Klein quartic

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Klein quartic

Postby quickfur » Fri May 23, 2014 8:33 pm

Today, I stumbled across the Wikipedia article on the Klein quartic, which says that there is no faithful 3D representation of it. The question is, are higher-dimensional faithful representations possible? What's the lowest dimension in which a faithful representation is possible? Is it possible in 4D? If so, what shape does it take?
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Re: Klein quartic

Postby Klitzing » Fri May 23, 2014 10:34 pm

A first answer to that question of what this symmetry of the Klein quartic would look like e.g. is given in an article by William Thurston at http://library.msri.org/books/Book35/files/thurston.pdf (1998):

Consider the regular dodecahedron, which is representable within 3D. It has incidence matrix
Code: Select all
20 |  3 |  3
---+----+---
 2 | 30 |  2
---+----+---
 5 |  5 | 12


The symmetry of the Klein quartic, taken geometrically in the similar way, then would be described by the following incidence matrix of some abstract regular polyhedron:
Code: Select all
56 |  3 |  3
---+----+---
 2 | 84 |  2
---+----+---
 7 |  7 | 24
That figure obviously cannot be represented faithful within 3D.

(Moreover, it not even can be represented faithful within flat hyperbolic space, as there the order 3 tiling of regular heptagons surely has an infinite number of vertices, edges, and faces.)

--- rk
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Re: Klein quartic

Postby Klitzing » Wed May 28, 2014 9:45 am

Just for further info, here is a representation of the dual configuration within 3D, so with curved edges and drilled cells:
Image
Spot the triangular prisms, placed at the vertices of the tetrahedral strut, and the drilled square-antiprisms, placed at its edges. The squares of those cells in this blend for sure cancel out. So we are left indeed with 4*2 + 8*6 = 8+48 = 56 triangles, as required. Note moreover that those are connected as needed: 7 per each vertex.

--- rk
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Re: Klein quartic

Postby quickfur » Wed May 28, 2014 7:02 pm

Interesting. So the image you posted is the dual of the Klein quartic? Doesn't that mean it's possible to construct a 3D model of the Klein quartic itself where arbitrary deformation is allowed for the heptagons? It's supposed to be a genus-3 surface, right? Is it possible to build a (possibly self-intersecting and arbitrarily complex) 2-manifold immersed in 3-space which is tiled by the 24 heptagons?
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Re: Klein quartic

Postby Klitzing » Wed May 28, 2014 8:10 pm

Sure it is! Just take the center points of these triangles and draw edges between those orthogonal to the existing ones. That's all. - Sure, it will not have planar regular heptagons.

And yes, it is genus 3. Just push the top vertex of that tetrahedral strut down into the bottom plane. Then you will get a "solid" with exactly 3 holes.

--- rk
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