So, now, equipped with this knowledge, I can have a proper look at the 5D graphotopes, based on my first post on the forum, over 9 years ago (
viewtopic.php?f=24&t=350&p=4436&hilit=longdome#p4436)
It's fun since I didn't specify the exact graphs (exercise to the reader indeed) and now I have to recreate them myself
Well, the good news is that now I got the understanding of longdome function and I think I can extend it to arbitrary graphotope.
Penteract: max(x^2, y^2, z^2, w^2, v^2) = 1. Tesseract/tesseract x5. Doesn't roll.
Cubicircle: max(x^2 + y^2, z^2, w^2, v^2) = 1. Cubinder/cubinder x3, Tesseract/cube x2. In tesseract/cube, can freely roll in one direction.
Domisquare: max(max(x^2, y^2) + z^2, w^2, v^2) = 1. Dominder/dominder x2, cubinder/cube x2, tesseract/square x1. From cubinder/cube, can roll in one direction to tesseract/square. From tesseract/square, can roll in two directions to cubinder/cube.
Dual cylinder: max(x^2 + y^2, z^2 + w^2, v^2) = 1. Duocylinder/duocylinder x1, cubinder/cylinder x4. From cubinder/cylinder, can freely roll in one direction.
Tridominder: max(max(x^2, y^2, z^2) + w^2, v^2) = 1. Tridome/tridome x1, dominder/cube x3, tesseract/line x1. From dominder/cube, can roll in one direction to tesseract/line. From tesseract/line, can roll in three directions to dominder/cube.
Spherisquare: max(x^2 + y^2 + z^2, w^2, v^2) = 1. Spherinder/spherinder x2, cubinder/square x3. From cubinder/square, can freely roll in two directions.
Longdominder: max(x^2 + y^2, y^2 + z^2, z^2 + w^2, v^2) = 1. Longdome/longdome x1, dominder/cylinder x2, cubinder/square x2. From dominder/cylinder, can roll in one direction to cubinder/square. From cubinder/square, can roll in one direction to dominder/cylinder or in second direction to cubinder/square.
Domicircle: max(max(x^2, y^2) + z^2, w^2 + v^2) = 1. Dominder/dome x2, duocylinder/cylinder x2, cubinder/circle x1. From dominder/dome, can freely roll in one direction. From duocylinder/cylinder, can roll in one direction to cubinder/circle. From cubinder/circle, can roll in two directions to duocylinder/cylinder.
Tetradome: max(x^2, y^2, z^2, w^2) + v^2 = 1. Tridome/cube x4, tesseract/point x1. From tridome/cube, can roll in one direction to tesseract/point. From tesseract/point, can roll in four directions to tridome/cube.
Spheridominder: max(max(x^2 + y^2, z^2) + w^2, v^2) = 1. Spheridome/spheridome x1, spherinder/cylinder x1, dominder/square x2, cubinder/line x1. From spherinder/cylinder, can roll in one direction to cubinder/line. From dominder/square, can freely roll in one direction, or can roll in second direction to cubinder/line. From cubinder/line, can freely roll in two directions to dominder/square, or can roll in third direction to spherinder/cylinder.
Branchdome: max(x^2 + y^2, y^2 + z^2, z^2 + w^2, z^2 + v^2) = 1. Tridome/dome x1, longdome/cylinder x2, dominder/square x1, cubinder/line x1. From tridome/dome, can roll in one direction to dominder/square. From longdome/cylinder, can roll in one direction to cubinder/line. From dominder/square, can roll in one direction to tridome/dome, or in second direction to cubinder/line. From cubinder/line, can roll in two directions to longdome/cylinder, or in third direction to dominder/square.
Sphericircle: max(x^2 + y^2 + z^2, w^2 + v^2) = 1. Spherinder/sphere x2, duocylinder/circle x3. From spherinder/sphere, can freely roll in one direction. From duocylinder/circle, can freely roll in two directions.
Cyclodominder: max(max(x^2, y^2) + max(z^2, w^2), v^2) = 1. Cyclodome/cyclodome x1, dominder/square x4. From dominder square, can roll in two directions to dominder/square.
