Findings:

Last update: 9-6-2004

Besides regular convex polytopes structures with stable equilibrium of [n][p] configurations where p=n+1; p=2*n; n=2, p>n; n=3, p=12; n=4 , p=24 or 120 , the current computational model v1.3 in java will generate regular structures for various configurations of maximal size [100][1000].
An equal number of lines of shortest distance and equal length connected to each point will be an indicator for symmetry.
Since the points all reside at an n-dimensional sphere the structures, mostly having triangular faces with edges of maximal mutual distance are forced to be convex.
At the time of publication the exact nature of some structures is unknown and needs to be further researched and classified.
Below are stable and multi-stable configurations generating regular structures with stable equilibria which were produced by the computational model.
Some of the structures found are so-called n dimensional semi-regular convex polytopes or Archimedean solids, being built up out of different (n-1) dimensional regular convex polytopes, such as the (n-1)-simplex and (n-1)-cross-polytope.
Most of such structures in 6-8D were first described by Gosset in the 1920's and later by Coxeter.
Other structures which as far as i know have not been published before, have been given a date on which they were found.
These structures found reach up to 10 dimensions. Beyond 10D a systematical computer search for regular structures of optimal distance and stable equilibrium conducted within scope [20][120], resulted in no other findings than simplices ([n][n+1]), crosspolytopes ([n][2*n]), diplo-simplices ([n][2*(n+1]) and [j*k][(j+1)*k] structures already known.
I would appreciate help to find out more about these structures or find similar structures unlisted in particular beyond the scope of [20][120];

Stable [n][p] configurations generating regular structures of optimal distance and stable equilibrium not belonging to the symmetry group of regular convex polytopes:
Besides the regular convex polytopes i sofar have found 6 configurations which seem to have only one stable structure with stable equilibrium, 3 of which are semiregular convex polytopes or Archimedean solids;

[4][12] the stable structure of which has 4 lines connected to each point forming 24 edges and 0 triangular/square faces and tetrahedral/octahedral cells. This configuration has one structure with stable equilibrium at distance 95.8109....
[5][16] the stable structure of which forms an Archimedean solid called hemipenteract¹, hemi-5-cube³ or Gosset polytope 1_21¹ and has 10 lines connected to each point forming 80 edges, 160 triangular faces, 120 tetrahedral cells has 26 ² 4D-volumes consisting of 16 4D-simplices, 5 joining per vertex and 10 4D-cross-polytopes also 5 joining per vertex. This configuration has one structure with stable equilibrium at distance 160*(1+sqrt(2))/sqrt(5) ²=172.7470....
[6][27] the stable structure of which forms an Archimedean solid or so called Schläfli polytope³, Delaunay polytope for E6³ or Gosset polytope 2_21⁴ and has 16 lines connected to each point forming 216 edges, 720 triangular faces, 1080 tetrahedral cells, 648 4-simplices and 99 5D-volumes consisting of 72 5D-simplices, 16 joining per vertex and 27 5D-cross-polytopes, 10 joining per vertex. This configuration forms besides 2*n-1 simplices the only known regular convex polytope with an odd number of points / vertices.
It has one structure with stable equilibrium at distance 498.3717....
[7][56] the stable structure of which forms an Archimedean solid called Hesse polytope³, contact polytope for E^*7 ³ or Gosset polytope 3_21 ⁴ and has 27 lines connected to each point forming 756 edges, 4032 triangular faces, 10080 tetrahedral cells, 12096 4D-simplices, 6048 5D-simplices and 702 6D volumes consisting of 576 6D-simplices, 72 joining per vertex and 126 6D-cross-polytopes, 27 joining per vertex. This configuration has one structure with stable equilibrium at distance 2163.4964....
[8][72] found on 15-12-2002, the stable structure of which has 14 lines connected to each point forming 504 edges, 1176 triangular faces, 1260 tetrahedral cells, 252 octahedral cells, 1008 4D-simplices, 504 5D-simplices, 144 6D-simplices and 18 7D-simplices, 2 joining per vertex.
The above count does not satify the modified Euler polyhedral formula (-138) and in order to belong to the symmetry group of semi-regular polytopes or Archimedean solids, the structure should have an unknown no. of other regular or semi-regular polytopes of 7 dimensions or less. This configuration has one structure with stable equilibrium at distance 3589.7166....
[8][240] the stable structure of which forms an Archimedean solid called Gosset polytope³, contact polytope for E8³ or Gosset polytope 4_21 ⁴ and has 56 lines connected to each point forming 6720 edges, 60480 triangular faces, 241920 tetrahedral cells, 483840 4D-simplices, 483840 5D-simplices, 207360 6D-simplices, 17280 7D-simplices, 576 joining per vertex and 2160 7D-cross-polytopes, 126 joining per vertex. This configuration has one structure with stable equilibrium at distance at distance 39982.2904....
The configuration will be generated by the applet but finding the high number of faces and cells will take ages, therefore no link is provided.
[10][40] found on 3-8-2003, the stable structure of which has 24 lines connected to each point, forming 480 edges, 2240 triangular faces 4080 tetrahedral cells, 2304 4D-simplices and 0 5D-9D simplices/ crosspolytopes.
This configuration has one structure with stable equilibrium at distance 1107.9221....

