Quickfur's renders

[quickfur]

I have finished my coordinate generator, which generates wireframes of polychora, and here's a sample.
Truth is, I'm not sure if it's a compound of 3 tesseracts or a compound of 3 16-cells.
Thnks to QF for coordinates

It would help if you could color each tesseract differently. It's kinda hard to tell from this picture.

[SharkRetriver]

Alright, code generator slightly updated. I think it's 16-cells actually, since it's sharper than this other compound I generated.

[quickfur]

Hmm, are you sure you got the edges right? The three components seem to be non-convex to me, but both the 16-cell and the tesseract are convex. Or did I get the coordinates wrong?

[quickfur]

OK, just checked, the coordinates are correct. But your edges look wrong. So you probably want to check your edge generation code to make sure it's doing the right thing.

[SharkRetriver]

The generator chooses the edges based on the preset distance. Do you know the distances for the edges for the compounds?
Also, since the cells are convex, I think this means I discovered a new compound, yay!

Did a check using my generator, here are the possible edge distances
2.82842712474619 //this: http://www.flickr.com/photos/sharkretriever/6294746064/in/set-72157627885891097
2 //normal 24-cell
3.46410161513775 //image above
4 //didn't try yet
0

[quickfur]

The generator chooses the edges based on the preset distance. Do you know the distances for the edges for the compounds?
Also, since the cells are convex, I think this means I discovered a new compound, yay!

Did a check using my generator, here are the possible edge distances
2.82842712474619 //this: http://www.flickr.com/photos/sharkretriever/6294746064/in/set-72157627885891097
2 //normal 24-cell
3.46410161513775 //image above
4 //didn't try yet
0

Actually, the correct edge length is 2. The reason it looks like a 24-cell is because each tesseract in the compound contains a subset of the 24-cell's edges, and all of them together contains all of the 24-cell's edges. (This peculiar coincidence of edge lengths between the 24-cell and an inscribed tesseract is what makes the 24-cell regular.) You'll have to color the edges in order to tell the tesseracts apart.

Now the edge length of the 16-cell of radius 2 is 2√2 which is roughly 2.8284..., so your first edge length there gives you the compound of 3 16-cells. (Nice job on the picture!)

[quickfur]

Argh... that will teach me to post renders without double-checking my results. I posted a render of the compound of 3 tesseracts, but i made a mistake in the script so the output was wrong. Will post later when I fixed it.

[quickfur]

OK, so sharkretriver inspired me to do my own render of the compound of 3 tesseracts. Due to my projection program's limitations, it can't handle non-convex objects, so it can't perform hidden surface removal, but it can render the 3 component tesseracts into a single povray scene to make the compound, so here it is:



Here are the component tesseracts:





Interestingly enough, although the compound itself is non-convex (its ridges and facets intersect each other inside the compound), it actually fills up the entire volume of its convex hull, which is a 24-cell.

Also interesting is how each vertex has two tesseracts meeting at it, each in tetrahedral configuration dual to each other. Quite amazing how it fits together.

[quickfur]

Argh, apparently yesterday was the day of embarrassing myself multiple times.

Just realized that what I said about convexity was wrong; the compound of 3 tesseracts does not fill up the entire volume of the 24-cell. There are slight concave indents where three cells meet. These aren't very obvious in the projection image due to the lack of hidden surface removal. Sigh... now I really have to get cracking on implementing full-scale non-convex handling in my projection program.

[quickfur]

Alright, I decided to do the compound of 3 16-cells as well:



Again, there is no hidden surface removal because my projection program can't handle non-convex objects yet. You can see how confusing the images get without HSR. The constituent 16-cells are:



Worthy of note is that these projections are, respectively, two cell-first projections (the two alternations of the tesseract) and the vertex-first projection.

[SharkRetriver]

Truth is, I don't know how the individual 16-cells on my pic are projected. The convex hull is vertex-first tho.
Also, a compound of 3 tesseracts is same arrangement as 24-cell?
I think the earlier picture I posted is a compound of stella octangula or stellated octahedra o_O
Lastly, when I try to colour the compound of 3 16-cells into dif colour per 16-cell, there are overlapping edges so the render is bad. It's probably the projection

[quickfur]

Truth is, I don't know how the individual 16-cells on my pic are projected. The convex hull is vertex-first tho.
Also, a compound of 3 tesseracts is same arrangement as 24-cell?

Well, the edges are the same as the 24-cell's edges, but the faces and cells are not. But if you're only rendering edges, then yeah, it would look exactly like a 24-cell.

I think the earlier picture I posted is a compound of stella octangula or stellated octahedra o_O

Well, i've no idea what it is, but it's certainly interesting, as it appears to have star polygons for faces. I'm not even sure if those star polygons are planar! Will have to investigate this a little deeper.

Lastly, when I try to colour the compound of 3 16-cells into dif colour per 16-cell, there are overlapping edges so the render is bad. It's probably the projection

Actually if you look closely at my render, there are coincident ridges (faces) in the image so there are some povray artifacts. It's actually a lot worse before, but I moved the 4D viewpoint slightly from <5,0,0,0> to <4.999,0.0001,0,0> which got rid of most of the artifacts, but still has some patches left.

