ICN5D wrote:quickfur wrote:And since in 5D, the circumradius of the 5-cube is sqrt(5)/2 = 1.11803..., that means any pyramid of the 5-cube cannot possibly have edge length 1, since that would require its height to be the square root of a negative number.
Hmm, that's interesting. Has anyone ever suggested an "imaginary height" for a true unit 5D-pyramid, non-degenerate? That would be a dimension within a dimension. Wow, I don't know, that's pretty crazy. With a complex conjugate volume? Just brainstorming here.....
Well, I didn't want to say it outright, but when I referred to the height being the square root of a negative number, that is, in essence, saying that the pyramid has imaginary height. But since you brought it up... the problem with allowing imaginary height is that then you have to allow some coordinates to be imaginary (namely, the coordinates of the apex of the pyramid), but then, by closure under rotation, all coordinates would need to permit complex numbers. So then we're no longer talking about Euclidean polytopes, but complex polytopes, which ... I admit these monsters are wayyyy over my head. They cease having geometrical properties like angles and bounding surfaces, but become abstract things defined by the structure of their incidence matrices. A complex polygon, for example, has 4 degrees of freedom, so you're talking about the equivalent of 4D here, except with complex number operations on vertices. Much more, I can't say, as they are completely out of my depth. This is where you get into dark corners of mathematics that defy all attempts at visualization, and I can't say that appeals to me very much.
Meaning to say that in sufficiently large n, it takes k times the amount of effort to get from its center to its corners than it takes to get from its center to the centroid of one of its facets,
Does this resemble hyperbolic space in a ways? Perhaps that's why the universe looks like it's accelerating while expanding. All of those extra Calabi-Yau extensions.
No, hyperbolic space is where the farther out you go, the more space there is; here, everything is packaged in nice little regular boxes... but these boxes are very unlike the 3D boxes that we know and love. Even though they're not expanding, and they have fixed measurements (I mean, nothing can be more fixed than having coordinates of the form (±1/2, ±1/2, ±1/2, ... ±1/2)), yet when you're lifting these things and moving them around, you discover that the center of the box is extremely light, but its corners are unbearably heavy.
If you like, you can imagine this counterintuitive situation as the n-cube becoming more and more "star-like" in higher dimensions, such that its corners are the peak of very long "spikes" protruding from a tiny core that contains almost none of its total volume.
I used to see them as more sphere-like, but you're right, it would be probably more star-like. I would see all of the corners smoothing out into a sphere-like shape. The corners would actually be protruding, and smoothing out into a cone as more and more axes (edges) were added to each one. I guess the over all outer envelope could be sphere-like, with lots of space in between.
The funny thing is, the n-cross, which we regard as "sharp and pointy" in 3D, becomes more and more like the n-sphere as you go up the dimensions, in the sense that it fills less and less volume relative to a commensurate n-cube, until the ratio of n-cube volume to n-cross volume approaches 0 as n increases without bound. Unlike the n-cube's vertices, which stretch farther and farther away from its center as n grows, the n-cross's vertices remain at a constant distance, and thus the n-cross pyramid of unit edge length always exists! (Which, of course, should be obvious since the (n+1)-cross is nothing but the n-cross bipyramid.) Meaning that the height of an n-cross pyramid is constant no matter what n is. Which is unremarkable in and of itself, but considering that the n-cross is the dual of the n-cube, this fact becomes a strange contrast to the n-cube pyramid's height which drops to 0 in 5D and ceases to exist thereafter.
One example is that while the volume of the n-cube is concentrated around its corners, the volume of the n-sphere is, on the contrary, concentrated in a narrow band around its equator, such that for sufficiently large n, almost all of the n-sphere's volume is contained in a tiny band within epsilon units of the hyperplane that bisects it, where epsilon is a number that shrinks to 0 as n grows without bound. This strange effect is almost impossible to intuitively visualize, since it's so contrary to our 3D-centric experience; perhaps one way to think of it is that the n-sphere is like a cast-iron disk embedded inside a hollow plastic ball: almost all of the mass is concentrated in the disk, and the rest of the ball contributes almost nothing more.
You know, you're right. It does have that effect. That's very strange, I never would have thought about that before. Well, it's not that difficult to see. Each n-sphere sort of bulges out from the poles and center, it's in the nature of the n-sphere. So, if we keep multiplying this bulging effect, over and over again, soon we get this flattened thing, where its essence is further away from the poles and center. Really, once again, this is an awesome website, where I have seen some of the most amazing things in my mind. Weird mathematical effects, crazy shapes, counter-intuitive phenomena, it's expanding my visual capacity really far.
If you really want to push your visual capacity, try visualizing the 11-cell one of these days. It's a 4D regular polytope, but it's not a polytope in any usual sense that we're used to: its cells exist in projective space. Let me know when you've successfully visualized it, 'cos I'm still having trouble with that one.
- The ending of many classes of CRF segmentochora with n-cube symmetry at 19D, after which there are only 3 classes. (Why 19D? Because the equation that determines it is a quadratic polynomial in √2, which crosses 0 at an irrational root between 19 and 20. On the surface, though, it seems really strange for the number 19 to appear seemingly out of nowhere.)
Yep, very strange indeed. I have no idea what any of those are, but maybe it has something with 19 being a prime number? I haven't computed a 19D shape yet, probably only up to 10. Only a few of them. I'll have to check those out.
