Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions
|
|
- Bethanie Hicks
- 5 years ago
- Views:
Transcription
1 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
2 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
3 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, 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)
4 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
5 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
6 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 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
7 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
8 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
9 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
10 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
11 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
12 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).
13 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 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 ().
14 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
15 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
16 (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
17 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
18 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
19 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
20 References [] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, Springer Verlag (965). []F.A. Cotton, G. Wilkinson, C.A. Murillo, M. Bochmann, Advanced norganic Chemistry, 6 th Ed., Wiley-nterscience, New York (999). [] D.L.D Caspar and A. Klug, Cold Spring Harbor Symp. Quant. Biol. 7, (96); R. Twarock, Phil. Trans. R. Soc. A64, (006) 57. [4] M. V. Jaric(Ed), ntroduction to the Mathematics of Quasicrystals, Academic Press, New York (989). [5] M. Koca, R. Koc and M. Al-Ajmi, J. Math. Phys. 48, (00) 54. [6] M. Koca, N. O. Koca and R. Koc, J. Math. Phys. 5, (007) [7] P.Huybers and H.S.M.Coxeter, C.R. Math. Reports Acad.Sci. Canada, (979) 59. [8] B. Weissbach and H Martini, Beitrage zur Algebra und Geometrie (Contributions to Algebra and Geometry) 4, (00). [9] R.W.Carter, Simple Groups of Lie Type, John Wiley & Sons Ltd, 97; J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 990. [0] M. Koca, R. Koc, M. Al-Barwani, J. Phys. A: Math. Gen. 4, (00) 0. [] M. Koca, R.Koc, M.Al-Barwani, J. M. Phys. 44, (00) 0; M. Koca, R. Koc, M. Al-Barwani, J. M. Phys. 47, (006) ; M. Koca, R. Koc, M. Al-Barwani and S. Al-Farsi, Linear Alg. Appl. 4, (006) 44. [] J.H.Conway and D.A. Smith, On Quaternion s and Octonions: Their Geometry, Arithmetics, and Symmetry, A.K.Peters, Ltd, Natick, MA (00). [] R. Slansky, Phys. Rep.79 (98). [4] M. Koca, M. Al-Ajmi and S. Al-Shidhani, arxiv: (to be published) [5] M. Koca, N.O. Koca and M. Al-Barwani, arxiv:
BMT 2014 Symmetry Groups of Regular Polyhedra 22 March 2014
Time Limit: 60 mins. Maximum Score: 125 points. Instructions: 1. When a problem asks you to compute or list something, no proof is necessary. However, for all other problems, unless otherwise indicated,
More informationAn eightfold path to E 8
An eightfold path to E 8 Robert A. Wilson First draft 17th November 2008; this version 29th April 2012 Introduction Finite-dimensional real reflection groups were classified by Coxeter [2]. In two dimensions,
More informationThe Geometry of Root Systems. Brian C. Hall
The Geometry of Root Systems A E Z S Brian C. Hall T G R S T G R S 1 1. I Root systems arise in the theory of Lie groups and Lie algebras, but can also be studied as mathematical objects in their own right.
More informationPOLYHEDRON PUZZLES AND GROUPS
POLYHEDRON PUZZLES AND GROUPS JORGE REZENDE. Introduction Consider a polyhedron. For example, a platonic, an archimedean, or a dual of an archimedean polyhedron. Construct flat polygonal plates in the
More informationA Highly Symmetric Four-Dimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey
A Highly Symmetric Four-Dimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey 7974 Abstract A quasiperiodic pattern (or quasicrystal) is constructed in
More informationSome notes on Coxeter groups
Some notes on Coxeter groups Brooks Roberts November 28, 2017 CONTENTS 1 Contents 1 Sources 2 2 Reflections 3 3 The orthogonal group 7 4 Finite subgroups in two dimensions 9 5 Finite subgroups in three
More informationSYMMETRIES IN R 3 NAMITA GUPTA
SYMMETRIES IN R 3 NAMITA GUPTA Abstract. This paper will introduce the concept of symmetries being represented as permutations and will proceed to explain the group structure of such symmetries under composition.
