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# Space Filling Polyhedra

Space Filling Polyhedra [message #26370] Fri, 22 April 2011 15:01 UTC
Hi,

I received time ago two space filling polyhedra, but before showing them, I needed to make some drawings to explain the concept.
So here is the explanation.
You can transform a cube into a dodecahedron by drawing on each squared face a new edge and then pushing it outside the cube.
There is an analogy with a square that you can transform into an octagon by drawing new vertices at mid-edge and then pushing them outside.
Squares can fill the plane and if you tranform one square out of two (checkerboard pattern) you obtain a tiling of the plane made of octagons and another pointy shape (let's call it an anti-octagon).

In the same way, cubes can fill space and applying the cube-to-dodecahedron transformation to one cube out of two you got a couple of space-filling polyhedra: the dodecahedron, of course, and a pointy shape that we can call an anti-dodecahedron.
Let's stop the explanations one second and see the results.
The dodecahedron:

The anti-dodecahedron:

Now, if you consider that you are filling the space only with dodecahedra (the anti-dodecahedra being replaced by void), you will find that dodecahedra can fill space with a ratio of 90.43% (for reference, this ratio is 74.05% for spheres). Not bad isn't it?

You can also check the video.

• Attachment: dodnet.jpg