by wendy » Mon Jul 24, 2006 7:54 am
Conway listed an awful lot of them in his novel "Quarterions and Octonions". The ones listed by Marek are some of the mirror-groups.
The ones i am familiar with are the likes of
[3,3,3+] 60 pentachoral rotational
[3,3,3] 120 pentachoral reflective
[(3,4,3)] 240 di-pentachoral
[3,3,A+] 96 halfcubic rotational
[3,3+,4] 192 pyritochoral
[3,3,4]+ 192 tesseract-rotational
[3,3,A] 192 half-tesseractal
[3+,4,3+] 288 (unnamed)
[3,3,4] 384 tesseractal
[3+,4,3] 576 great pyritochoral
[3,4,3]+ 576 24choral-rotational
[3,4,3] 1152 24-choral
[(3,4,30] 2304 octagonnical
[3,3,5]+ 7200 twelftychoral-rotational
[3,3,5] 14400 twelftychoral
The next relate to the chiral groups, formed by adjoining a swirlybob with a polygonal rotation. These correspond to the rotational groups of the complex polygons where CE2 becomes E3. It becomes then:
swrily-bobs of orders 8, 24, 48 and 120, adjoined with assorted polygons.
8 x, cx, cc, ccx
24 y, x, yx, z, yc, yz, cz, ycx, ycz
120 f2, f3, f5, f3f2, f5f2, f5f3, f5f3f2
where each x, y, f2 = 2, c, f3 = 3, z = 4, f5 = 5, eg f3f2 = 120*3*2 = 720.
There are more groups in the book, but i do not understand them...
W