non-negative integers if it represents the highest weight vector. Here we are not restricted to the integer values of the Dynkin indices. They can be any real number. When the components of the vector in the dual space are non integers 6

values we will separate them by commas otherwise no commas will be used. For an arbitrary Coxeter diagram of rank we define the fundamental orbits as O( ) O(00), O( ) O(00), and O( ) O(00). (8) Any linear combination of the basis vectors i over the real numbers will, in general, lead to quasi regular polyhedra under the action of the Coxeter group. n the next four sections we discuss a systematic construction of chiral polyhedra and their dual solids. n our construction tetrahedron and icosahedron will also occur but we prove that they are not chiral polyhedra. The orbit O( ) ( C C )( aa a ) as tetrahedron The proper rotation subgroup of the Coxeter group W( A ) A A applies on an arbitrary vector ( ae ae ) ae as follows: rr ( ae ae ae ), rr ( ae ae ae ), rr ( ae ae ae ). To obtain a tetrahedron from these four vertices the Dynkin indices should satisfy the relations a a a a, ai a, i,,. (9) We take a and start with a vector ( ee e)then the orbit O( ) will be given by O( ) { ( ee e), ( ee e), ( ee e), ( ee e)}. (0) The tetrahedron with these vertices is shown in Fig.5. Figure 5. The tetrahedron with the vertices given in equation (0). 7

These are the vertices of a tetrahedron invariant under the rotation group given in (8). Of course the full symmetry of the tetrahedron is a group of order 4 isomorphic to the permutation group S4 generated by reflections of the Coxeter-Dynkin diagram A [6]. Now the mirror image of the tetrahedron of (0) can be determined applying the same group of elements in (8) on the vector r ( ee e ). Then the second orbit which is the mirror image of the set in (0) is determined to be O( ) { ( ee e), ( ee e), ( ee e), ( ee e)}. () Of course we know that the union of two orbits in (0) and () determines the vertices of a cube. The point here is that if we were restricted to the group CC of (8) then the tetrahedron in (0) would be a chiral solid. However this is not true because there exist additional rotational symmetries which exchange these two orbits of (0) and () proving that the tetrahedron is not a chiral solid. Now we discuss these additional symmetries. t is obvious that the Coxeter diagram in Fig. has an additional S symmetry which permutes three A diagrams. ndeed this symmetry extends the group CC to the proper octahedral rotation group as will be explained now. One of e e e e the generators of this symmetry d [, ] is a -fold rotation leading to the transformation e e, e e, e e. t is straightforward to see that do( ) O( ).This proves that by a proper rotation tetrahedron can be transformed to its mirror image therefore it is not a chiral solid. The generator d and those elements in (8) enlarge the symmetry to a group of order 8 which can be concisely written as the set of elements {[ V0, V 0] [ V, V]} [] where the sets V and 0 V are defined by e e e V0 {, e, e, e }, V {, }. () A cyclic subgroup C of the symmetric group S permutes three sets like those in () extending the group CC of order 4 to a group of order 4. Actually the larger group obtained by this extension is the chiral octahedral group of order 4 which can be symbolically written as {[ TT, ] [ T, T]} S4 WB ( ) C. () This is the proper rotational symmetry of the octahedron whose vertices are represented by the set of quaternions ( e, e, e) and the cube whose vertices are the union of the orbits O( ) O( ). 4 The icosahedron derived from the orbit O( ) ( W( A) C)( aa a) 8

The proper rotational subgroup of the Coxeter group W( A) S 4 is the tetrahedral group, the even permutations of the four letters, of order. They can be generated by A 4 the generators arr and b rr which satisfy the generation relation a b ( ab). Let (aaa ) be a general vector. The following sets of vertices form equilateral triangles (, rr, ( rr ) ), (, rr, ( rr ) ) (4) with the respective square of edge lengths ( a aa a ) and ( a a a a ). We have another vertex rrrras shown in Fig.6. r r r r r r 4 5 r r r r Figure 6. The vertices connected to the general vertex. Then one can obtain three more triangles by joining rr to the vertices rr and rr and by drawing a line between rr and rr. f we require that all these five triangles be equal each other then we obtain the relations ( a aa a ) ( a a a a ) ( a a ). (5) Factoring by and defining a a x and y one obtains x y and a cubic equation a a a x x x 0. Assuming x 0 we get the solutions x and x.this leads to two vectors ) a( + ) a nd a( +. Here a is an overall scale factor which can be adjusted accordingly. The x 0 solution represents an octahedron which is not a chiral solid anyway. Let us study the orbit which is obtained from the vector. When expressed in terms of quaternions it will read a ( e e).we choose the scale factor a for convenience. To obtain the orbit O( ) ( W( A) C) we use the generators of the tetrahedral group of interest in terms of quaternions, namely, 9

