Configurations [n][2*(n+1)]:

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/cross-polytopes in 2D till (n-1)D.
Below a number of these configurations are listed;

• [5][12] the optimal structure of which has 5 lines connected to each point forming 30 edges and 0 simplices/cross-polytopes in 2D, 3D and 4D, succeeds about twice every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 96.4231... and 96.4168....
• [6][14] the optimal structure of which has 6 lines connected to each point forming 42 edges and 0 simplices/cross-polytopes in 2D till 5D, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 132.3778... and 132.3593....
• [7][16] the optimal structure which seems to form a so-called diplo simplex or Delaunay polytope for E^*7¹ and has 7 lines connected to each point forming 56 edges and 0 simplices/cross-polytopes in 2D till 6D, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 173.9852... and 173.9592....
• [8][18] the optimal structure of which has 8 lines connected to each point forming 72 edges and 0 simplices/cross-polytopes in 2D till 7D, succeeds about once every 10 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 221.2470..., 221.2155... and 221.2139....
• [9][20] the optimal structure of which has 9 lines connected to each point forming 90 edges and 0 simplices/cross-polytopes in 2D till 8D, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 274.1640... 274.1275... and 274.1282....
• [10][22] the optimal structure of which has 10 lines connected to each point forming 110 edges and an unknown no. of non-simplices/non-cross-polytopes, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 332.7368... 332.6980... and 332.7975....
• [11][24] the optimal structure of which has 11 lines connected to each point forming 132 edges and an unknown no. of non-simplices/non-cross-polytopes, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 396.9656..., 396.9248... and 396.9240....
• [12][26] the optimal structure of which has 12 lines connected to each point forming 156 edges and 0 simplices/cross-polytopes in 2D till 11D, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 466.8507... and 466.8085... and and 466.8082....
• [13][28] the optimal structure of which has 13 lines connected to each point forming 182 edges and 0 simplices/cross-polytopes in 2D till 12D, succeeds about once every 5 times. This configuration is multi-stable configuration with a minimum of 2 stable structures at distances 542.3922... and 542.3487....
• [14][30] the optimal structure of which has 14 lines connected to each point forming 210 edges and an unknown no. of non-simplices/non-cross-polytopes, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 623.5902..., 623.5458... and 623.5455....
• [15][32] the optimal structure of which has 15 lines connected to each point forming 240 edges and an unknown no. of non-simplices/non-cross-polytopes, succeeds about once every 5 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 710.4448... and 710.3994....
• [16][34] the optimal structure of which has 16 lines connected to each point forming 272 edges and 0 simplices/cross-polytopes in 2D till 15D, succeeds about once every 8 times. This configuration is multi-stable with a minimum of 4 stable structures at distances 802.956067..., 802.956062..., 802.909832 and 802.910003....
• [17][36] the optimal structure of which has 17 lines connected to each point forming 306 edges and an unknown no. of non-simplices/non-cross-polytopes, succeeds about once every 8 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 901.1239... and 901.0772....
• [18][38] the optimal structure of which has 18 lines connected to each point forming 342 edges and 0 simplices/cross-polytopes in 2D till 17D, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 3 stable structures at distances 1004.9485..., 1004.9012... and 1004.9011....
• [19][40] the optimal structure of which has 19 lines connected to each point forming 380 edges and 0 simplices/cross-polytopes in 2D till 18D, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 1114.4298... and 1114.3820....
• [20][42] the optimal structure of which has 20 lines connected to each point forming 420 edges and 0 simplices/cross-polytopes in 2D till 19D, succeeds about once every 3 times. This configuration is multi-stable with a minimum of 2 stable structures at distances 1229.5678... and 1229.5195....

¹ The Cell Structures of Certain Lattices by J.H. Conway and N.J.A Sloane.

2003, Symen H. Hovinga