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Configurations [jk][(j+1)k]:

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.

Below a number of these configurations are listed where the quantity of n-dimensional simplices and numerical distances were calculated by the applet under law 1/R. Law 1/R^(n-1) which has a faster convergence will in most cases not produce optimal distances, so please switch it off;

[2*k][3*k] configurations:
The optimal structures are sums of k equilateral triangles where distances match: d(2,k)=3*k*sqrt(3)+9/2*k*(k-1)*sqrt(2)

[4][6] the optimal structure of which is a sum of 2 equilateral triangles and has 3 lines connected to each point with a total of 9 edges, succeeds about twice every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 6*sqrt(3)+9*sqrt(2)=23.1202... and 23.1116....
[6][9] the optimal structure of which is a sum of 3 equilateral triangles and has 6 lines connected to each point forming 27 edges and 27 triangular faces, succeeds about once every 4 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 9*sqrt(3) + 27*sqrt(2)¹=53.7722... and 53.7636....
[8][12] the optimal structure of which is a sum of 4 equilateral triangles and has 9 lines connected to each point forming 54 edges, 108 triangular faces and 81 tetrahedral cells, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 12*sqrt(3)+54*sqrt(2)=97.1521... and 97.1435....
James M. Gibbs¹ found on 2-12-2002 this configurations has also another stable structure which is a sum of two tetrahedra and one square at distance 52*sqrt(2) + 24*sqrt(2/3) + 4 = 97.13502....
The last being the least optimal structure has not been generated by the applet yet.
[10][15] the optimal structure of which is a sum of 5 equilateral triangles and has 12 lines connected to each point forming 90 edges, 270 triangular faces, 405 tetrahedral cells and 243 4D simplices, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 15*sqrt(3)+90*sqrt(2)=153.2599..., 153.2514 and 153.2228....
[12][18] the optimal structure of which is a sum of 6 equilateral triangles and has 15 lines connected to each point forming 135 edges, 540 triangular faces and 1215 tetrahedral cells 1458 4D-simplices and 729 5D-simplices, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 18*sqrt(3)+135*sqrt(2)=222.0957...,222.0871 and 222.0786....
[14][21] the optimal structure of which is a sum of 7 equilateral triangles and has 18 lines connected to each point forming 189 edges, 945 triangular faces and 2835 tetrahedral cells, 5103 4D-simplices, 5103 5D-simplices and 2187 6D-simplices, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 21*sqrt(3)+189*sqrt(2)=303.6594..., 303.6508... and 303.6423....
[16][24] the optimal structure of which is a sum of 8 equilateral triangles and has 21 lines connected to each point forming 252 edges, 1512 triangular faces, 5670 tetrahedral cells, 13608 4D-simplices. 20412 5D-simplices, 17496 6D-simplices and 6561 7D-simplices, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 24*sqrt(3)+252*sqrt(2)=397.9510..., 397.9424... and 397.9339....
[18][27] the optimal structure of which is a sum of 9 equilateral triangles and has 24 lines connected to each point forming 324 edges, 2268 triangular faces, 10206 tetrahedral cells, 30618 4D-simplices, 61236 5D-simplices, 78732 6D-simplices, 59049 7D-simplices and 19683 8D-simplices, succeeds about once every 8 times. This configuration is multi-stable with a minimum of 4 stable structures at distances 27*sqrt(3)+324*sqrt(2)=504.9705..., 504.9620... and 504.9534... and 504.9448....
[20][30] the optimal structure of which is a sum of 10 equilateral triangles and has 27 lines connected to each point forming 405 edges, 3240 triangular faces, 17010 tetrahedral cells, 61236 4D-simplices, 153090 5D-simplices, 262440 6D-simplices, 295245 7D-simplices, 196830 8D-simplices and 59049 9D-simplices, succeeds about once every 10 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 30*sqrt(3)+405*sqrt(2)=624.7180..., 624.7094... and 624.7008....

[3*k][4*k] configurations:
The optimal structures are sums of k tetrahedra where distances match: d(3,k)=4*k*sqrt(6)+16/2*k*(k-1)*sqrt(2)

[6][8] the optimal structure of which is a sum of 2 tetrahedra and has 4 lines connected to each point forming 16 edges, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 8*sqrt(6) + 16*sqrt(2)¹=42.2233... and 42.2207....
[9][12] the optimal structure of which is a sum of 3 tetrahedra and has 8 lines connected to each point forming 48 edges and 64 triangular faces, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 12*sqrt(6)+48*sqrt(2)=97.2761... and 97.2735....
[12][16] the optimal structure of which is a sum of 4 tetrahedra and has 12 lines connected to each point forming 96 edges, 256 triangular faces and 256 tetrahedral cells, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 16*sqrt(6)+96*sqrt(2)=174.9563... and 174.9537....
[15][20] the optimal structure of which is a sum of 5 tetrahedra and has 16 lines connected to each point forming 160 edges, 640 triangular faces, 1280 tetrahedral cells and 1024 4D-simplices, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 20*sqrt(6)+160*sqrt(2)=275.2639..., 275.2613... and 275.2587....
[18][24] the optimal structure of which is a sum of 6 tetrahedra and has 20 lines connected to each point forming 240 edges, 1280 triangular faces, 3840 tertahedral cells 6144 4D-simplices, 4096 5D-simplices, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 24*sqrt(6)+240*sqrt(2)=398.1990... and 398.1964....

