by ICN5D » Wed Jul 23, 2014 6:46 pm
Introducing Toratopic Notation to the New Reader
• The capitalized " I " is a dimension marker
- The amount of 'I' is how many dimensions the shape has
• The parentheses " () " is a distinct diameter, that divides up the dimensions into distinct parts
- A () makes the shape round in some way
- The number of '()' is how many diameters the shape has
• For Open Toratopes:
- n-cubes don't have any at all, because they aren't round, i.e. I , II , III , IIII
- For Open-T rounded prisms:
-- a cylinder (II)I has both with 2 flat sides and 1 rounded side
-- a spherinder (III)I also has 2 flat, 1 round
-- a duocylinder (II)(II) has 2 rounded and no flat
-- a triocylinder (II)(II)(II) has 3 rounded and no flat
-- a cubinder (II)II has 4 flat and 1 round
-- a tesserinder (II)III has 6 flat and 1 round
-- a duocylinder diprism (II)(II)II has 4 flat and 2 round
• For Closed Toratopes:
- N-spheres have just one pair, around all dims, i.e. (II) , (III) , (IIII) , (IIIII)
- Toroids ( donut-shapes): these have two or more, making different size diameters, i.e. ((II)I) , ((III)I) , ((II)II) , (((II)I)I) , ((II)(II))
- Totatopes with Multiple Inner () : these have multiple equal-sized diameters, i.e. ((II)(II)) , (((II)(II))(II)) , ((III)(II)) , (((II)I)(II)) , (((II)I)((II)I))
• To make a cross section down to N-1 dimensions, we simply remove only the " I " markers
- 3D ((II)I) cuts to ((I)I) or ((II)) or, 1D ((I))
- 3D (III) cuts to (II) or (I)
- 4D (((II)I)I) cuts to (((I)I)I) or (((II))I) or (((II)I)) in 3D ,
• For Base-Species Toratopes, when cut into cross sections, the notation has extra parentheses around the dimension markers, i.e. (((I))) , (((I))I) , ((((I))((I))) , etc
- These extras represent a particular number and arrangement of a lower-D toratope
- If no extras are left, it is from a cut diameter that is higher-D than a 1-sphere/circle, " non-base-species" i.e. ((III)I) , ((II)II) , ((IIII)III) , ((III)(III)) , (((III)II)(III))
- There are as many Lower Dimensional Toratopes from cuts as there are total number in that dimension
-- For 2D : only cuts to circles (II)
-- For 3D : only cuts to spheres (III) and tori ((II)I)
-- For 4D : only cuts to glomes (IIII) , spheritori ((II)II) , torispheres ((III)I) , ditori (((II)I)I) , and tigers ((II)(II))
-- For 5D : only cuts to any of the 12
-- For 6D : only cuts to any of the 33
-- For 7D : only cuts to any of the 90
etc
• There are as many ways to arrange a toratope as there are dimensions and diameters of it
• For every extra () pair around a single dimension marker, there are 2^{n} shapes along that dimension
• For every extra () pair around a diameter, there are 2^{n} shapes paired concentrically in that diameter
• A circle (II) has TWO dimensions and ONE diameter, orientation (XY)
- It can be stacked in two directions, paired concentrically in one diameter
• Stacking a (II) :
((I)I) is 2 along X
(((I))I) is 4 along X
((((I)))I) is 8 along X
(((((I))))I) is 16 along X
(I(I)) is 2 along Y
(I((I))) is 4 along Y
(I(((I)))) is 8 along Y
(I((((I))))) is 16 along Y
((I)(I)) is a 2x2 array in XY plane
(((I))(I)) is 4x2 array in XY plane
(((I))((I))) is 4x4 array in XY
((((I)))((I))) is 8x4 array in XY
• Concentric Pairing a (II) :
((II)) is 2 pair
(((II))) is 4 concentric , a quartet
((((II)))) is 8 concentric, and octet
(((((II))))) is 16 concentric , a 16-plet
((((((((II)))))))) is 32 concentric, a 32-plet
• Stacking and Pairing a (II)
(((I)I)) is 2 along X, paired by 2, 4 total
((((I)I))) is 2 quartets along X, 8 total
(((I)(I))) is 2x2 array of pairs, 8 total
(((I)(I))) is 2x2 array of quartets, 16 total
((((I))((I)))) is 4x4 array of quartets, 64 total
(((((((I)))((I)))))) is 8x4 array of octets, 256 total
more to come on 3D toratope arrangements
Wow, that 10D list is insane Marek!!! I haven't got around to translating 9D yet, into bundling terms. Looks awesome!
Last edited by
ICN5D on Thu Jul 24, 2014 12:45 am, edited 1 time in total.
in search of combinatorial objects of finite extent