Visualizing 3-hyperplanes

[quickfur]

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.

[wendy]

Actually, if the circumdiameter is greater than 2, you don't get a 'complex' polytope, but a hyperbolic one.

Complex polytopes are in essence just real euclidean ones, but the dimensions are all doubled (eg a line is 2d, a hedrix is 4d, &c). That space is very useful for demonstrating that it is possible to rotate all-space around a point in an even dimensuon

The {3,4,3,3,3} is a polytope that has a unit-edge pentact as vertex figure. Have not really figured out what imaginary lengths are, but they sure act strange. You can have a 1:i:0 right-angle triangle. Now there's a clue.

The 11 dimensional polytope that Quickfur talks of is a {3,5,3}. There's a 57-cell polytope {5,3,5} that you can try your luck on too. If you want some fun, have a look at this little beauty. It's convex, it has infinite faces, none of which intersect, and it's convex. I knew what it was straight up, because i have played around with models of it. But take a gander, Ok. viewtopic.php?f=3&t=1842

[ICN5D]

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.

Those sound rather interesting. And, bizarre. Right up my alley, as you can tell.

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 11 cell? Sounds MC Escher-ish, nice and impossible!

So the sequence goes: 1, infinity, 5, 6, 3, 3, 3, 3, 3, 3, ... . Why such an odd sequence? Who knows.

Really? Very strange, indeed. It doesn't make any immediate sense, but I'm sure there's some hidden equation in there somewhere. Some underlying principle, like the golden spiral. Perhaps that's what it is, an analog of it? In some very bizarre way. An indescribable way for us poor humans.

Only the tesseract pyramid is degenerate.

I do believe the cylconinder, and many more exhibit this degenerate property. If we relax the definition of base and vertex to include high-D elements lacing to lower-D, then the cylconinder and others are rather pyramid-like in nature. The circle could have been anywhere above the duocylinder, but it ends up being inside, or having zero height to the XY toratope surcell. Same with the contrianglinder, the triangle-vertex laces to the cyltrianglinder-base as if it has no height above the cyltriang. All three cylinder-cells lace to the edge of the triangle, and make three coninders. The triangle torus lies flat on ZW, and laces to the triangle by collapsing its major radius all the way to the subshape triangle. Both lacings are evident in the cartesian product with a cone and triangle. Since I'm pretty sure about these, it's worth an investigation!

--Philip

[Klitzing]

Other counterintuitive things in higher dimensions include:
- The ending of the uniform antiprisms after 3D;

Hugh?

• xo3oo...oo3ox&#x, i.e. having both, unit-edged bases and unit-edged lacings, does exist for every dimension. In fact this is nothing but the corresponding cross-polytope x3o...o3o4o (one node plus). That one surely has positive height in every arbitrary large, but still finite dimension, and obviously is uniform!
• At least xo3oo4ox&#x (oct || cube) does exist within 4D. It still has positive height. But yes, top and bottom vertices would differ. So the total shape is no longer uniform.
• xo3oo4oo3ox&#x (ico || dual ico) then again would be a shape which has both vertex layers being identical. The height here likewise is positive. Thus it would exist as a unit-edged figure of which a symmetry acts transitive on the vertices. That is, it would be a scaliform polytope. (It is not completely uniform, because of the hierarchical part of the definition of that term: all subelements too should be uniform as well. But that ico-ap clearly uses oct-pyrs for some of its cells.)

--- rk

[Klitzing]

Other counterintuitive things in higher dimensions include:
- ...
- The ending of the pentagonal regular/uniform polytopes after 4D, which includes the ending of all regular star polytopes after 4D;

Wrong again.

You would have to consider not only x5/2o (pentagram), but also x4/3x (octagram)!
That one natively occurs in any number of dimensions: e.g. within o3o...o3x4/3x, the conjugate of o3o...o3x4x .
(E.g. in 3D: x4x3o = tic = truncated cube with 2 octagons and 1 triangle per vertex. x4/3x3o = quith = quasitruncated cube with 2 octagrams and 1 triangle per vertex.)

And even x5/2o does occur within any dimension as well. But then by means of cross products: prisms, duoprisms, etc.

--- rk

[Klitzing]

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.....

Hehe, you would not need any imaginary stuff, nor abstrusely defined internal dimensions, for that purpose at all!

It is more that your tapering is considered to be immersed within an euclidean space of one dimension more. If you would have considered an embedding space of hyperbolical geometry instead, then all these "forbidden" things would re-occur. It then is just a matter of the size of the (negative) curvature of the embedding space.

E.g. just consider pyramids above regular polygons. Whenever the lacing edge length has to be the same as that of the base polygon, then you can have only 3 such ones: the tetrahedron (triangle-pyramid), the upper half of the octahedron (square-pyramid), and the cap of the icosahedron (pentagonal pyramid). Nothing else (for convex bases). - The hexagonal pyramid already would become flat.

But within hyperbolic space you could have pyramids above virtually any polygon!

