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Showing posts from January, 2018

Compound of five 24-cells

Compound of five 24-cells

An animation motivated by John Baez, http://bit.ly/2DXpPwL

We first see the 720 edges of a vertex-centered 600-cell in light gray, then five 24-cells fade in one by one. The first, in red, is also vertex-centered and covers 24 of the 120 vertices of the 600-cell. The next four 24-cells arrange like the faces of a tetrahedron, covering the remaining 96 vertices. The 600-cell fades out at the end so that we can focus on the five 24-cells and their combined 480 edges.

Through all this, one 600-cell vertex has been projected to infinity. Edges connected to that vertex (from the 600-cell and red 24-cell) are cut short.
https://youtu.be/t5i5nym05qI

Spherical sections of the {3,3,7} hyperbolic honeycomb

Spherical sections of the {3,3,7} hyperbolic honeycomb

We start inside a white tetrahedron, with colors changing as we intersect further layers of tetrahedra. The section radius grows at a constant hyperbolic rate.

hyperbolichoneycombs.org
https://youtu.be/ebNe9G0FElg

Spherical sections of the 120-Cell in S^3

Spherical sections of the 120-Cell in S^3

We start inside a white dodecahedron and end at its antipode, with colors changing by layers of dodecahedra. The section radius grows at a constant rate.
https://youtu.be/BG_rgn2Bmpg

Superflip for the Klein Quartic Rubik Analogue

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Superflip for the Klein Quartic Rubik Analogue

This is the only non-trivial element in the center of the permutation group of this puzzle.

For details and a conjecture:
https://groups.yahoo.com/neo/groups/4D_Cubing/conversations/messages/3976

I first saw Vladimir give a talk in 2010 and his work and images were totally inspiring to me.

I first saw Vladimir give a talk in 2010 and his work and images were totally inspiring to me. That has continued ever since!

Originally shared by Vladimir Bulatov

I've posted slides from my talk on hyperbolic tilings
http://bulatov.org/math/180110
http://bulatov.org/math/180110

"That's the sexiest model you've made"

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"That's the sexiest model you've made"

Sarah Nemec-Nelson just told me that when I showed her these photos. The shape is a helicoid in hyperbolic 3-space. You can read a little more about this ruled surface in a previous post and the comments there.

https://plus.google.com/+RoiceNelson/posts/8heVohDiBsH

For the physical model, I also included the helicoid axis as a Dupin Cyclide, specifically a horn cyclide. That is the banana shape that a thickened hyperbolic line takes in the ball model. You can produce a horn cyclide from a cone by inverting it in a sphere!

https://en.wikipedia.org/wiki/Dupin_cyclide

Here is a fun and heavily visual paper about cyclides titled Ortho-Circles of Dupin Cyclides.

https://pdfs.semanticscholar.org/4e86/13b074369af1263efe97e299b90a7d81807f.pdf

To make the model more interesting, I chose an axis avoiding the origin and an asymmetric orientation of the helicoid twisting. You can purchase a copy for yourself (again, with no markup) here: htt…