As for the edges, you might want to shift your 3D viewpoint a bit so that they don't overlap.

[SharkRetriver]

I can only change the w value of my viewpoint, and that's it XD
Also, I think I actually rendered the edges wrong for the compound of 3 16-cells, since when I rendered one of the 16-cells I got a compound of 2 16-cells T_T
EDIT: found problem. Anyways,


Happy Halloween! XD

[quickfur]

What are your projection parameters? Looks like your 4D viewpoint is very close to the object (perspective distortion is very high). Not that it's a problem; just wondering.

[SharkRetriver]

w=5, that's it.
I'm rendering compound of three tesseracts, there's edge overlap on that.
three 16-cells, i don't think there's any overlap

[quickfur]

OK, w=5 is about half the distance I use. The tesseract coordinates I gave you has radius 2, whereas I usually scale everything down to radius 1 for my renders, so even though my 4D viewpoint is also <5,0,0,0> it is effectively twice the distance because the object is half its original size.

The reason for scaling to radius 1 is mainly for consistency, 'cos the canonical coordinates for the 120-cell is huge, and the 600-cell is tiny. Standardizing on radius 1 makes it possible to use exactly the same 3D viewport without worrying whether the resulting image will be too small or too big.

When I first started out, I chose <5,0,0,0> as my default viewpoint because 5 units is approximately the typical eye-to-object distance when you're looking at a medium-sized ball (of unit radius) on your desk. It seems to be a good distance where perspective distortion is noticeable but not extreme. My goal was to obtain insight into how a native 4D being would perceive its native space, so I tried to model the typical viewing parameters a hypothetical 4D being would have when examining an object on its desk, based on generalizations from the 3D case, rather than choosing parameters to simplify programming or to satisfy some abstract mathematical condition.

[Mrrl]

There is image of compound of 3 tesseracts in my viewer:



Model is here:
http://astr73.narod.ru/4DView/Examples/ ... scene1.zip

and viewer for it is here:
http://astr73.narod.ru/4DView/4DView.zip (it's the same as for 4D car). It is only for Windows, sorry for that...

[quickfur]

There is image of compound of 3 tesseracts in my viewer:
[...]

Cool! I like it! Мне понравилось!

[quickfur]

Alright, it's that time of the month again. The polytope of this month is:



That's the truncated 16-cell doing a double rotation; yellow octahedra lie on the near side and green octahedra on the far side of the polytope.

Truncated regular polytopes have the nice property that they have cells of the same kind separated by edges, so they are good subjects for animations (it's easier to make a color scheme that shows the rotating elements clearly). Same goes with runcinated polytopes.

[SharkRetriver]

Yay! Truth is I like the tesseract truncates (n-cubes) the most since I like the easy coordinates...
@Mrrl AKA Andrey lololol awesome pic!

[quickfur]

Yay! Truth is I like the tesseract truncates (n-cubes) the most since I like the easy coordinates...
@Mrrl AKA Andrey lololol awesome pic!

The 24-cell truncates also have rather nice coordinates, though most of them need apacs of more than a single point.

The simplex coordinates are also very nice, if you work in (n+1)-space instead of n-space. But projecting those coordinates back to n-space will give you a headache 'cos lots of square roots will appear, though for the the case of the regular n-simplex there is a neat pattern which involves square numbers and triangular numbers.

The messiest coordinates are the icosahedral polytopes, and the sporadic special polytopes (like the two 3D snubs, the 4D grand antiprism, and the truncations of the Gosset E polytopes).

[SharkRetriver]

the 24 is still strange because of the analogue problem, but it's a cool polychoron.
About the simplexes, they're odd as well. I don't like
I looked up E polytopes on the wiki. I think that the trirectified 4_21 should be the next polytope of the month. There are only 4.8 million edges, lol
Also, why is there only the snub 24-cell, and no other snub polychora?
I'm also considering changing w to 10 before "mass-rendering" for various reasons...

[quickfur]

the 24 is still strange because of the analogue problem, but it's a cool polychoron.

Well that's what makes it cool. Analogue-wise, it's essentially a rhombic dodecahedron that just happens to be regular. It's in a series of "stellated" hypercubes, which are constructed by cutting one n-cube into (n-1)-cubical pyramids and attaching them onto the facets of another n-cube. In 1D, it is identical to the line segment (well, everything is, so that says nothing). In 2D, it is identical to the 2-cross (diamond, which happens to be the same as the square). In 3D, it starts to become distinct, as the rhombic dodecahedron. In 4D, it's the 24-cell, which, due to a set of interesting coincidences (such as the fact that 2^2 = 1+1+1+1) just happens to be regular. Then in 5D, it starts to become distinct from the maximal projection of the (n+1)-cube into n-space.

About the simplexes, they're odd as well. I don't like

But their mesotruncates and omnitruncates are even.

