Re: first shape discovered by a computer

On 8/21/2024 5:20 AM, sobriquet wrote:

Op 21/08/2024 om 08:36 schreef Chris M. Thomasson:

On 8/20/2024 11:33 PM, Chris M. Thomasson wrote:

On 8/20/2024 10:55 PM, sobriquet wrote:

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Hi!
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https://www.desmos.com/3d/t5fsaljsmh
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The shape is comprised of 12 pyramids that come in two varieties, but they all appear to have the same volume.

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It looks like a unit 5-gon in the xy plane at (x, y, 0) with each vertex connected to two points, at (0, 0, -1) and (0, 0, 1)?

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Not quite... Like two 5 gons one on the xy plane and one on the xz plane rotated by pi / 2. I need to take a closer look and try to recreate it on my end. Fun. Thanks for the post. Might have some more time tonight.

 It's the shape with the biggest volume one can obtain with 8 vertices on the unit sphere.
 I got the coordinates for the vertices from the paper referenced in the recent Matt Parker video about the shape.
 https://www.youtube.com/watch?v=XZy3rXr2yeM
 https://www.ams.org/journals/mcom/1963-17-082/S0025-5718-63-99183-X/S0025-5718-63-99183-X.pdf
  When I computed the volume of the shape that has an isosceles
triangle as the base for the pyramid, it seemed to give the
impression that it's a twelfth of the total volume, implying
that the other pyramid shape with a scalene triangle base has
the same volume.
 https://www.desmos.com/calculator/stjrx6qsxt

Okay. Well, the fun part is that I can use it to plot one of my experiential 3d von Kochs on. My algorithm uses a single triangle to do its thing. Here is an older low-res example:
https://youtu.be/AIrP4KeSIjI