scalar Idot Jetinitions a c IN product Given a
n dimension Euclidean space R equipped with standardinner product I T fju 5 x y day at any u n usual The between I z e IR Same as distance in R R2 C concept IR d i Iri z z E not necessarily z linear
Jetinition An a isometry of R is map 7 R R Sam that d FLI 7 d Iz I z VI I C R T preserves distances The set at all isometries of R is denoted Isom R
Remand E Isom IR and 7 E IsomCIR C Isom R Id pu g Fog FCI A For E Isom 112 F and Fce e c orthogonal A E On R new matrices For example 7 I judo Isis IT anticlockwise A f rotationby O
F c Isom R a C translation I y IR such that 7 Lz y ta Fixes translation c IR f c Isom t Ri FLI A z z In some A C OnCR 1ER
Definition Let X CIR be a subset
Sym X FE Isom Ri 17 permutes X Called Symmetrygroup A X CR Observation Sym x naturally acts on X Sym ex xx X F z Fix
Dihedral Groups ME IN M 3 A Az Am Vertices at a m centered Xm regular you at E C R2 E A X Ai Az As A µ Az Definition
Dm Sym Xm C Isom R2 with Dm subgroup at Symon dihedral group
Important observation Dm acts Faithfully on A A Am
Rotan Rotational symmetries subgroup i e o E om C Don T p Rotation rotation clockwise by o cycle notation in about E by about E 2 GA Az Ad to Example CA Azaz 3 2 F 0 A Az Az A K DE E 0 m i Move generally Td c A de reflection in on CA Am Dm acts transitively wire through E and i E Stab A Observe that given A Xm e t
I Dnt forb Ail 1 Stats Ai 2in any reflection IRotul m Dm Rotm I t Ratu All reflective symmetries Example T to to A Tz h H D e F E ti te t's
Az rotationby about I T2 z reflective Any Symmetry Hence is Dm generated by and I with relations ord o m ord T z m to f t
1123 equal Symmetries in All Faces m gous regular 1123 Platonic solid in Regular convex polyhedron in 1123
are Five Amazing Fact There only possibilities
X X platonic solid centered at E C R Sym Important observation Sym x acts faithfully Sym X Subgroup A ICvertisot lets SymCX C 2 Verities of Rot X rotational symmetries just say TeX t ra h e d ro i r Ac
x C Sym 2 Ai Az Az A 4 As µ What can µ permutations show up Az Let I Reflection in plane containy A and Az I Asay EE A Az Az A43
Ai E LX A j Sym t i j Sym x 2 EA 43 _Syme
Curie contains e and Hy Rotation by about
6 A A Az C 2 Ai Ai Az Ky We could do this For any AIA Aia A Cycles length 3 generate Atty Rotc X Alte
Amazing Fact Any Finite subgroup of Isom 1123 must be a subgroup of a dihedral group or Sym X where X is a platonic solid
Even more There amazing fact are versions of platonic solids in IR For a 3
n 4 There are 6 For example the hypercube Tesseract
5 come hour 1123 Theres one unique to 112
n 4 There are 3 A High dimensional version st tetrahedron cube and octahedron