Quickfur's renders

[quickfur]

Hi all,

For your viewing pleasure, I present:



Octagonny!

What's that?!

Octagonny is a 4-dimensional polytope with 48 identical cells. Each cell is a truncated cube (a cube with its corners cut off). It is properly known as the "bitruncated 24-cell", and, as this name implies, is derived from the 24-cell by truncating its vertices at half the depth it takes to get to its dual.

Octagonny is very cool because it is not among the 6 regular polychora, yet has identical cells. The 4D Catalan solids (the duals of the uniform polychora) also have identical cells, but this one is special because its cells are uniform, whereas the cells of the Catalan solids are non-uniform. The only other 4D polytope with this property is the bitruncated 5-cell, a 10-celled creature having truncated tetrahedra as cells.

Erm, but isn't that picture just a 3D object?

Indeed, the picture is a 3D object... of the image of the octagonny projected into 3-space via a perspective projection. In other words, it's the image that would form in the retina of a Tetronian when she looks at an octagonny. The rotation is happening entirely in 3D, so you don't see any of the cool rotation effects when something rotates in 4D. I added in the rotation because it helps to see the 3D structure of the projected image better.

Aha! I see missing/wrong ridges in your animation!

Yes, yes, I know, something isn't quite right with the face lattice I used to render the image. Some of the ridges are missing some vertices for some reason, and some have been omitted because they appear to be truncated (only 2 vertices, not sure why). I generated the polytope data from a 24-cell using Komei Fukuda's cdd solver, by computing the dual 24-cell, then hand-merging the vertices of both together, and then computing the resulting face lattice. (The centres of the octagonny's cells are precisely the vertices of a 24-cell and its dual, so once I have the vertices of a 24-cell and its dual, suitably scaled, of course, I have the normal vectors to the octagonny's cells. From there, I run cdd to compute the resulting polytope.) Obviously, something went wrong somewhere in the process. However, due to the large number of vertices and ridges (288 vertices, 576 edges, and 336 ridges), I can't possibly sort through everything manually to find out what's wrong.

I probably have to regenerate the data files, and merge them more carefully to find out what exactly has gone wrong. It might also be a problem in the program I wrote to do the projection and generate the povray polygons.

But at any rate, I've been dying to see the octagonny for years... now, finally I can confirm that what I saw in my mind's eyes was correct. In fact, just looking at the animation gives me new insights into the structure of the octagonny. It's amazing how a picture is really worth more than a thousand words.

[quickfur]

I've written a script to consistently convert cdd's output format to the internal representation my projection program uses, so now the wrong faces are fixed. Here is the new animation of octagonny:



I've highlighted the central cell (the one closest to the 4D viewpoint), so that the structure of the rest of the projection is more easily distinguished.

The central cell is a truncated cube, lying at the "north pole" of the octagonny, if you will. Around it are (a) 8 truncated cubes in the positions of the cube corners (the octants), and (b) 6 truncated cubes in the positions of the cube faces (the cardinal directions). I've culled the hidden surfaces so that the result is not an incomprehensible tangle of polygons; these 15 cells precisely tile the 3D projection envelope without overlap, and lie on the "north hemiglome" of the octagonny.

Not seen here are (c) 6 other truncated cubes attached to the (b) cells, which may be thought of as lying at the "equator" (3-quator?) of the octagonny.

On the far side of the octagonny (the "south hemiglome") are (d) another 8 truncated cubes in the octants positions, attached to (e) the opposite central truncated cube (lying at the "south pole"), and (f) another 6 truncated cubes in the cardinal directions position, also attached to the south pole cell. The (a) cells on the "north hemiglome" are attached directly to the (d) cells on the "south hemiglome", while the (b) cells are attached firstly to the (c) cells, which are attached to the (f) cells.

(Sigh, the structure is so obvious in 4D, but the description is so wordy 'cos we don't have convenient terminology to refer to different parts of the 3-sphere. Any help from the PG, wendy? )

At any rate, this is a really beautiful structure. Now, I just have to work on my script a little more to deal with numerical precision problems in a saner way, and I should be able to project the 120-cell soon!

