A long overdue discussion about the future of rotopes

Higher-dimensional geometry (previously "Polyshapes").

Re: A long overdue discussion about the future of rotopes

Postby Keiji » Sun Nov 16, 2008 12:40 am

What does symmetry have to do with tapering, though?
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: A long overdue discussion about the future of rotopes

Postby quickfur » Mon Nov 17, 2008 12:51 am

Hayate wrote:What does symmetry have to do with tapering, though?

Case in point: it doesn't. :)

In order to represent highly symmetric polytopes in a nice way, we need to somehow incorporate symmetry into the picture. Or rather, tapering is like a "sub-operation" of a more general symmetric representation system, in the sense that it can be a part of a larger symmetric whole (see Conway's kis operator, for example).
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A long overdue discussion about the future of rotopes

Postby wendy » Mon Nov 17, 2008 7:53 am

A good number of figures, especially those involving cones, spheres, cylinders, &c, are not really polytopes. They're solids. None the same, one can usefully treat figures in many of the ways that polytopes work.

The things like lace-prisms, products, truncations &c are 'constructions'. Some polytopes have several constructions. For example, I constructed the general case o3m3...3o4o, "double-cube" and concentrated on the 5d case. I correctly enumerated the 4d case as having 24v, 96e, 96h, 24c, and even that the faces were x4o #* q, (ie a square * sqrt(2) tegum), without realising this is {3,4,3}. In the polygloss, constructions are "presentations" of the polytope.

Products are freely appliable to any thing. The crind, prism and pyramid products rely on content, and can be applied to any set of shapes (even a picture). Who has not seen something like a picture jumping out of a point (pyramid) or cut as a solid slab (prism). Tegums and combs require a surface to operate on.

Tapers are a special instance of "progression". This is based on a notion that the Dynkin symbols that I use is really a coordinate system, and that it really is possible to make something that is 80% cube + 20% octahedron. One can take something like x3o4q, as a coordinate (1,0,1.4142), applied to the vectors (r2,0,0), (r2,r2,0), (1,1,1). One reflects this by all permutations and change of sign to get this polytope.

Progressions also allow one to determine cross-sections. For example, there is a section of the 24-choron that goes from x3o4o to o3o4q, and eighty percent of the way gives at least these vertices. Lace cities, like the presentation of the {3,3,5} given earlier, allows one to find sections for unpublished directions. For example, one can do sections "pentagon-first", or "edge first", even though these are not among the published ideas.

Some sections like the one i gave for 2_21 (/4B), are quiclky derived from lattice-vertices, none the same, would not be apparent if one had only ever seen the vertex / half-penteract / pentacros sections of this figure. An entirely different lace city is in 2_21 as

Code: Select all
              pt         3./3 --  3/.3
         /2A            /    \   /     \
     pt       2A/    3/.3 -- /3.3/ -- 3./3
         2/A             \   /    \   /
              pt         3./3 --  3/.3

   pt    o3o3oAo    3./3    o3o x3x @ o3x
   /2A   x3o3oAo    3/.3    o3x o3o @ x3o
   2/A   o3o3xAo    /3.3/   x3o o3x @ o3o
   2A/   o3o3oAx


The first lace city shows one of the 40 great triangles q3o around the 2_21. This shows, eg that the long diagonals of the cross-faces form triangles around the figure. The second presentation shows the same figure, where the outside rim is a set of six hexapeta, sharing three vertices at each end with the next, forming a perfect hexagon. Stellating just the simplexes gives rise to the 4B\, the dual of 1_22. This particular figure is the cell of one of the most efficient packing of spheres in six dimensions.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: A long overdue discussion about the future of rotopes

Postby wendy » Mon Nov 17, 2008 11:50 am

I'll try to give a simple overview of my notation. It covers an awful lot of stuff in many dimensions.

The basic description is by a linearised trace of the dynkin symbol, tested against all of the spheric, euclidean, and hyperbolic symplexes of finite content.

1. The symbol is divided into "structural" and "decorative" elements. Structure defines the kaleidoscope or symmetry, and decorations are the motif.

2. Quotes "", ' ' are used to mark off a symbol from text, where it is elsewhise not clear. Brackets () and [] enclose elements to create a lower or upper case symbol, so (p+q) = r, but [p+q] = R.

STRUCTURAL RULES

1. The mirror-group is implemented as nodes (mirrors), and branches (angles between mirrors). Mirrors at right angles are not listed, unless there is no other way to connect the group, thus o-----o o is o3o&o.

