Visualizing the 24-cell

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Visualizing the 24-cell

Postby quickfur » Fri Dec 10, 2004 1:36 am

Thanks to pat's polytope applet that supports clipping (even if it's imperfect), I think I now have a good handle on how to visualize the 24-cell.

The basic idea is to project the 24-cell into 3-space using a vertex-first viewpoint. The envelope of such a projection is the rhombic dodecahedron. (Google for it if you're not sure what exactly a rhombic dodecahedron is).

The first thing to observe is that in 3D, if you have 6 octahedra, each lying along the coordidate axes and touching at a common vertex, and if you can somehow "squish" them together towards the origin so that they touch each other, they will form a rhombic dodecahedron. Of course, the resulting octahedra would not be regular, but would be slightly flattened. But now, imagine that these 6 slightly-flattened octahedra are actually regular, "unsquished" octahedra projected from 4-space. As you know, when you rotate a 3D object into the W-axis, the projection of the object into 3D will "flatten". So each of these flattened octahedra are actually regular octahedra slightly bent into the W-axis, and touching each other. This, in fact, is the first 6 cells of the 24-cell, as viewed vertex-first.

At this point, it's important to understand that there are two ways to bend these cells into the W-axis so that they form a rhombic dodecahedral projection. One way is to rotate them into +W, the other way is to rotate them into -W. Both will give a 4D object that projects on the same volume, but in 4D, one would be curved inwards (into +W) whereas the other would be curved outwards (into -W). So although the 3D projection is a convex rhombic dodecahedron, it's important to understand that in 4D it is actually an open shell (like a hemisphere). For now, let's say we bent the octahedra into +W, and let's say that the vertex projected onto the origin is the "north pole". So these first 6 cells are the cells surrounding the "north pole" of the 24-cell.

Secondly, note that the surface of the rhombic dodecahedron consists of 12 rhombuses. If you take 12 octahedra and rotate them into the W-axis at a 90-degree angle, they will flatten into 12 rhombuses. With the proper arrangement, you can attach them together in 4D such that their projection is a rhombic dodecahedron. Each edge of the rhombic dodecahedron represents a triangular face shared by 2 of the 12 octahedra. This triangular face happens to lie at 90 degrees to our viewpoint, so they appear collapsed into single lines. This leaves 48 "exposed" faces that aren't joined to anything yet: each rhombus represents 2 triangular faces on the octahedron, and the two sides of the rhombus represent opposite pairs of faces on the octahedron. Since there are 12 octahedra, this gives us 48 exposed faces.

Now, note that 24 of these faces are facing "outwards" (i.e., facing -W) and the other 24 are facing "inwards" (facing +W). Because the outward-facing faces form a rhombic dodecahedron, they exactly match the exposed faces on the first 6 cells (the "north pole" cells). Remember that each rhombus on the boundary of the first 6 cells represents two exposed triangular faces of two adjacent octahedra. We can join these faces to the outward-facing faces of the 12 octahedra. This attaches the 12 octahedra to the first 6 octahedra in 4-space. This gives us 2/3 of the 24-cell: the 12 cells we just made are the "equitorial" cells of the 24-cell attached to the "north pole" cells.

Finally, we take another 6 octahedra, and bend them into the W-axis just like that first 6 cells, except that this time, we bend them into -W instead of +W. This gives us a concave arrangement of cells, which form the "south pole" cells of the 24-cell. The outer envelope of these "south pole" cells also project onto a rhombic dodecahedron. The "inward"-facing exposed faces of the equitorial cells match up perfectly, so we join them together, and now we have the complete 24-cell.

So there we have it, the 24-cell, with 6 cells in the first layer (the "north pole" cells), 12 cells in the second layer (the "equitorial" cells - observe how they are precisely at 90-degree angles to our viewpoint, parallel to the W-axis), and 6 cells in the third, final layer (the "south pole" cells).