Infact it is just the other way round (considering convex shapes): All pyramids with a positive squared height could be mapped by projection e.g. from their body center onto a spherical space, providing there a corresponding tiling. Flat pyramids (i.e. zero height) straight forward become tilings of the euclidean space (the base then could be considered as being represented by the "outer" realm of that space). And pyramids which would evaluate to a negative squared height could be similarily considered as tilings of some flat hyperbolic space. - If such a geometrical manifold then becomes embedded into some space of 1 dimension plus, having the same or lower curvature value, then these tilings even can be reused as cells to build some polytopal structure with. Completely in the same way as polygons (considered as finite, i.e. spherical geometry 1D tilings) can be used to build polyhedra within an euclidean surroundings.

That is, |||||> (penteract-pyramid) is a completely valid structure. It just is, that it cannot reside within 6D euclidean space. But for sure within 6D hyperbolical space!

The same holds true for my segmentochora as well. Cf. to http://bendwavy.org/klitzing/explain/segmentochora.htm. There the segmentochora with some higher axial symmetry (tetrahedral, octahedral, resp. icosahedral axis) are displayed having different background colors: White if possible to tesselate the glome (spherical, i.e. having a positive height when represented within embedding euclidean space of 1 dimension plus), yellow if possible to tesselate flat 3D euclidean space (zero height when being represented within 4D euclidean space), respectively light green if only possible to tesselate some 3D hyperbolic space.

--- rk

[Klitzing]

Even another is cube atop tesseract, the triangle triprism. It's mathematically destined by unbreakable rules to become this way.

Hugh?

cube || tes  =  ox oo3oo4xx&#x  obviously has height  H = sqrt(3)/2 = 0.866,
the same as the 2D triangle  =  ox&#x.

--- rk

[Klitzing]

I want to compile a list of all degenerate tapertopes/rotopes in 5D, I'm very interested in this.

Wrt. | and > only, you'd get in 5D:

Code: Select all

||||| - pent        (R > 1)
||||> - tespy       (H = 0)
|||>| - cubpyp      (R > 1)
|||>> - cubasc      (H = 0)
||>|| - squasquippy (R = 1)
||>|> - squippyippy
||>>| - squascop
||>>> - squete
|>||| - tracube     (R > 1)
|>||> - tisdippy
|>|>| - trippyp
|>|>> - trippasc
|>>|| - squatet
|>>|> - tepepy
|>>>| - penp
|>>>> - hix


where

• H = 0 denotes a zero height within 5D euclidean space, or being flat. Accordingly you could try to extrude such shapes. But a tapering, still subject to unit-edges only, would be impossible.
• R = 1 denotes cases with a circumradius being unity as well. Those can be extruded for sure. But a tapering clearly would produce a degenerate flat figure (H = 0) within the next dimension.
• R > 1 denotes cases with a circumradius exceeding unity. Those still can be extruded. But a tapering would not even be possible (within the next dimensional euclidean space embedding), not even as degenerate figure.

--- rk

[Klitzing]

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.

I already have written a neat little intro to that 11-cell (as well as to the 57-cell) at my website. Just cf. http://bendwavy.org/klitzing/explain/gc.htm. Perhaps this might serve?

--- rk

[quickfur]

Other counterintuitive things in higher dimensions include:
- The ending of the uniform antiprisms after 3D;

Hugh?

• xo3oo...oo3ox&#x, i.e. having both, unit-edged bases and unit-edged lacings, does exist for every dimension. In fact this is nothing but the corresponding cross-polytope x3o...o3o4o (one node plus). That one surely has positive height in every arbitrary large, but still finite dimension, and obviously is uniform!

Sorry, I worded that very poorly. What I meant is the ending of an infinite family of uniform antiprisms. And yes the n-simplex||dual_n-simplex exists in every dimension, but it coincides with the (n+1)-cross and is the only member of its family.

At least xo3oo4ox&#x (oct || cube) does exist within 4D. It still has positive height. But yes, top and bottom vertices would differ. So the total shape is no longer uniform.

Not to mention that even n-cube||n-cross ceases to exist after a while, due to the increasing difference in circumradius between the n-cube and n-cross.

xo3oo4oo3ox&#x (ico || dual ico) then again would be a shape which has both vertex layers being identical. The height here likewise is positive. Thus it would exist as a unit-edged figure of which a symmetry acts transitive on the vertices. That is, it would be a scaliform polytope. (It is not completely uniform, because of the hierarchical part of the definition of that term: all subelements too should be uniform as well. But that ico-ap clearly uses oct-pyrs for some of its cells.)

--- rk

Again, this is a sporadic occurrence of a single member, but as an infinite family, the antiprisms basically stop after 3D, with a few isolated stragglers in 4D and 5D, but after that it converges with the n-cross.

Other counterintuitive things in higher dimensions include:
- ...
- The ending of the pentagonal regular/uniform polytopes after 4D, which includes the ending of all regular star polytopes after 4D;

Wrong again.

You would have to consider not only x5/2o (pentagram), but also x4/3x (octagram)!
That one natively occurs in any number of dimensions: e.g. within o3o...o3x4/3x, the conjugate of o3o...o3x4x .
(E.g. in 3D: x4x3o = tic = truncated cube with 2 octagons and 1 triangle per vertex. x4/3x3o = quith = quasitruncated cube with 2 octagrams and 1 triangle per vertex.)