More information2. On integer geometry (22 March 2011)
2. On integer geometry (22 March 2011) 2.1. asic notions and definitions. notion of geometry in general can be interpreted in many different ways. In our course we think of geometry as of a set of objects
More informationE 8. Robert A. Wilson. 17/11/08, QMUL, Pure Mathematics Seminar
E 8 Robert A. Wilson 17/11/08, QMUL, Pure Mathematics Seminar 1 Introduction This is the first talk in a projected series of five, which has two main aims. First, to describe some research I did over the
More informationPrototiles and Tilings from Voronoi and Delone cells of the Root
Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A n Nazife Ozdes Koca a), Abeer Al-Siyabi b) Department of Physics, College of Science, Sultan Qaboos University P.O. Box 36, Al-Khoud,
More informationarxiv:physics/ v1 [physics.chem-ph] 14 Nov 2005
A NEW APPROACH to ANALYSE H (h g) JAHN-TELLER SYSTEM for C 60 Ramazan Koç and Hayriye Tütüncüler Department of Physics, Faculty of Engineering University of Gaziantep, 710 Gaziantep, Turkey arxiv:physics/0111v1
More information5. Atoms and the periodic table of chemical elements. Definition of the geometrical structure of a molecule
Historical introduction The Schrödinger equation for one-particle problems Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical elements
More information1 Fields and vector spaces
1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary
More informationLecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More informationThe Fundamental Group of SO(n) Via Quotients of Braid Groups
The Fundamental Group of SO(n) Via Quotients of Braid Groups Ina Hajdini and Orlin Stoytchev July 1, 016 arxiv:1607.05876v1 [math.ho] 0 Jul 016 Abstract We describe an algebraic proof of the well-known
More informationLINEAR ALGEBRA AND iroup THEORY FOR PHYSICISTS
LINEAR ALGEBRA AND iroup THEORY FOR PHYSICISTS K.N. SRINIVASA RAO Professor of Theoretical Physics (Retd) University of Mysore, Mysore, INDIA JOHN WILEY «SONS NEW YORK CHICHESTER BRISBANE TORONTO SINGAPORE
More informationRepresentation Theory
Frank Porter Ph 129b February 10, 2009 Chapter 3 Representation Theory 3.1 Exercises Solutions to Problems 1. For the Poincare group L, show that any element Λ(M,z) can be written as a product of a pure
More informationGROUP THEORY PRIMER. New terms: so(2n), so(2n+1), symplectic algebra sp(2n)
GROUP THEORY PRIMER New terms: so(2n), so(2n+1), symplectic algebra sp(2n) 1. Some examples of semi-simple Lie algebras In the previous chapter, we developed the idea of understanding semi-simple Lie algebras
More informationIntroduction to Group Theory
Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationL(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that
ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive
More informationCOLOR AND ISOSPIN WAVES FROM TETRAHEDRAL SHUBNIKOV GROUPS
11/2012 COLOR AND ISOSPIN WAVES FROM TETRAHEDRAL SHUBNIKOV GROUPS Bodo Lampe Abstract This note supplements a recent article [1] in which it was pointed out that the observed spectrum of quarks and leptons
More informationShort- course on symmetry and crystallography. Part 1: Point symmetry. Michael Engel Ann Arbor, June 2011
Short- course on symmetry and crystallography Part 1: Point symmetry Michael Engel Ann Arbor, June 2011 Euclidean move Defini&on 1: An Euclidean move T = {A, b} transformabon that leaves space invariant:
More informationOn certain Regular Maps with Automorphism group PSL(2, p) Martin Downs
