## Quickfur's renders

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### Re: Quickfur's renders

I’d just like to give a public shoutout to Quickfur. Today I built an omnitruncated 120-cell in GeoGebra with the coordinates from their site. GeoGebra crashed when trying to open it, but at least it’s theoretically a cool visualization.

Actually, managed to render it! Here's a screenshot:

(much higher quality version here: https://drive.google.com/open?id=1lkvs2rjzHBAnwqynmjLAkytJ69GXV30g)
URL
Dionian

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### Re: Quickfur's renders

Congrats! Looks great. Yeah omni120cell is a beast. With 14400 vertices, 28800 edges, 17040 faces, and 2640 cells, you need some serious computing power just to be able to handle it.
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### Re: Quickfur's renders

Errmm, Because there is now POMs for a while, can u do animations for the existing ones? Thanks!
Dekeract
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### Re: Quickfur's renders

Have you seen the existing animations yet? I didn't make an index of them, so probably not many people know where they are. There are animations for the 3,3-duoprism, 3,4-duoprism, the tesseract, bidex (bi24dim600cell), the cantitruncated 5-cell, the duocylinder, the grand antiprism, rectified tesseract, runcinated 120-cell, runcitruncated 16-cell, runcitruncated tesseract, truncated 16-cell, truncated 5-cell, and truncated tesseract. There's also an animation of J32 transmogrifying into an icosahedron, that you might be interested in, as well as J91 transmogrifying into an icosahedron.
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### Re: Quickfur's renders

Yes I have seen them. Oh and btw can u maybe work on allowing your program to render a ten-dimensional cube?
Dekeract
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### Re: Quickfur's renders

Dekeract wrote:render a ten-dimensional cube?
A 10-dimensional cube is bound to be a mess; 10 dimensions is just too much as you can see in e.g. this image
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### Re: Quickfur's renders

Do u hav ethe new POMs back up?
Dekeract
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### Re: Quickfur's renders

No.
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### Re: Quickfur's renders

Am I the only one here who thinks this guy's repeated posting in this thread is approaching harassment territory? I mean, quickfur might be busy, he'll do the polytope of the month again whenever he wants to.
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### Re: Quickfur's renders

username5243 wrote:Am I the only one here who thinks this guy's repeated posting in this thread is approaching harassment territory? I mean, quickfur might be busy, he'll do the polytope of the month again whenever he wants to.

Don't worry about it, I don't mind.
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### Re: Octagonny!

Quickfur's site refers to this:

wendy wrote:Octagonny, o3x4x3o and its isomorph octagrammy o3x4/3x4o, are fairly important figures, that rate similar to the regular figures.

There are, by clifford rotations, figures that correspond to the symmetries in three dimensions.

[2,2] 8 tesseract
[3,3] 24 24choron
[3,4] 48 octagonny
[3,5] 120 twelftychoron

All of these produce very interesting symmetries, where the 'arrow-rotations', project to six-dimensional figures, being the prism-product of euclidean figures, and their doubles.

The vertices of the duals of these, correspond to the units of quaterions (both integral and finite-dense).

Class-2 isomorphism can be presented in terms of replacing a+bx by a-bx, usually where x² is an integer. Since we have here "partial inversion", we can replicate the lattices

{12,12/5} by {3,6} + {3,6} (dual hexagons)
{8,8/3} by {4,4} + {4.4} dual squares
octagonny by {3,3,4,3}, dual 24chora.

The tiling of octagony, 288/73 at a vertex, is the first nonhyperbolic group that has no expression in Coxeter-Dynkin symbols. It requires six mirrors to start.

There are class=2 tilings that need only five: eg {5,3,3,5/2}.

It is in light of these special alignments with Z4 (octagonal numbers), that S/Q/S was titled 'octagonny'.

Wendy

I know this is 12 years old. But what do you mean by it?