And even x5/2o does occur within any dimension as well. But then by means of cross products: prisms, duoprisms, etc.

--- rk

But I'm not talking about uniform stars here, I'm talking about regular stars. Is o3o...o3x4/3x regular?

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.....

Hehe, you would not need any imaginary stuff, nor abstrusely defined internal dimensions, for that purpose at all!

It is more that your tapering is considered to be immersed within an euclidean space of one dimension more. If you would have considered an embedding space of hyperbolical geometry instead, then all these "forbidden" things would re-occur. It then is just a matter of the size of the (negative) curvature of the embedding space.

E.g. just consider pyramids above regular polygons. Whenever the lacing edge length has to be the same as that of the base polygon, then you can have only 3 such ones: the tetrahedron (triangle-pyramid), the upper half of the octahedron (square-pyramid), and the cap of the icosahedron (pentagonal pyramid). Nothing else (for convex bases). - The hexagonal pyramid already would become flat.

But within hyperbolic space you could have pyramids above virtually any polygon!

Hmm, you're right, it does work in hyperbolic space! So I take that back.

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.

I already have written a neat little intro to that 11-cell (as well as to the 57-cell) at my website. Just cf. http://bendwavy.org/klitzing/explain/gc.htm. Perhaps this might serve?
[...]

Well, I understand how the 11-cell "works", that's not the problem at all. The problem is how to visualize it, as a complete entity in itself. What kind of space can you immerse it in?? Its surcells are in projective space but what does that imply for the ambient space of the entire polytope? AFAIK, you can only really deal with it in an abstract sense, as an incidence complex, but you can't have any physical realization of it. But maybe you have a better explanation of it.

[Klitzing]

Other counterintuitive things in higher dimensions include:
- ...
- The ending of the pentagonal regular/uniform polytopes after 4D, which includes the ending of all regular star polytopes after 4D;

Wrong again.

You would have to consider not only x5/2o (pentagram), but also x4/3x (octagram)!
That one natively occurs in any number of dimensions: e.g. within o3o...o3x4/3x, the conjugate of o3o...o3x4x .
(E.g. in 3D: x4x3o = tic = truncated cube with 2 octagons and 1 triangle per vertex. x4/3x3o = quith = quasitruncated cube with 2 octagrams and 1 triangle per vertex.)

And even x5/2o does occur within any dimension as well. But then by means of cross products: prisms, duoprisms, etc.

--- rk

But I'm not talking about uniform stars here, I'm talking about regular stars. Is o3o...o3x4/3x regular?

Sorry, but despite your (newer?) intend you had written in your above cited post "regular / uniform polytopes" -

---rk

[Klitzing]

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.

I already have written a neat little intro to that 11-cell (as well as to the 57-cell) at my website. Just cf. http://bendwavy.org/klitzing/explain/gc.htm. Perhaps this might serve?
[...]

Well, I understand how the 11-cell "works", that's not the problem at all. The problem is how to visualize it, as a complete entity in itself. What kind of space can you immerse it in?? Its surcells are in projective space but what does that imply for the ambient space of the entire polytope? AFAIK, you can only really deal with it in an abstract sense, as an incidence complex, but you can't have any physical realization of it. But maybe you have a better explanation of it.

In fact, for these figures you have 2 possibilities to look at: either bottom-up or top-down.

The bottom-up POV first identifies opposite points of potential to be used polytopes, and uses such ones then within the incidence structure both for the vertex figure(s) and for the facet(s). When considering such structures, which are regular, then you especially can restrict here to a single such vertex figure type and also a single such facet type. - The formal incidence relations so will remain intact. But the space they live in, in this view surely gets completely obscure.

OTOH there is also a quite different top-down view. Instead of first doing that identification separately for several elements, and joining those figures thereafter into some abstract figure, you well could consider some much larger already pre-asembled structure to start with, and apply to that one some color subsymmetry in such a way, that provides both, an identification of vertices and of facet elements. - Under this view you just have to consider rather normal structures only. It is only the operation of the final identifications, which inserts finally that twist of mind.

This latter POV in fact is, which relates that regular abstract 11-cell to the hyperbolic tesselation / infinite polytope x3o5o3o. By way of the same idea then the regular abstract 57-cell can likewise be obtained from the hyperbolic x5o3o5o. - This resulting correspondence between finite and infinite structures is, what Wendy too already tried to tell you...

Both POVs moreover have been detailed within the above provided link...

--- rk

[quickfur]

Other counterintuitive things in higher dimensions include:
- ...
- The ending of the pentagonal regular/uniform polytopes after 4D, which includes the ending of all regular star polytopes after 4D;

Wrong again.

You would have to consider not only x5/2o (pentagram), but also x4/3x (octagram)!
That one natively occurs in any number of dimensions: e.g. within o3o...o3x4/3x, the conjugate of o3o...o3x4x .
(E.g. in 3D: x4x3o = tic = truncated cube with 2 octagons and 1 triangle per vertex. x4/3x3o = quith = quasitruncated cube with 2 octagrams and 1 triangle per vertex.)

And even x5/2o does occur within any dimension as well. But then by means of cross products: prisms, duoprisms, etc.