[wendy]

While it is common to give "pet" names to four and five dimensional things, the matter becomes more complex as the dimensions get higher. One has many words to juggle for basic meaning, when a regular structure gives many words and many meanings easily.

Words are divided into 'edge-form' (of fixed dimensions) and "mark-forms" (relation to solid space).

Edge-forms are given in terms of 'fabric of space' (eg hedrix, pl hedrices) and "patches of space" (eg hedron, pl. hedra). A term exists for each kind of fabric, these end in -ix. 0d TEELix, 1d LATRix, 2d HEDRix, 3d CHORix, 4d TERix, 5d PETix, 6d ECTix, 7d ZETTix, 8d YOTTix. In this series, PLANix refers to N-1 dimensions, plane = PLANIX, while SOLix is same-space (eg SOLID = SOL + ID, refers to a fabric in all-space with an exact boundary). Note one can talk of solid regions in 2d (ie a solid region of ink on a page, meaning that the ink and the space it is embedded in are the same dimension), or a solid range in a line.

One prefixes a patch-term to designate a figure formed of such patches or fabrics. A fabric is taken to mean the entirity of the thing, while a patch is part of a larger thing, so

POLY + HEDR + ON poly = many (+closure) + 2d + patches = (surface of) 3d solid
POLY + CHOR + ON poly = many (closed) + 3d + patches = (surface of) 4d solid

MULTI + HEDR + ON multi = many (no implied closure) + 2d patches = eg net of a cube [which is not closed]

Using the fabric stem with -ID or -OUS, refers to the notion that the shape described is like that shape exactly (eg a peice of paper, represents 2d fabric almost exactly, even when scrunched up, is HEDRID, while a mattress only roughly approximates it, is HEDROUS. In general, the -ID refers to a more exact approximation than -OUS, so where SOLID refers to something that has an exact boundry in all-space (like a cube or sphere), SOLOUS might refers to a less-than-exact boundary (like a cloud). A HEDRID is -any- object of 2d with an exact boundry, eg a hexagon or circle, regardless of the embedding space.

Spheres are refered to by the GLOMO- symbol

GLOMO+ HEDR + ON = globe-shaped + 2d + patch (bounded) = surface + content of a 3d sphere
GLOMO + HEDR + IX = globe-shape 2d space = "S2" (just the surface as "all-space"

Sphere can then be applied to any dimension, usually taken as solid. Ball is taken as a solid sphere of N dimensions, DISK is of N-1 dimensions.

HORO- is used of Euclidean space, eg we live in a horochorix (euclidean + 3d + fabric). BOLLO refers to hyperbolic space.
PLATO- refers to "flat" space (ie of the same curvature as all-space, while GEO- refers to an embedded space (geodesic).

The dual of an edge (line between points) is a margin (margin between faces). Just as n-edge refers to a surtope of n dimensions, so does an n-margin refers to a surtope of N-n-1 dimensions. A 0-edge is a vertex, a 0-margin is a face.

MARGIN something that divides a surface, eg the limits of a face.
Margin-angle = angle or cornering of an angle (ie "dihedral angle").

One allocates then stems (word-peices) to margin-dimensions.

-2: N+1 hyper (used when one needs to go 1d higher than all-space: eg hyperspace in 4d is actually 5d)
-1 N solid, sphere,
0 N-1 plane, face, facing, facet,
-1 N-2 margin,

For tilings of M dimensions, we have

M cell
M-1 wall
M-2 sill

These correspond to, eg using cell, wall to refer to hexagons in a hexagonal tiling. The terms are not dependent on cells being "solid". For "solid" space, M becomes N, so a "solid wall" divides "solid cells".

W

[quickfur]

Actually, I was thinking more of terms that describe the relative positions of surtopes in a projection. E.g., in 3D we have north/south poles, east/west hemispheres, equators, etc.. In 4D we also have ana/kata poles/hemispheres thrown into the mix. Which is not so much a problem, except that when speaking of a projection, one needs mix terms that refer to the n-sphere and the (n-1)-sphere, and it would be nice to have short terms that unambiguously refer to some part of either sphere. Preferably, one should be able to speak of the north pole of the 2-sphere in the projection image as corresponding with the ana hemiglome in the original space.

