Message #3470

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] 4D puzzle math
Date: Wed, 20 Jul 2016 14:52:24 -0500

Yep, Ashton is right. Those are Schläfli symbols
<https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol>. Here are a few
articles discussing them (in the context of honeycombs, but the info is
there):

• Hexagonal Hyperbolic Honeycombs
  <https://johncarlosbaez.wordpress.com/2014/05/14/hexagonal-hyperbolic-honeycombs/>,
  by John Baez
  
• The dual {5,3,4} and {4,3,5}
  <http://roice3.blogspot.com/2013/09/the-dual-534-and-435.html>

In the case of the hypercube…

• {4} designates a square.
  
• {4,3} means squares with 3 squares surrounding each vertex (i.e. a
  cube).
  
• {4,3,3} means {4,3} cubes with 3 cubes surrounding each edge, the
  hypercube.

Sometimes the symbols are not unique, e.g. the {4}x{4} (a duoprism
<https://en.wikipedia.org/wiki/Duoprism>of square prisms) is also a
hypercube, as is {4,3}x{} (a cubical prism).

I don’t think there is a Schläfli symbol for a rectangular cuboid, so
unfortunately you won’t be able to make what you are wanting in the "invent
my own" functionality of MC4D. Andrey’s magic puzzle ultimate supports
these though, and includes some by default like the 2x2x2x3 and the
2x3x4x5. (Note that to make a 4D equivalent of a 3x3x5, you’ll need to
choose a 4th number!) You could take a look at his configuration for those
in the MPUlt_puzzles.txt file, and adapt to make your own new cuboid
puzzles of whatever size.

Roice

On Wed, Jul 20, 2016 at 2:25 PM, Ashton Santee ashtonsantee@gmail.com
[4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:

>
>
> I believe you are looking for more information about that specific
> notation. I don’t know enough about it to tell you how to read it but there
> is a Wikipedia page on this notation.
> https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol
> "Regular n-polytopes" is the section that has the definition of a
> hypercube so that might be a good place to start then see how this notation
> can be used for other multi dimension shapes.
> I am not sure what inputs MC4D will accept and not accept in this
> notation. You will have to experiment with it and report back and let us
> know what the limits of this notation are in MC4D.
>
> On Wed, Jul 20, 2016 at 11:28 AM tytheguy246@gmail.com [4D_Cubing] <
> 4D_Cubing@yahoogroups.com> wrote:
>
>>
>>
>> Does anyone know about the numbers of four-dimensional puzzles? For
>> example, why is {4,3,3} a hypercube? Also, how do I "invent my own" as the
>> button says? Would a 4D equivalent to the 3x3x5 be possible using this
>> code? If anyone knows about an article about this or would just like to
>> tell me, please let me know!
>>
>> Thanks,
>>
>> Ty
>>
>> –
> Ashton Santee
> 916-7766-775
>
>
>
>