BULLETIN OF THE GREEK MATHEMATICAL SOCIETY Volume 53, 2007 (59 67) On certain Regular Maps with Automorphism group PSL(2, p) Martin Downs Received 18/04/2007 Accepted 03/10/2007 Abstract Let p be any prime
More informationON THE PUZZLES WITH POLYHEDRA AND NUMBERS
ON THE PUZZLES WITH POLYHEDRA AND NUMBERS JORGE REZENDE. Introduction The Portuguese Mathematical Society (SPM) published, in 00, a set of didactical puzzles called Puzzles com poliedros e números (Puzzles
More informationmsqm 2011/8/14 21:35 page 189 #197
msqm 2011/8/14 21:35 page 189 #197 Bibliography Dirac, P. A. M., The Principles of Quantum Mechanics, 4th Edition, (Oxford University Press, London, 1958). Feynman, R. P. and A. P. Hibbs, Quantum Mechanics
More information7. The classical and exceptional Lie algebras * version 1.4 *
7 The classical and exceptional Lie algebras * version 4 * Matthew Foster November 4, 06 Contents 7 su(n): A n 7 Example: su(4) = A 3 4 7 sp(n): C n 5 73 so(n): D n 9 74 so(n + ): B n 3 75 Classical Lie
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, 2017 5 Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification
More informationHOMEWORK Graduate Abstract Algebra I May 2, 2004
Math 5331 Sec 121 Spring 2004, UT Arlington HOMEWORK Graduate Abstract Algebra I May 2, 2004 The required text is Algebra, by Thomas W. Hungerford, Graduate Texts in Mathematics, Vol 73, Springer. (it
More informationAbstract Algebra Study Sheet
Abstract Algebra Study Sheet This study sheet should serve as a guide to which sections of Artin will be most relevant to the final exam. When you study, you may find it productive to prioritize the definitions,
More informationDefinition: A vector is a directed line segment which represents a displacement from one point P to another point Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH Algebra Section : - Introduction to Vectors. You may have already met the notion of a vector in physics. There you will have
More informationPhysics 251 Solution Set 1 Spring 2017
Physics 5 Solution Set Spring 07. Consider the set R consisting of pairs of real numbers. For (x,y) R, define scalar multiplication by: c(x,y) (cx,cy) for any real number c, and define vector addition
More informationMATH 433 Applied Algebra Lecture 22: Review for Exam 2.
MATH 433 Applied Algebra Lecture 22: Review for Exam 2. Topics for Exam 2 Permutations Cycles, transpositions Cycle decomposition of a permutation Order of a permutation Sign of a permutation Symmetric
More information(Kac Moody) Chevalley groups and Lie algebras with built in structure constants Lecture 1. Lisa Carbone Rutgers University
(Kac Moody) Chevalley groups and Lie algebras with built in structure constants Lecture 1 Lisa Carbone Rutgers University Slides will be posted at: http://sites.math.rutgers.edu/ carbonel/ Video will be
More informationSymmetries, Groups, and Conservation Laws
Chapter Symmetries, Groups, and Conservation Laws The dynamical properties and interactions of a system of particles and fields are derived from the principle of least action, where the action is a 4-dimensional
More informationIntroduction to Modern Quantum Field Theory
Department of Mathematics University of Texas at Arlington Arlington, TX USA Febuary, 2016 Recall Einstein s famous equation, E 2 = (Mc 2 ) 2 + (c p) 2, where c is the speed of light, M is the classical
More informationLittle Orthogonality Theorem (LOT)
Little Orthogonality Theorem (LOT) Take diagonal elements of D matrices in RG * D R D R i j G ij mi N * D R D R N i j G G ij ij RG mi mi ( ) By definition, D j j j R TrD R ( R). Sum GOT over β: * * ( )
More informationContinuous symmetries and conserved currents
