by wendy » Sat May 28, 2005 7:29 am
One can quite easily, and usefully, project four dimensions onto two dimensions. I often look at higher dimensional things by heavy dimension reduction.
In essence, a point (x,y,z) contains x, y and z coordinates, which can be presented as numbers.
A thing in (w,x,y,z) evidently projects as well onto (x,y) or (w,x) or (x,z) planes. Many of the uniform and regular figures in four dimensions project quite elegantly into (w,x) + (y,z) planes, where these are in some cases the same design [ie like the cube, plan = elevation].
The 24-choron, on the other hand does not (rather like the hexagonal, the view down perpendiculars is different).
The vertices of this is x3x o3o + x3o q3o + o3x o3q
What this means, is that we have the 24-choron breaking down into three duoprisms, which present thenmselves as one base on one axis, and the other base on the other axis.
x3x o3o = hexagon * point
x3o q3o = triangle ^ * triangle (r2) ^
o3x o3q = triangle v * triangle (r2) V
The left plot consists of a hexagon, with an inscribed star-of-david. The outer hexagon-vertices are "red", while the inner hexagon is alternately white and blue. In the alternate projection, we see the central dot as red, and the the outer hexagon being alternately white and blue vertices.
In the first projection, we see the edges as
outer perimeter
star of david on outer hexagon (makes three edges each)
star of david on inner hexagon (makes one edge each)
You can then see the faces (octahedra), as a central hexagon [which is what one gets face first] and outer rectangles [squares shortende by prospective] representing vertex-first views.
On the second projection, one sees the octahedra as two triangles (agani forshortened by prospective) and the outer six edges as "directly on the edge). When you draw a cube face-first, you see faces become lines.
Most of the other figures look identical projected down both axies.
W