The limitation of current terms is also very obvious when one has to resort to such convoluted expressions as "upper-left-front octant of the 2-sphere" or "lower-right-kata hexadectant of the 3-sphere". It gets worse when one wishes to refer collectively to, say, the 12 regions of the 3-sphere corresponding with the ana edges of the tetracube, etc.. Turning these already convoluted expressions into adjectives makes for quite a mouthful to express something that is really very simple in its native space.

[wendy]

Where projection looses dimension (such as rendering eight dimensions to two), one might regard this as a "division", that is, each space of the divisor is rendered as a point. For example, the ground-plan of a house is the house "divided" by the height. For cartesian product, this reverses the multiplication.

The present models represent E3 to E3, because the initial mapping is in four dimensions, as S3 (surface of a solid), to E3 (its projection on a choric page). As with S2 to E2, or E2 to E2, we need to look at maps, etc to see analogs. So a chorix (eg S3, H3, E3), can be projected to a chorix (eg E3 paper), by way of projections like orthographic, stereographic, gnomic, inversive, beltrami-klein, poincare, co-circumference, etc. All of these are point to point functions on the same space (eg Hn -> En).

We see E3 as E2, with the depth from the eye being a learnt device. One imagines in 4d, that E3 is seen as we see E2: every point without "going through" its neighbours. For us, this means holographic vision. The way we implement this is to suppose we can "see through" structure, or see bits of it removed. It's kind of like looking at a plan from the edge of the page (across it).

In any case, the implementation of holographic vision can be done by one of the following:

a) One can show the latroframe (vertices + edges), selectively with assorted faces (3d) and margins (2d) filled in. The latroframe can be right (straight lines between vertices), or true (edges projected correctly).

b) One can project it as a translucent foam of cells. Cells can then be coloured differently.

c) One can show it as a series of pictures showing interior to exterior (of the model), as successively added shells.

The present models are orthographic projections, rendered as a translucent foam.

Motion is shown, this represents the 'wheel' rotation in four dimensions. This represents rotation in two axies only.

Since one is presenting a three-dimensional object, it is correct to use 3d terminology. The main difference is that the model is intended to be viewed holographically, (ie each point separate), not translucently (ie look through points to other points). It is none the less best to rely on the users past experience here, and suppose that the model has poles and a axis through these poles. So North and South is appropriate here.

Even were the model one of a clifford rotation, one would still use these reference points. In clifford rotation, a cell at the north pole would elongate and contract in the same manner that a cell at the west would in going east. So you have a rotation that is linear from the north-pole through the centre to the south-pole, and other cells would orbit the centre in assorted ellipses around the centre. (in clifford rotation, all rotations are circular, and these present as circles to ellipses to a line, because we are "dividing" 4-space by the lines radiating from the eye to get 3-space).

[quickfur]

Here's another pretty animation of octagonny:



This one has several enhancements: the central cell, closest to the 4D viewer, has been highlighted (in fact, rendered in opaque texture to reduce visual clutter), and edges and vertices are shown except for those lying on the central cell.

Oh, also, this is a parallel projection instead of a perspective one.

[Keiji]

Octagonny?

I call it Koshire

[quickfur]

"Octagonny" is Wendy's pet name for it. I myself have no name for it, although its more formal name is "bitruncated 24-cell".

I (re)discovered it while trying to visualize what truncated polychora would look like in projection. (My usual method of working with polychora and other 4D shapes is to visualize them in 3D projection, which I think is the best way of understanding them.) I was ecstatic to realize that it must be cell-transitive because the 24-cell is self-dual---facet-transitive figures appeal to me a lot for some reason. (One of these days I should run a 4D Catalan through my program just to see what it looks like.) Anyway, after some consideration, I managed to get a little glimpse in my mind of what Octagonny's 3D projection would look like. I made a 2D projection of it, which turns out to be a figure with lots of octagons: a bounding octagon, with 8 smaller octagons surrounding a central octagon. You can sorta see this figure in the animation in passing. The name "octagonny" does seem quite fitting in this context.