Continuous symmetries and conserved currents based on S-22 Consider a set of scalar fields, and a lagrangian density let s make an infinitesimal change: variation of the action: setting we would get equations
More informationThe groups SO(3) and SU(2) and their representations
CHAPTER VI The groups SO(3) and SU() and their representations Two continuous groups of transformations that play an important role in physics are the special orthogonal group of order 3, SO(3), and the
More informationON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS
ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,
More informationFrom Wikipedia, the free encyclopedia
1 of 6 8/28/2011 1:45 PM From Wikipedia, the free encyclopedia Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationHecke Groups, Dessins d Enfants and the Archimedean Solids
arxiv:1309.2326v1 [math.ag] 9 Sep 2013 Hecke Groups Dessins d Enfants and the Archimedean Solids Yang-Hui He 1 and James Read 2 1 Department of Mathematics City University London Northampton Square London
More informationA Brief Introduction to Tensors
A Brief Introduction to Tensors Jay R Walton Fall 2013 1 Preliminaries In general, a tensor is a multilinear transformation defined over an underlying finite dimensional vector space In this brief introduction,
More informationClass Equation & Conjugacy in Groups
Subject: ALEBRA - V Lesson: Class Equation & Conjugacy in roups Lesson Developer: Shweta andhi Department / College: Department of Mathematics, Miranda House, University of Delhi Institute of Lifelong
More informationGALOIS GROUPS ATTACHED TO POINTS OF FINITE ORDER ON ELLIPTIC CURVES OVER NUMBER FIELDS (D APRÈS SERRE)
GALOIS GROUPS ATTACHED TO POINTS OF FINITE ORDER ON ELLIPTIC CURVES OVER NUMBER FIELDS (D APRÈS SERRE) JACQUES VÉLU 1. Introduction Let E be an elliptic curve defined over a number field K and equipped
More informationIRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS. Contents
IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS NEEL PATEL Abstract. The goal of this paper is to study the irreducible representations of semisimple Lie algebras. We will begin by considering two
More informationSymmetric spherical and planar patterns
Symmetric spherical and planar patterns Jan van de Craats version August 3, 2011 1 Spherical patterns 1.1 Introduction Many objects in daily life exhibit various forms of symmetry. Balls, bowls, cylinders
More informationMath 120A: Extra Questions for Midterm
Math 120A: Extra Questions for Midterm Definitions Complete the following sentences. 1. The direct product of groups G and H is the set under the group operation 2. The symmetric group on n-letters S n
More information4 Arithmetic of Segments Hilbert s Road from Geometry
4 Arithmetic of Segments Hilbert s Road from Geometry to Algebra In this section, we explain Hilbert s procedure to construct an arithmetic of segments, also called Streckenrechnung. Hilbert constructs
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationHomework 7 Solution Chapter 7 - Cosets and Lagrange s theorem. due: Oct. 31.
Homework 7 Solution Chapter 7 - Cosets and Lagrange s theorem. due: Oct. 31. 1. Find all left cosets of K in G. (a) G = Z, K = 4. K = {4m m Z}, 1 + K = {4m + 1 m Z}, 2 + K = {4m + 2 m Z}, 3 + K = {4m +
More information1. Appendix A- Typologies
geometry 3D soild typology geometry 3D soild type 3D geomtry with a focus point cone (1/2) cc, f1602 cone (2/2) [...] (see left column) right cone cc, f1615 circular right cone cc, f1616 elliptical right
More informationA simple example of a polyhedral network or polynet, constructed from truncated octahedra and hexagonal prisms.