Multi-stable [n][p] configurations generating regular structures of optimal distance and stable equilibrium not belonging to the symmetry group of regular convex polytopes:

All configurations [j*k][(j+1)*k] where k>1 and j>1 seem to be multi-stable and have optimal structures which are regular.
The structures have (k-1)*(j+1) lines of equal length connected to each point forming (k-1) n-dimensional simplices, the quantity of which satisfy S(j,k,n)= (j+1)^(n+1)*k!/((k-(n+1))!*(n+1)!).
The distance matches d(j,k)=(j+1)*k*sqrt(j*(j+1)/2)+(j+1)^2/2*k*(k-1)*sqrt(2).
The essential configuration is [j][j+1], which forms a j-simplex.
The coordinates of the structure can be written as a sum of k j-simplices of configuration [j][j+1] where each simplex is located in j separate dimensions.
Click here to see [j*k][(j+1)*k] configurations.

Also all configurations [n][p] where p=2*(n+1) and n>4 seem to be multi-stable and have regular structures which are optimal. These structures which have a double no. of points as their simplex counterparts are sometimes also referred to as diplo-simplices.
Each optimal [n][2*(n+1)] structure has n lines of equal length connected to each point forming n*(n+1) edges and 0 simplices/ crosspolytopes in 2D till (n-1)D.
Click here to see [n][2*(n+1] configurations.

Furthermore other configurations generating regular structures of optimal distance which have other stable equilibriums at a near optimal distance exist as well, such as;

[4][13] the optimal structure of which has 4 lines connected to each point forming 26 edges and 0 of triangular/square faces and tetrahedral/octohedral cells, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 112.6764... and 112.6665....
[4][48] found on 25-8-2003, the optimal structure of which has 2 lines connected to each point forming 48 edges and 6 intersecting octagons, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 4 stable structures at distances 1559.8157..., 1559.8067..., 1559.8042... and 1559.8025....
[4][50] found on 25-8-2003, the optimal structure of which has 2 lines connected to each point forming 50 edges and 0 regular faces and cells, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 4 stable structures at distances 1692.7835..., 1692.7720..., 1692.7678 and 1692.7644....
[6][42] found on 14-12-2002, the optimal structure of which has 10 lines connected to each point forming 210 edges and 0 2D-5D simplices/cross-polytopes. This configuration is multi-stable with a minimum of structure at distances 1210.5308... and 1210.5140....
[6][54] the stable structure of which seems to form a so called Diplo-Schläfli polytope or Voronoi polytope for E6³ and has 4 lines connected to each point forming 108 edges and 0 simplices / crosspolytopes in 2D till 5D. This configuration is multi-stable with a minimum of structure at distances 2003.8361..., 2003.8316..., 2003.8164..., 2003.8056... and 2003.7937....
[7][72] found on 2-11-2003, the optimal structure of which has 4 lines connected to each point forming 756 edges, 2016 triangular faces, 1134 tetrahedral cells and 0 simplices / crosspolytopes in 4D till 6D. This configuration is multi-stable with a minimum of structure at distances 3579.93.85..., 3579.9121... and 3579.8228... the last under 1/R^3 only.
[9][24] found on 26-8-2003, the optimal structure of which has 2 lines connected to each point forming 24 edges and 6 squares, succeeds about once every 4 times. This configuration is multi-stable with a minimum of 4 stable structures at distances 395.9482..., 395.9393..., 395.9368... and 395.9314....
[10][32] found on 13-12-2002, the optimal structure of which seems to form two of the above configurations [5][16] or hemipenteracts¹ and has 10 lines connected to each point, forming 160 edges, 320 triangular faces 240 tetrahedral cells, 32 4D-simplices and 20 4D-cross-polytopes. This configuration is multi-stable with a minimum of 2 stable structures at distances 707.5327.... and 707.4897....

Multi-stable [n][p] configurations generating regular structures of non-optimal distance and stable equilibrium not belonging to the symmetry group of regular convex polytopes:
Sofar i found only two configurations, which have an irregular structure which is optimal and a regular structure which is not optimal;

[4][10] this is a [n][2*(n+1)] configuration, however the regular structure generated are not optimal, the optimal structure has 2 lines connected to each point forming 2 pentagonal faces. The less optimal structure has 4 lines connected to each point forming 20 edges and succeeds about once every 3 times, another less optimal structure found on 9-6-2004 will generate an Archimedian polytope with 30 edges, 30 faces, 5 tetrahedra and 5 octahedra at distance 66.1159 under 1/R^3 law. This configuration is multi-stable with a minimum of 4 stable structures at distances 66.1321..., 66.1176..., 66.1159 under 1/R^3 law. respectively.
[5][20] the optimal structure of which is irregular, the less non-optimal structure of which has 5 lines connected to each point forming 30 edges and succeeds about twice every 5 times. Another less optimal structure under 1/R^4 law has 60 edges. This configuration is multi-stable with a minimum of 4 stable structures at distances 271.0034...,271.0030...,271.0026 and 271.0000... the last two under 1/R^3 law. respectively.

¹ George Olshevsky inventor of the name 'polychora', who maintains a splendid website about 4D solids, pointed to earlier publications of a number of structures generated.
² James M. Gibbs creator of a 4D polytope viewer in Java, has made valuable contributions regarding the nature of specific structures found.
³ The Cell Structures of Certain Lattices by J.H. Conway and N.J.A Sloane.
⁴ Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. pp 202-204.

2003, Symen H. Hovinga

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