Does this description make sense to anyone? :-)

(My next task is to try to visualize the 120-cell. Somehow I get the feeling that this won't be quite as easy... :lol: Someday, I might try visualizing the 600-cell, someday, if I manage to get a good handle on the 120-cell first.)
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Addendum

Postby quickfur » Tue Jan 04, 2005 9:15 pm

I just noticed I'd only described the vertex-first projection of the 24-cell.

Well, here's the cell-first projection of the 24-cell: it would be a cuboctahedron, with an embedded octahedron. The apices of the embedded octahedron touch the center of the 6 square faces of the cuboctahedron. The embedded octahedron is the closest cell facing you. If you draw the wireframe of the cuboctahedron with the embedded octahedron, you'll notice that each of the triangular faces of the embedded octahedron faces a triangular face of the cuboctahedron, and the triangles are facing each other in a dual orientation (the vertices match up with the edges and vice versa). These pairs of triangles are actually opposite faces of octahedral cells, which are rotated into 4D in such a way as to fit precisely between the central octahedron and the cuboctahedral envelope.

So, for each face of the central octahedron, there is an octahedral cell attached. This gives us a total of 9 cells. Now, the square faces of the cuboctahedron are actually octahedral cells that happen to be rotated 90 degrees into the W axis, and the apices of the central octahedron is touching the apices of these octahedral cells. On the "other side" of the cells are another 9 cells, in exactly the same arrangement as the first 9, except that they are "behind" in the W direction and so can't be seen.

Since there are 6 square faces in the cuboctahedron, corresponding with 6 octahedral cells, this gives us 9 + 6 + 9 = 24 cells.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

120-cell

Postby wendy » Tue Jan 18, 2005 1:39 pm

There's a really good and easy-to-understand picture of the twelfty-cell in coxeter's 'regular complex polytopes'..

This one projects it so that a girthing band of dodecahedra fall as one.

i rather like it. drew it up many times for the practice.

W

On the other hand, i am trying to get my head around some recent pictures of the 3_21 and 2_31, which are seven-dimensional.
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby Paul » Tue Jan 18, 2005 5:46 pm

Hello Wendy,

Yes... I too have Coexter's Regular Complex Polytopes

Perhaps you can provide a more specific page reference to the "twelfty-cell"...

On the other hand, i am trying to get my head around some recent pictures of the 3_21 and 2_31, which are seven-dimensional.


Fascinating, Wendy. Do this polytopes reside in octonion imaginary space? What system of description of these polytopes is '3_21' and '2_31' referencing? Perhaps you could expand on what you've told us about these polytopes? It sounds most interesting.
Paul
Trionian
 
Posts: 74
Joined: Sat Sep 04, 2004 10:56 pm

Postby wendy » Tue Jan 18, 2005 11:00 pm

twelftycell = twelftychoron = polychoron of 120 faces.

3_21 and 2_31 are the gosset-polytopes having 56 and 126 vertices respectively. The terminology is Coxeter's form, based on the dynkin-symbol. They live in real space, like any other polyecton (7D polytope).

By my short notation (derived from these) we have 3_21 = 5B and 2_31 as 5/B.

Code: Select all
<pre>

  {---- five 3-brabches in chain -}   B     

   @------o-----o-----o-----o-----o     o
                              \---------------/     <-- branch type B 

    x   3   o    3   o   3  o    3 o   3   o   B o    =  /5B = 5B
                                               as with coxeter, no slash makes
                                               the first implied marked.)
   
.                            /------o-----o     Coxeter = chain of 3
.   @----o-----o-----o                        + branchs of 2 and 1
.                            \-------o             =  3_21   (_ means sub)

</pre>


One should note that the octonion-space OE1 is a subset of the euclidean space, E8, but it is apparantly a superset of the quaterion and complex euclidean spaces QE2 and CE4.