c[ ( ee e), ( ee e)], d [ e, e]. (6) They act on the quaternionic units as follows: c: e e e e; d: e e, e e, e e. (7) Applying the generators c and d several times on the vector ( e e ) we obtain the set of vectors O( ) { ( ee), ( e e), ( e e)} (8) which constitute the vertices of the icosahedron shown in Fig. 7. Figure 7. The icosahedron obtained from tetrahedral symmetry A 4. Similarly if we use the solution x we will get the vector ( e e ) after a suitable choice of the factor a. Acting the generators in (7) repeatedly on the vector we get the orbit O( ) { ( ee), ( e e), ( e e) }. (9) This is another icosahedron which is the mirror image of the icosahedron of (8). ndeed one can show that ro ( ) O( r) O( ). Before we proceed further we note the fact that the Coxeter-Dynkin diagram A has the diagram symmetry and, in other words, a nd. t is clear that this symmetry does not alter the orbits since under the diagram symmetry and remain intact. The diagram symmetry acts on the quaternions as e e, e e and e e. This transformation can be obtained by the action of the element [ e, e ] which is not an element of the groupw( A ) S4. The proper rotation group A4 [ T, T] can then be extended by the generator [ e, e ] to the 0

pyritohedral groupt A4C {[ T, T] [ T, T] }. Actually in the paper [6] we have h shown that the orbit O( ) is invariant under a larger group h of (7) which admits the pyritohedral group as a maximal subgroup. t is straightforward to see that the element e e d [, ] exchanges the two icosahedral orbits; d : O () O( ). One can see 0 that the element d represents a rotation around the first axis by 90 and extends the group A4 [ T, T] to the proper octahedral rotation group S 4 A 4 : C {[ T, T] T, T]}. This proves that two mirror images of the icosahedron are transformed to each other by rotations therefore the icosahedron is not a chiral solid rather it is achiral. When two orbits of (8) and (9) are combined one obtains a quasi regular polyhedron which can be obtained as the orbit of the group W( B )(,,0)[4]. The quasi regular polyhedron represented by the combined vertices of (8-9) is shown in Fig. 8.t consists of two types of faces, squares of side and isogonal hexagons of sides and. Figure 8. The quasi regular polyhedron represented by the vertices of (8-9). Although we know that the dual of an icosahedron is a dodecahedron [6] here we show how the vertices of the dodecahedron can be obtained from the vertices of the icosahedron, say, from the vertices of O( ) given in (8). We have to determine the centers of the planes in Fig.6. We can choose the vector as the vector representing the center of the face # because it is invariant under the rotation represented by rr. n other words the triangle # is rotated to itself by a rotation around the vector. With the same reason the center of the face # can be taken as the vector. We note that the line joining these vectors is orthogonal to the vector, namely, ( ). 0. The centers of the faces #, #4 and #5 can be determined by averaging the vertices representing these faces: b ( rr rr ), b4 ( rr rr ), b5 ( rr rr ). (0) Since we have the following relations among these three vectors rrb b4, rrb 4 b5 and rrb b 5 they are in the same orbit under the group action A4( b) A4( b4) A4( b5). Therefore it is sufficient to work with one of these vectors, say, with. n terms of the b

quaternionic units it readsb ( e 6 e ). A quick check shows that b is not orthogonal to the vector rather ( b ). 0 provided. Then we obtain three orbits A4( b) { ( ee ), ( e e), ( e e )} () A4( ) { ( ee e), ( ee e), ( ee e), ( e e e )} A e e e e e e e e e 4( ) { ( e e ), ( ), ( ), ( e )}. Note that the last two orbits represent the vertices of two dual tetrahedra, when combined, represent a cube. These 0 vertices which decompose as three orbits under the tetrahedral group represent the vertices of a dodecahedron as shown in the Fig.9 which is also achiral solid. So far we have shown that, although, tetrahedron and icosahedron can be obtained as chiral solids there exists additional proper rotational group elements that convert them to their mirror images. Therefore they are not chiral solids. Figure 9. Dodecahedron represented by the vertices of (). Although our main topic is to study the chiral objects systematically using the Coxeter diagrams, here with a brief digression, we construct the pyritohedron, a non regular dodecahedron, made by irregular pentagons. f we plot the solid represented by the orbit A4( b) in the first line of () we obtain an irregular icosahedron as shown in Fig.0. Figure 0. rregular icosahedron represented by the vertices of the orbit A4( b).

