by **wendy** » Sat Jan 17, 2015 9:25 am

Using complex numbers as coordinates is an ideal way to introduce clifford rotations, and the various regular complex polytopes are a good introduction to some of the subsymmetries of the regular polytopes.

If you take the line y = Ax + B, it is easy to show some interesting things in 4D. For example, where B=0, there is a series of points x,y where wy = A wx.

Put w = cis(vt) where v is a velocity and t is a time, and you can rotate the whole of 4-space where the point follows a path where A never changes. That is, one can have rigid rotation of all of space around a point. It also means for a set swirly space, two points define an argand diagram.

If you now plot the coordinates of A as X+iY, and set up a sphere whose diameter is (0,0,0) to (0,0,1) [a diameter-1 sphere sitting on 0 on the argand diagram], you get the sphere of rotations from where the line from (X,Y,0) to (0,0,1) passes through this sphere. When we say the decagons of the (3,3,5) are in icosahedral arrangment, they actually produce the vertices of an icosahedron on this sphere. Likewise, the order-six swirls produce the 20 vertices of a dodecahedron, and the 30 vertices of an icosadodeca make the 30 order 4 rotations.

The complex polygons like 3(5)3 and 5(3)5, are swirl-rotations of order 3 and 5 on top of the poincare dodecahedron. All of these are various kinds of swirl symmetries.

One should note that where the two outside numbers are different, ie p(2q)r, where p,q,r is one of 2,3,3, or 2,3,4 or 2,3,5, that the resulting thing is not a subgroup of any of the regular polyhedra, because if the numbers on either side (ie p\= r ) then the class doubles.