Superdome: max(x^2 + y^2, y^2 + z^2, z^2 + w^2, w^2 + v^2) = 1. Longdome/dome x2, dominder/circle x2, duocylinder/square x1. From longdome/dome, can roll in one direction to dominder/circle. From dominder/circle, can roll in one direction to longdome/dome, or in second direction to duocylinder/square. From duocylinder/square, can roll in two directions to dominder/circle.
Spheritridome: max(x^2 + y^2, z^2, w^2) + v^2 = 1. Spheridome/cylinder x2, tridome/square x2, cubinder/point x1. From spheridome/cylinder, can roll in one direction to cubinder/point. From tridome/square, can freely roll in one direction, or can roll in second direction to cubinder/point. From cubinder/point, can roll in two directions to spheridome/cylinder, or can freely roll in two other directions to tridome/square.
Semiglominder: max(max(x^2, y^2) + z^2 + w^2, v^2) = 1. Semiglome/semiglome x1, spherinder/square x2, dominder/line x2. From spherinder/square, can freely roll in two directions to dominder/line. From dominder/line, can roll in two directions to spherinder/square, or can freely roll in third direction.
Bicesphere: max(x^2 + y^2 + z^2, y^2 + w^2, z^2 + v^2) = 1. Spheridome/dome x2, longdome/square x1, dominder/line x2. From spheridome/dome, can roll in one direction to dominder/line. From longdome/square, can roll in two directions to dominder/line. From dominder/line, can roll in one direction to spheridome/dome, in second direction to longdome/square, or in third direction to dominder/line.
Longspheridome: max(x^2 + y^2 + z^2, z^2 + w^2, w^2 + v^2) = 1. Spheridome/sphere x1, spherinder/circle x1, longdome/circle x2, duocylinder/line x1. From spheridome/sphere, can roll in one direction to spherinder/circle. From spherinder/circle, can roll in one direction to spheridome/sphere, or in second direction to duocylinder/line. From longdome/circle, can freely roll in one direction, or can roll in second direction to duocylinder/line. From duocylinder/line, can roll in one direction to spherinder/circle, or can freely roll in two other directions to longdome/circle.
Branchcyclodome: max(x^2 + y^2, x^2 + z^2, y^2 + w^2, z^2 + w^2, x^2 + v^2) = 1. Cyclodome/dome x1, tridome/circle x1, longdome/square x2, dominder/line x1. From cyclodome/dome, can roll in one direction to dominder/line. From tridome/circle, can roll in two directions to longdome/square. From longdome/square, can roll in one direction to tridome/circle, or in second direction to dominder/line. From dominder/line, can roll in one direction to cyclodome/dome, or in two other directions to longdome/square.
Cyclongdome: max(x^2 + y^2, x^2 + z^2, y^2 + w^2, z^2 + v^2, w^2 + v^2) = 1. Longdome/circle x5. From longdome/circle, can roll in two directions to longdome/circle.
Bispheridome: max(max(x^2, y^2) + z^2, w^2) + v^2 = 1. Semiglome/dome x1, spheridome/square x2, tridome/line x1, dominder/point x1. From semiglome/dome, can roll in one direction to dominder/point. From spheridome/square, can roll in one direction to tridome/line, or in second direction to dominder/point. From tridome/line, can roll in two directions to spheridome/square, or in third direction to dominder/point. From dominder/point, can roll in one direction to semiglome/dome, in two other directions to spheridome/square, or in a fourth direction to tridome/line.
Duospheridome: max(x^2 + y^2, z^2 + w^2) + v^2 = 1. Spheridome/circle x4, duocylinder/point x1. From spheridome/circle, can freely roll in one direction, or can roll in second direction to duocylinder/point. From duocylinder/point, can freely roll in two pairs of directions to spheridome/circle.
Glominder: max(x^2 + y^2 + z^2 + w^2, v^2) = 1. Glome/glome x1, spherinder/line x4. From spherinder/line, can freely roll in three directions.
Sesquispheridome: max(x^2 + y^2 + z^2, y^2 + z^2 + w^2, w^2 + v^2) = 1. Semiglome/sphere x1, spheridome/circle x1, spherinder/line x1, longdome/line x2. From semiglome/sphere, can roll in one direction to spherinder/line. From spheridome/circle, can freely roll in two directions to longdome/line. From spherinder/line, can roll in one direction to semiglome/sphere, or can freely roll in two other directions to longdome/line. From longdome/line, can roll in one direction to spheridome/circle, or in second direction to spherinder/line, or can freely roll in a third direction.