Polyhedral Nets A simple example of a polyhedral network or polynet, constructed from truncated octahedra and hexagonal prisms. The building process indicated produces a labyrinth. The labyrinth graph
More informationTHE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL LIE ALGEBRAS
Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 67 (2006) 2006, Pages 225 259 S 0077-1554(06)00156-7 Article electronically published on December 27, 2006 THE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL
More information2.3 Band structure and lattice symmetries: example of diamond
2.2.9 Product of representaitons Besides the sums of representations, one can also define their products. Consider two groups G and H and their direct product G H. If we have two representations D 1 and
More informationChapter 1. Crystal structure. 1.1 Crystal lattices
Chapter 1 Crystal structure 1.1 Crystal lattices We will concentrate as stated in the introduction, on perfect crystals, i.e. on arrays of atoms, where a given arrangement is repeated forming a periodic
More information1 Classifying Unitary Representations: A 1
Lie Theory Through Examples John Baez Lecture 4 1 Classifying Unitary Representations: A 1 Last time we saw how to classify unitary representations of a torus T using its weight lattice L : the dual of
More informationIIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1
IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1 Let Σ be the set of all symmetries of the plane Π. 1. Give examples of s, t Σ such that st ts. 2. If s, t Σ agree on three non-collinear points, then
More informationA Study on Kac-Moody Superalgebras
ICGTMP, 2012 Chern Institute of Mathematics, Tianjin, China Aug 2o-26, 2012 The importance of being Lie Discrete groups describe discrete symmetries. Continues symmetries are described by so called Lie
More informationThe Golden Section, the Pentagon and the Dodecahedron
The Golden Section, the Pentagon and the Dodecahedron C. Godsalve email:seagods@hotmail.com July, 009 Contents Introduction The Golden Ratio 3 The Pentagon 3 4 The Dodecahedron 8 A few more details 4 Introduction
More informationarxiv: v1 [math-ph] 6 Dec 2018
arxiv:1812.02804v1 [math-ph] 6 Dec 2018 From the Trinity (A 3, B 3, H 3 ) to an ADE correspondence Pierre-Philippe Dechant To the late Lady Isabel and Lord John Butterfield Abstract. In this paper we present
More informationPart II. Geometry and Groups. Year
Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),
More informationGroup Theory in Particle Physics
Group Theory in Particle Physics Joshua Albert Phy 205 http://en.wikipedia.org/wiki/image:e8_graph.svg Where Did it Come From? Group Theory has it's origins in: Algebraic Equations Number Theory Geometry
More informationThe mod-2 cohomology. of the finite Coxeter groups. James A. Swenson University of Wisconsin Platteville
p. 1/1 The mod-2 cohomology of the finite Coxeter groups James A. Swenson swensonj@uwplatt.edu http://www.uwplatt.edu/ swensonj/ University of Wisconsin Platteville p. 2/1 Thank you! Thanks for spending
More informationFIRST EIGENVALUE OF SYMMETRIC MINIMAL SURFACES IN S 3
FIRST EIGENVALUE OF SYMMETRIC MINIMAL SURFACES IN S 3 JAIGYOUNG CHOE AND MARC SORET Abstract. Let λ 1 be the first nontrivial eigenvalue of the Laplacian on a compact surface without boundary. We show
More informationLie Algebra of Unit Tangent Bundle in Minkowski 3-Space
INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 12 NO. 1 PAGE 1 (2019) Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space Murat Bekar (Communicated by Levent Kula ) ABSTRACT In this paper, a one-to-one
More informationSOME SPECIAL KLEINIAN GROUPS AND THEIR ORBIFOLDS
Proyecciones Vol. 21, N o 1, pp. 21-50, May 2002. Universidad Católica del Norte Antofagasta - Chile SOME SPECIAL KLEINIAN GROUPS AND THEIR ORBIFOLDS RUBÉN HIDALGO Universidad Técnica Federico Santa María
More informationChapter 1 Fundamental Concepts
Chapter 1 Fundamental Concepts 1-1 Symmetry Operations and Elements 1-2 Defining the Coordinate System 1-3 Combining Symmetry Operations 1-4 Symmetry Point Groups 1-5 Point Groups of Molecules 1-6 Systematic
More informationCentralizers of Coxeter Elements and Inner Automorphisms of Right-Angled Coxeter Groups
International Journal of Algebra, Vol. 3, 2009, no. 10, 465-473 Centralizers of Coxeter Elements and Inner Automorphisms of Right-Angled Coxeter Groups Anton Kaul Mathematics Department, California Polytecnic
More informationLongest element of a finite Coxeter group
Longest element of a finite Coxeter group September 10, 2015 Here we draw together some well-known properties of the (unique) longest element w in a finite Coxeter group W, with reference to theorems and
More informationRepresentations of Lorentz Group
Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the
More informationGroup theory - QMII 2017
Group theory - QMII 7 Daniel Aloni References. Lecture notes - Gilad Perez. Lie algebra in particle physics - H. Georgi. Google... Motivation As a warm up let us motivate the need for Group theory in physics.
