Re: first shape discovered by a computer

Op 21/08/2024 om 14:20 schreef sobriquet:

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.

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It's the shape with the biggest volume one can obtain with 8 vertices on the unit sphere.
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I got the coordinates for the vertices from the paper referenced in the recent Matt Parker video about the shape.
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https://www.youtube.com/watch?v=XZy3rXr2yeM
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https://www.ams.org/journals/mcom/1963-17-082/S0025-5718-63-99183-X/S0025-5718-63-99183-X.pdf
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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.
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https://www.desmos.com/calculator/stjrx6qsxt

 I think I got the exact coordinates for the vertices now.
 https://www.desmos.com/3d/xaxwkqwgw3