I've never heard of "octagonal numbers", except as the integer sequence 3n2 - 2n = n*(3*n - 2);

(1*1, 2*4, 3*7, 4*10, 5*13, 6*16, 7*19, ...) = (1, 8, 21, 40, 65, 96, 133, ...)

octagonalNumbers.png (15.86 KiB) Viewed 920 times

This is part of a general pattern of polygonal numbers:

linear:
(0/2)n2 + (2/2)n = n,
triangular:
(1/2)n2 + (1/2)n = n*(n+1)/2,
square:
(2/2)n2 + (0/2)n = n*n,
pentagonal:
(3/2)n2 - (1/2)n = n*(3*n - 1)/2,
hexagonal:
(4/2)n2 - (2/2)n = n*(2*n - 1),
heptagonal:
(5/2)n2 - (3/2)n = n*(5*n - 3)/2,
octagonal:
(6/2)n2 - (4/2)n = n*(3*n - 2).

Also, to me "Z4" means the integers modulo 4; that's arithmetic where "4 = 0", for example 0+1=1, 1+1=2, 2+1=3, 3+1=0, 2*1=2, 2*2=0, 2*3=2, 3*3=1, etc.

Or did you mean "Z4"? That would be the grid of points in 4D space with integer coordinates, such as (-3, 8, 0, 3). I presume you would interpret these as quaternions, to be able to multiply them. (Ordinarily they can only be added and subtracted.)

I know that the quaternion group of symmetries of the cube (or octahedron {3,4}) has 48 elements: 8 like (0,1,0,0), 16 like (1/2, 1/2, -1/2, 1/2), and 24 like (1/sqrt2, -1/sqrt2, 0, 0). And it's apparent from quickfur's illustrations that these, as vectors in 4D, point to the 48 cells of the bitruncated 24-cell. But they don't have integer coordinates.

So I'm still wondering what "Z4 (octagonal numbers)" refers to, and how it relates to the bitruncated 24-cell.
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### Re: Quickfur's renders

You should probably ask Wendy directly; I'm not too familiar with the details myself and am not confident I can give an accurate explanation.
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### Re: Quickfur's renders

Excuse me. I thought I was asking Wendy directly; I guess the "you" wasn't clear.

I see the equivalences of the 2D lattices she mentions: {8, 8/3} (overlapping octagons placed edge-to-edge) has vertices at any integer combination (linear combination with integer coefficients) of the vectors at angles 2πk/8 (k=0,1,2,3,...); while {4, 4} has vertices at integer combinations of the vectors at angles 2πk/4 = 2π(2k)/8 (the even multiples of 2π/8); and the "dual" {4, 4} is rotated, so its angles are 2πk/4 + 2π/8 = 2π(2k+1)/8 (the odd multiples of 2π/8). So indeed {4, 4} + {4, 4} = {8, 8/3}.

I also see that the bitruncated 24-cell has dichoral angle 3π/4 (at the octagonal faces), the same as the octagon's angle, so these polychora could be fitted around a face in the same way that octagons can be fitted around a vertex in {8, 8/3}. (And its other dichoral angle is 2π/3, at the triangular faces.) I don't yet see that 288/73 of these can fit around a vertex, but I'm not really interested in that.

I'm guessing "octagonal numbers" is just the lattice {8, 8/3}. It seems odd to call these "numbers", though.

This 2D lattice is isomorphic to the 4D lattice Z4, since it has 4 basis vectors: (1,0), (0,1), (1/sqrt2, 1/sqrt2), (-1/sqrt2, 1/sqrt2). (This is only a basis when considering integer or rational combinations, not irrational combinations.) Z4 itself has basis vectors (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1).

Or maybe "octagonal numbers" are numbers of the form a+b*sqrt2 where a,b are integers. Any point in 2D with such coordinates is in the lattice {8, 8/3}, but not all points in the lattice have such coordinates; for example, (1/sqrt2, 1/sqrt2) = ((1/2)*sqrt2, (1/2)*sqrt2), since 1/2 is not an integer.
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### Re: Quickfur's renders

I'm pretty sure the connection Wendy has in mind has to do with the symmetry group of the bitruncated 24-cell, not just the general extension field with elements of the form a+b√2. (The latter would include the coordinates of all uniform polytopes with n-cube symmetry in all dimensions, so to specifically mention the bitruncated 24-cell in connection with the octagonal numbers would seem an odd thing to do if this were the case.) The permutation group of the vertices probably also factor into this, though I have to admit the exact connection is unclear to me.
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### Re: Quickfur's renders

The 'span' of a set, is the dot-product of the set and an array of integers. If a different span is called for, then one should note it, eg pentagonal span.

