4d symmetry

Higher-dimensional geometry (previously "Polyshapes").

4d symmetry

Postby papernuke » Mon Jul 24, 2006 1:40 am

Do 4d objects have symmetry? For example a tesserac?
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Re: 4d symmetry

Postby Marek14 » Mon Jul 24, 2006 6:41 am

Icon wrote:Do 4d objects have symmetry? For example a tesserac?


Yes.

OK, that wasn't too informative :)

In 2D, the basic symmetries you can have are rotations around a point or reflections along a line. Often-used symmetry groups are those of regular polygons. A regular n-gon has a symmetry group of size 2n.


In 3D, you can have reflections along a plane or rotation around a line. Symmetries around a POINT in 3D are more difficult to describe.

In 4D, the basic symmetries are reflection along a hyperplane, or rotation around a plane. However, in this second case, there can also be a second, separate rotation WITHIN that plane. Generally, I'm not sure if the possible symmetry groups are classified. I just know the sizes of the basic groups that occur in polychora:

Pentachoric - 120
Hexadecachoric - 384
Icositetrachoric - 1152
Hexacosichoric - 14400

Tetrahedral prismatic - 48
Octahedral prismatic - 96
Icosahedral prismatic - 240

m-gonal-n-gonal duoprismatic - 4mn
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Postby 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...

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