--- rk

But I'm not talking about uniform stars here, I'm talking about regular stars. Is o3o...o3x4/3x regular?

Sorry, but despite your (newer?) intend you had written in your above cited post "regular / uniform polytopes" -

---rk

Was my original sentence really that unclear? I said "the ending of pentagonal regular/uniform polytopes after 4D", meaning after 4D there are no more regular pentagonal polytopes and their uniform counterparts. Then: "ending of all regular star polytopes after 4D": nowhere did I claim that there are no more uniform star polytopes.

[quickfur]

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.

I already have written a neat little intro to that 11-cell (as well as to the 57-cell) at my website. Just cf. http://bendwavy.org/klitzing/explain/gc.htm. Perhaps this might serve?
[...]

Well, I understand how the 11-cell "works", that's not the problem at all. The problem is how to visualize it, as a complete entity in itself. What kind of space can you immerse it in?? Its surcells are in projective space but what does that imply for the ambient space of the entire polytope? AFAIK, you can only really deal with it in an abstract sense, as an incidence complex, but you can't have any physical realization of it. But maybe you have a better explanation of it.

In fact, for these figures you have 2 possibilities to look at: either bottom-up or top-down.

The bottom-up POV first identifies opposite points of potential to be used polytopes, and uses such ones then within the incidence structure both for the vertex figure(s) and for the facet(s). When considering such structures, which are regular, then you especially can restrict here to a single such vertex figure type and also a single such facet type. - The formal incidence relations so will remain intact. But the space they live in, in this view surely gets completely obscure.

OTOH there is also a quite different top-down view. Instead of first doing that identification separately for several elements, and joining those figures thereafter into some abstract figure, you well could consider some much larger already pre-asembled structure to start with, and apply to that one some color subsymmetry in such a way, that provides both, an identification of vertices and of facet elements. - Under this view you just have to consider rather normal structures only. It is only the operation of the final identifications, which inserts finally that twist of mind.

This latter POV in fact is, which relates that regular abstract 11-cell to the hyperbolic tesselation / infinite polytope x3o5o3o. By way of the same idea then the regular abstract 57-cell can likewise be obtained from the hyperbolic x5o3o5o. - This resulting correspondence between finite and infinite structures is, what Wendy too already tried to tell you...

Both POVs moreover have been detailed within the above provided link...
[...]

This still doesn't answer the question of what space the 11-cell lives in, or what it "looks like" as a complete entity. That's what I mean by "visualizing" it. Understanding the abstract structure isn't all that difficult: the mathematics is relatively straightforward -- you just identify certain elements with others according to some mapping -- but what this means, geometrically, is far from clear. The symbols are easy to manipulate according to the known rules, but the meaning behind the symbolism eludes the mind, and ultimately you're just dealing with the abstract incidence complex without any physical interpretation.

[ICN5D]

I think I'll leave the 11-cell alone, for a while. I don't know what the notation means, nor the incidence matrices. They look cool, and very mathematical, but it's mandarin Chinese to me right now!


I am still interested in the zero-height pyramid phenomenon. I still believe the cylconinder has it, but a mathematical proof of it is beyond me. It just feels right, in an indescribable way. I hope someone who is reading this has the tenacity to look into it. Though, it could be simply laced in a similar fashion to a zero-height pyramid structure. Once the degenerate property was brought to my attention, it made a little too much sense for me to dismiss that easily.

[quickfur]

[...]I am still interested in the zero-height pyramid phenomenon. I still believe the cylconinder has it, but a mathematical proof of it is beyond me. It just feels right, in an indescribable way. I hope someone who is reading this has the tenacity to look into it. Though, it could be simply laced in a similar fashion to a zero-height pyramid structure. Once the degenerate property was brought to my attention, it made a little too much sense for me to dismiss that easily.

Well, nothing dictates that the pyramid must have height zero; that depends on what requirements you place on it! Even a tesseract pyramid can have non-zero height -- if you permit longer edge lengths from the base to the apex. In the case of the tesseract, it's easy to identify an edge length that could be used to impose length restrictions on the lacing edges, but what length should be used when rounded shapes are involved is less clear.

For example, what height should a cone |O> have? Should it have a height equal to the radius of its base, or the diameter of its base? Or something else altogether? Deciding one way or another will change what will/will not have zero height.

The "lazy" way out is to just specify that the tapering operator > always extrudes by 1 unit into the next higher dimension, then all valid constructions will have non-zero height by definition. But if you wish to impose some additional requirements on things -- such as unit edge length or whatever the equivalent for rounded shapes is -- then some shapes may be eliminated because it becomes impossible for certain shapes to obey the length requirements.

So the zero height phenomenon is really an artifact of imposing length requirements on the constructions; it's not an inherent property of the construction itself. Crossing a 100m river is always possible if you're permitted to use any length of bridge you wish, but if you're limited to using a 10m bridge, then the task may become impossible.

[ICN5D]

Oh, my bad, I meant a unit edge cylconinder! And, as for how to apply the unit lacing with round things, that's an interesting question. Haven't thought about that in detail. The unit tapering height would, of course, make for a tesseract pyramid with height. But, you're right, it's a cheap cop-out and not as interesting anymore.