The 'span of an equation' sets the set to 1, x, x², x³, ... as far as required. If the equation is integer and unitary (ie all values in Z, and the high term is \pm 1), than the span defines an integer number system. All polygon-shortchords solve such equations.

Zn is the span of chords of a polygon of n sides. I have evaluated the polynomials as high as the 81st power. If you mean the modulus set, you should write the set Z//n (that is, Z mod n, // is mod in rexx). Z4 is thus $$Z[1, \sqrt{2}]$$, the mathematicians often leave out the '1' here, but that defines the coset $$Z*\sqrt{2}$$, being q, 2q, 3q, etc.

The expression for a system over 12 dimensions is eg RD12, or ZD12. Writing Z^12 essentially defines the numbers that are the twelfth powers of Z (0. 1, 4096, 531441, 16777216, &c). Number systems might be described as something like C2D4 (that is four axies of class-2 numbers).

Polygon-class numbers have a realisation of a sparse skew lattice in RDn. Sparse means that there is a distance around a member which no other member occurs, skew means it does not follows the axies, and lattice means that it is closed to addition and subtraction. In this presentation, multiplication of two vectors is the same as the product of each coordinate separately, so A_i B_i = (a1 b1, a2 b2, a3 b3,,,).

The hypergeometric numbers are formed of a+bj, where j²=+1. This presents a D2 construct (a+b, a-b), In place of unit circles, there are unit hyerbeola, and numbers form along these hyperbola, except for (0,0), which is at the crossing of the zero and alt-zero lines (ie (0,y) and (yx,0)). In practice, we take a lattice that projects every point onto a separate dot on each of the three axies.

The cyclotomic integers CZn, is the span of the polynomial r^n+1 = 0. These form an integer system CZn for every n, but is sparce when n=1 (regular integers), 2 (gaussian) and 3 (eisenstein) integers. They correspond to the span Zn[1, r]. The set CZ7, for example includes substrates of Z [1, \sqrt{-7}]bc.

Z4 = octagonal numbers, Z5 = pentagonal numbers, Z6 = dodecagonal numbers are all class-2 systems.

The Quarterions have integers based around the regular figures and the octagonny-dual in 4D.

OZ2 = tesseract OZ3 = 24choron OZ4 = octagonny, and OZ5 = twelftychoron. The eutactic stars are the duals of these, the lattice is the span of the stars.

One should understand that the level of precision here, makes what I have seen of the mathematician's notation as (a) sloppy, and (b) as subtle as a blunt axe.
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### Re: Quickfur's renders

Okay, thanks for explaining.

wendy wrote:Z4 is thus $$Z[1, \sqrt{2}]$$, the mathematicians often leave out the '1' here, but that defines the coset $$Z*\sqrt{2}$$, being q, 2q, 3q, etc.

I think they do that because Z[q] is supposed to allow not only addition, but also multiplication, including q2 = 2 and the "empty product" q0 = 1.

To restrict to addition, they might write something like spanZ(1, q), or <1, q> (angle brackets), though that would be more specific as Z<1, q>. And the angle brackets might have other unintended meanings.

wendy wrote:The expression for a system over 12 dimensions is eg RD12, or ZD12. Writing Z^12 essentially defines the numbers that are the twelfth powers of Z (0. 1, 4096, 531441, 16777216, &c). Number systems might be described as something like C2D4 (that is four axies of class-2 numbers).

That makes sense, though of course it conflicts with the "standard" mathematical notation.

It would help if you'd define such notation like Zn each time you first use it (or provide a link to somewhere else it's defined).

wendy wrote:The hypergeometric numbers are formed of a+bj, where j²=+1. This presents a D2 construct (a+b, a-b), In place of unit circles, there are unit hyerbeola, and numbers form along these hyperbola, except for (0,0), which is at the crossing of the zero and alt-zero lines (ie (0,y) and (yx,0)). In practice, we take a lattice that projects every point onto a separate dot on each of the three axies.

