Geometry Examples #3

Here we have some exercises in depth perception. You only have one eye in the game, so you can't perceive depth directly, you have to slide around and look for it. Go try it! You don't need to read the rest of this.

But, just for reference, here's the complete explanation. In all the 3D examples the view from the starting position looks like a cube with an X on its front face. But, the reason for the X varies.

• In "a" the front face is concave. As you slide left and right, the center of the X moves in the same direction within the square of the front face.
• In "b" the front face is flat. The X is just painted on and its center doesn't move.
• In "c" the front face is convex. As you slide left and right, the center of the X moves in the opposite direction.
• In "d" the front face is even more convex, and the center of the X moves even more.

Similarly, in all the 4D examples, the view from the starting position looks like a tesseract with an X on its cubical front face. OK, it's not really an X, more of an eight-legged caltrop, but I'm sure you can see the resemblance. As you slide left and right, the center of the caltrop moves in exactly the same way as in 3D.

Another useful exercise is to move around and look at these examples from the side.

geom3-construct

Imagine filling space with cubes and then coloring the cubes with two alternating colors, like squares on a checkerboard. I can't show you the whole thing in game, so I'll just show a stack of three cubes (in "3a") and let you imagine the rest.

In "b" I've broken the red cubes into six pyramids each. Move them around and take a look, or just delete the front pyramid if you're impatient. Notice how the top pyramid in the bottom cube and the bottom pyramid in the top cube fit nicely onto the green cube in the middle.

That green cube has imaginary red cubes adjacent to it on its other four sides. In "c" I've included one pyramid from each of those cubes so that the green cube is surrounded by six pyramids. See how the sides of the pyramids on different faces line up?

In "d" I've welded the pyramids onto the green cube to make a single shape. If you want to play with it, you'll have to move or delete some of the remaining pyramids first.

That shape is a rhombic dodecahedron. It's not a regular polyhedron because the faces are diamonds (rhombuses), not squares, but it's still a nice shape with lots of symmetry.

Now, here's the fun part. Did you remember to imagine all the other cubes? When we break the red cubes into pyramids, every pyramid is next to a green cube, so we can weld all the pyramids onto all the cubes simultaneously and thereby fill space with rhombic dodecahedra!

Now let's do the same thing in 4D.

In "4a" we have a stack of three tesseracts. In "b" I've broken the red tesseracts into eight pyramids each. That puts pyramids on two faces of the green tesseract. In "c" I've added the other six, and in "d" I've welded the pyramids onto the tesseract to make a single shape.

What do the faces of that shape look like? Well, they start at one pyramid tip, pass through a square "edge" of the tesseract, and end at another pyramid tip. In all other dimensions that makes a kind of diamond shape, but in 4D it so happens that the distance from tip to tip is exactly equal to the diagonal distance across the square. So, the faces are octahedra, and the shape is a regular polychoron, the 24-cell!

This construction is the best way to understand the 24-cell that I know of. I learned about it from Coxeter's book Regular Polytopes. It's essentially the same as Gosset's construction: take two tesseracts, disassemble the red one, and attach the pyramids to the green one. You can also get a 24-cell by truncating a hexadecachoron (Cesàro's construction) or a tesseract.

A few notes:

• To finish the analogy, yes, we can fill space with 24-cells.
• I hope the depth perception exercises were useful. Moving or deleting the front pyramid of the red cube creates the same concave shape, and adding a pyramid to the front face of the green cube creates the same convex shape.
• If the red cubes broken into pyramids seem familiar, maybe it's because you saw them in puzzle-redcube3b and 4b.

geom3-24cell

There's a lot more to say about the 24-cell! For one thing, as you can see in "a", it can be colored with three colors in a nice symmetrical pattern. (Don't use texture 0 here!) For some intuition about that, see Shape Reference.

See how the visible part of the 24-cell has a cubical structure, with one vertex in the center adjacent to eight vertices in the corners? Of course we know why that is … there's a cubical face of a tesseract underneath that pyramid! But, now suppose we take the dual of the 24-cell, replacing the vertices with faces and vice versa. The center vertex becomes a center face, and the adjacent vertices become adjacent faces, and since there are eight of them, the center face has to be an octahedron. You can see that in "b". (If you move forward a bit the adjacent faces will go away and give you a better view.)

Here's another way to see that the center face is an octahedron. Flip back to "a" again. The center vertex is connected to the six octahedral faces around it. Find the center points of those octahedra. When we take the dual, those center points become the vertices of the center face.