2. Branches connect "subject" to "object". The branches G,E, connect to a later object, ie in gGeEx3o, G connects g to o, E connects e to o, and 3 connects x to o. Likweise, "subject" branches o3xAaBbCc the branch (in upper case) connects the corresponding the o to the lower case letter. This allows us to write something like 2_21 as "o3o3o3o3oAo".

3. The special nodes z and zz repeat the first and second mirror after an &. This allows the forming of loops. One can follow a z or zz node with further branches, eg o3o3o5/2z3o gives a triangle with a branch, one of the sides (not opposite the junction) marked 5/2.

4. The special branch O, occurs by itself or connected to other O's, designate a sphere. It is used in products to generate figures like cones, cylinders &c.

5. A point has no symmetry (all of its symmetry is from the arroundings), is designated by the connection of the first-node to itself, ie z. This is needed in some products, eg a simplex is z ** @ (n+1) = the n+1 th (** pyramid) (@ = power) of a point (z). x and o are lines, already assume a direction, while z does not.

6. For the purpose of finding the first node, the count may cross a branch marked '2' but not one marked &.

DECORATIVE RULES

1. Wythoff's mirror-edge constructions are implemented by placing an edge-node, by replacing o as required. Edges of 0.618 (v), 1.000 (x), 1.414 (q), 1.618 (f), 1.732 (h) and 2.000 (u) are supported. The value (y) designates a freely variable node independent of x. eg x2y is a rectangle, x2x a square, and x2f a golden rectangle.

2. The mirror-margin construction is implemented by placing a letter 'm' at those walls of the fundemental region to be retained. In practice, replacing x by m creates the dual of a figure, eg CO = o3x4o, rhombic dodeca = o3m4o.

3. The special symbols s, g, j are used to create the snubs (by alternation of vertices), gyrates (alternation of faces), and a special grand antiprism. Generall, s and g occupy all nodes connected by odd branches, separately for each combination, eg

s3s4o = icosahedron, o3o4s = tetrahedron, s3s4s = snub cube, are alternate vertices of x3x4o, o3o4x, and x3x4x.

The special cases are s3s4s (snub Cube), s3s5s (snub Dodeca), s3s4o3o snub 24ch, and j5j2j5j grand antiprism.

4. When & appears in a symbol over a mirror-edge construction, a prism is implied, when in a mirror-margin, a tegum is implied.

5. For spheres, the placement of x or m implies a new increasing or decreasing axis-value. This allows for the forming of ellipsoids of all kinds, eg

xOxOo has x<y=z oblate elipsoid, while xOoOx has x=y<z is a prolate elipsoid. x is increasing, m is decreasing: this allows the inclusion in prism/tegum-products, the x/m dual rule. xOo&x is a cylinder, mOo&m is a "bi-cone" circular-tegum.

6. The special node 'r' is used to represent 'adjacent vertices', for representing microscopic polytopes at a point. For example, o4o3o is a point, while r4o3o is a miniscule cube.

COMPOUNDS.

1. A Wythoff-compound is two or more figures discribed in the same kaleidoscope. These may be implemented by making nodes carry two, three, ... nodal letters. eg q3o4o + o3o4x is the compound of an octahedron of edge 1.414 and a cube of edge 1, it may be written as qo3oo4ox. Each element of a compound is a 'base'.

2. When &#o (hull), &#x (prism), &#m (tegum) or &#t (tower) are used, this produces a single figure covering these compounds.
&#o creates a convex hull, eg qo3oo4ox&#o gives the rhombic dodecahedron o3m4o.
&#t eg xfo3oox&#t creates a tower of sections x3o, f3o, o3x. This is a tri-diminished icosahedron, but a form of this occurs in every dimension, eg
xfo3oox3ooo&#t is the vertex-figure of the snub 24ch.
&#x creates a lace-prism. When more than two bases are supplied the result forms a simplex in the altitude.
&#n creates a lace-tegum. (ditto.) Lace-tegums are defined in terms of intersecting pyramids.

The special node operator '$' is used to create a lace prism that appears as the vertex-figure of a uniform polytope. For example, the vertex-figure of x4x3o is $4$3o. There are rules for reading this symbol directly too, eg $8$3o8$ can be read without recourse to the hyperbolic frieze 8,3,8.