How to define unit circular lengths? Perhaps it can be the diameter of the circle base is equal to the lacing length to the vertex? That would follow the same procedure with unit edge n-cube pyramids. Is that simple pythagoras equation H² = 1 - R² adaptable to round things, in a general way? Or, more specifically "rotapertopes" ? Now that I think about it, if rotating a unit edge triangle makes a unit edge cone, this can be the method! Same with rotating a unit square pyramid, to make a unit cylindrone. The unit cylconinder can start with a line, then triangle in the sequence |>O|O

I have become very familiar with round shapes, it's sort of been my "field of study", while developing on the spin operator. Extrusions and taperings are pretty straightforward, where spins seem to be more complex, and more interesting. In light of defining it for you and Richard, I have become even more familiar in its intricate ways. Now I know why only one surcell pairing becomes a torus, it's the axis left over from a bisecting N-1 plane. All surcells bisected by the N-1 remain stationary and spin in place. Before, I simply took notice of this effect and generalized it as a rule, without really knowing why.

It's just, I'm REALLY interested in this effect, as you can tell! It would be rather amazing if it were true, with a unit edge cylconinder having no height. Even more-so, if true, it would be reflected in other shapes, as well. Not necessarily only with n-cube pyramids or flat sided tapertopes. But with round shapes, where the math still holds true. It would really explain some of them in greater detail, as to how and why they lace up a certain way, if entirely unit edged. I'm totally fixated on this, now that it has been brought to my attention. My curiosity is unstoppable (and hopefully infectious) sometimes, I must apologize. It's in my nature

--Philip

[quickfur]

Oh, my bad, I meant a unit edge cylconinder! And, as for how to apply the unit lacing with round things, that's an interesting question. Haven't thought about that in detail. The unit tapering height would, of course, make for a tesseract pyramid with height. But, you're right, it's a cheap cop-out and not as interesting anymore.

How to define unit circular lengths? Perhaps it can be the diameter of the circle base is equal to the lacing length to the vertex? That would follow the same procedure with unit edge n-cube pyramids. Is that simple pythagoras equation H² = 1 - R² adaptable to round things, in a general way? Or, more specifically "rotapertopes" ? Now that I think about it, if rotating a unit edge triangle makes a unit edge cone, this can be the method! Same with rotating a unit square pyramid, to make a unit cylindrone. The unit cylconinder can start with a line, then triangle in the sequence |>O|O

This sounds like a good idea. I like the connection of the unit equilateral triangle with the cone, that does seem to be a well-motivated choice. It does make the height of the cone irrational (√3/2), though. Also, it would clearly differentiate the cone from the polygonal pyramids, because the series of n-gon pyramids has decreasing height (and even stops at n=6 because there the height is exactly 0). If height were kept constant, you could rationalize the cone as the limiting shape of n-gonal pyramids as n approaches infinity, but since we're looking for unit edges here, this clearly doesn't apply.

I have become very familiar with round shapes, it's sort of been my "field of study", while developing on the spin operator. Extrusions and taperings are pretty straightforward, where spins seem to be more complex, and more interesting. In light of defining it for you and Richard, I have become even more familiar in its intricate ways. Now I know why only one surcell pairing becomes a torus, it's the axis left over from a bisecting N-1 plane. All surcells bisected by the N-1 remain stationary and spin in place. Before, I simply took notice of this effect and generalized it as a rule, without really knowing why.

It's just, I'm REALLY interested in this effect, as you can tell! It would be rather amazing if it were true, with a unit edge cylconinder having no height. Even more-so, if true, it would be reflected in other shapes, as well. Not necessarily only with n-cube pyramids or flat sided tapertopes. But with round shapes, where the math still holds true. It would really explain some of them in greater detail, as to how and why they lace up a certain way, if entirely unit edged. I'm totally fixated on this, now that it has been brought to my attention. My curiosity is unstoppable (and hopefully infectious) sometimes, I must apologize. It's in my nature

OK, I'm still not clear what exactly your cylconinder is, at least not from the notation |O>|O, because it's not clear what plane of rotation should be chosen for the second O. I got as far as |O>|, which gives the cone prism (4D), but I'm not sure how to proceed. Since the last operation is a rotation, I doubt it would have 0 height, because no matter what your n-dimensional base shape is, lathing it in some orientation will always yield a full-volumed (n+1)-dimensional solid.

There are several ways to rotate the cone prism into 5D that I can see -- one would produce a spherone (i.e., spherical cone) prism, another produces a shape that looks like the convex hull of a duocylinder and a parallel circle (disc). Yet another case produces a bicircular tegum prism (prism of a bicircular tegum, that is, prism of the dual of a duocylinder). There may be other cases that I didn't consider. AFAICT none of them have zero height according to our requirement of lacing edge length = diameter of circle.