I'm familiar with these, which I call "perplex numbers"; there are maybe a dozen other names. I think the most common is "split-complex numbers".

I don't understand the last sentence. Are the three axes for (1 + j)/2, (1 - j)/2, and 1? Or are you talking about a 3D lattice?

wendy wrote:The cyclotomic integers CZn, is the span of the polynomial r^n+1 = 0. These form an integer system CZn for every n, but is sparce when n=1 (regular integers), 2 (gaussian) and 3 (eisenstein) integers. They correspond to the span Zn[1, r]. The set CZ7, for example includes substrates of Z [1, \sqrt{-7}]bc.

Z4 = octagonal numbers, Z5 = pentagonal numbers, Z6 = dodecagonal numbers are all class-2 systems.

Why isn't Z4 called square or tetragonal numbers, and Z8 octagonal numbers? Why are you using rn + 1 instead of rn - 1?

(I don't know a lot about chords or cyclotomic numbers. The answer might be obvious if I read or think about it more.)

Your notion of "span of a polynomial" seems incomplete. Is the variable supposed to satisfy only that polynomial equation and no lower-degree ones? In that case, the perplex numbers would be the span of j2-1 = (j+1)(j-1) = 0, and j+1 and j-1 are both not 0. On the other hand, the cyclotomic number r=exp(iπ/3) is a root of r3+1 = (r+1)(r2-r+1); it doesn't satisfy r+1=0, but it does satisfy r2-r+1=0. The polynomial rn+1, or rn-1, does not tell us which, if any, of its factors also have r as a root.
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### Re: Quickfur's renders

'square' and 'hexagon' are already taken with numbers in {4,4} and {3,6} resp.

In any case, the even chords of an even polygon can not be expressed as a sum of the odd chords, or their closure to multiplication. So for example, you can not get sqrt(2) or sqrt(3) from the odd chords of the square (ie 1), or hexagon (1,3). So the octagon and dodecagon are the first polygons to exhibit Z4 and Z6.

The span of an equation is part of a proof that an equation has integer coordinates, and the first term is 1, then every number in the span divides an integer, and such an integer is the determinate of the third-order tensor (trimex), times the vector. This happens if the equation is 'prime', ie without further factors in Z. Then you can divide the matrix described before, into the unit-matrix times the determinate.

The short-chords of polygons (ie the side of a triangle with the other two as edges), divide a series of equations corresponding to a class i call J(2n), which are invariable prime, when evaluated for x=integer, gives numbers that are coprime, except when p|J(2n), then p|J(2pn) etc, and which are generally in the nature of a cosh function. The fibonacci numbers correspond to the repunits in J(-3).

The reason that I call the hypercomplex plane such, is that where trig functions happen in the complex plane, hypertrig functions are in the hypercomplex plane.

The span of an equation is a construction, is not necessarily complete. What it does show is some body-centering of lattices etc, but the complete sense is there, in that K < S < nK where < is 'contains'. For some uses, this is sufficient. An example where body-centering occurs is x^3=12, which is a subset of the Z[1, cbrt(12), cbrt(18)]. The solution to Pell's equation a²N-b²=1, actually has a body-centering when N mod 4 = 1, the solution has the nature of a cube in these cases.

The span of an equation is the carry-left on an abacus. I mean, I do calculations in x^4 = 4x^2-1 quite often, because you get sqrt(2), sqrt(3) and sqrt(6) exact.

The equation r^n+1 etc, has quite well-knownn factors, which you can often find on a calculator (with r=10). If one is supposing that r^n+1=0, it is assumed that n is the first instance of zero, and so r^m+1=0, where n is an odd-number times m, are excluded. So for n=9, we exclude solutions in n=3 and n=1. The actual program to do this is related to the process at the Cunningham project, for instance.