An n-dimensional segmentotope is simply the convex hull of two (n-1)-dimensional elements that lie in parallel hyperplanes. Sorta like a generalization of prisms, where the top and bottom faces don't have to be the same. A CRF segmentotope is one where all edges (including lacing edges) are unit length, and a CRF segmentotope with n-cube symmetry is simply one where the top and bottom facets are in the shape of a uniform (n-1)-polytope derived from the (n-1)-cube via various truncations, rectifications, etc. (i.e., Stott expansions). In 3D, you get interesting shapes like the square cupola (= square||octagon), octagonal prism (octagon||octagon), square antiprism (square||diamond), etc.. In 4D, the possibilities increase quite a lot: you can make things like cube||octahedron, cuboctahedron||cube, truncated_cube||cube, truncated_cube||cuboctahedron, etc..
Since in n dimensions there are 2n-1 uniform polytopes with n-cube symmetry, with each increasing dimension there is a combinatorial explosion of possible CRF segmentotopes with n-cube symmetry... but something else also happens alongside, subtly in earlier dimensions, but becoming prominent in higher dimensions: the circumradius of the n-cube starts to grow out of control, and pretty soon, many combinations become impossible because they require imaginary height -- the n-cube pyramid is but one of the early casualties of this (an n-cube pyramid is just (n-1)-cube||point, after all). As n grows, more and more classes of segmentotopes become impossible, and 19D is where the last of the classes die off. Starting with 20D onwards, only 3 families of CRF segmentotopes with n-cube symmetry exist: the prisms of (n-1)-uniform polytopes (height 1), Stott expansions of the (n-1)-cross pyramid (height 1/√2), and a third class (also height 1/√2). The earlier segmentotopes that exhibit various other heights are all no longer possible starting with 20D. So here you have the unexpected pattern that for all n>19, segmentotopes with n-cube symmetry can only have two heights: either 1 or 1/√2, even though for n≤19 many other heights are possible.
This unusual convergence on a small fixed set of possibilities happens also with the regular polytopes: in 1D the only regular polytope (indeed, the only polytope) is the line segment; in 2D, there are an infinite number of regular polygons, but in 3D, there are only 5 regular polyhedra (the Platonic solids). In 4D, there are 6 regular polyhedra, thanks to the unexpected appearance of the 24-cell, but just when you'd expect the number to climb up again, in 5D the pentagonal polytopes drop out, and the number of regular polytopes falls to 3 and remain at 3 for all n≥5. So the sequence goes: 1, infinity, 5, 6, 3, 3, 3, 3, 3, 3, ... . Why such an odd sequence? Who knows.
This is a well-known fact. The circumradius of an n-cube (i.e., the radius of the n-sphere that contains its vertices) is proportional to sqrt(n). If we fix edge length at 1, then the circumradius is sqrt(n)/2. So in 4D, the circumradius is equal to the edge length, which is a happy coincidence that makes the 24-cell a regular polytope. However, this same fact also means that in 5D, you can taper a tesseract with equal edge length, but it will be completely flat!
Well, you're definitely right about that one. And all along it's been staring me in the face. The cylconinder, |O>|O or (II)'(II) , has no height, and I didn't notice this strange effect before. I mean, I saw that one of the lacing elements was a cone-torus, but I never once guessed why the circle would have been inside or in the same plane as the duocylinder. It's a very beautiful representation of this effect, with curved surfaces instead. Two torii bound orthogonally lace to a circle at origin, laced by a cone torus and cylinder torus ( torinder ). The cone torus is on the same plane as the circle, the cylinder torus is orthogonal. But, I'm only repeating myself now! So, where does that leave us with 6-D tapertopes? Like the penteract pyramid? Are they impossible, or is the unit height degenerate indefinitely? Surely we can still shrink something to a point indefinitely, right? Where is the concentration of volume in one of these?
Only the tesseract pyramid is degenerate. The penteract pyramid doesn't exist (unless you permit non-unit edge lengths) because the height becomes imaginary. You can, of course, make pyramids out of anything; just don't expect the lacing edge length to stay at unit length. In a sense, we do have an analogue of this effect in 3D: you can make unit-edge-length pyramids out of the triangle, square, and pentagon, but the hexagonal pyramid of unit edge length has height 0, and the heptagonal pyramid cannot have unit edge length since a unit edge is too short to reach the center from the heptagon's vertices. So you could draw a crude analogy between cube <-> pentagon, tesseract <-> hexagon, penteract <-> heptagon, in the sense that the cube pyramid exists, and it relatively shallow (like the pentagonal pyramid), the tesseract pyramid is degenerate (like the hexagonal pyramid of height 0), and the penteract pyramid of unit edge doesn't exist (there is no such thing as a heptagonal pyramid of unit edge).
As for pyramids of rotatopes: the n-sphere pyramid always exists, if you define it as having apex-to-base length equal to the radius of the base. The n-sphere prism pyramid (i.e., cylinder pyramid, spherinder pyramid, etc.) always exists, if you let the distance between the lids of the (n-sphere)-inder be equal to the radius of the n-sphere. But higher-dimensional analogues of the cubinder pyramid (i.e., pyramids of n-times extruded cylinder) eventually will stop having non-negative height, my rough estimate is somewhere around 7D or so, and so only exist in limited number. Of course, if you start with an n-spherinder, then you can extrude it about 4-5 times before you get a shape that has no unit-edge pyramid. So these kinds of shapes are possible in higher dimensions, but you do have to make them "sufficiently spherical" in order for the pyramid to exist. If there's too much extrusion and not enough sphere-ness, then the pyramid won't exist. The limit is about 4-5 extrusions from a completely round n-sphere. The reason is actually the same as that for the non-existence of a unit-edge penteract pyramid: each extrusion sweeps out a shape that essentially projects to an (n+1)-cube (where the spherical elements project to a point); once you do about 4-5 extrusions, you basically have the Cartesian product of the tesseract with some n-sphere, and then you can't make a pyramid anymore because one of the lacing elements will be a tesseract pyramid of height zero, which makes it impossible to reach the apex of the pyramid to be constructed.