# Message #2689

From: schuma <mananself@gmail.com>

Subject: Re: About me

Date: Wed, 20 Mar 2013 05:59:15 -0000

> What version of the 24 cell would you like me to solve? There are quite a few!

In MPUlt, the 24-cell_FT is the shallow-cut puzzle that supports all the moves. This is THE 24-cell puzzle that got most attention, I think.

> Maybe even a 120 cell pentultimate would be an idea for a future 4d puzzle!

In the latest MPUlt, there is a 120-cell_halfcut. It’s the Face turning 120-cell where the cutting plane passes the center. I think it can be considered as the Pentultimate in 4D. I don’t think anyone has attempted it. But I’m sure you have enough courage to make the move.

> My mind sometimes cannot sleep because of ideas running through my head, but it is sooo discouraging that I don’t have funds or proper equipment to make and sell them…

I also like thinking about puzzles. For me, some ideas end up being simulators people can play on computers. (see http://nanma80.github.com/) Compared to physical puzzles, it’s easy to test the concept, easy to share among friends all over the world, easy to incorporate feedbacks and suggestions and iterate to the next version, and of course, go beyond physics, like allowing reflection moves. About the ideas of physical puzzles, some people go to twistypuzzles.com to share some concepts, or, even internal mechanism. Sometimes, other builders bring them into reality, and give credit to the original contributor. Oskar van Deventer, for example, made a lot of puzzles. But he acknowledged many people for their original ideas. I think the users of this forum have a pretty healthy attitude on recognizing the ideas, thanks to good management by David Litwin and others.

About the regular polytopes in high dimensional space, Melinda said,

> Cracking my bible (Regular Polytopes) to chapter and verse 7-8 gives …

Believe it or not, the first thing after I got home today was also checking this book. Before this verse, Coxeter was calculating the dihedral angles. I think the argument can be explained using this example.

The dihedral angle of a 3D cube is 90 degrees. When you use some cubes to make a 4D regular polytope, how many cubes can you fit around an edge? It’s necessary that the sum of dihedral angles is less than 360 degrees to make a bounded 4D polytope, or equal to 360 degrees to make an unbounded tessellation. So, two cubes are too few (the outcome is flat). Three cubes around an edge are OK (hypercube). Four cubes around an edge will form the tessellation. So three cubes around an edge is the only valid way to make a 4D bounded polytope using cubes as faces, in Euclidean space. The necessary condition is that the dihedral angle must be small enough. As I understand, equation (7.77) formalizes this argument in math.

To make 5D regular polytopes, we need to pick some 4D regular polytopes to be the faces (that’s part of the definition of regularity). It turns out, the dihedral angles of all the special 4D polytopes are just too large to fit around an edge. So they cannot be used. So in 5D, we only have {3,3,3,3}, {4,3,3,3} and {3,3,3,4}.

In 6D, the Schlaefli symbol needs to be {p,q,r,s,t}, where {p,q,r,s} may be {3,3,3,3}, {4,3,3,3} or {3,3,3,4}, and {q,r,s,t} may be {3,3,3,3}, {4,3,3,3} or {3,3,3,4} (this is the definition of regularity). So q,r,s have to be 3,3,3. And you can’t make anything but {3,3,3,3,3}, {4,3,3,3,3} and {3,3,3,3,4}. Without fancy building blocks, you just can’t make fancy stuff. By the same argument (induction), these polytopes are the only regular ones in higher dimensions.

You may argue that the definition of regularity is too strong. If we relax that, maybe we get more interesting things.

Nan

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:

>

> In dimension 5 and above, there are only 3 kinds of regular polytopes: the

> n-simplex, the n-cube, and the n-orthoplex.

>

> http://en.wikipedia.org/wiki/List_of_regular_polytopes#Five-dimensional_regular_polytopes_and_higher

>

> So dimension 4 is very special, having 6 different flavors of regular

> polytopes. Dimension 3 is also special, having 5. If anyone could give

> insight into *why* things change for dimension 5, please do share.

>

> Even though there is no "5D dodecahedron", there are 5D polytopes that are

> at least reminiscent of the dodecahedron. They just aren’t regular. For

> instance, you could make a prism based on the 120-cell, aka a {5,3,3}x{}.

> I bet it’d be a horrific puzzle though!

>

> Cheers,

> Roice

>

>

> On Tue, Mar 19, 2013 at 8:18 PM, Philip Strimpel <iamrubikman@…>wrote:

>

> >

> > Hello Melinda,

> > Many thanks for the kind welcome! :) What version of the 24 cell would

> > you like me to solve? There are quite a few! Also, I don’t know if it is

> > because my computer is two slow or not, but I can’t seem to understand how

> > to twist anything besides 3^4 and 120 cell. I would really enjoy attempting

> > some of the 5 and 6d puzzles. My computer is waaay too slow to even try the

> > 600 cell though. :( Now THAT would be awesome to solve! Maybe even a 120

> > cell pentultimate would be an idea for a future 4d puzzle! : I am curious

> > though… How come there can’t be a 5d dodecahedron? I know it would

> > probably have hundreds or thousands of cells to it, but does nobody know of

> > it because it would be too big to comprehend, or is it really virtually and

> > physically impossible? Has anybody else thought of this? Just something to

> > bring to the table…

> >

> > Best regards,

> > Philip

> >

> >

> > ————————————

> >

> > Yahoo! Groups Links

> >

> >

> >

> >

>