Yet another improved notation for shapes

Higher-dimensional geometry (previously "Polyshapes").

Yet another improved notation for shapes

Postby Keiji » Sun Oct 26, 2008 6:48 pm

This is an effort to sort out the mess that is the shape list on HDDB.

[link removed, new version in next post]

Any crit, questions, suggestions etc on this notation?

I'd like to eventually enumerate shapes using it in some way. This would bridge the gaps between bracketopes, rotopes, uniform shapes and all other kinds of shapes on HDDB and eliminate the immeasurable rotopes and the totally and partially useless bracketopes.
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Re: Yet another improved notation for shapes

Postby Keiji » Tue Oct 28, 2008 9:40 pm

I made several fixes to the notation and created a page properly describing it on HDDB:

http://teamikaria.com/wiki/SSC2
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Re: Yet another improved notation for shapes

Postby Keiji » Fri Nov 07, 2008 10:57 pm

I've removed the matrix transformation and boolean operations, since they don't really fit in with the ideals of SSC2, are hardly ever used and just plain complicate things.

However, I've also devised a second form of notation for SSC2, which is the matrix notation. Take a look at the matrix notation section of the page.

I'm now trying to group together shapes by their matrix order and family (the matrix family is what you get when you highlight all fundamental entries blue, padding red and any leftover cells, which will be information for operations, blue). I've then sorted them ascending by the literal read value going left to right top to bottom within each family. After manually working out the matrices for all rotopes up to 4D excluding immeasurable ones (which are undefined in SSC2), I've found some quite nice patterns:

Order 1: Digon
Order 2: Circle, Triangle, Square
Order 3 family A: Sphere, Torus, Tetrahedron, Cube
Order 3 family B: Cone, Square pyramid
Order 4: Triangular prism, Cylinder
Order 5 family A: Glome, Toraspherinder, Toracubinder, Tiger, Ditorus, Pentachoron, Tesseract
Order 5 family B: Sphone, Toric pyramid, Cubic pyramid
Order 5 family C: Dicone, Square dipyramid
Order 7 family A: Spherinder, Torinder, Tetrahedral prism
Order 7 family B: Coninder, Square pyramidal prism
Order 7 family C: Cylindrone
Order 10: Duocylinder, Cubinder, Cyltrianglinder, Triangular diprism

I must add that this misses out some of the 4D rotopes because they don't have pages on HDDB and I unintentionally skipped over them. However, you should be able to see the patterns.
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Re: Yet another improved notation for shapes

Postby wendy » Fri Nov 21, 2008 10:41 am

I had a look at these tables, to see what can be seen.

It covers a very large range, similar to my own notations. Some things missing here

Ellipses - my treatment is to deal with these as "truncated spheres".
Composite lace figures (with several progressions). eg xoo3ooo3oxoAoox&#t. In six dimensions, this is a progression between three symmetric orientations of a 16choron, each at the vertices of a triangle.

Some things here, but not in my notations

Rototopes (still trying to get this one)

Some things covered differently.

Bracketotopes.

Note that the products prism/tegum/crind are all associative, ie a#b#c = (a#b)#c = a#(b#c). This is because they can be defined in terms of associative functions (max/sum/rss) over a radial product. The intensity at X, Y, Z is eg max/sum/rss(X,Y,Z).

You could use the . to represent an end-branch, eg cube = [.] (products always have at least two elements, the dot signifies a third element).

Prism, Tegum.

These words were set up to specifically give matched forms, eg 'prism product', or prismic or prismatic, vs tegum product, or tegmic or tegmatic.

The three products tegum/prism/crind are coherent, that is, you can use any of these as a measure polytope, for which the volume of the product is the product of the volumes. For this, I use Pn, Tn, Cn for their measures, and the special names

C1 = diametric, C2 = circular, C3 = cubic, C4 = glomic C5 = crindapetic
T1 = diagonal, T2 = rhombic, T3 = octahedral = biquadrate T5 = tegmapetic
P1 = linear, P2 = square, P3 = cubic, P4 = tesseractic. P5 = prismapetic &c

One can use the product with any unit of any dimension: a "cubic acre" is a cubic L, where a "square acre" = acre. This is common usage, and the reason that C1, T1, P1 have different names. (a linear acre is the edge of a square of 1 acre, while a diagonal acre is the diagonal of such a square. A diametric acre is the diameter of a circle of area one acre = all different).


Kanitopes.