[Polyhedron Dude]

I've been thinking of the 11-cell and 57-cell lately. The way I visualize them is by imagining that you are inside of the cells, but you see them looking like normal ikes and does respectively - however with a weird twist, there is another "you" on the opposite side of the room standing on the ceiling upside down. Assuming we decorate the room with furniture, stairways, catwalks, and doors, etc. We could climb up to a catwalk towards the center of the ike shaped room (assuming we're in the 11-cell). As you walk towards the other you, who is walking upside down on the identical catwalk above - you notice that he/she does everything you do, but mirror image and upside down - no doubt a freaky situation. This other you is actually you, but the space inside the ike is curved back on itself to produce this situation. Imagine you gave the other you a one dollar bill, he will do likewise. You look at the dollar he gave you and you notice it is mirror imaged! for this room is non-orientable! You decide to explore adjacent rooms and notice that opposite sides of each room are identical and lead to identical rooms - actually they are the same rooms. There are ten different rooms that are adjacent to each room, added to the room you are in adds to 11. I would love to see a video game version of this.

[ICN5D]

another produces a shape that looks like the convex hull of a duocylinder and a parallel circle (disc)

That's it, that's exactly the one I'm talking about. Also, a strange way to put it, from my perspective. I know, it's not very clear in the notation. I am applying the prismic spin, which rotates the last constructed axis W, of the prism. This will rotate the cone ends around that are separated along W, leaving the rest of the lacing elements to rotate stationary. But, I suppose there are many more cylconinders that can be made from a spin, and one name for them all won't be good enough. So, that's what convex hull means? Nice, now that explains nearly 90% of the shapes on this website to me. I had no idea what that meant.

Since the last operation is a rotation, I doubt it would have 0 height, because no matter what your n-dimensional base shape is, lathing it in some orientation will always yield a full-volumed (n+1)-dimensional solid.

I had a feeling about this, when I started thinking about the projections of a coninder. It's also line atop cylinder along W, and if Z is rotated into V, it becomes circle atop duocylinder. This rotation is identical to turning a triangle into a cone, no height is lost. Well, that means the cylconinder really does have height, and it's very cone-like with a big duocylinder base, tapering down to a circle, which feels edge-like . Well, glad we cleared that one up. And, it also clears up the contrianglinder, which is the convex hull of a cyltrianglinder and a parallel triangle. It really seems like it, though. The way the lacing patterns are. So, then, at which dimension would it happen? It seems like spins can hold off the degenerate pyramid before the cartesian product with a tesseract. It is sensitive to extrusions, but how about taperings?

[ICN5D]

I've been thinking of the 11-cell and 57-cell lately. The way I visualize them is by imagining that you are inside of the cells, but you see them looking like normal ikes and does respectively - however with a weird twist, there is another "you" on the opposite side of the room standing on the ceiling upside down. Assuming we decorate the room with furniture, stairways, catwalks, and doors, etc. We could climb up to a catwalk towards the center of the ike shaped room (assuming we're in the 11-cell). As you walk towards the other you, who is walking upside down on the identical catwalk above - you notice that he/she does everything you do, but mirror image and upside down - no doubt a freaky situation. This other you is actually you, but the space inside the ike is curved back on itself to produce this situation. Imagine you gave the other you a one dollar bill, he will do likewise. You look at the dollar he gave you and you notice it is mirror imaged! for this room is non-orientable! You decide to explore adjacent rooms and notice that opposite sides of each room are identical and lead to identical rooms - actually they are the same rooms. There are ten different rooms that are adjacent to each room, added to the room you are in adds to 11. I would love to see a video game version of this.

My god, what are you doing to my brain!!!! First the tiger...., now this? You have the most ridiculous clairvoyance into these crazy things. I don't understand how you can understand it. But, you make me understand it. Which makes me really appreciate how you can translate it into layman's terms. That's my goal as well, make it understandable to my friends. So, this crazy 11-cell, it seems sort of like a tiny hypersphere, but you wouldn't be able to interact with that mirror image of yourself, nor any other of the infinite ones. This 11-cell has a weird tiling effect, where everything is reflected off of each other. Like some sort of pinched off space, in a really bizarre way I can't grasp yet. Almost like it's within its own hyperplane, and the tiling is bounded by the one single room. That's why it feels so 4-sphere like.

Or, is it some sort of self-contained mobius strip space?

[Klitzing]

Was my original sentence really that unclear? I said "the ending of pentagonal regular/uniform polytopes after 4D", meaning after 4D there are no more regular pentagonal polytopes and their uniform counterparts. Then: "ending of all regular star polytopes after 4D": nowhere did I claim that there are no more uniform star polytopes.

Read that bit indeed as "no regulars and no uniforms (with stars)".
--- rk

[wendy]

Complex Euclidean Space

Complex Euclidean space is what you get, when you do ordinary algebraic geometry (like a line is y = ax + b), when the numbers belong to the complex numbers. One of the interesting things is that in ordinary space, one can only keep things co-linear by a change of sign. In Complex space, things remain co-linear even under a rotation around the unit circle. That is, an equation y/x = a, has a solution for +x, +y and -x, -y. In complex space, you can have x cis(wt), y.cis(wt), where 'w' is omega, and t is time. Yes, that's right all of space can be bisily rotating around -b, and still observe y=ax+b.

