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From: David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz>
Newsgroups: rec.puzzles
Subject: Re: Orange stacks
Date: Tue, 15 Jul 2025 06:32:04 -0000 (UTC)
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On Mon, 14 Jul 2025 11:54:06 -0000 (UTC), Richard Tobin wrote:

>>If I have this right, if I build my pyramids and glue the twelve
>>touching elements to the reference element, making two shapes, both of
>>thirteen elements each,
> 
> Yes.
> 
>>then I finish up with two quite different shapes with quite different
>>symmetries.
> 
> But - hard though it may seem to believe - no!

I'm still struggling with the idea the structures of the square-based and 
triangular-based pyramids are the same.

I think we can agree that there are 90 degree angles between the centres 
of the cannonballs in the layers of the square based pyramid.

In my mind the triangular based pyramid is entirely regular and based on 
the tetrahedral structure with a tetrahedral angle of (approximately) 
109.5 degrees between all centres. There are no right angles.

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

In my mind the square based and triangular based pyramids are not the same 
packing structure.

-- 
David Entwistle