Deriving the vertex figure of the tesseract

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

Deriving the vertex figure of the tesseract

Postby quickfur » Sat Oct 16, 2010 3:56 pm

This is a reply to a PM, but I'm posting it here 'cos it seems to be generally relevant:

For the vertex figure of the tesseract, it is easiest to look at the vertex-first projection (the one with the rhombic dodecahedron envelope) instead of the usual cell-first projection (the cube-within-a-cube):

Image

The vertex in yellow, at the center of the rhombic dodecahedron, is the vertex closest to the 4D viewpoint. You can see that it has 4 edges joined to it in tetrahedral symmetry. When you cut it with a hyperplane, you can imagine that the hyperplane intersects with the edges, say, half way. Then you just connect the intersection points with each other, and you get a tetrahedron.

In fact, if you carefully trace the edges as you go, you can actually derive all the shapes that the tesseract forms when it intersects with a hyperplane vertex-first. I don't have the diagrams scanned into my computer, but I derived all these figures by hand just by tracing them on the rhombic dodecahedron projection, before I wrote a program to do it for me:

Image

The way to derive it is easy: start at the yellow vertex, then send out 4 points along the 4 edges, moving at the same speed. From the start until they reach the 4 vertices at the corners, they form a tetrahedron. Now at the corners, each point splits into 3 points moving along edges that follow those 4 vertices. That makes it 12 points, and if you trace them out, they form a truncated tetrahedron. Keep going and eventually these points will merge in pairs at 6 vertices, forming an octahedron. Now they split up again in twos, but this time along the two other edges that haven't yet been traversed, thus forming a truncated tetrahedron in dual orientation. Keep going and eventually the points merge in 3's at the top and form a tetrahedron in dual orientation.

Finally, there are 4 edges I omitted from this image, because they lie on the far side of the tesseract; but they essentially connect those last 4 points to the center of the projection, but this time not to the nearest vertex but to the farthest vertex (which coincides with the nearest in projection). Tracing the points along these last 4 edges give you a shrinking tetrahedron that eventually becomes a point again.
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Re: Deriving the vertex figure of the tesseract

Postby Secret » Sun Oct 17, 2010 4:26 am

Hmm that should be helpful

Well can we visualize the 3-plane together with the vertex first cross section of the tesseract?
cause videos out in the internet only show the cross sections in progress but without the 3-plane

if this is possible we can obtain cross sections of various 4 polytopes (polychron) simply from its 3D->2D projection
And watch how the 3-plane slice through the shapes
It can also give us a more natural feel of 4D shapes to bring in more insights
eventually getting the interpretation right to perceive 4D naturally
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Re: Deriving the vertex figure of the tesseract

Postby quickfur » Sun Oct 17, 2010 6:38 pm

Secret wrote:Hmm that should be helpful

Well can we visualize the 3-plane together with the vertex first cross section of the tesseract?
cause videos out in the internet only show the cross sections in progress but without the 3-plane

In this case, the 3-plane could be represented by a cube in which the initial tetrahedron is inscribed. Since the vertex-first projection is, by definition, the projection looking straight at the vertex, so the cutting hyperplane must be orthogonal to the line-of-sight. Which means that it will be a cube with no foreshortening (since it is viewed at face-on).

The problem is that even with 3D polyhedra, the way we visualize the vertex figure is usually from an angle, with the cutting plane moving across our field of vision as it cuts through the polyhedron. This is because it's rather hard to imagine a plane we're looking at directly moving forwards towards the polyhedron and cutting it; most of the intersecting movement lies in the Z axis (if you take Z as the line-of-sight), which is collapsed in the scene-to-eye projection. So we usually imagine looking at the intersecting process from a slightly angled viewpoint, so that we can see how the plane moves as it intersects with the object.

However, in the 4D case, it is not as easy to understand the resulting projections if we're looking at the object and the cutting hyperplane from an angle. It's still possible, though; and the idea is certainly seems to be a promising one. Maybe one of these days I'll actually tune my polytope projector program to render these sort of scenes, and see what the results look like.

if this is possible we can obtain cross sections of various 4 polytopes (polychron) simply from its 3D->2D projection
And watch how the 3-plane slice through the shapes
It can also give us a more natural feel of 4D shapes to bring in more insights
eventually getting the interpretation right to perceive 4D naturally

IMNSHO, taking cross sections isn't really the easiest way to approach 4D visualization. I much prefer working with projections, since that corresponds with the way our own eyes see 3D objects, and so is easier to develop an intuitive feel for.