A complex line can have any number of vertices. In a regular edge-2 line, the vertices are spaced around the unit circle in the argand diagram. There are two infinite classes (polygons and simplexes), and two bi-infinite classes (prism and tegum powers of the regular complex edges). Outside of these, you have a motley collection of special instance ones (like {3,4,3}, {3,3,5}, u. {5.3.3}). There's some families of polygons, three extra polyhedra and a polychoron.

CE space is what Norman Johnson calls 'unitary space', for some totally absurd reason.

Imaginary Distances

Imaginary distances never arise in CE space. The measure here is always real, being eg x x* (the product of a number and its conjucate).

When you see imaginary distances appearing, it is a reference to Hyperbolic Space. What is happening is that you are not using a ruler that is uncurved enough to pass through all of the points. You have to pick one with a lesser curvature. In hyperbolic space, a zero-curvature thing is a bent line (a horocycle). Distances along this are quite dandy because circles are parts of both the euclidean horocycle, and the underlying hyperbolic space. So you can do euclidean geometry as long as you carry bits from circle to circle to circle.

In practice, i never try to enumerate the distance as an imaginary number. Numbers like 2+i never occur. Instead, one thinks of then as negative squares, and try to figure out what the things might mean in space. That is there are 'ideal points' and 'real points', &c, and things like 'ideal points' have a real reflex (ie a plane). An ideal distance is the distance between two ideal points, the sole example that we know both the ideal and real distance tells us that the real distance is 2.828 and the indicated ideal distance is 6.828, but there appears to be a measure of 4 that is 'discardable'. Alternately, there is a different scale, where both of these numbers appear as 4.828.

There's a picture of a polyhedron with ideal vertices on the laminatope topic.

[Klitzing]

I think I'll leave the 11-cell alone, for a while. I don't know what the notation means, nor the incidence matrices. They look cool, and very mathematical, but it's mandarin Chinese to me right now!


I am still interested in the zero-height pyramid phenomenon. I still believe the cylconinder has it, but a mathematical proof of it is beyond me. It just feels right, in an indescribable way. I hope someone who is reading this has the tenacity to look into it. Though, it could be simply laced in a similar fashion to a zero-height pyramid structure. Once the degenerate property was brought to my attention, it made a little too much sense for me to dismiss that easily.

Well, we know that ||||>, i.e. the pyramid on the tesseract, is degenerate (has zero height) - all unit edges assumed, and being evaluated within euclidean space.

Your way of spin implementation now tells, that the spinning object itself will use the same circumradius, as would its spin. E.g. the circumradius of | and |O are the same, both having R = 1/2. Same for || and ||O, both having R = 1/sqrt(2). Etc. Thus esp. for |||| and ||||O you'd get R = 1. And therefore their respective taperings would become flat (in euclidean space implementations).

--- rk

[quickfur]

Was my original sentence really that unclear? I said "the ending of pentagonal regular/uniform polytopes after 4D", meaning after 4D there are no more regular pentagonal polytopes and their uniform counterparts. Then: "ending of all regular star polytopes after 4D": nowhere did I claim that there are no more uniform star polytopes.

Read that bit indeed as "no regulars and no uniforms (with stars)".
--- rk

Sorry, the way I worded it was admittedly ambiguous. So it's really my fault.

[quickfur]

I've been thinking of the 11-cell and 57-cell lately. The way I visualize them is by imagining that you are inside of the cells, but you see them looking like normal ikes and does respectively - however with a weird twist, there is another "you" on the opposite side of the room standing on the ceiling upside down. Assuming we decorate the room with furniture, stairways, catwalks, and doors, etc. We could climb up to a catwalk towards the center of the ike shaped room (assuming we're in the 11-cell). As you walk towards the other you, who is walking upside down on the identical catwalk above - you notice that he/she does everything you do, but mirror image and upside down - no doubt a freaky situation. This other you is actually you, but the space inside the ike is curved back on itself to produce this situation. Imagine you gave the other you a one dollar bill, he will do likewise. You look at the dollar he gave you and you notice it is mirror imaged! for this room is non-orientable! You decide to explore adjacent rooms and notice that opposite sides of each room are identical and lead to identical rooms - actually they are the same rooms. There are ten different rooms that are adjacent to each room, added to the room you are in adds to 11. I would love to see a video game version of this.

I like this.

It still doesn't give me a way to visualize the entire 11-cell as a complete entity in itself, though. Sure I can see individual cells this way, but what does the whole look like? That part still eludes me.

[ICN5D]

Complex Euclidean Space

Complex Euclidean space is what you get, when you do ordinary algebraic geometry (like a line is y = ax + b), when the numbers belong to the complex numbers. One of the interesting things is that in ordinary space, one can only keep things co-linear by a change of sign. In Complex space, things remain co-linear even under a rotation around the unit circle. That is, an equation y/x = a, has a solution for +x, +y and -x, -y. In complex space, you can have x cis(wt), y.cis(wt), where 'w' is omega, and t is time. Yes, that's right all of space can be bisily rotating around -b, and still observe y=ax+b.

This reminds me of learning about quadratic equations that have no x-intercepts on an XY plane. The intercepts actually exist in this complex space, outside of the coordinate grid. If using the 3rd dim, one could plot these intercepts as above the grid.