So where does that leave us? The faces of the 24-cell are octahedra, and at least some of the faces of its dual are octahedra too. In fact it turns out that they all are, and that the 24-cell is self-dual! I haven't really proved that, at least I think I haven't, but I hope I've convinced you that it's plausible.

Speaking of duals, the dual of the rhombic dodecahedron is the cuboctahedron, which you can also find in Shape Reference.

Here's another fun bit of trivia. In "a" the 24-cell is oriented as in Gosset's construction, but when we take the dual in "b", that's oriented as in Cesàro's construction. The center face is where the corner of the hexadecachoron was cut off, and the eight adjacent faces are the leftover parts of the front eight hexadecachoron faces.

There's one last thing I'd like to tell you about the 24-cell. Remember how in Gosset's construction each octahedron passed through a square "edge" of the tesseract? Well, as a result, there's a one-to-one correspondence between squares and octahedra. So, if you want to visualize how the 24-cell is colored, but (like me) you can't visualize all the octahedra at once, you can think about the squares instead.

Where are all the squares? The front face of the tesseract has six, and the back face that looks like a smaller cube inside the front face has six more, and then there are twelve that connect the edges of those two faces.

Now take another look at "a". There are cyan octahedra on the side squares of the front face, so those two squares are cyan. And, if you pick up the 24-cell and turn it from left to right through 360 degrees, you'll see that the side squares of the back face are cyan too, so that there's a ring of four cyan octahedra. In the same way, the magenta and yellow octahedra on the front face also form rings through the back face.

Hold that thought for a minute …

geom3-hidden

Example "3a" is a cube. You know the back face is in there, but you can't see it, right? Well, now hit control-N! That removes the front face so that you can see inside.

"3b" is also a cube, just rotated so that you can see two faces at once. Try control-N on that, too.

Now check out the same two things in 4D! Be sure to try it with different textures. Also, here's a nice experiment. If you slide sideways with normal vision, when the side face appears, it looks like a small cube pushing through the side of the larger cube. Well, if you do the same thing with X-ray vision, you can see that that's exactly what's happening!

Next I want to look at how the faces of the tesseract are connected. Load "4a" and turn on X-ray vision. The now-invisible front face makes a ring with the left, right, and back faces, while the four leftover faces make another ring. But, there's nothing special about left and right. We can make two more rings through the front face, and another two out of the leftovers, for a total of six rings.

Now, each of those six rings contains four cubic faces and four squares in between, and the squares in different rings don't overlap. So, in the 24-cell, the octahedral faces can be organized into six disjoint rings! We already saw three of them earlier, but now we know there are three more. And, with X-ray vision we can see them! Try switching to texture 8 and moving up close to get a good view.

(Although I want to stay focused on the rings, I feel compelled to point out that the division of e.g. the cyan faces into two rings is artificial. The faces are actually connected in the same way as the faces of a tesseract. In fact, they could be the faces of a tesseract that we truncated.)

That's all I have about the 24-cell for now.

Although I'm excited about the 24-cell, I'd also like to emphasize the importance of X-ray vision. It's really very helpful for understanding all kinds of shapes. You might try it on the pentachoron in Shape Reference.

geom3-torus

The place to start here is with the 4x3 example "torus4-4x3". It's sort of like a tesseract. The front face is an elongated cube, but if you turn on X-ray vision you can see that there's no back face—the left and right sides just connect directly to each other. Together with the front face, they form a ring of three cubes. The leftover faces also form a ring, but it's a ring of four triangular prisms.

Hit "page down" to go to the next example, the 4x4 one. It's just a normal tesseract made of two rings of four cubes.

Hit "page down" again to go to the 4x5 example. Move forward so that the side faces drop out of sight, then switch to X-ray vision and see that now there are two back faces! Those plus the front, left, and right faces form a ring of five cubes, while the leftover faces form a ring of four pentagonal prisms.

Those pentagonal prisms don't look very regular, do they? But try this: press "space" to pick the shape up, "U" or "O" to turn it 90 degrees in either the in or out directions, and "space" again to put it down. Now the pentagons are clearly regular, and the ring of pentagonal prisms has become the primary ring!

I imagine you can see where this is going, but let's do one more. Hit "page down" to go to the 4x6 example. Here it's not very satisfying to make the side faces drop out of sight, so just approach to a middle distance and turn X-ray vision on and off. We have a single back face once again, but now there are two left faces and two right faces, and together they form a ring of six cubes. The leftover faces form a ring of four hexagonal prisms.