For example, a cone is xoOoo&#t, while a fulstrum is xyOoo&#t

PRODUCTS

1. The four products are symboled according to the inclusion or exclusion of a '1' at the front and back of the surtope-string. A leading 1 is a content-product, a trailing 1 is a draft-product. 0's in these give surface, and repetition products:

*# prism #* tegum, ** pyramid ## comb.

2. The general operator @ is used to mark power, lies between the operator and the integer to which the power is to be takem. eg 5 +@ 3 is 5 + 5 + 5 = 15, while 5 *@ 3 = 3 @* 5 = 5 * 5 * 5. So x5o *#@ 3 is the third (3) prism (*#) power (@) of a pentagon (x5o).

3. When several products are inferred, the inner ones must be enclosed in matching brackets.

A cone might be represented as xOo ** z

CLOSE

It must always be recalled that there are ever-more figures than can ever be listed here. Laminates are barely discussed, save by way of lace-prisms. This includes a vast number of laminate tilings, like LPC1 (the layers of triangles and squares in 2d), or the 3d cases LB2, LPA2, LPB2, LC2, LPC2, etc. Figures not governed by Wythoff's rules (like the Mobius tilings in hyperbolic geometry) are not covered in this, nor are many of the figures constructed by the Conway-Hart notation (although this is easy to implement, eg by using the ! operator).
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: A long overdue discussion about the future of rotopes

Postby quickfur » Tue Nov 18, 2008 10:52 pm

wendy wrote:[...]1. The mirror-group is implemented as nodes (mirrors), and branches (angles between mirrors). Mirrors at right angles are not listed, unless there is no other way to connect the group, thus o-----o o is o3o&o.

For Hayate's benefit: the idea behind this is to generate the points of a polytope by transitively reflecting it using a certain setup of mirrors (like a kaleidoscope). So you could say, start with 3 mirrors perpendicular to each other, and a point equidistant from them, and they would reflect the point into 3 other points. If you've ever looked at multiple mirrors facing each other, you'll know that mirrors also reflect mirrors into "virtual" mirrors, and that these virtual mirrors can also reflect images of reflected points, as well as other mirrors (virtual or otherwise). So if we take this process to its logical conclusion, we see that the single point we started with has mapped to the 8 vertices of a cube. Other arrangements of mirrors and points will produce different polytopes (e.g., if a pair of mirrors make an angle of pi/n with each other, you get a 2n-gonal prism).

Now, only certain arrangements of the mirrors will produce a closed polytope (other arrangements will result in an infinitely dense set of points). In particular, the angle between the mirrors must be commensurate with pi, since otherwise a single point and a mirror will just continuously reflect into new points and virtual mirrors, and the process never terminates (the images become dense).

The arrangements of mirrors can be represented by graphs, based on how they are positioned relative to each other; this is what Wendy is representing in her notation.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A long overdue discussion about the future of rotopes

Postby Keiji » Tue Nov 18, 2008 11:23 pm

Yes, I understand Wythoff symbols and Wythoffian Coxeter groups. What I don't understand are (what I am assuming are) wendy's extensions to the above.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: A long overdue discussion about the future of rotopes

Postby wendy » Wed Nov 19, 2008 9:05 am

The Dynkin symbol is not just a kaleidoscope for presenting a polytope: it is a generally oblique coordinate system. That is, one is not restricted to using o and x, but can make the values any number: eg

1 [3] 0 [5] 0 is a unit-edge icosahedron, could be written as 1,0,0.
0 [3] 1 [5] 0 is a unit edge ID
0 [3] 0 [5] 1 is a unit edge Dodecahedron

Of course, instead of having 8 octants, you have 120 sectors. You can still do 'even chnage of sign', but instead of four values (1,1,1), (1,-1,-1), (-1,1,-1) and (-1,-1,1), you have sixty. The snub dodecahedron, then is some x3y5,z for some x,y,z with even change of sign.

So,a point P gives not only a posotion-vector (o..P), but a "position polytope", whose edges connect P to P' in each of the mirrors. It may take as many as 120 reflections to get around the whole of space though!

Once you get a method for working with oblique coordinate systems. In particular, the usual dot product of vectors now involve a matrix to do this. The special matrix for this is the Stott matrix. Since the vertex of the polytope 'defines' the polytope, the dot-product of the vector and itself gives the length of the vector, and if the edge is set to 2 (the usual standard), the circum-radius.

We use a simpler example.