For 2d polygons, one has Bp. In 3d, prisms are ABp, antiprisms are Cap eg Ca6. Powers of line segments A are written as A2=AA, A3=AAA &c. The primary representation of the n-cube is An (rather than Nn.c1)

I use a similar system, based on using letters A, B, C, D, E, F, G, H, and Nn to designate a dimension, and then a letter for the shape (based on the 3d form, ie, t, o, c, q, i, d, h, g, e for the simplex, cross, cube, 343, icoas, dodeca, half-cube, gosset (k_21), and elte (1_2k). Numbers are supplied from the PS trace, counting from 1, 2, 4, 8 ... and summed. Because prism-product preserves uniformity, one can concatinate dimensions, eg ACd1 is the line (A) Dodecahedron (3d + {5,3..} ) + xoo... This tends to minimise the values for the more discussed figures, and allows for discussion of figures referenced to a particular source, eg: Norman Johnson designates x3x3o4x as Dt11 "runcitruncated tesseract" , while George Olshevsky designates it as Do13 "primorhombic 16ch.

I also designate snubs at position 2^n, eg sC = Cc8, sD = Cd8, snub 24c = Dq16, GAP = Di16.

The sequence of these is without any A's, since any 3d figure gives a 4d figure by adding an A. Line, square, cube, tesseract ... comes first by this.

The polytopes for the cubes, B4, Ca2, Ca3, Cc1, Dc1, Do2, Do5, Do7, Ec1 &c, are listed but not given a sequence number. These have different places, eg

B4=A2, Ca2 = Ct1, Ca3 = Co1, Cc1 = A3, Dc1 = A4, Do2 = Dq1, Do5 = Dq2, Do7 = Do3, Ec1 = A5, &c.

"kanitopes" sound like a useful name for /uniform/ polytopes, which has always been up for rename. It would not been the first idea borrowed from here!
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Re: Yet another improved notation for shapes

Postby Keiji » Fri Nov 21, 2008 11:31 am

wendy wrote:I had a look at these tables, to see what can be seen.

It covers a very large range, similar to my own notations. Some things missing here

Ellipses - my treatment is to deal with these as "truncated spheres".


An ellipse is a squashed circle. SSC2 does not define the transformations of a shape such as position, rotation or stretching, so an ellipse would just be a circle.

wendy wrote:Composite lace figures (with several progressions). eg xoo3ooo3oxoAoox&#t. In six dimensions, this is a progression between three symmetric orientations of a 16choron, each at the vertices of a triangle.


Well, I didn't understand a word you said there :P

Rototopes... Bracketotopes...

Are spelled "rotopes" and "bracketopes".

Note that the products prism/tegum/crind are all associative, ie a#b#c = (a#b)#c = a#(b#c). This is because they can be defined in terms of associative functions (max/sum/rss) over a radial product. The intensity at X, Y, Z is eg max/sum/rss(X,Y,Z).


Brick products are commutative and they are associative over themselves, so P[A,B,C] = P[C,A,B] = P[P[A,B],C] = P[A,P[B,C]] etc, where P could be square (prism/max), diamond (tegum/sum), circle (crind/rss) or any other brick. What are not are the relationships between products and arguments, so P[A,B] is not necessarily A[P,B] and P[A,Q[B,C]] is not necessarily P[Q[A,B],C].

The three products tegum/prism/crind are coherent, that is, you can use any of these as a measure polytope, for which the volume of the product is the product of the volumes. For this, I use Pn, Tn, Cn for their measures, and the special names

C1 = diametric, C2 = circular, C3 = cubic, C4 = glomic C5 = crindapetic
T1 = diagonal, T2 = rhombic, T3 = octahedral = biquadrate T5 = tegmapetic
P1 = linear, P2 = square, P3 = cubic, P4 = tesseractic. P5 = prismapetic &c

One can use the product with any unit of any dimension: a "cubic acre" is a cubic L, where a "square acre" = acre. This is common usage, and the reason that C1, T1, P1 have different names. (a linear acre is the edge of a square of 1 acre, while a diagonal acre is the diagonal of such a square. A diametric acre is the diameter of a circle of area one acre = all different).


Interesting and a nice idea, but units aren't part of SSC2 nor need to be, as as mentioned it does not define transformations.

"kanitopes" sound like a useful name for /uniform/ polytopes, which has always been up for rename. It would not been the first idea borrowed from here!


Kanitopes cannot be used as a synonym for uniform polytopes; the set of kanitopes is a subset of the uniform polytopes. Non-convex uniform polytopes are not kanitopes; nor are prisms and antiprisms (other than the respective special cases cube and octahedron). Regular polygons are also not kanitopes as there are no kanitopes below 3D. While SSC2 matrix notation can represent non-convex uniform polytopes these cannot be defined in the string notation, and the non-Wythoffian polytopes such as the grand antiprism cannot be defined even in the matrix notation (unless there is an alternative construction).
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