However, since the imaginary part of OE1 corresponds roughly to the seven-dimensional world, there is an existance of these in 7D.
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby wendy » Wed Jan 19, 2005 2:30 am

Coxeter's regular-complex-polytopes book is not an easy read. You can get most of the information you need from the vertex-angle (ie from his &sigma; ) in the appendix.

I had to make up the models of the assorted complex polyhedra before any of it made any kind of sense. Still, having a 3{3}3{3}3 lieing around makes it a little easier to visualise, although i can't figure out the right-angles as yet.

I suppose it would have been easier if he put models of the things in an envelope in the back cover or something...

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby wendy » Wed Jan 19, 2005 9:23 am

If you have Coxeter's "regular complex polytopes" , the relevant diagram to look at is figure 4,6 on page 41 of the first edition.

The one on the left is the {3,3,5} or fifhundchoron.

Some of the tetrahedra are quite clear, but as they get towards the outer ring they get quite squashed.

The one on the right is a {5,3,3} or twelftychoron. You see the central dodecahedron (actually a ring of 10), face forward. The next two layers (50 apeice), appear edge-forward, so you only see four pentagons of these. The last 10 appear as the girthing decagon. The equatorial zigzag falls in the middle segment of the line.

None the same, they are hard to visualise without the picture.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby Paul » Wed Jan 19, 2005 3:45 pm

Hello Wendy,

Well,... I have the second edition, and I don't know if there are significant differences between the first and second editions... but, this is a scan of p. 41 of Coxeter's Regular Complex Polytopes.

Image

Page 41 does indeed have a Figures 4.6D and 4.6E, with a {3,3,5} and {5,3,3} depicted, respectively.

Is this what your referencing?
Paul
Trionian
 
Posts: 74
Joined: Sat Sep 04, 2004 10:56 pm

Postby wendy » Wed Jan 19, 2005 11:50 pm

That's the one. I had to look through much of the book to find it though.

Wendy
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby quickfur » Thu Aug 11, 2005 5:54 am

Just another quick note that my 24-cell page has just been updated. Now it has two cool (perspective) projections of the 24-cell.

In the course of producing the second diagram, I rediscovered the incredibly cool fact that the symmetry of the 24-cell corresponds with the union of the symmetries of the tetracube in 3 different orientations. Each of these orientations corresponds with one of the 3 axes of the octahedron (or equivalently, the 3 principle planes of 3-space which intersect the origin-centered octahedron in 3 squares).

If you look at said diagram carefully, you may be able to discern two somewhat flat frustum-like volumes attached to the horizontal plane of the central octahedron. If you examine the diagram even more closely, you may be able to find the equivalent frustum pairs attached to the other two planes of the central octahedron. These pairs each correspond with a face-first view of the tetracube. (In case you're not aware, the face-first perspective projection of the tetracube yields a pair of frustums joined at their bases. We're omitting cells obscured from the 4D viewpoint, of course.) There are precisely 3 possible principal orientations of such views.

The other cool thing about this diagram is that its envelope is an augmented cuboctahedron: with the square faces augmented by flat pyramids. There are two types of vertices, one with degree 4 and the other 6. This shape is closely related to the triakis hexahedron, which is the envelope for the vertex-first perspective projection of the 24-cell. Roughly speaking, one may rotate the pyramidal caps of the triakis hexahedron by 45 degrees, and attaching triangles to the resulting holes, and one would get this shape. This is cool because rotating the 24-cell in 4D also causes this transformation in the projection image (albeit not in the same manner).

Also, I noticed that the self-duality of the 24-cell is different from that of the n-simplex; the simplices are dual simply because every vertex is directly opposite a facet, and vice versa. Not so with the 24-cell; its facets are opposite other facets. One may think of simplex self-duality as being analogous to that of an odd polygon, whereas 24-cell self-duality is analogous to that of an even polygon. The 24-cell earns the distinguishing title of being the only higher-dimensional polytope equivalent to a self-dual even polygon.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North


Return to Other Polytopes

Who is online

Users browsing this forum: No registered users and 18 guests