I'll let you check out the rest by yourself. For ease of paging, I made a complete array of shapes from 3x3 to 6x6 even though the MxN and NxM ones are just the same shape in different orientations. The NxN shapes are particularly nice, with lots of symmetry. The 3x3 shape even has the unusual property that all of its faces are adjacent to one another! I thought the only things that had that property were simplexes (plane triangle, tetrahedron, pentachoron, and so on).

The 8x8, 12x12, and 16x16 examples are there to help you imagine the limiting case, which looks in X-ray vision like some kind of torus. The reality is a bit more complicated, however. What a 4D person sees, remember, is solid volumes. The lines that we see are just the boundary frames of those volumes. So, in fact the entire shape ought to be a solid block of color. In the simplified game world it would be a solid block of exactly the same color (green), but in a more realistic 4D world lighting effects would give slightly different colors to different faces, and even to different parts of the same face.

Anyway, the point is, the torus that we see is just a kind of edge of the true 4D shape. If it helps, here's an analogy. Imagine that you're in 3D, there's a cylinder standing on the ground, and you're looking at it from above. And that it's a line drawing. What you see is a circle, or two circles with X-ray vision; but the 3D shape is not a circle.

Or, how about this. Suppose instead of a solid cylinder we have a tin can, and then we take a can opener and remove the lid. The lid is a flat 2D surface with a circle as boundary, and the can is a can-shaped 2D surface with a circle as boundary. If we weld them back together, the boundaries join and disappear, leaving a closed 2D surface that contains some empty 3D space.

Now let's try that in 4D. In the limiting case, the rings of faces we've been talking about become solid 3D doughnuts. The boundary of a doughnut is a torus. Because 4D is awesome, we can take two of these doughnuts and fit them together so that the boundaries line up. Then we can weld them together. The boundaries will join and disappear, leaving a closed 3D surface that contains some empty 4D space. That's the shape we're talking about, except that I always imagine the blocks in the game to be solid, not empty inside.

(The doughnuts look like linked rings, so is it really possible to fit them together without making a cut in one? I'm not sure, but my best guess is, yes. Remember you can't link two rings in 4D, only a ring and a sphere; see Geometry Examples.)

The shared boundary of the two doughnuts is a flat torus. I have to be careful with my words here, though. Most of the time when I say "flat", I just mean a thing that has one dimension less than the space it lives in. So, if you take a sphere from our 3D world and move it to 4D, it's flat, because you can put it on your four-dimensional desk and use it as a coaster.

When mathematicians say "flat", though, they usually mean a surface that has no intrinsic curvature, so that for example the angles of triangles still add up to 180. So, if you take a square piece of paper and roll it up into a tube, as far as mathematicians are concerned it's still flat because any triangles you've drawn on it are unchanged. In 4D you can take that tube and roll it up in the other direction without deforming it, and that makes a flat torus.

It's not flat in the first sense, though, since it extends in all four directions, and if you look at it, it definitely looks pretty curvy. Take "torus4-12x12", for example. From a distance it looks like a cylinder, but if you move closer, it turns into a barrel because the equator is closer to you.

It's too bad the game doesn't have physics, because the torus shape would be a fun one to play with. I'm not sure I understand how it would behave, but here's my best guess. It can roll in one direction. That leaves a two-dimensional plane of directions in which it can be tipped over. When you pick one of those directions, the perpendicular direction in that plane is the new roll direction. So, even if you can only tip it over once, you should be able to roll it to wherever you want.

geom3-project

Surprise! I got sidetracked, but there's still more to say about the rhombic dodecahedron.

Let me start with something I wrote a while ago: Tesseract Model. Please read at least the first half, up to but not including the paragraph that starts with "Then, once you have a dodecahedron".

For the record, the phrase "back in college, when I was thinking about four-dimensional objects" is misleading in two ways. First, I was interested in four-dimensional objects even before that, in high school. Second, I don't think I even knew about any four-dimensional objects except the tesseract. Also, this was all back before the internet. I didn't learn that the projected shape was called a rhombic dodecahedron until much later.

Let me recapitulate. Here are some familiar facts.

1. We can project a cube into 2D to get a solid hexagon. (This kind of projection is called vertex-first because the cube is oriented so that a vertex is in front.)
2. The center of the hexagon has triangular symmetry.
3. We can break the hexagon into three pieces corresponding to the three visible faces of the cube.
4. The pieces are distorted squares.