I discovered other things about it, such as the fact that its cells can be partitioned into 2 sets of 24, corresponding to the cells of a 24-cell and its dual, such that cells within one group only connected to the other group via their triangular faces, and to each other via their octagonal faces. If you start with one cell and move along a straight line (well, a "great circle" on the glome) through its octagonal face, and do the same until you return to the starting cell, you get a cycle of 8 cells. If you do this passing through triangular faces instead, you get a cycle of 6 cells.

But it was a long time before I could finally render it in full 3D projection, confirming that what I saw in my mind's eye was correct.

[wendy]

Octagonny, o3x4x3o and its isomorph octagrammy o3x4/3x4o, are fairly important figures, that rate similar to the regular figures.

There are, by clifford rotations, figures that correspond to the symmetries in three dimensions.

[2,2] 8 tesseract
[3,3] 24 24choron
[3,4] 48 octagonny
[3,5] 120 twelftychoron

All of these produce very interesting symmetries, where the 'arrow-rotations', project to six-dimensional figures, being the prism-product of euclidean figures, and their doubles.

The vertices of the duals of these, correspond to the units of quaterions (both integral and finite-dense).

Class-2 isomorphism can be presented in terms of replacing a+bx by a-bx, usually where x² is an integer. Since we have here "partial inversion", we can replicate the lattices

{12,12/5} by {3,6} + {3,6} (dual hexagons)
{8,8/3} by {4,4} + {4.4} dual squares
octagonny by {3,3,4,3}, dual 24chora.

The tiling of octagony, 288/73 at a vertex, is the first nonhyperbolic group that has no expression in Coxeter-Dynkin symbols. It requires six mirrors to start.

There are class=2 tilings that need only five: eg {5,3,3,5/2}.

It is in light of these special alignments with Z4 (octagonal numbers), that S/Q/S was titled 'octagonny'.

Wendy

[quickfur]

I'm in the process of a major facelift of my 4D website, and in the process, I've added a page that describes where exactly each and every one of the 600-cell's cells fit into the overall shape:

http://eusebeia.dyndns.org/4d/600-cell.html

Let me know what you think.

[Keiji]

Very nice description

Do you think you could add

Code: Select all

img {border: 0;}

to your stylesheet, though? Blue borders on the footer images are ugly.

[quickfur]

Very nice description

Do you think you could add

Code: Select all

img {border: 0;}

to your stylesheet, though? Blue borders on the footer images are ugly.

Done. It didn't show up in my browser (Opera), although it does show up in Firefox. Did it show up in IE? (I don't have IE to test on.)

[Keiji]

Me neither, but I know from experience that it would have done.

[quickfur]

I've added a new page on my 4D website to show off the projections of octagonny, also known as the bitruncated 24-cell:

Bitruncated 24-cell

Both parallel and perspective projections are featured. Any comments?

In particular, I hope people like my approach of dissecting a 4D polytope by comparing it to the globe: you have a reference point designated the "north pole", which defines a northern hemisphere separated from the southern hemisphere by an "equator" (perhaps more appropriately, a 3-quator, being a spherical divider). From this, we can draw analogies from latitudes and longitudes to describe the placement of the cells.

So far, this approach has worked fairly well for the more complicated polychora like the 600-cell, but we'll have to see if this is still true for something at the level of complexity of the omnitruncated 600-cell (which has 2640 cells, 17040 faces, 28800 edges, and 14400 vertices. Woohoo!).

[wendy]

The model is very good.

The comment at the end about vertex+face transitivy for the S/Q/S and S/S/S should also note that the regular figures and the p*# p prisms are also so, but the p *# q prisms are not (since they have faces of two different kinds!). S/Q/S = octagonny, S/S/S = bitruncated pentachoron.

It goes with note, that the truncated-cubes in the octagonny are like the dodecahedra in the twelfty-chora (5,3,3), in so far that they form a "reduced spheric", the dodecahedra being 'poincare dodecahedra'. [cubes in 4,3,3, and octahedra in 3,4,3 also do this].

For what it's worth, the projection is a 'layered orthogonal projection over a latroframe'.

An orthogonal projection is usually implemented by removing a co-ordinate, eg project w-x-y-z onto x-y-z. Since this gives a solid, and not a plane object, a second projection is to remove a second axis, eg x-y-z to x-y.