More informationSome Observations on Klein Quartic, Groups, and Geometry
Some Observations on Klein Quartic, Groups, and Geometry Norfolk State University November 8, 2014 Some Historic Background Theorem. (Hurwitz, 1893) Let X be a curve of genus g 2 over a field of characteristic
More informationSemi-Simple Lie Algebras and. Their Representations. Robert N. Cahn. Lawrence Berkeley Laboratory. University of California. Berkeley, California
i Semi-Simple Lie Algebras and Their Representations Robert N. Cahn Lawrence Berkeley Laboratory University of California Berkeley, California 1984 THE BENJAMIN/CUMMINGS PUBLISHING COMPANY Advanced Book
More informationGroup theory applied to crystallography
International Union of Crystallography Commission on Mathematical and Theoretical Crystallography Summer School on Mathematical and Theoretical Crystallography 7 April - May 8, Gargnano, Italy Group theory
More informationSUMS PROBLEM COMPETITION, 2000
SUMS ROBLEM COMETITION, 2000 SOLUTIONS 1 The result is well known, and called Morley s Theorem Many proofs are known See for example HSM Coxeter, Introduction to Geometry, page 23 2 If the number of vertices,
More informationRank 4 toroidal hypertopes
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 67 79 https://doi.org/10.26493/1855-3974.1319.375 (Also available at http://amc-journal.eu) Rank
More informationIntroduction to Group Theory
Introduction to Group Theory Ling-Fong Li (Institute) Group 1 / 6 INTRODUCTION Group theory : framework for studying symmetry. The representation theory of the group simpli es the physical solutions. For
More informationSymmetry Operations and Elements
Symmetry Operations and Elements The goal for this section of the course is to understand how symmetry arguments can be applied to solve physical problems of chemical interest. To achieve this goal we
More informationAssignment 3. A tutorial on the applications of discrete groups.
Assignment 3 Given January 16, Due January 3, 015. A tutorial on the applications of discrete groups. Consider the group C 3v which is the cyclic group with three elements, C 3, augmented by a reflection
More information12-neighbour packings of unit balls in E 3
12-neighbour packings of unit balls in E 3 Károly Böröczky Department of Geometry Eötvös Loránd University Pázmány Péter sétány 1/c H-1117 Budapest Hungary László Szabó Institute of Informatics and Economics
More informationSchool of Mathematics and Statistics. MT5824 Topics in Groups. Problem Sheet I: Revision and Re-Activation
MRQ 2009 School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet I: Revision and Re-Activation 1. Let H and K be subgroups of a group G. Define HK = {hk h H, k K }. (a) Show that HK
More informationParticles I, Tutorial notes Sessions I-III: Roots & Weights
Particles I, Tutorial notes Sessions I-III: Roots & Weights Kfir Blum June, 008 Comments/corrections regarding these notes will be appreciated. My Email address is: kf ir.blum@weizmann.ac.il Contents 1
More informationBackground on Chevalley Groups Constructed from a Root System
Background on Chevalley Groups Constructed from a Root System Paul Tokorcheck Department of Mathematics University of California, Santa Cruz 10 October 2011 Abstract In 1955, Claude Chevalley described
More informationVector spaces. EE 387, Notes 8, Handout #12
Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is
More informationSymmetries and Polynomials
Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we ll learn how to solve a cubic. We ll also sketch how to solve a quartic. We ll explore the connections
More informationConway s group and octonions
Conway s group and octonions Robert A. Wilson School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E 4NS Submitted 7th March 009 Abstract We give a description of the
More informationGeometry. Separating Maps of the Lattice E 8 and Triangulations of the Eight-Dimensional Torus. G. Dartois and A. Grigis.
