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Replies: 17   Last Post: Jan 11, 1995 6:43 AM

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 John Conway Posts: 2,238 Registered: 12/3/04
Posted: Dec 17, 1994 7:57 AM

This classification of quadrilaterals is almost, but not quite,
the same as their classification by symmetry. The only exception
is that the generic trapezoid has no symmetry, like the generic

I worked out the classification of all N-gons by symmetry
only a few weeks ago. The group of the regular polygon is
the dihedral group D2n of order 2n. If m times i is 2n,
then m is the order of either just 3 subgroups (if m and i are
both even), or just 1 (if either is odd), up to conjugacy.

The subgroups are Dm if i odd

Cm if m odd

Dm-, Dm+, Cm if m and i are even,

where the reflections of Dm+ permute the vertices evenly,
while those of Dm- permute them oddly. (alternatively,
one type of reflection fixes a vertex, the other an edge).

Every group except Cn is the group of some type of polygon.

For n = 4 we have D8 square

D4+ C4 D- rect'l (none) rhombus

D2+ C2 D2- iso-trap para kite

Sorry - those corresponding arrays were cramped a bit too closely.

For n = 6 we get

D12

D6+ C6 D6-

C3

D4

D2- C2 D2+

C1

for the groups. D12 - regular

D6+ edges same length, vertices form two equilat
triangles of distinct sizes

D6- edges alternate in length, alternate sides
produced would give two equilat triangles
of distinct sizes

C6 can't happen

C3 vertices form two equilat triangles of difft
sizes (but same center) in most general way

D4 "rectangular" symmetry

D2- "vertex-isosceles"

D2+ "edge-isosceles"

C2 the par-hexagon (opp sides parallel)

C1 conmpletely irregular.

John Conway

Date Subject Author
12/15/94 E7M2WAT@TOE.TOWSON.EDU
12/16/94 W Gary Martin
12/16/94 Henri Picciotto
12/17/94 John Conway
12/17/94 roitman@oberon.math.ukans.edu
12/18/94 John Conway
12/20/94 Chenteh Kenneth Fan
12/21/94 John Conway
12/21/94 Walter Whiteley
12/21/94 William T. Webber
12/21/94 James King
12/21/94 Chenteh Kenneth Fan
12/21/94 John Conway
12/21/94 John Conway
12/21/94 John Conway
12/21/94 Chenteh Kenneth Fan
12/21/94 Chenteh Kenneth Fan
1/11/95 joe malkevitch