A 10-dimensional cube is bound to be a mess; 10 dimensions is just too much as you can see in e.g. this imageDekeract wrote:render a ten-dimensional cube?
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.
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
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.
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).
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.
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.
wendy wrote:'square' and 'hexagon' are already taken with numbers in {4,4} and {3,6} resp.
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.
mr_e_man wrote:what "Z4 (octagonal numbers)" refers to, and how it relates to the bitruncated 24-cell
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.
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.
wendy wrote:One prefixes a patch-term to designate a figure formed of such patches or fabrics. A fabric is taken to mean the entirity of the thing, while a patch is part of a larger thing, so
POLY + HEDR + ON poly = many (+closure) + 2d + patches = (surface of) 3d solid
POLY + CHOR + ON poly = many (closed) + 3d + patches = (surface of) 4d solid
MULTI + HEDR + ON multi = many (no implied closure) + 2d patches = eg net of a cube [which is not closed]
quickfur wrote:Hehe... forgot to post the Polytope of the Month here:
This is the rectified 24-cell. Which happens to be the same as the cantellated 16-cell:
Basically, 24-cell symmetry is just tesseractic symmetry where the 8-cell elements and the 16-cell elements are identical.
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