The latroframe is the frame formed by the vertices + edges. This can be seen through in 3d, and provides a reference for the subsequent pictures.

The faces of the figure are then shown from the innermost (the one with the greatest w), to the outermost in successive layers. Usually, this is done with a translucent material, so that faces behind it are shown (as gel-cells in a edge-frame).

[Keiji]

Very nice, I added all the information to the HDDB page.

[quickfur]

The model is very good.

The comment at the end about vertex+face transitivy for the S/Q/S and S/S/S should also note that the regular figures and the p*# p prisms are also so, but the p *# q prisms are not (since they have faces of two different kinds!). S/Q/S = octagonny, S/S/S = bitruncated pentachoron.

Good point, I completely forgot about the n,n-duoprisms.

It goes with note, that the truncated-cubes in the octagonny are like the dodecahedra in the twelfty-chora (5,3,3), in so far that they form a "reduced spheric", the dodecahedra being 'poincare dodecahedra'. [cubes in 4,3,3, and octahedra in 3,4,3 also do this].

What's a reduced spheric?

For what it's worth, the projection is a 'layered orthogonal projection over a latroframe'.

An orthogonal projection is usually implemented by removing a co-ordinate, eg project w-x-y-z onto x-y-z. Since this gives a solid, and not a plane object, a second projection is to remove a second axis, eg x-y-z to x-y.

One thing I always like to emphasize here is that the viewpoint of the second projection need not, and often should not, be related to the first projection. I am convinced that the best way to understand 4D objects is to study their projections into 3D, which means that the second projection is really a "probe" used by us poor 3D beings to understand the full structure of the 3D image. Hence, the second projection should be able to be freely varied completely independently of the 4D viewpoint. In other words, the projection is really only to 3D; the second projection is only there so we 3D beings can use it to understand the resulting 3D image.

The latroframe is the frame formed by the vertices + edges. This can be seen through in 3d, and provides a reference for the subsequent pictures.

I settled on this method after trying various other alternatives, such as render only ridges with transparency (what you'd call a "foam" I suppose), or rendering the cells being described with ridges and the rest in edge/vertex outlines, etc.. Eventually I found it most helpful to leave the edges/vertices in all the renders, and include/exclude ridges from cells to show different layers of the structure.

The faces of the figure are then shown from the innermost (the one with the greatest w), to the outermost in successive layers. Usually, this is done with a translucent material, so that faces behind it are shown (as gel-cells in a edge-frame).

Right. I did find that a lot of manual adjustment to transparency levels was necessary in order to convey the intended information; each successive layer often needed to be more (or less) transparent in order to show the structure more clearly.

The other thing I was thinking of including was to show a different viewpoint for each layer next to the current ones, which would show the layers as latitudes going from top to bottom (thus the analogy with the globe would be much clearer). Maybe I'll add these images to, say, the 600-cell, and see what you people think.

[quickfur]

[...]The other thing I was thinking of including was to show a different viewpoint for each layer next to the current ones, which would show the layers as latitudes going from top to bottom (thus the analogy with the globe would be much clearer). Maybe I'll add these images to, say, the 600-cell, and see what you people think.

OK, I've added latitudinal projections of the 600-cell to the 600-cell page, to show how the layers of cells correspond with different latitudes on the 600-cell:

http://eusebeia.dyndns.org/4d/600-cell.html

Any comments?

[Keiji]

I don't really understand how things disappear in the side view. I can see why tetrahedra get flattened, but e.g. the very first step containing the icosahedron of 20 tetrahedra becomes just five tetrahedra in the side view; why?

[wendy]

In the first instance, the viewer is not necessarily familiar with four dimensional objects. You are generally suggesting something they have never experienced.

You may understand that a simple projection from 4d (the source) to 2d (the target) involves creating and viewing a 3d object.

The projections from 4d to 3d are similar to that of cartography: projections like orthographic, stereographic, gnomic all have been used.

The viewing of the created 3d object involves one of several techniques to see the interior of the created subject: ie to emulate holographic vision. One method might be to create a framework, of vertices + edges, or an axial system. The framework is a better option here. Into this, you could overlay translucent cells of various intensities, so one gets the impression of looking through glass cells. Alternately, you could produce a series of pictures that ripple from the centre of the projection to the outside.

