by 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!