"... In PAGE 2: ... Because the triangular faces no longer contain mirror symmetry con- straints, the tile boundaries can now be deformed to create intricate interlocking tile shapes. The straight and oriented tetrahedron are just two of the seven groups derived from the Platonic solids ( Table1 ). In addition to the straight and oriented octahedron/cube and icosahedron/dodecahedron, there is also the double tetrahe- dron group.... ..."

"... In PAGE 6: ...Ie 21 26 The overall distributions into symmetry groups for HSs are 25 known completely for /r lt;16 according to another recent review. Table5 contains supplements to these data. For h gt; 17.... ..."

"... In PAGE 19: ...resulting CSP by finding all the solutions and finding the automorphisms of the graph that has an n-ary hyperedge for every solution. Table2 shows the size of the solution symmetry group. Table 2.... ..."

"... In PAGE 3: ...3 Upon compacti cation on a torus Td, 11D SUGRA exhibits a group of continuous global symmetry G, as well as a (composite) gauge invariance under its R-symmetry maximal compact subgroup K, as listed in Table1 [8]. The symmetry can be seen among the scalar elds, which take value in the symmetric space KnG.... In PAGE 3: ... The symmetry group Gd acts on the right on the coset KnG, and induces a compensating moduli-dependent K-rotation to preserve the gauge K = 1. The massless spectrum also includes a number of p-forms, given for p = 1; 2 in the last two columns of Table1 , where dualization into forms of lower degree has been carried out. They transform linearly under G and induce charges for particles (m) and strings (n) respectively.... In PAGE 5: ... This is consistent with the fact that E9 is the a ne Lie group associated to E8, and implies that the representations are in nite dimensional. For d gt; 9, the symmetry is even more dramatic, with hyperbolic E10 or Kac-Moody E11, while for d lt; 9 we recover the groups in Table1 . The representations corresponding to the extremal nodes indicated in (13) correspond to a particle multiplet with highest weight M = 1=Rd (a KK mode) , a string multiplet with highest weight T1 = R1=l3 p (a singly wound M2-brane), and a membrane multiplet with highest weight 1=l3 p.... In PAGE 5: ... The representations corresponding to the extremal nodes indicated in (13) correspond to a particle multiplet with highest weight M = 1=Rd (a KK mode) , a string multiplet with highest weight T1 = R1=l3 p (a singly wound M2-brane), and a membrane multiplet with highest weight 1=l3 p. The particle and string multiplets are precisely the ones charged under the one- and two- form potentials in Table1 . The other members in the same multiplet are obtained by applying Weyl re ections on these highest weights, and e.... ..."

"... In PAGE 6: ...5 Suppose ? is a nite subgroup of O(2), and ?. Then all sub- groups which are admissible for ows ft : IR2 7! IR2 with reversing symmetry group ? are listed in Table 1 and Table2 . The !-limit sets that realize admissibility can be taken to be Liapunov stable periodic orbits.... In PAGE 13: ... symmetric periodic orbit with Zn symmetry, cf. Fig. 1(c). The results are summarized in Table 1. Together with the results of [5] { which are summarized in Table2 { they cover all admissible nite subgroups of O(2) for ows in IR2. 6 Symmetric !-limit sets in xed point subspaces In this paper we have concentrated mainly on symmetric !-limit sets with trivial instantaneous symmetry, i.... ..."