Triquad: max(x^2 + y^2 + z^2, y^2 + w^2, z^2 + v^2, w^2 + v^2) = 1. Spheridome/circle x2, cyclodome/circle x1, longdome/line x2. From spheridome/circle, can roll in one direction to spheridome/circle, or in a second direction to longdome/line. From cyclodome/circle, can roll in two directions to longdome/line. From longdome/line, can roll in one direction to spheridome/circle, in second direction to cyclodome/circle, or in a third direction to longdome/line.
Tricycle: max(x^2, y^2, z^2) + max(w^2, v^2) = 1. Cyclodome/square x3, tridome/line x2. From cyclodome/square, can roll in two directions to tridome/line. From tridome/line, can roll in three directions to cyclodome/square.
Sesquiglome: max(x^2, y^2, z^2) + w^2 + v^2 = 1. Semiglome/square x3, tridome/point x2. From semiglome/square, can freely roll in two directions to tridome/point. From tridome/point, can roll in three directions to semiglome/square, or can freely roll in the fourth direction.
Glomidome: max(x^2 + y^2 + z^2, w^2) + v^2 = 1. Glome/sphere x1, spheridome/line x3, spherinder/point x1. From glome/sphere, can roll in one direction to spherinder/point. From spheridome/line, can freely roll in two directions, or can roll in third direction to spherinder/point. From spherinder/point, can roll in one direction to glome/sphere, or can freely roll in the remaining three directions to spheridome/line.
Trialong: max(x^2 + y^2 + z^2, x^2 + z^2 + w^2, x^2 + w^2 + v^2) = 1. Semiglome/circle x2, spheridome/line x2, longdome/point x1. From semiglome/circle, can roll in one direction to spheridome/line, or in a second direction to longdome/point. From spheridome/line, can roll in one direction to semiglome/circle, in second direction to spheridome/line, or in a third direction to longdome/point. From longdome/point, can roll in two directions to semiglome/circle, or in the remaining two directions to spheridome/line.
Siamese spheridome: max(x^2 + y^2, z^2) + max(w^2, v^2) = 1. Semiglome/circle x1, spheridome/line x2, cyclodome/line x2. From semiglome/circle, can roll in two directions to spheridome/line. From spheridome/line, can roll in one direction to semiglome/circle, or can freely roll in two other directions to cyclodome/line. From cyclodome/line, can roll in two directions to spheridome/line, or can freely roll in a third direction.
Spheriglome: max(x^2 + y^2, z^2) + w^2 + v^2 = 1. Glome/circle x2, semiglome/line x2, spheridome/point x1. From glome/circle, can roll in two directions to spheridome/point. From semiglome/line, can freely roll in one direction, or can freely roll in two other directions to spheridome/point. From spheridome/point, can roll in one direction to glome/circle, or can freely roll in two other directions to semiglome/line, or can freely roll in the fourth direction.
Pyraglome: max(x^2, y^2) + max(z^2, w^2) + v^2 = 1. Semiglome/line x4, cyclodome/point x1. From semiglome/line, can roll in two directions to semiglome/line, or in a third direction to cyclodome/point. From cyclodome/point, can roll in four directions to semiglome/line.
Semipentaglome: max(x^2, y^2) + z^2 + w^2 + v^2 = 1. Glome/line x2, semiglome/point x3. From glome/line, can freely roll in three directions to semiglome/point. From semiglome/point, can freely roll in two directions to glome/line, or can freely roll in two other directions.
Pentaglome: x^2 + y^2 + z^2 + w^2 + v^2 = 1. Glome/point x5. From glome/point, can freely roll in four directions.
So, the general formula for graphotope turns out to be a maximum of:
x1^2, x2^2, ..., xn^2
xa^2 + xb^2, where nodes a and b are joined by edge
xa^2 + xb^2 + xc^2, where nodes a, b and c are all joined
xa^2 + xb^2 + xc^2 + xd^2, where nodes a, b, c and d are all joined (they form a K4 subgraph)
and so on. It can be imagined as starting with a hypercube and then "rounding" some of its elements. The equation can then be simplified by omitting any sum that is absorbed in a higher sum and by distributing sum over maximum.