Prashantkrishnan wrote:Why is it that for more than 4 dimensions there are only three regular polytopes? Why can't there be more of them?
d/n + b/m > 1/2
sin(pi.d/n).sin(pi.c/k) < cos(pi.b/m).
cos²(pi.b/m)/sin²(pi.d/n) + cos²(pi.c/k)/sin²(pi.a/l) < 1.
Klitzing wrote:{3,3,5/2} (grand hexacosachoron),
{3,5,5/2} (faceted hexadecachoron), - should be faceted hexacosachoron
{3,5/2,5} (great faceted hexadecachoron), - should be great faceted hexacosachoron
{5/2,3,3} (great grand stellated hecatonicosachoron),
{5/2,5,3} (small stellated hecatonicosachoron),
{5,5/2,3} (great grand hecatonicosachoron),
{5/2,3,5} (great stellated hecatonicosachoron),
{5,3,5/2} (grand hecatonicosachoron),
{5,5/2,5} (great hecatonicosachoron),
{5/2,5,5/2} (grand stellated hecatonicosachoron).
Polyhedron Dude wrote:Found a couple typos, corrected above.
If we really want to get a bit sinister, there are also lots of regular abstract polytopes like the abstract polyhedron with 24 heptagons as faces, AKA the Klein quartic - I imagine these abstract polytopes would make excellent mazes where all the rooms have the same shape, here heptagons, where each wall has a door that leads to another room. There seems to be an endless array of these sort of polytopes, some are non-orientable.
Polyhedron Dude wrote:Klitzing wrote:{3,5,5/2} (faceted hexadecachoron), - should be faceted hexacosachoron
{3,5/2,5} (great faceted hexadecachoron), - should be great faceted hexacosachoron.
Found a couple typos, corrected above.
If we really want to get a bit sinister, there are also lots of regular abstract polytopes like the abstract polyhedron with 24 heptagons as faces, AKA the Klein quartic - I imagine these abstract polytopes would make excellent mazes where all the rooms have the same shape, here heptagons, where each wall has a door that leads to another room. There seems to be an endless array of these sort of polytopes, some are non-orientable.
cos²(pi/h) = cos²(pi/p) + cos²(pi/q)
o3o7x (N → ∞)
. . . | 14N | 3 | 3
------+-----+-----+---
. . x | 2 | 21N | 2
------+-----+-----+---
. o7x | 7 | 7 | 6N
x5o5o (N → ∞)
. . . | 2N | 5 | 5
------+----+----+---
x . . | 2 | 5N | 2
------+----+----+---
x5o . | 5 | 5 | 2N
Marek14 wrote:Polyhedron Dude wrote:Found a couple typos, corrected above.
If we really want to get a bit sinister, there are also lots of regular abstract polytopes like the abstract polyhedron with 24 heptagons as faces, AKA the Klein quartic - I imagine these abstract polytopes would make excellent mazes where all the rooms have the same shape, here heptagons, where each wall has a door that leads to another room. There seems to be an endless array of these sort of polytopes, some are non-orientable.
Probably not excellent mazes, since a good maze should be solvable -- unless you count the maze's success by how many of your enemies it has successfully dispatched.
So, a mental exercise: what would you do if you found yourself in heptagonal maze like this? Oh, and you don't know its topology beforehand. What strategy would you use?
Polyhedron Dude wrote:Marek14 wrote:Polyhedron Dude wrote:Found a couple typos, corrected above.
If we really want to get a bit sinister, there are also lots of regular abstract polytopes like the abstract polyhedron with 24 heptagons as faces, AKA the Klein quartic - I imagine these abstract polytopes would make excellent mazes where all the rooms have the same shape, here heptagons, where each wall has a door that leads to another room. There seems to be an endless array of these sort of polytopes, some are non-orientable.
Probably not excellent mazes, since a good maze should be solvable -- unless you count the maze's success by how many of your enemies it has successfully dispatched.
So, a mental exercise: what would you do if you found yourself in heptagonal maze like this? Oh, and you don't know its topology beforehand. What strategy would you use?
What I had in mind would go something like this:
You start off in a heptagon shaped area around 80 to 100 feet across. The area can be filled with all sorts of terrain, objects, rooms, pools, etc to make each of the 24 areas unique and interesting. Each of the seven walls of the heptagon has a locked door or gate, and you must solve riddles and puzzles as well as search for keys and levers to open the various gates and doors. So the first goal would be to unlock all 24 heptagonal areas to explore. Next would be to unlock every gate and door, once all seven gates/doors in an area is unlocked, it gives you access to one of the 24 key items. Once all of the 24 items are found, then you put them together like a puzzle and use this as a key to gain access to a transporter to let you exit the entire maze - and then you enter the next maze - the 57-choron .
A B
D C
Klitzing wrote:Putting these ideas together, we now get for lace city display generally a rhomb (sometimes becoming a square) of the following form:
- Code: Select all
A B
D C
where the perp-dimensional space is represented as A_{n} x A_{d-n-2} for any 0 <= n <= (d-2)/2 according to:
- A = x3o...o3o o3o...o3o
- B = o3o...o3o x3o...o3o
- C = o3o...o3x o3o...o3o
- D = o3o...o3o o3o...o3x
o x
x o
o3o x3o
o3x o3o
x o o x
o x x o
o3o3o x3o3o
o3o3x o3o3o
x o3o o x3o
o o3x x o3o
o3o3o3o x3o3o3o
o3o3o3x o3o3o3o
x o3o3o o x3o3o
o o3o3x x o3o3o
x3o o3o o3o x3o
o3o o3x o3x o3o
o3o3o3o3o x3o3o3o3o
o3o3o3o3x o3o3o3o3o
x o3o3o3o o x3o3o3o
o o3o3o3x x o3o3o3o
x3o o3o3o o3o x3o3o
o3o o3o3x o3x o3o3o
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