Let us recall that the vector b ( ) differs from the vector by the sign in front of. n Fig. we show the faces joining to the vector b. 4 5 b Figure. Faces of the irregular icosahedron joined to the vertex b. There we see that two of the triangles are equilateral and the rest three are isosceles triangles. We determine the centers of the faces of this irregular icosahedron. The faces # and # can be represented again by the vectors and respectively. The centers of the faces of the #, #4 and #5 can be determined, up to a scale factor, by averaging the vectors representing the vertices of the isosceles triangles. They can be obtained, up to a scale factor, as d ( ) e e, d4 ( ) e e, d5 ( ) e e. () These three vertices determine a plane which can be represented by its normal vector D5 e (7 ) e up to a scale factor. Now, we can determine the scale factor so that ( d ). D 0 determines the five vertices lying in the same plane. We 05 80 obtain. The particular edge represented by the vector d4 d5 e 0 leads to an orbit of size 6 given by { e, e, e}. This shows that Pyritohedral group transforms this type of edges to each other. The vertices of the pyritohedron are given by the set of quaternions: { ( ) e e, ( ) e e, ( ) e e}, O( ) O( ) { ( ee e )} () which leads to the pyritohedron as shown in Fig.. ts symmetry is represented by pyritohedral group A 4 C {[ TT, ] [ TT, ] } of order 4. The 0 vertices of the pyritohedron lie in three orbits 0=+4+4 as shown in ().

Figure. The pyritohedron consisting of irregular pentagonal faces. A variety of pyritohedron can be constructed. f two orbits of the tetrahedron leading to the vertices of a cube determined are chosen to be the set of quaternions ( ee e) then one can build the orbit of size- which depends on a single parameter. ndeed the following sets of quaternions are invariant under the pyritohedral group : T h { ae be, ae be, ae be } (4) where a and bare arbitrary real parameters. Here now, three vectors ae be, aebe determine a plane whose normal can be represented by the vector ( ab) e ae. The condition that five points represented by the vectors ( ee e), ( ee e) and ae be, ae be are in the same plane determines thatba a. Therefore the set of vertices of a pyritohedron has an arbitrary parameter and includes also dodecahedron and the rhombic dodecahedron [6], a Catalan solid, as members of the family for a and a respectively. The pyritohedron is facetransitive since the normal vectors of the faces form an orbit of size under the pyritohedral group. t is an achiral solid. 5 The snub cube derived from the orbito( ) ( W( B) C)( aa a). The snub cube is an Archimedean chiral solid. ts vertices and its dual solid can be determined employing the same method described in Sec. and Sec.4. The proper rotational subgroup of the Coxeter group W( B) S4C is the octahedral groupw( B) C S 4, isomorphic to the symmetric group of order 4. They can be generated by the generators a rr and b rr which satisfy the generation 4 relation a b ( ab). Let ( aaa ) be a general vector. The following sets of vertices form an equilateral triangle and a square respectively (, rr, ( rr ) ), (, rr, ( rr ), ( rr ) ), (5) 4

with the respective square of edge lengths ( a aa a ) and ( a a a a ). We have another vertex rrrras shown in Fig.. rr r r r r r r 5 4 rr ) ( r r Figure. The vertices connected to the vertex. Similar to the arguments discussed in Sec.4 we obtain four equilateral triangles and one square sharing the vertex (aaa ) (see Fig. ) provided the following equations are satisfied ( a aa a ) ( a a a a ) ( a a ). a a x Factoring by a and defining again x and y one obtains y and the a a cubic equation x x x0. This equation has one real solution x.89. Now the first orbit can be derived from the vector ) a( x + y and its mirror image can be defined as r a( x( - ) + y). n terms of quaternionic units the vectors read ( x ) ( xe +e x e ) a, a( x ) ( exe x e ). (6) Deleting the overall scale factor in (6) the set of vectors constituting the orbits can be easily determined [6] as O( ) {( xe e x e ), ( xe e x e ), ( xe e x )}, e O( ) {( e xe x e ), ( e xe x e ), ( e xe x )}. (7) e The snub cubes represented by these sets of vertices are depicted in Fig.4. Note that no proper rotational symmetry exists which transforms these two mirror images to each other so that they are truly chiral solids. 5

