by wendy » Thu Feb 13, 2020 6:51 am
The swirlprism as such ultimately calls on a 3D symmetry, which is converted into 4D. The connection is through complex analytical geometry.
You start off with a line \( y = ax +b\), where these fall into the complex plane, so y, x, a and b are all complex numbers. This equates to that in 4D, you can impose a restriction to space, that between any two points, there is exactly one 2D space that contains them both.
If we now suppose \(b=0\), you get a line passing through the origin: \( y = ax\). We now introduce a variable \(w = cis(\omega t)\), which is a rotation in the complex plane, around the origin. In the case of \(wy = awx \), we see that the point (x,y) will orbit the centre, but the slope of the line never changes. This means that you can have orbits around the origin, none of which cross.
We now take the complex plane representing \(a = r, i \), and a sphere whose diameter is (r=0, i=0, 0) to (r=0, i=0, 1), and map the points \(a\) onto this sphere through a ray from (0, 0, 1), to (r, i, 0), and where it intersects is the representation of this slope. Points opposite on this sphere represent perfectly orthogonal spaces.
This sphere, for a given radius, represents a bundle of equal-radius circles in 4D, whose centre is the origin and which entirely smothers the glome. A swril, so to speak.
For a given point in 3D, we have concentric spheres around it, and this maps onto 4D as every other point of 3D is a circle whose radius is preserved. Swirling then is something that you can do at a photographic level: swirl-trees and swirl-cows, so to speak.
The swirl-symmetries correspond to the 3D symmetries: prismatic, tetrahedral, octahedral, and dodecahedral.
The Poincare symmetries are related, and involve swirl symmetry, but I have not been able to figure out the connection.
If you swirl a poincare cell by one of its symmetry angles, it will move onto a copy of itself, rotated by that angle. That is, a poincare-cell is similar to the toric projection of a plane, such as in games, where you go off the north-screeen, and come back on the south. There are poincare-cells that match every swirl symmetry, but the swirl-prism is complete circles, whereas the poincare-cells are portions of the surface.
Poincare-cells are known for 2p, 24, 48 and 120, being duoprisms, the 24choron, the octagonny, and the twelftychoron. The Poincare dodecahedron is an example of the 120 group, but as I showed John Conway, you can easily use a pentagonal tegum, made of five cells of a {3,3,5}.
The symmetries of the complex polytopes are a subdivision of the poincare-cells, in a way that the additional "mirrors" are further rotations of 2, 3, 4, 5 (as these are allowed in 8, 24 and 120), on top of these numbers. So something like 3{6}2 has two mirrors, which has a rotation order separately of 3*8 and 2*8, the composite has 6*8 = 48 for its symmetry. But numbers arise in these symmetries that are not seen in {3,3,4}, {3,4,3} and {3,3,5}. The 3{10}2 for example, has a number-system that is seen elsewhere seen in {5,12} or {12,10}, and its vertices include inscribed dodecagons, which never occur in [3,3,5].
So the 120 symmetries of the swirl-prism are 120 triangle-prism hoops around the centre of the figure, and is not related to the 120 Poincare cells. Instead, it can be thought of, that the order of the symmetry of [3,3,5] is 120 triangle-prism hoops × 120 poincare-cells.