And here's the analogy.

1. We can project a tesseract into 3D to get a solid rhombic dodecahedron.
2. The center of the dodecahedron has tetrahedral symmetry.
3. We can break the dodecahedron into four pieces corresponding to the four visible faces of the tesseract.
4. The pieces are distorted cubes.

Now, guess what … since we have a program that can display 4D objects, we can take the analogy one step further!

1. We can project a 5D cube into 4D to get some kind of solid object.
2. The center of the object has pentachoral symmetry.
3. We can break the object into five pieces corresponding to the five visible faces of the 5D cube.
4. The pieces are distorted tesseracts.

Now we can go see all that inside the game! Just for completeness, let's start with the upper end of the projection from 3D to 2D. In "3a" we have a standard cube. If you slide up and to the left, you can get to an angle where it looks like a solid hexagon.

Next is the lower end of the projection from 4D to 3D. In "3b" we have a rhombic dodecahedron. Once again, if you slide up and to the left, you can get to an angle where it looks like a solid hexagon. That's a feature of the analogy that I forgot to mention, that the Nth projected shape looks like the (N-1)th from certain angles.

In "3c" I've broken the dodecahedron into the four pieces that correspond to the four faces of the tesseract. Go ahead and take the dodecahedron apart and see how the pieces look like flattened cubes.

In "3d" there's just one piece by itself, for ease of manipulation.

In "4a" we have a standard tesseract so that we can examine the upper end of the projection from 4D to 3D. If you slide up, to the left, and inward or outward, you can get to an angle where it looks like a solid rhombic dodecahedron.

Next is the lower end of the projection from 5D to 4D. In "4b" we have … well, actually I don't know what it's called, but it's a nice symmetrical shape with 20 faces. I've pre-oriented it for you so that it looks like a solid rhombic dodecahedron, but if you pick it up and spin it around, it'll be clear that it's a more complicated shape.

In "4c" I've broken the shape into the five pieces that correspond to the faces of the 5D cube. I think the best way to understand what's going on here is to focus on the front piece, which is the only one that's initially visible. So, switch to texture 0 and take a good look. The piece is a flattened tesseract, remember. You can see four of its faces. The other four fit into the rhombic dodecahedron outline in the same way, just with the tetrahedral axes reversed. Or, to put it another way, you can take any one of the visible faces and slide it over to the other side of the dodecahedron, and that's where one of the back faces is. Or, even better, you can just turn on X-ray vision and see them.

So far so good? Now switch to texture 8 and try the X-ray vision again. Those are the back faces of the front piece, right? But, those faces are in the interior of the whole shape, and they connect to faces on the other four pieces. So, if you turn off X-ray vision and delete the front piece, you'll see exactly the same thing! But, now you're looking at four flattened tesseracts, with just one face visible each. If you turn X-ray vision on again you'll see the other 4x7 = 28 faces.

So, that's five pieces: one we deleted plus four more underneath. In fact, the four pieces underneath form a little nest for the fifth piece. Go back and delete a piece in "3c" to see what I'm talking about.

In "4d" there's just the front piece by itself. If you pick it up and rotate it you can see how flat it is.

Here are a few last random thoughts.

Can you fill space with the projected 5D shape? I have no idea. It would make sense, though. In 3D you can fill space with hexagons to make a Q*bert board, and in 4D you can fill space with rhombic dodecahedra.

Because of the "construct" examples above, I got the idea that the 24-cell was the unique analogue of the rhombic dodecahedron, and as a result I spent a lot of time trying to figure out how the 24-cell was the projection of a 5D cube. That was remarkably stupid. The number of faces isn't just wrong, it's not even divisible by 5, so the 24-cell can't be broken into five equal parts. Lesson: the analogue depends on the analogy.

However, I did learn one interesting fact in the process: there's no way to break the 24-cell into nice pieces at all. Here's the proof. If you construct a 24-cell on top of a unit tesseract, then the distance from the center to the vertices is 1. In other words, that's the radius of the circum-3-sphere. But, the length of the diagonals of the octahedral faces is root 2. So, there's no way to break the 24-cell into pieces with octahedral faces that are symmetrical about the origin, because any octahedron that starts at the origin is going to poke out through the surface of the 24-cell.

Finally, all five of these shapes are fairly pointy … the rhombic dodecahedron, the projection of the 5D cube, their pieces, and the 24-cell. I guess one could quantify that somehow.