It helps if the picture is accompanied by a different view in 4d. For example, one could do the same progression from the top as from the centre: a series of views that gradually fill the thing from top to bottom of the view. This would to some part help suggest that filling from the inner to the outer is really viewing from the top to the bottom. Putting an eye above the top might help show this view (the top-to-bottom system does not have to be the same as the centre to rim view).

You might also in some introduction page, show something like a polygon (like a pentagon on a dodecahedron), being tilted away from the viewer in a series, and then show parallel, the dodecahedron being 'squashed' into a flat surface. It's an idea.

[quickfur]

I don't really understand how things disappear in the side view. I can see why tetrahedra get flattened, but e.g. the very first step containing the icosahedron of 20 tetrahedra becomes just five tetrahedra in the side view; why?

Because only 5 are visible from that angle of view (well, actually 15: the other 10 being projected to triangular faces). It's just like when you look at an icosahedron from the side: not all of the triangular faces surrounding the "north pole" are visible.

Of course, the fact that this view doesn't really show you where each cell is, is one reason I initially chose the layer-by-layer method in the beginning, since in those views, most of the cells would be visible until you get to the equator.

[quickfur]

[...]
It helps if the picture is accompanied by a different view in 4d. For example, one could do the same progression from the top as from the centre: a series of views that gradually fill the thing from top to bottom of the view. This would to some part help suggest that filling from the inner to the outer is really viewing from the top to the bottom. Putting an eye above the top might help show this view (the top-to-bottom system does not have to be the same as the centre to rim view).

Right, that's another way of showing it, I guess.

You might also in some introduction page, show something like a polygon (like a pentagon on a dodecahedron), being tilted away from the viewer in a series, and then show parallel, the dodecahedron being 'squashed' into a flat surface. It's an idea.

Well, somewhere in my todo list is a new chapter in the 4D visualization document that describes how to interpret these projections. Currently, I've already written one chapter that describes how to interpret a 4-cube projection; I think a few more chapters in between might be necessary for the reader to fully understand what these projections are.

In the meantime, however, I'm slowly working to replace those old XFig figures with the nicer POVRay-rendered images, so it may be a while before I get to writing new stuff.

[Keiji]

Because only 5 are visible from that angle of view

Wow I'm stupid, how did I not realize that

[quickfur]

Because only 5 are visible from that angle of view

Wow I'm stupid, how did I not realize that

No worries, nobody said visualizing 4D was trivial.

[Keiji]

No, but I usually just understand those things instantly =p

[quickfur]

I've updated my website's cubinder page with brand new POVRay-rendered projections. There's even an image of the square torus that bounds the cubinder's rounded surface, with a section cut out so you can see its cross-section!

http://eusebeia.dyndns.org/4d/cubinder.html

Go get a fresh look at the cubinder while it's still hot!

[kingmaz]

That was a most helpful explanation.
Indeed, this whole website is very good and has helped me understand things better - thank you.

[quickfur]

That was a most helpful explanation.

Except I just realized that one part is dead wrong: the cubinder does not roll on its cylindrical surfaces, but on its square torus! Just as a 3D cylinder does not roll on its circular lids, but on its curved side, so the cubinder cannot roll on its cylindrical surfaces, which lie in a flat hyperplane, but only on its square torus. The square torus is only curved in one direction, and that's why a cubinder can only cover the space of a line by rolling.

I've made the correction on the page.

[quickfur]

And now, a little snack for those of you who've been delving into complicated higher-dimensional polytopes with hundreds of thousands of facets:

The bitruncated 5-cell!

This little cutie is cell-transitive, consisting of 10 truncated tetrahedra. Can you count them on your fingers? I knew you could! I've even oriented it in such a way that two of the cells map to your thumbs, too. So hold out your left thumb for the nearest cell, your right thumb for the farthest cell, and clap your hands together to fit both halves of the bitruncated 5-cell into one!

(Oops. I hope you know how to disentangle your fingers from 4D back into 3D... )

[Keiji]

Can you say "comic relief"?

Ehh. Would you mind changing "pentatope" to "pentachoron"? A pentatope is a 5D shape with flat facets. Very different to a pentachoron