(a) (b) Figure 4. Two snub cubes (a) O( ) and (b) O( ) (mirror image of each other). One can combine the vertices of these two chiral solids in one solid which is achiral and it is depicted in Fig.5. This quasi regular solid can be obtained from the vector a( x + y) by applying the octahedral group W( B )( ). Figure 5. The quasi regular polyhedron consisting of two chiral orbits O( ) and O( ). The dual of the snub cube can be determined by determining the centers of the faces as shown in Fig.. Similar arguments discussed in Sec. 4 can be used to determine the centers of the faces in Fig.. The faces # and # are represented by the vectors and respectively. The vectors representing the centers of the faces #, #4 and #5 can be determined and they lie in the same orbit under the proper octahedral group. The vector representing the center of the face # can be given, up to a scale factor, in terms of the quaternionic units as c (x) e +e x e. (8) x The scale factors multiplying the vectors, and c can be determined as x and when represents the normal of the plane containing these five points. Then 8 vertices of the dual solid of the snub cube, the pentagonal icositetrahedron, are given in three orbits as follows 6

x O( ) { e, e, e} O( ) ( ee e) x O( c) {[ (x) ee x e],[ (x) e e x e],[ (x) e ex e]} (9) The pentagonal icositetrahedron is shown in Fig.6. Figure 6. The pentagonal icositetrahedron, a Catalan solid, dual of the snub cube. 6 The snub dodecahedron derived from the orbit O( ) ( W( H) C)( aa a) The snub dodecahedron is the second Archimedean chiral solid. ts vertices and its dual solid can be determined employing the same method described in section 5. The proper rotational subgroup of the Coxeter group W( H) A5 C is the icosahedral groupw( H) C A 5, which is the simple finite group of order 60 representing the even permutations of five letters. They can be generated by the generators arr and b rr 5 which satisfy the generation relation a b ( ab). Let ( aaa ) be a general vector. The following sets of vertices form a pentagon and an equilateral triangle respectively 4 (, rr, ( rr ), ( rr ), ( rr ) ), (, rr, ( rr ) ), (40) with the respective square of edge lengths ( a aa a ) and ( a aaa ). We have another vertex rrrr which is depicted in Fig.7. ) ( r r ) ( r r r r r r 4 r r 5 r r r r Figure 7. The vertices connected to the vertex. 7

The only difference of this from the one in Fig. is that in the present case the face # is an equilateral triangle whose center is represented by the vector and the face # is a regular pentagon whose center is represented by the vector. Assuming that the face # face #, face #4 and face #5 are equilateral triangles which lie in the same orbit of size 60 one obtains the following equations: ( a aa a ) ( a aa a ) ( a a ). (4) a a Factoring by a and defining y and x a a one obtains x y and the x cubic equation is x x x 0. This equation has the real solution x.945. Now the first orbit can be derived from the vector ( + x ) a and its mirror x image can be defined as r a( ( - ) + x). n terms of the x quaternionic units these vectors read a a [ ( x ) exe ( x ) e], [ ( x ) exe ( x ) e]. (4) Two snub dodecahedra obtained using these vectors are shown in Fig.8 (a) and (b). One can combine the vertices of these two chiral solids in one solid which is achiral and it is depicted in Fig.8 (c). This quasi regular solid (quasi regular great rhombicosidodecahedron) can be obtained from the vector a( + x x ) by applying the icosahedral groupw( H)( ). (a) (b) (c ) Figure 8. The snub dodecahedron (a) O( ) and (b) its mirror image O ( ), (c) quasi regular great rhombicosidodecahedron. The vertices of the dual solid (pentagonal hexecontahedron) of the snub dodecahedron represented by O( ) can be given as the union of three orbits of the groupw( H ) C A5. The orbit O( ) consists of 0 vertices of a dodecahedron. The second orbit consists of vertices of an icosahedrono( ) where 8

x x x ( ) x. (4) The third orbit Oc ( ) involves the vertices including the centers of the faces #, #4 and #5 where the vector c is given by c x x {[( ) x x ] e ( 0) x ( 7) x ( x x) e ( x ) e } (44) Applying the group A5 [, ] on the vector c one generates an orbit of size 60. The 9 vertices consisting of these three orbits constitute the dual solid (pentagonal hexecontahedron) of the snub dodecahedron represented by the orbit O( ). The pentagonal hexecontahedron is shown in Fig. 9. Figure 9. The pentagonal hexacontahedron, the dual of the snub dodecahedron. The pentagonal hexacontahedron is one of the face transitive Catalan solid which has 9 vertices, 80 edges and 60 faces. 7 Concluding Remarks n this work we presented a systematic construction of the chiral polyhedra, the snub cube, snub dodecahedron and their duals using proper rotational subgroups of the octahedral group and the icosahedral group. We used the Coxeter diagrams B and Hrespectively. Employing the same technique for the diagrams AAA and A we obtained the vertices of tetrahedron and icosahedrons which are not the chiral solids because they can be transformed to their mirror images by the proper rotational subgroup of the octahedral group. As a by-product we also constructed the orbit of the pyritohedron using the pyritohedral group which is the symmetry of the iron pyrits. This method can be extended to the higher dimensional Coxeter groups to determine the chiral polytopes. For example, the snub 4-cell, a chiral polytope in the 4D Euclidean space can be determined using the Coxeter diagram [5]. D 4 9

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