I seem to have stumbled upon a bit of a problem...

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I seem to have stumbled upon a bit of a problem...

Postby The_Tydra » Sat Mar 10, 2007 6:50 am

Alright, this is a bit of a paradox I came upon concerning 4 dimensional geometry quite some time ago and since then I haven't been able to explain if or where I made a mistake in logic. Allow me to explain.

First, take a look at this image:
Image
This, as you all know, is a hexagram. A hexagram is created when you cross two triangles, as you can clearly see a regular hexagon is formed in the center where the two triangles intersect. Moving on one dimension higher...
Image
we have a stella octangula, formed by the crossing of two tetrahedra. While it isn't obvious at first glance, a regular octahedron is formed in the center where the two intersect. This of course simply makes sense, so much so that we don't even go to the trouble of thinking about it, two self dual shapes, each with four faces and triangular vertices, forms a shape with eight triangular faces. Finally, let's take the process illustrated in the past two images up to the fourth dimension. So now we have two 5-cells crossing each other, but what is it that would be formed by such an intersection? From what we have covered so far we can conclude that it would create a regular polychoron with 10 tetrahedral cells. This wouldn't be seen as a problem for someone without knowledge of 4 dimensional shapes, however for those of us who have studied the subject we know that there are only supposed to be three regular polychorons with tetrahedral cells, the 5-cell, the 16-cell, and 600-cell, with the Schlafli symbols of (3,3,3), (3,3,4), and (3,3,5) respectively. Our 10 celled polychoron, while being the natural consequence of the existence of the 5-cell, is in direct contradiction with the concept of the other two tetrahedron celled polychorons.

Any thoughts on this? Did I somehow screw up royally? Did I come upon something of true signifigance? Post your views.
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Postby wendy » Sat Mar 10, 2007 7:14 am

In small dimensions, a lot of constructions give the same effect. But as one gets to higher dimensions, these separate into different figures.

The figure of inverted simpleces xo3oo..3ox do give the vertices of the hexagon and the octahedron in 2d, 3d. But in 4D and higher, these are always a separate thing.

There is in 4D the rather interesting o3x4x3o (octagonny), which is the intersection of dual {3,4,3} as well.

A rather interesting thread to follow is the one that in 3d gives the triangular prism, and in 4d, the o3x3o3o, being a figure bounded by 5 tetrahedra and 5 octahedra. It emerges as a separate construction in 6d (giving a polytope with 27 vertices), and works through to a tiling in eight dimensions. Thorald Gosset discovered the series in 1886.
The dream you dream alone is only a dream
the dream we dream together is reality.
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