If you are unfamiliar with polyhedra and n-dimensional space please read
Regular
convex Polytopes a short historical overview first.
From 1977 age 15, i have been interested what regular bodies would look like
in dimensions higher than 3 and started to calculate the properties of an n-dimensional
Tetrahedron, Cube and Octahedron.
Being unaware of work already published and having no clue to start transposing
Dodecahedra and Icosahedra to n-space i left these out of my calculations. Deep
in my mind this kept me puzzled for a long time.
In begin of 1991 i wondered what would happen if one would put a number of p
equally charged particles or points, distributed at random in an n-dimensional
space confined by a n-dimensional sphere. Each particle will push the other
particles away until the whole will form a neutral state within the sphere.
It was obvious that for 2 dimensions the model would generate structures where
the particles form the vertices of any p-gon. But would this model also generate
structures of any n-dimensional body with Platonic properties or so-called regular
convex polytopes, i wondered.
It took me 2 months of free time to work out a mathematical model.
After implementing it in Zbasic which allowed a high decimal precission, indeed
regular structures showed up for certain what i call [n][p] configurations and
i was amazed about the speed that line symmetry suddenly showed up during iteration
of regular configurations.
I emperically concluded that that for 3 dimensions the model will form regular
structures such as the Tetrahedron, Octahedron and Icosahedron each time, regardless
of initial positions and in fact the n-dimensional variants of the Tetrahedron
and Octahedron are reproducible in higher dimensions as well.
Despite my initial thoughts, the Dodecahedron will not show up and for 8 points
in 3 dimensions the model will not generates a Cube but a square anti-prism
with its square top 45 degrees tilted.
In 1994 i implemented the model in C gaining speed and graphical possibilities.
With this version it was possible to generate the 120 edges of a stable hypericosahedron
after +/- 100,000 iterations.
Not Infinity,5,6,3,3,3,3... but Infinity,3,4,2,2,2,2...?
For 3 dimensions only 3 structures with Platonic properties are generatable,
namely the tetrahedron, octahedron and icosahedron all having edges or particles
which are optimally distributed, whereas the Cube and Dodecahedron do not.
Instead other non-regular structures with better distances exist and are consequently
generated, making the set of 5 to my opinion not as "cosmic" as Plato
followers might believe them to be.
Even when regular structures of non-optimal distance are forced upon the model,
where due to the symetry all force vectors are zero or near zero, the limited
precision of the computational model** will force the structure out of balance,
resulting in a very short live and transformation to non-regular structures
with a more optimal distance.
In 4 dimensions only 4 structures with Platonic properties are generatable,
namely the Hypertetrahedron, the Hyperoctahedron, the Hypericosahedron and the
24-cell which all have optimal spatial distribution.
An execptional case exists for the 24 cell which seems to be the only regular
Platonic polytope structure which has different states with a what i call
stable equilibrium.
Structures with a non stable equilibrium can be easily torn apart by moving
any of the particles or points or due to the limited precission of the model,
it is so to say like a pencil standing right up on it's point.
Instead structures with stable equilibrium will forced back to the original
state when any of the particles or points is fractionally moved.
Configuration [4][24] possesses a bi-stable state i.e. depending on the initial
point distribution two different structures will show up.
In the model about every 4 out of 5 times the structure will not be a regular
polytope and show up as 6 "claws" of 3 lines segments each going a
different direction, the total sum of mutual distances being 388.0972964082708,
in all other cases a polytope known as the '24-cell' will be generated, the
total sum of mutual distances being 388.1002501747, which is a difference of
only +0.00076%, but still means the 24-cell has optimal distance.
The Hypercube and Hyperdodecahedron* do not have an optimal spatial distribution
and other non-regular structures with better distances will be generated instead.
In 5 dimensiones and more only the n-cell and n-crosspolytope are stable and
have optimal distribution.
The 5 and more dimensional cousins of the cube or n-cube, all have similar to
the cube a non optimal distribution.
This let me to conclude that any multi-dimensional regular convex polytope with
Platonic properties, which is self-dual has optimal spatial distribution, and
that all Platonic structures which are dual have one structure with optimal
spatial distribution and one which has not. (i.e. the Dodecahedron, Hyperdodecahedron*
and all n-cubes)
Other findings
In the first model in 1991, i noticed that there where highly regular results
with less well-known structures. For example for 16 points in 5 dimensions or
configuration [5][16], seemed very regular but could not be a 5D-Platonic solid
as proven by Schläfli back in 1850.
With little information about what had been published and what not, i emailed
with some people who had published about hyper dimensional geometry, with no
outcome however.
The exact nature was actually still a mystery to me at first publication of
the web-site until James
M. Gibbs calculated it's excact properties making it semiregular and therefor
belonging to the category of Archimedean solids.
Furthermore George Olshevsky
inventor of the name 'polychora', who maintains a splendid website about 4D
solids, pointed me to Gosset who in the 1920's probably first published about
this structure.
It is likely that the model is capable to generate structures previously unknown
and in fact one structure mentioned in 'Findings'
has to my knowledge not published before. (See [8][72] which generates a highly
regular body, the exact properties of which are still unknown.)
The improved model
Due to unforeseen circumstances the software and notes were lost, but end of
2002 coincidentally i found back the original Zbasic model from 1991.
Since i am not a mathematician and i do not deliver mathematical proof of any
kind, it was hard to communicate with anyone about this topic, hence my idea
to implement the model in java and by putting it on the web let everybody see
for themselves.
In release v1.3 an n-simplex and n-crosspolytope search algorithms were added
to be able to specify symmetrical features of structures generated.
In the latest release v1.4 both 1/R and faster converging 1/R^(n-1) law are
implemented and currently i'm doing a giant search for semi-regulars beyond
10 dimensions within the scope of [20][1000].
While people interested in hyper dimensional geometry can have fun with the
model, and are invited to comment on it, i hope some die-harts will find new
regular structures and if possible improve software and /or find mathematical
proof or foundations for some of my assumptions, all which if found relevant
i will publish on this site.
Furthermore i hope this publishment will inspire others and might one day contribute
to theoretical mathematics and topology in general and perhaps is applicable
somewhere such as in the field of improved error correction algorithms.
* Attempt in December 2002: 600 points in 4 dimensions generate a non-regular
structure after 975,000 iterations.
** In the C and Java implementation of the model 64 bit double precission is
being used.
2004, Symen H. Hovinga