quickfur wrote:Hmm, you're also missing sporadic uniform polytopes like the snub cube, snub dodecahedron, snub 24-cell, grand antiprism.
Keiji wrote:I do believe I was working on an SSC3 at some point. I forgot what I was doing with it, though...
quickfur wrote:Oh really? I would've expected Ko0 to be just a single point. But I guess might as well reuse it for something useful.
What about distinguishing between binary and decimal notation for the truncates?
Keiji wrote:I do believe I was working on an SSC3 at some point. I forgot what I was doing with it, though...
quickfur wrote:One thing to keep in mind if you're going to start using abstract polytopes for representation: many abstract polytopes cannot be realized geometrically at all.
What's the definition of a brick product? I've heard it before but never really understood what it was. I tried searching in the wiki but couldn't find an actual definition.
quickfur wrote:Also, is a hexagon considered to have brick symmetry? Because its coordinates can be written (0,±a), (±b,±c) where c<a.
Why are the octogoltriates the "simplest" powertopes?
Keiji wrote:quickfur wrote:One thing to keep in mind if you're going to start using abstract polytopes for representation: many abstract polytopes cannot be realized geometrically at all.
This doesn't bother me: the ASCII string "5Gxv{3" doesn't represent a valid shape in SSC2, but each individual symbol does have a meaning in the notation. Similarly, there are a lot of symmetric matrices which don't represent valid shapes.
Keiji wrote:[...]
I'm not the first person to use Dx 0 for snubs.
Keiji wrote:[...]
Because I wrote that before realizing that hexagons were bricks.
quickfur wrote:Keiji wrote:I'm not the first person to use Dx 0 for snubs.
You could assign Ks0 to be the grand antiprism, since it's remotely related to the 600-cell (having a subset of the 600-cell's vertices), and there are no snub-like uniform members of the 600-cell family.
quickfur wrote:Keiji wrote:Because I wrote that before realizing that hexagons were bricks.
Actually, all even polygons have brick symmetry, now that I think about it. I think the wiki says that they must be powers of 2, but that's wrong.
quickfur wrote:OK, first, an incidence matrix has binary entries. So you can interpret it as a bitmap.
if you are already deriving the incidence matrix by some other means, then why bother with the matrix representation at all? Wouldn't it be better to use the most convenient representation (presumably the one you used to derive it) instead?
Keiji wrote:quickfur wrote:[...]
if you are already deriving the incidence matrix by some other means, then why bother with the matrix representation at all? Wouldn't it be better to use the most convenient representation (presumably the one you used to derive it) instead?
It gives a generalized notation. The problem with using specific notations is that when you want to add an operator to them, you suddenly have to invent new syntax, and then you discover that you can write something that may or may not be a valid shape, and potentially end up in a big mess like my infamous rotope construction chart. With a generalized notation, you don't have to worry about all that.
quickfur wrote:Plus, doing operations with compressed incidence matrices adds complexity to algorithms because you need to decompress, recompress, etc.
Keiji wrote:[...] thus, one can compare for all such possible permutations, or use a canonical ordering.
On the brick product front, I have discovered that the torus is a valid value for P.
For example, torus{square, digon, digon} gives a 4D shape where the 3D cross-section is a crind with a smaller crind removed from the center, and as you move the cross section, the outer crind becomes smaller and the inner one becomes larger, until they meet at the ridge of the shape. The order of the arguments matters; assuming torus is defined as ((II)I) as opposed to (I(II)), then although torus{square, digon, digon} and torus{digon, square, digon} are the same, torus{digon, digon, square} is totally different, although I haven't yet worked out what this shape would be.
quickfur wrote:Keiji wrote:[...] thus, one can compare for all such possible permutations, or use a canonical ordering.
And this is where you will have trouble. Comparing all possible permutations is infeasible for large polytopes, because the number of permutations is exponential, O(n!).
the numbers reference row/column numbers
it's not obvious what exactly is canonical for a given polytope. You may end up having to examine the entire structure of the polytope before you can decide what is canonical, and then you have to go and swap those rows/columns around and renumber things.
On the brick product front, I have discovered that the torus is a valid value for P.
And not just the regular torus, if my understanding is correct, any affine transform of the torus qualifies too. So you can, for example, do brick products with a slant-squished torus with a 0-radius hole, for example. I'm sure that 0-radius hole will cause headaches, if the slant-squish transform doesn't.
Hmm. I still don't understand how exactly the shape of the torus/square/digon correspond to each other in the product.
And I'm still in the dark about why P must have brick symmetry.
Keiji wrote:quickfur wrote:Keiji wrote:[...] thus, one can compare for all such possible permutations, or use a canonical ordering.
And this is where you will have trouble. Comparing all possible permutations is infeasible for large polytopes, because the number of permutations is exponential, O(n!).