Discrete Comput Geom 3:555 567 (000) DOI: 0.007/s004540000 Discrete & Computational Geometry 000 Springer-Verlag New York Inc. Separating Maps of the Lattice E 8 and Triangulations of the Eight-Dimensional
More informationPERFECT COLOURINGS OF REGULAR GRAPHS
PERFECT COLOURINGS OF REGULAR GRAPHS JOSEPH RAY CLARENCE DAMASCO AND DIRK FRETTLÖH Abstract. A vertex colouring of some graph is called perfect if each vertex of colour i has exactly a ij neighbours of
More informationPermutation groups/1. 1 Automorphism groups, permutation groups, abstract
Permutation groups Whatever you have to do with a structure-endowed entity Σ try to determine its group of automorphisms... You can expect to gain a deep insight into the constitution of Σ in this way.
More informationHandout 2 for MATH 323, Algebra 1: Permutation groups and abstract groups
Handout 2 for MATH 323, Algebra 1: Permutation groups and abstract groups Laurence Barker, Mathematics Department, Bilkent University, version: 30th October 2011. These notes discuss only some aspects
More informationON THE CHIEF FACTORS OF PARABOLIC MAXIMAL SUBGROUPS IN FINITE SIMPLE GROUPS OF NORMAL LIE TYPE
Siberian Mathematical Journal, Vol. 55, No. 4, pp. 622 638, 2014 Original Russian Text Copyright c 2014 Korableva V.V. ON THE CHIEF FACTORS OF PARABOLIC MAXIMAL SUBGROUPS IN FINITE SIMPLE GROUPS OF NORMAL
More informationRegular Cyclic Coverings of the Platonic Maps* GARETH A. JONES and DAVID B. SUROWSKI**
Regular Cyclic Coverings of the Platonic Maps* GARETH A. JONES and DAVID B. SUROWSKI** We use homological methods to describe the regular maps and hypermaps which are cyclic coverings of the Platonic maps,
More informationarxiv: v1 [math-ph] 18 Feb 2016
THE BIRTH OF E 8 OUT OF THE SPINORS OF THE ICOSAHEDRON PIERRE-PHILIPPE DECHANT arxiv:160.05985v1 [math-ph] 18 Feb 016 ABSTRACT. E 8 is prominent in mathematics and theoretical physics, and is generally
More informationHighest-weight Theory: Verma Modules
Highest-weight Theory: Verma Modules Math G4344, Spring 2012 We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or,
More informationPlan for the rest of the semester. ψ a
Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and
More informationAPPLICATION OF HOARE S THEOREM TO SYMMETRIES OF RIEMANN SURFACES
Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 307 3 APPLICATION OF HOARE S THEOREM TO SYMMETRIES OF RIEMANN SURFACES E. Bujalance, A.F. Costa, and D. Singerman Universidad
More informationA method for construction of Lie group invariants
arxiv:1206.4395v1 [math.rt] 20 Jun 2012 A method for construction of Lie group invariants Yu. Palii Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia and Institute
More informationROOT SYSTEMS AND DYNKIN DIAGRAMS
ROOT SYSTEMS AND DYNKIN DIAGRAMS DAVID MEHRLE In 1969, Murray Gell-Mann won the nobel prize in physics for his contributions and discoveries concerning the classification of elementary particles and their
More information