Part 1 - Supplement

Tetrahedral Symmetry as A(4)

  E R12 R13 R14 R12 R22 R32 R42 R1 R2 R3 R4
E E (12)(34) (13)(24) (14)(23) (243) (134) (142) (123) (234) (143) (124) (132)
 R12  (12)(34) E (14)(23) (13)(24) (142) (123) (243) (134) (132) (124) (143) (234)
R13 (13)(24) (14)(23) E (12)(34) (123) (142) (134) (243) (143) (234) (132) (124)
R14 (14)(23) (13)(24) (12)(34) E (134) (243) (123) (142) (124) (132) (234) (143)
R1 (234) (124) (132) (143) E (13)(24) (14)(23) (12)(34) (243) (142) (123) (134)
R2 (143) (132) (124) (234) (13)(24) E (12)(34) (14)(23) (123) (134) (243) (142)
R3 (124) (234) (143) (132) (14)(23) (12)(34) E (13)(24) (134) (123) (142) (243)
R4 (132) (143) (234) (124) (12)(34) (14)(23) (13)(24) E (142) (243) (134) (123)
R12 (243) (123) (134) (142) (234) (132) (143) (124) E (14)(23) (12)(34) (13)(24)
R22 (134) (142) (243) (123) (124) (143) (132) (234) (14)(23) E (13)(24) (12)(34)
R32 (142) (134) (123) (243) (132) (234) (124) (143) (12)(34) (13)(24) E (14)(23)
R42 (123) (243) (142) (134) (143) (124) (234) (132) (13)(24) (12)(34) (14)(23) E
  E R12 R13 R14 R12 R22 R32 R42 R1 R2 R3 R4
E E R12 R13 R14 R12 R22 R32 R42 R1 R2 R3 R4
R12 R12 E R14 R13 R32 R42 R12 R22 R4 R3 R2 R1
R13 R13 R14 E R12 R42 R32 R22 R12 R2 R1 R4 R3
R14 R14 R13 R12 E R22 R12 R42 R32 R3 R4 R1 R2
R1 R1 R3 R4 R2 E R13 R14 R12 R12 R32 R42 R22
R2 R2 R4 R3 R1 R13 E R12 R14 R42 R22 R12 R32
R3 R3 R1 R2 R4 R14 R12 E R13 R22 R42 R32 R12
R4 R4 R2 R1 R3 R12 R14 R13 E R32 R12 R22 R42
R12 R12 R42 R22 R32 R1 R4 R2 R3 E R14 R12 R13
R22 R22 R32 R12 R42 R3 R2 R4 R1 R14 E R13 R12
R32 R32 R22 R42 R12 R4 R1 R3 R2 R12 R13 E R14
R42 R42 R12 R32 R22 R2 R3 R1 R4 R13 R12 R14 E


These twelve rotations clearly correpsond to even permutations of four elements, and thus the set of rotational symmetries of regular tetrahedra is equivalent to A(4).

For a proof that the full set of symmetries corresponds to S(4), click here.

Note that the order is changed on the top row of the table so that the diagonal is entirely the identity, E.   Though the group is non-Abelian, this arrangement presents a pseudo-symmetry where the three boxes along the diagonal are commutative and the six boxes not along this diagonal are across the diagonal from their inverse.

This can be done because every because for the main diagonal to be the identity, each row must be inverted by its corresponding column.  As we all know, (AB)-1 = B-1A-1,   so   AB-1 = (BA-1)-1.

Part 1 - Proper Tetrahedral Symmmetry

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Part 2 - Full Tetrahedral Symmmetry