Even though it's exponential, it should still be fast enough for any polytope whose compressed incidence matrix fits on my (1080p) screen! And I can't really think of a reason why we'd want to mess with polytopes more complicated than that, at least in the near future.
the numbers reference row/column numbers
Sorry, what? I'm pretty sure they don't. The numbers inside the matrix have nothing to do with row and column numbers; the information is combined by such methods as picking a starting cell, then reading cells with the same row number as the original cell's column number. Row and column numbers are never combined or compared with the number that actually occupies a cell.
it's not obvious what exactly is canonical for a given polytope. You may end up having to examine the entire structure of the polytope before you can decide what is canonical, and then you have to go and swap those rows/columns around and renumber things.
And in existing notations we already have to do this anyway, so how is it a disadvantage?
[...] So you can, for example, do brick products with a slant-squished torus with a 0-radius hole, for example. I'm sure that 0-radius hole will cause headaches, if the slant-squish transform doesn't.
You can stretch it along an axis, yes, but you can't do something like slanting, because that would break the brick symmetry. Remember, brick symmetry mandates all sign permutations, not just the one with all coordinates negated.
[...] The torus appears as the crindal hole through the product. The square and digon correspond to the same parts of the crind-like product that they do in the original crind - remember, a crind is the RSS (circle as P) product of a square and a digon.
And I'm still in the dark about why P must have brick symmetry.
I never really arrived at a proper explanation of this, but I think it's something to do with uniqueness of the product. If you used a non-brick as P, say, a pentagon, the product would not be well defined, there would probably be multiple different possibilities (like how the square root has both a positive and negative result), and they would probably be rather ugly shapes too.
quickfur wrote:What about the omnitruncated 120-cell, which has 14400 vertices? And the omnitruncated 10-cube ...?
Obviously I still don't quite understand how the notation works. I tried looking at your link but it's not obvious what exactly the numbers refer to.
What about the 0-radius torus hole though?
Or, for that matter, what about a random cloud of points (e.g. a cantor set) in one octant reflected across the coordinate planes so that they have brick symmetry?
I'll have to admit I'm not really following here. Why does the torus produce a hole, and why is the hole a crind?
I guess I won't fully understand this before I understand just what exactly the brick product does.
Keiji wrote:I think you ought to re-read the Wikipedia page on this subject that I linked to before, because you seem to be misunderstanding something to do with the compressed incidence matrix format.
[...snipped lots of good stuff...] You're not the only one who wants to understand exactly what the brick product does. Bowers could really help here.
quickfur wrote:But speaking of Bowers and notations, have you ever read about his regular polytwisters? They are essentially duocylinders where the circular tori are replaced by polygonal spiral tori. They have no vertices but do have spiral ridges and two surchorixes that are completely symmetrical, so they are regular objects. Now there's something that cannot be represented by incidence matrices.
Bowers is an absolute genius, but unfortunately he's not very good at communicating his brilliant ideas to others. Have you seen his array notation? It's an absolutely amazing piece of work for someone with only elementary math education, transcending the best notational systems professional mathematicians invented for large numbers, and on the verge of reaching the finite equivalent of the so-called "large Veblen ordinal". However, his explanations take a lot of effort to understand, because he uses his own, non-standard terminology, and often fails to state the assumptions/definitions he's working from. Sometimes I have a hard time understanding what exactly he's trying to say. For example, for a long time I thought his array notation didn't even reach the finite equivalent of Gamma_0, and thought I could do better, but upon closer inspection, I discovered that it far transcends it.
Deedlit wrote:Bowers is an absolute genius, but unfortunately he's not very good at communicating his brilliant ideas to others. Have you seen his array notation? It's an absolutely amazing piece of work for someone with only elementary math education, transcending the best notational systems professional mathematicians invented for large numbers, and on the verge of reaching the finite equivalent of the so-called "large Veblen ordinal". However, his explanations take a lot of effort to understand, because he uses his own, non-standard terminology, and often fails to state the assumptions/definitions he's working from. Sometimes I have a hard time understanding what exactly he's trying to say. For example, for a long time I thought his array notation didn't even reach the finite equivalent of Gamma_0, and thought I could do better, but upon closer inspection, I discovered that it far transcends it.
Hello, quickfur. I am very curious to see your analysis of Bowers' notation. I find his system quite interesting, but I do not believe it transcends the notations used by professional mathematicians. One such notation system is the Gregorczyk-Wainer hierarchy, defined by ...
[...] As for Bowers' notation, his original array notation is equivalent to the Wainer hierarchy below the ordinal omega^omega. His extended array notation in n dimensions is equivalent to the Wainer hierarchy below the ordinal omega^omega^n.
As for his BEAF, his functions do not seem to be sufficiently well defined. They depend on exactly what constitutes a structure, and for each structure, what constitutes the set of previous structures. For example, is (X+1)^X a structure? What about (X^^^X)^X? If the previous structure to X^^^X is X^^^p, what is the previous structure or structures to X^(X^^^X)? (Note that X^(X^^^p) doesn't work, since this would drop the function below X^^^X.) I can make sensible rules for his tetrational structures - this corresponds to the Wainer hierarchy below epsilon0 - but trying to make sense of pentational structures and beyond turned into a great big mess. We're still way below Gamma0. I see no reason to believe his notation reaches up to the large Veblen ordinal, so the Wainer hierarhcy goes very much farther with known ordinal notations.
Can you help decipher pentational structures and beyond?
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