On the other hand, though, to really visualize 4D we ultimately need a way of intuitively working with intersections and assembly (the joining of 4D objects together---such as 8 cubical pyramids with a tesseract to form a 24-cell). I've already developed a feel for dealing with convex vs. convex objects, non-convex objects, and partial obscuring of objects (such as things partly hidden behind walls, or viewed through a small window). But I've yet to develop these fully in a way that can be presented to others in a coherent way.
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Re: Deriving the vertex figure of the tesseract

Postby Secret » Mon Oct 18, 2010 10:22 am

quickfur wrote:
Secret wrote:Hmm that should be helpful

Well can we visualize the 3-plane together with the vertex first cross section of the tesseract?
cause videos out in the internet only show the cross sections in progress but without the 3-plane

In this case, the 3-plane could be represented by a cube in which the initial tetrahedron is inscribed. Since the vertex-first projection is, by definition, the projection looking straight at the vertex, so the cutting hyperplane must be orthogonal to the line-of-sight. Which means that it will be a cube with no foreshortening (since it is viewed at face-on).

The problem is that even with 3D polyhedra, the way we visualize the vertex figure is usually from an angle, with the cutting plane moving across our field of vision as it cuts through the polyhedron. This is because it's rather hard to imagine a plane we're looking at directly moving forwards towards the polyhedron and cutting it; most of the intersecting movement lies in the Z axis (if you take Z as the line-of-sight), which is collapsed in the scene-to-eye projection. So we usually imagine looking at the intersecting process from a slightly angled viewpoint, so that we can see how the plane moves as it intersects with the object.

However, in the 4D case, it is not as easy to understand the resulting projections if we're looking at the object and the cutting hyperplane from an angle. It's still possible, though; and the idea is certainly seems to be a promising one. Maybe one of these days I'll actually tune my polytope projector program to render these sort of scenes, and see what the results look like.

if this is possible we can obtain cross sections of various 4 polytopes (polychron) simply from its 3D->2D projection
And watch how the 3-plane slice through the shapes
It can also give us a more natural feel of 4D shapes to bring in more insights
eventually getting the interpretation right to perceive 4D naturally

IMNSHO, taking cross sections isn't really the easiest way to approach 4D visualization. I much prefer working with projections, since that corresponds with the way our own eyes see 3D objects, and so is easier to develop an intuitive feel for.

On the other hand, though, to really visualize 4D we ultimately need a way of intuitively working with intersections and assembly (the joining of 4D objects together---such as 8 cubical pyramids with a tesseract to form a 24-cell). I've already developed a feel for dealing with convex vs. convex objects, non-convex objects, and partial obscuring of objects (such as things partly hidden behind walls, or viewed through a small window). But I've yet to develop these fully in a way that can be presented to others in a coherent way.


Speaking of intersections and assembly, check out my illustrations at my other thread, which includes the assembly of the pentachoron and the tesseract step by step. (still working on the others though)

For manual hidden suface removal, i only manage to done on the tesseract and obscuring 4D objects using a cubic wall so far

Also I'm not just using the cross sections or the projections to show a 4D object, I'm trying to see whether it is possible to use both to render a full sequence of operations on the 4D objects. We can start on the simpler shapes. Once will finally correctly interpret the 4 depth (which is the core of 4D visualization), we can extend the process to more complicated shapes such as the grand antiprism and the truncated 24-cell

For the 3-plane together with intersections shown, I'm actually meaning to cut from the vertex but not from the vertex figure, instead seeing the process from an angle in 4D. Anologuous to what we render cubes cut by a plane starting from its corner. I think the combination of all 4D visualization methods in video form would greatly help us in interperting the projections correctly
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Re: Deriving the vertex figure of the tesseract

Postby quickfur » Mon Oct 18, 2010 3:07 pm

Speaking of the grand antiprism, here's an animation I did just yesterday:

Image

This shows the two rings of 10 antiprisms each, rotating in the plane that the blue/cyan ring lies in. I've omitted the tetrahedral cells for clarity's sake; you can see their structure on my grand antiprism page.

The blue/cyan ring is circular, just like the red/purple ring, but since the plane it lies in is orthogonal to the 4D viewpoint, it collapses into a "tube-within-a-tube" in projection. The "inner tube" is the far half of the ring, and the "outer tube" is the near half.
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Re: Deriving the vertex figure of the tesseract

Postby Klitzing » Wed Oct 24, 2012 9:15 am

There is an easy way to derive the vertex figure of any Wythoffian polytope (i.e. one which can be derived by his kaleidoscopical construction - which mostly comes down to: having a Dynkin symbol with ringed/unringed nodes only) right from the mere Dynkin symbol!

In fact, this is why Wendy once invented the topics of lace prisms etc.
The actual technique already was outlined in a post of an other thread.
(The background on Wendys lace thingy zoo, including an explanation of the used notations, likewise was outlined in an other post of that thread.)

--- rk
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