I looked up E polytopes on the wiki. I think that the trirectified 4_21 should be the next polytope of the month. There are only 4.8 million edges, lol

Oh don't worry about that one... I do have the omnitruncated 120-cell lined up. Among the 4D polytopes, it's the grand-daddy of them all. It has 14400 vertices, 28800 edges, 17040 faces, and 2640 cells. One of these days, it's gonna be up. Of course, that's nothing compared to the trirectified 4_21. That's still a long way off. After I finish with 4D, I'm going to 5D, which has a lot more truncates, even though it only has 3 regular polytopes. If I'm still alive after that, it's 6D time, and if I outlive that, then I'll get to the 4_21 in 7D.

Also, why is there only the snub 24-cell, and no other snub polychora?

Well, the concept of "snub", in the strict sense of surrounding the faces of the base polyhedron by triangles, really only applies to 3D. A more generally applicable operation is alternation. The 3D snubs are topologically equivalent to the alternations of the omnitruncate of their respective platonic solid. In a sense, you may think of the icosahedron as the "snub tetrahedron", since the "omnitruncated tetrahedron" is A111, which is equivalent to C011 (truncated octahedron), the alternation of which is topologically equivalent to the icosahedron. So in a sense, "snub" in 3D means "alternated omnitruncate".

However, the snub 24-cell in 4D is not the alternation of an omnitruncate. It is topologically equivalent to the alternation of the truncated 24-cell (alternation of F0011), and its icosahedral facets are not completely surrounded by tetrahedra, unlike the analogous 3D snubs. Four of each icosahedron's faces actually touch adjacent icosahedral cells. So in a sense, the snub 24-cell isn't really a snub. It's just that there are no other uniform snubs in 4D, and it does after all consist of facets of the same kind with the gaps filled in by regular tetrahedra.

Now, the key here is that alternation as an operation does not, in general, give you a uniform polytope. The snub cube cannot be produced by alternating a uniform great rhombicuboctahedron, because even though the result is topologically equivalent to a snub cube, its edge lengths are not equal and its faces are not regular. You need some unspecified amount of distortion in order to get the "real" snub cube. So nothing stops you from, say, taking the alternation of F1111 instead. You'll end up with a polytope with 48 snub cubes for facets plus a whole bunch of octahedra and probably some irregular shapes. However, it would not be possible to make the polytope uniform. Only the alternation of F0011 can be made uniform.

Alternation in 3D also gives rise to the infinite family of antiprisms: alternating a (2n)-gonal prism gives you an n-gonal antiprism. (Again, modulo some arbitrary distortion to make the result uniform.) Given this fact, it seems reasonable to suppose that alternating 2n,2n-duoprisms in 4D should give us some kind of analogue of the antiprisms. But alas, none of them can be made uniform except one: the alternation of the 4,4-duoprism (aka tesseract), which is the 16-cell, which also happens to be regular.

Given the large amount of polytopes that can be alternated in 4D, you could say that actually there are quite a good number of snubs; but only the snub 24-cell is uniform.

I'm also considering changing w to 10 before "mass-rendering" for various reasons...

Sounds good to me.

[quickfur]

Hehe... forgot to post the Polytope of the Month here:



This is the rectified 24-cell. Which happens to be the same as the cantellated 16-cell:



Basically, 24-cell symmetry is just tesseractic symmetry where the 8-cell elements and the 16-cell elements are identical.

[SharkRetriver]

lol I don't think that you got my email reply, so again:
Thanks for posting it on my birthday XD
Also, the generator is much too hard for me to code, so for now I'll be focusing on simpler things.

[quickfur]

It's that time of the month again! Here's the polytope of the month:



This is the cantellated 600-cell, also known as H0101 or o5x3o3x. It is basically a rectified 120-cell with its icosidodecahedral cells expanded outwards radially.

Have at it!

[quickfur]

It's been awfully quiet here for a while. So it's time to post another mind-bending animation from my Polytope of the Month:



This is the cantitruncated 5-cell doing an unequal double rotation, with a 1:3 rotation rate ratio. The rotation in the XY plane (projected to the horizontal plane here) is at 1/3 the rate of the rotation in the WX plane (projected to a vertical line here).

If you look carefully, you can see the spiralling motion of the green truncated tetrahedra as they rotate, simultaneously "turning inside-out" due to the view angle of the projection. That's the motion of the general point on the 3-sphere when it undergoes a double-rotation. Only points that lie on either rotation plane directly have actual circular motion; in general the motion is spiralling in a corkscrew fashion.

[quickfur]

It's that time of the month again! This month, I have a special treat: two polytopes of the month, doing 3:1 and 4:1 double rotations, no less!



Click on either image to find out what it is. ;-) (OK OK, who am I kidding, right? People on this forum can tell just by looking at it. But just in case. )

[Oschkar]

You also need to update the cubinder, spherinder and spherone pages...

[quickfur]

You also need to update the cubinder, spherinder and spherone pages...

I think the cubinder page is OK for now (I think?), but yeah, spherinder and spherone, etc., need to be overhauled. Those crufty old XFig diagrams really need to be replaced by nice POVRay renders. Only thing is, I've yet to be satisfied by a reasonably-good sphere approximation that I can use to model these things. It's just a matter of sitting down and coding it, I suppose, I just need some motivation to actually do it. The present limitations of my polytope renderer makes it a bit painful to do this.