[4*k][5*k] configurations:
The optimal structures are sums of k 4-simplices where distances match: d(4,k)=5*k*sqrt(10)+25/2*k*(k-1)*sqrt(2)

[8][10] the optimal structure of which is a sum of 2 4D-simplices and has 5 lines connected to each point forming 25 edges, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 10*sqrt(10)+25*sqrt(2)=66.9781... and 66.9769....
[12][15] the optimal structure of which is a sum of 3 4D-simplices and has 10 lines connected to each point forming 75 edges and 125 triangular faces, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 15*sqrt(10)+75*sqrt(2)=153.5001..., 153.4990... and 153.4973....
[16][20] the optimal structure of which is a sum of 4 4D-simplices and has 15 lines connected to each point forming 150 edges, 500 triangular faces and 625 tetrahedral cells, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 20*sqrt(10)+150*sqrt(2)=275.3775..., 275.3764... and 275.3726....
[20][25] the optimal structure of which is a sum of 5 4D-simplices and has 20 lines connected to each point forming 250 edges, 1250 triangular faces, 3125 tetrahedral cells and 3125 4D-simplices, succeeds about once every 8 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 25*sqrt(10)+250*sqrt(2)=432.6103..., 432.6092... and 432.6080....

[5*k][6*k] configurations:
The optimal structures are sums of k 5-simplices where distances match: d(5,k)=6*k*sqrt(15)+36/2*k*(k-1)*sqrt(2)

[10][12] the optimal structure of which is a sum of 2 5D-simplices and has 6 lines connected to each point forming 36 edges, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 12*sqrt(15)+36*sqrt(2)=97.3874... and 97.3868....
[15][18] the optimal structure of which is a sum of 3 5D-simplices and has 12 lines connected to each point forming 108 edges and 216 triangular faces, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 18*sqrt(15)+108*sqrt(2)=222.4487... and 222.4481....
[20][24] the optimal structure of which is a sum of 4 5D-simplices and has 18 lines connected to each point forming 216 edges, 864 triangular faces and 1296 tetrahedral cells, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 24*sqrt(15)+216*sqrt(2)=398.4217... and 398.4211....

[6*k][7*k] configurations:
The optimal structures are sums of k 6-simplices where distances match: d(6,k)=7*k*sqrt(21)+49/2*k*(k-1)*sqrt(2)

[12][14] the optimal structure of which is a sum of 2 6D-simplices and has 7 lines connected to each point forming 49 edges, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 14*sqrt(21)+49*sqrt(2)=133.4525..., 133.4521... and 133.4217....
[18][21] the optimal structure of which is a sum of 3 6D-simplices and has 14 lines connected to each point forming 147 edges, 343 triangular faces and an unknown no. of 3D-17D non-simplices, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 5 stable structures at distances 21*sqrt(21)+147*sqrt(2)=304.1234..., 304.1231..., 304.1225..., 304.1219... and 304.1186....

[7*k][8*k] configurations:
The optimal structures are sums of k 7-simplices where distances match: d(7,k)=7*k*sqrt(28)+64/2*k*(k-1)*sqrt(2)

[14][16] the optimal structure of which is a sum of 2 7D-simplices and has 8 lines connected to each point forming 64 edges, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 4 stable structures at distances 16*sqrt(28)+64*sqrt(2)=175.1737..., 175.1734..., 175.1727 and 175.1713....

[8*k][9*k] configurations:
The optimal structures are sums of k 8-simplices where distances match: d(8,k)=8*k*sqrt(36)+81/2*k*(k-1)*sqrt(2)

[16][18] the optimal structure of which is a sum of 2 8D-simplices and has 9 lines connected to each point forming 81 edges, succeeds about once every 2 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 18*sqrt(36)+81*sqrt(2)=225.5512..., 225.5511... and 225.5506....

[9*k][10*k] configurations:
The optimal structures are sums of k 9-simplices where distances match: d(9,k)=9*k*sqrt(45)+100/2*k*(k-1)*sqrt(2)

[18 ][20] the optimal structure of which is a sum of 2 9D-simplices and has 10 lines connected to each point forming 100 edges, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 20*sqrt(45)+100*sqrt(2)=275.5854... and 275.5849....

[10*k][11*k] configurations:
The optimal structures are sums of k 10-simplices where distances match: d(10,k)=10*k*sqrt(55)+121/2*k*(k-1)*sqrt(2)

[20][22] the optimal structure of which is a sum of 2 10D-simplices and has 11 lines connected to each point forming 121 edges, succeeds about once every 8 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 22*sqrt(55)+121*sqrt(2)=334.2762..., 334.2761... and 334.2758....

¹ James M. Gibbs has made valuable contributions regarding the nature of specific structures found.

2003, Symen H. Hovinga

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Configurations [j*k][(j+1)*k]:

Configurations [jk][(j+1)k]: