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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: first shape discovered by a computer
Date: Wed, 21 Aug 2024 12:30:59 -0700
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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:
>>>>
>>>> Hi!
>>>>
>>>> https://www.desmos.com/3d/t5fsaljsmh
>>>>
>>>> The shape is comprised of 12 pyramids that come in two varieties, 
>>>> but they all appear to have the same volume.
>>>
>>> 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)?
>>
>> 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