The standard presentation of [3,4] (the octahedral group) is to give x,y,z, all change of sign, all permutations. When these are all different and non zero, you get 48 points. The notional point for this coordinate can be set as the sorted value of abs(x), abs(y) abs(z). eg -12, 3, 18 is a reflection of (18,12,3)

One can next write the standard verticex of a polyhedron, of edge 2, which, with all change of sign and permutations give the vertices. This is like in the 'primary octant x,y,z > 0.

O = r2, 0, 0 ; CO = r2, r2, 0 ; C = 1,1,1.

All other values can be constructed from a sum of some O + some CO + some C

in particular these form figures of edge r2. Note these form a parallelohedron in the primary sector, bounded by unit-thickness planes.

tO = O + CO = x3x4o = 2r2, r2, 0
tC = C + CO = o3x4x = r2+1, r2+1, 1
rCO = O + C = x3o4x = r2+1, 1, 1
tCO = O + CO + C = x3x4x = 2r2+1, r2+1, 1

Of course, since this is a coordinate system in the commas of the schläfli symbol, one can just as easily provide varying intensities between the 'construct' and the 'decorations'.

Code: Select all
    @---o-4-o   Coxeter Dynkin Graph  O
    @---@-4-o   Coxeter Dynkin Graph tO

       O       tO
     @-o4o   @-@4-o   extra dashses removed
    { 3,4 }   --      Schläfli symbol: just the structure
   t_0     t_0,1      Coxeter's truncate notation (based on Stott).
    {;3,4 } {;3;4}    Semicolon for @.
     x3o4o   x3x4o    Inline dynkin symbol  (allow measures, duals)
     1S0Q0   1S1Q0    Inline dynkin coordinate
     /S Q    /S/Q     Simplified inline
     /1 Q    /1/Q     Further simplified for high dimensions.
    (1,0,0) (1,1,0)   stott coordinate.


One can see that the Schläfli symbol can not express the truncated octahedron: it's only good for regular figures. Also the standard way of representing the CO is to put 3 over 4, in a coxeter-curtail. However, if you want to run these in line, you need to deal with any mark on any node. Coxeter's solution is to use the t_ (ie t sub ...). However, i prefer full height solutions, of types listed following.

Depending on how much you need to deal with coordinates, etc, you move up and down the list. So if you are doing intense work (eg on a geodesic dome), you may want to use the 1 S 0 Q 0 form. This can take real values, eg 5.8132 S 2.24577 Q 88.12356. If you are wrangling high dimensions, with relatively few nodes, then something like 1/28Q is more in order. This thing has 30 branches and 31 nodes, being in Coxeter's system t_1{3^29,4}, 3 repeated 29 times, not tothe 29th power.

The notation is meant to be a one-to-one substution. That is, you can readily translate between /S/Q asnd x3x4o, because symbols are not generally used in different meanings in different notations. You could write xSxQo for example.

Lace-prisms can as readily be analised in terms of the wythoff process as ordinary figures: one notes that the face of a lace-prism is itself a lace-prism. One needs just to check the "progressions" by removing nodes singularly. We see by removing the first node, the figure becomes ".. ox&#t", that is a point expanding into a line (triangle), and the second node is xo .. &#t (line expanding into a point = triangle). The top and bottom remain the same, ie x.Po. = Pgon while .oP.x is another P-gon.

COMPOUNDS.

Having defined a coordinate system, you can then create compounds by using several vertices, eg this is a P-gonal antiprism.

Code: Select all
     1 P 0            polygon     compound    as tower
      \               x. P o.
       \                           xo P ox     &#t
        \             .o P .x
     0 P 1            dual polygon


The mirrors are vertical through 1, 0. The diagonal creates an edge of an antiprism. When this is view in its full, it zigzags from top to bottom like lacing in a drum: hence the name 'lace-prism'. This figure can be expressed as s2sPs, (alternating vertices of all points at 1, 0), but others can not.

In line, we write these in succession at each coordinate, eg x_1 x_2 x_3 [P] y_1 y_2 y_3. Since we have the construct P is a capital, we use upper-case brackets: [] to designate it. The coordinates are then sequenced between these.

Since it is after all, a coordinate system, we could go from the centre at 0,0. The progression of the line from 0,0 to 0,1 over this height gives rise to a pyramid. Progressing from 1,0 to 1,1 creates a cupola. The square face forms at 1,- to 1,-, and the triangle at -,0, to 0,1. These have no representation in simpler notations. The triangular cupola, or half - cuboctahedron is xo3xx &#x.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Previous

Return to Other Geometry

Who is online

Users browsing this forum: No registered users and 8 guests