Thread: "Regular Polytopes in 4D"

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There are only 6 regular polytopes in 4D.

* The 5-cell corresponding to the tetrahedron in 3D
* The 8-cell corresponding to the cube in 3D
* The 16-cell corresponding to the octahedron in 3D
* The 24-cell corresponding to nothing in 3D
* The 120-cell corresponding to the dodecahedron in 3D
* The 600-cell corresponding to the icosahedron in 3D

Which of these are offered as sliced playable programs?

I know 5-cell, 8-cell and 120-cell.


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There are only 6 regular polytopes in 4D.

• The 5-cell corresponding to the tetrahedron in 3D


• The 8-cell corresponding to the cube in 3D


• The 16-cell corresponding to the octahedron in 3D


• The 24-cell corresponding to nothing in 3D


• The 120-cell corresponding to the dodecahedron in 3D


• The 600-cell corresponding to the icosahedron in 3D

Which of these are offered as sliced playable programs?

I know 5-cell, 8-cell and 120-cell.

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Yep, the 5-cell, 8-cell, and 120-cell are the only ones offered by MC4D
currently. These are the 4D regular polytopes having simplex vertex figures
(4 cells meeting at each vertex). That restriction simplifies the coding
situation (and puzzle difficulty for that matter).

I'm also not aware of any software which offers the others. Andrey has
talked about making 4D twisty puzzles without simplex vertex figures though,
and at the prodigious rate he's been producing puzzles, I wouldn't be
surprised to see one or more of the others realized in the not too distant
future.

Despite their nonexistence, there has still been lively discussion here
lately about the nature of these puzzles. See this
threadfor
instance.

Cheers,
Roice

On Wed, Feb 16, 2011 at 3:54 PM, Eduard wrote:

>
>
> There are only 6 regular polytopes in 4D.
>
> - The 5-cell corresponding to the tetrahedron in 3D
> - The 8-cell corresponding to the cube in 3D
> - The 16-cell corresponding to the octahedron in 3D
> - The 24-cell corresponding to nothing in 3D
> - The 120-cell corresponding to the dodecahedron in 3D
> - The 600-cell corresponding to the icosahedron in 3D
>
> Which of these are offered as sliced playable programs?
>
> I know 5-cell, 8-cell and 120-cell.
>

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That's not entirely correct. Don't forget the four Kepler--Poinsot star
polytopes and
the three regular skew polyhedr
a
. These are every
bit as regular as the regular convex polyhedra and deserve equal legitimacy.

-Melinda

On 2/16/2011 1:54 PM, Eduard wrote:
>
> There are only 6 regular polytopes in 4D.
>
> * The 5-cell corresponding to the tetrahedron in 3D
> * The 8-cell corresponding to the cube in 3D
> * The 16-cell corresponding to the octahedron in 3D
> * The 24-cell corresponding to nothing in 3D
> * The 120-cell corresponding to the dodecahedron in 3D
> * The 600-cell corresponding to the icosahedron in 3D
>
> Which of these are offered as sliced playable programs?
>
> I know 5-cell, 8-cell and 120-cell.
>
>
>
> __._,_.__

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That's not entirely correct. Don't forget the four Kepler–Poinsot href="http://en.wikipedia.org/wiki/Kepler-Poinsot_polyhedra">star
polytopes and the three href="http://en.wikipedia.org/wiki/Infinite_skew_polyhedron">regular
skew polyhedr href="http://en.wikipedia.org/wiki/Infinite_skew_polyhedron">a.
These are every bit as regular as the regular convex polyhedra and
deserve equal legitimacy.



-Melinda



On 2/16/2011 1:54 PM, Eduard wrote:

There are only 6 regular polytopes in 4D.

• The 5-cell corresponding to the tetrahedron in 3D


• The 8-cell corresponding to the cube in 3D


• The 16-cell corresponding to the octahedron in 3D


• The 24-cell corresponding to nothing in 3D


• The 120-cell corresponding to the dodecahedron in 3D


• The 600-cell corresponding to the icosahedron in 3D

Which of these are offered as sliced playable programs?

I know 5-cell, 8-cell and 120-cell.

__._,_.__

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Okay: 6 regular convexe finite polytopes.
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>

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Oops, I just realized that I was mixing up my dimensions and was talking
about regular 3D polytopes. In 4 dimensions there are regular star and
infinite polytopes but I don't know how many there are.

I'll just say one more thing about infinite polytopes: Although they
include an infinite number of repeated units when realized in a flat
infinite space, they are more naturally considered as /finite /polytopes
that live in finite, repeating spaces, exactly as Roice has shown with
his Magic Tile program. His images appear to have an infinite number of
polygons but they really have a finite number which is why you can solve
them. So what we call infinite might better be called "repeating", and
they deserve to be considered as first-class regular polytopes along
with the regular convex and star polytopes. I think that we tend to
disparage these varieties because they are harder to get our heads
around, but the math is just as elegant when spaces repeat or polygons
intersect with each other or with themselves.

-Melinda

On 2/17/2011 3:21 AM, Eduard Baumann wrote:
>
>
> Okay: 6 regular convexe finite polytopes.

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Oops, I just realized that I was mixing up my dimensions and was
talking about regular 3D polytopes. In 4 dimensions there are
regular star and infinite polytopes but I don't know how many there
are.



I'll just say one more thing about infinite polytopes: Although they
include an infinite number of repeated units when realized in a flat
infinite space, they are more naturally considered as finite polytopes
that live in finite, repeating spaces, exactly as Roice has shown
with his Magic Tile program. His images appear to have an infinite
number of polygons but they really have a finite number which is why
you can solve them. So what we call infinite might better be called
"repeating", and they deserve to be considered as first-class
regular polytopes along with the regular convex and star polytopes.
I think that we tend to disparage these varieties because they are
harder to get our heads around, but the math is just as elegant when
spaces repeat or polygons intersect with each other or with
themselves.



-Melinda



On 2/17/2011 3:21 AM, Eduard Baumann wrote:

type="cite">

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Okay: 6 regular convexe finite
polytopes.