When you see imaginary distances appearing, it is a reference to Hyperbolic Space. What is happening is that you are not using a ruler that is uncurved enough to pass through all of the points. You have to pick one with a lesser curvature. In hyperbolic space, a zero-curvature thing is a bent line (a horocycle). Distances along this are quite dandy because circles are parts of both the euclidean horocycle, and the underlying hyperbolic space. So you can do euclidean geometry as long as you carry bits from circle to circle to circle.

A circular ruler, that makes more sense now. I'm not sure if I totally get it ( though I'm familiar with circles ), it's the concept being applied. But, it does shed some light on being able to interact with yourself in the 11-cell, as Polyhedron Dude puts it. Applying the infinite vertices effect of a single line, traced out into a circle, will make everything curve back onto itself, this "non-orientable" property. One's up is another's down, one's left is another's right, in the same place. That's why it feels Mobius strip like, where the half-twist is sort of combined within, around the center, without having to reach the twisted part. It's a really bizarre concept, for sure!

Your way of spin implementation now tells, that the spinning object itself will use the same circumradius, as would its spin. E.g. the circumradius of | and |O are the same, both having R = 1/2. Same for || and ||O, both having R = 1/sqrt(2). Etc. Thus esp. for |||| and ||||O you'd get R = 1. And therefore their respective taperings would become flat (in euclidean space implementations).

Yes, nice definition. This bisecting rotation will sort of "inflate" the shape in place, without really "extending" it, as extrusions and taperings would do. Now, when I come across a shape with four extrusions, I now know that a single taper operator will create a degenerate pyramid. Though, I'm still not clear on other combinations of | and > and how they stack up. I see |||>> as being degenerate in your table. The equation for surtope hypervolume can be derived directly out of my notation computations, as long we assume unit edge lacing. After the degenerate height effect, it would probably be best to switch to unit height taperings, which would change the math slightly.


--Philip

[Klitzing]

Your way of spin implementation now tells, that the spinning object itself will use the same circumradius, as would its spin. E.g. the circumradius of | and |O are the same, both having R = 1/2. Same for || and ||O, both having R = 1/sqrt(2). Etc. Thus esp. for |||| and ||||O you'd get R = 1. And therefore their respective taperings would become flat (in euclidean space implementations).

Yes, nice definition. This bisecting rotation will sort of "inflate" the shape in place, without really "extending" it, as extrusions and taperings would do. Now, when I come across a shape with four extrusions, I now know that a single taper operator will create a degenerate pyramid. Though, I'm still not clear on other combinations of | and > and how they stack up. I see |||>> as being degenerate in your table. The equation for surtope hypervolume can be derived directly out of my notation computations, as long we assume unit edge lacing. After the degenerate height effect, it would probably be best to switch to unit height taperings, which would change the math slightly.

Sure you could. But then consider, that under my definition (unit lacings) |> gets a higher symmetry than under yours (unit height)!
And then too, |>|| still is the uniform 3,4-duoprism, if mine is applied, but it would become some non-uniform variant under yours.

--- rk

[quickfur]

Your way of spin implementation now tells, that the spinning object itself will use the same circumradius, as would its spin. E.g. the circumradius of | and |O are the same, both having R = 1/2. Same for || and ||O, both having R = 1/sqrt(2). Etc. Thus esp. for |||| and ||||O you'd get R = 1. And therefore their respective taperings would become flat (in euclidean space implementations).

Yes, nice definition. This bisecting rotation will sort of "inflate" the shape in place, without really "extending" it, as extrusions and taperings would do. Now, when I come across a shape with four extrusions, I now know that a single taper operator will create a degenerate pyramid. Though, I'm still not clear on other combinations of | and > and how they stack up. I see |||>> as being degenerate in your table. The equation for surtope hypervolume can be derived directly out of my notation computations, as long we assume unit edge lacing. After the degenerate height effect, it would probably be best to switch to unit height taperings, which would change the math slightly.

Sure you could. But then consider, that under my definition (unit lacings) |> gets a higher symmetry than under yours (unit height)!
And then too, |>|| still is the uniform 3,4-duoprism, if mine is applied, but it would become some non-uniform variant under yours.

--- rk

Alternatively, you could make the notation purely topological, so the tapering height is unspecified and you can apply any appropriate value in a realization of the notation. Then all pyramids will exist, topologically, and the question of whether a particular shape can be made to have unit edge length becomes a secondary concern.

[ICN5D]

Yes, this would work. An interesting algorithm would be one that computes a degenerate height based on the combination of operators in unit edge lacing.

What is the general effect on n-simplices, as n grows? What happens to the clustering of vertices? Where is the volume concentrated?

The topological method is the original rubric in my notation, in a tesselationist method. The unit edges are a new thing, and interesting as well.

I think I'm going to include a subscript for the axis in motion with the spin operator. So, the |O>|O_w means to hold plane XYZ stationary, allowing the cone endcaps stretched along W to rotate in a non-bisecting manner into V. This will clear up all of the "prismic, triangular, toric" ambiguity. Does that work better?

This should allow the cone atop cone and line atop cylinder projections to work as one, and create the duocylinder-circle convex hull.