by wendy » Tue Jan 07, 2014 9:03 am
It's a laminatope: it's bounded by unbounded planes.
Look at the pink bit of the attached picture. That's a crosssection along any of the circles you see.
In essence, you are sitting in the green bit to the left, looking right. The green things you see on the right are the x3o8o. The line between the green bit and the purple bits, are supposed to be straight, but Tyler plots poincare points, and makes straight lines between them. But you can only see the boundary between them, these are the etchings on the x3o8o.
If you follow a circle on the first picture, you will see that some are projected 'vertexfirst' (ie they have a vertex in the middle). These points centrally invert to one of the crossings outside the picture, this is represented by the line from the big green bit to the little green bits on the right.
If you follow the line along, you will see some projections are 'edge first'. They have an edge in the middle. On the second figure, these are the ones on the opposite side of the octagons to the big green area. You see they're edge first too.
The next set you see are pairs of circles between the vertex and edge first x3o8o. You see that there are octagons not directly attached to the green bit on the left, and they have three little pink legs. Between these little pink legs comes a little green dot, not resolved, this leads to a pair of x3o8o, each edgefirst, but too far for one to see that detail.
Each of those little legs are actually more octagons, ad infinitum.
It's not a mosaic. It's a convex thing, like a 2d version of the pink thing in the second diagram. And you wonder why i invent new words for all this stuff. Layer is a bit simplistic, although it is exactly what you get in x2x3o6o (two layers of triangular tilings). Just a larger kind of 2.
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The dream you dream alone is only a dream
the dream we dream together is reality.