Using superscripts and subscripts is best avoided. Some processes smash the case and the ex-scripts. A cm described as 10^8 atoms across, comes to be 108 atoms across. Gosh, you get things like (2 cos a)^^n = (2 cos na) is something that we make heavy use of. Except we don't use cos or cosh here.
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### Re: Quickfur's renders

wendy wrote:'square' and 'hexagon' are already taken with numbers in {4,4} and {3,6} resp.

{4,4} is a tiling or a lattice, not a set of numbers. Are you referring to the Gaussian integers? Obviously those already have a name, and needn't be called "square numbers".

wendy wrote:In any case, the even chords of an even polygon can not be expressed as a sum of the odd chords, or their closure to multiplication. So for example, you can not get sqrt(2) or sqrt(3) from the odd chords of the square (ie 1), or hexagon (1,3). So the octagon and dodecagon are the first polygons to exhibit Z4 and Z6.

I see that the span of chord lengths of an n-gon is the same as the span of squared chord lengths of a 2n-gon.

It seems unfair to the square {4}, to define the numbers Z4 based on it, then not allow it to have those numbers!

Well, you've answered the first part of my question:

mr_e_man wrote:what "Z4 (octagonal numbers)" refers to, and how it relates to the bitruncated 24-cell

Z4 is all numbers of the form a+b*sqrt2 where a,b are integers. This set is generated (using addition and subtraction) by the chords of a square, or by the squared chords of an octagon.

Now let's focus on the second part. I haven't checked, but I guess that the vertices of the self-intersecting "tiling" of octagonnies is exactly the lattice generated by octagonny's edge vectors; those are the same as its normal vectors, which are:

mr_e_man wrote:I know that the quaternion group of symmetries of the cube (or octahedron {3,4}) has 48 elements: 8 like (0,1,0,0), 16 like (1/2, 1/2, -1/2, 1/2), and 24 like (1/sqrt2, -1/sqrt2, 0, 0). And it's apparent from quickfur's illustrations that these, as vectors in 4D, point to the 48 cells of the bitruncated 24-cell.

This lattice is not exactly Z4D4, since Z4 does not include 1/2; but its intersection with any of the four coordinate axes is Z4. There may be a better way to describe the relation, similar to the relation of the Hurwitz quaternions to Z.

wendy wrote:The Quarterions have integers based around the regular figures and the octagonny-dual in 4D.

OZ2 = tesseract OZ3 = 24choron OZ4 = octagonny, and OZ5 = twelftychoron. The eutactic stars are the duals of these, the lattice is the span of the stars.

So what is "O" doing?
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### Re: Quickfur's renders

The O's ought be Q's for Quarterion. Without outside help, i ended up with a different set of letters. The same holds true for symmetries.

The letters for the symmetries A, B, C, D, E, F, G come from group theory, particularly the Lie groups. The Lie group B_n and C_n map onto the same mirror-group, the difference is the direction of the arrow on the '4' branch. They use a parallel line with an arrow, eg o=>=o for o-4-o. The Cartlan matrices correspond to the stott matricies, except again the treatment of the sqrt(2) or sqrt(3).

I use letters that denote the separable mirrors, so {3,3,....3,3} is 's', {3,3,3...3,4} is 'hr', the 3's are the h, and the single node to the right of the 4 is the r (rectangular). {3,3,..5} is 'f', as are all of its stellations, {3,3,...,3,3,B} = k_21 are 'g', {3,4,3} is hh. The euclidean tilings are t (the ring), q, qr, qrr, qq (for E3..3A. E3..3Q, Q3..3Q, and 3343.), y for the gossets, eg 2_21, 3_31, 5_21. and v for pentagonal tilings (5,3,3,5/2, etc). The tiling of octagonny is 5bb, being an inverted q.

These are translated using a known list.

Body-centering of numbers is more a result of using the wrong coordinates, although it does give the correct sense, which is sufficient enough.

QZ4 is defined in terms of xo3oo4oo3ox as the eutactic star. The exact numerical representation in other people's systems is for them to find.
QZ5 is defined as x3o5/2o5o3o as the eutactic star. It is 5v. 5v is VFA, ie a eutactic star o3v5o.
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### Re: Quickfur's renders

When will the next POM update be??
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