Visualization of hypercubes and simplexes in high dimensions.

Wlodzislaw Duch

Simplex: 2D representation of simplexes in dimensions 6-20, found by minimization of MDS stress measure using simmulated annealing.

In 10D the point in the middle is removed and two points appear in the inner circle; from 10-19D points are added to the inner circle.

In 13D all points are on the circle, but in 7D one point is put in the middle; this seems to be the best configurations if simple stress measure is used. From 7-9D there is only one point in center and points are added to the circle.

In 13D there are 3 points in the inner circle, in 14D four, and 15D adds a point to the outer circle.

17D, 18D and 19D solutions show similar patterns, adding points either to inner or outer cirlces.

In 20D there are two circles and a points in the middle appears for the first time.
These pictures are slightly messy because it is hard to find hig precision solution using simulated annealing.

Fro higher number of dimensions we have not tried to find the solutions, they will probably show similar patterns, but as usually, it is difficult to predict how high-D geometry will look like.

We have published these pictures in: Duch W and Naud A (1996) Simplexes, Multi-Dimensional Scaling and Self-Organized Mapping
Proc. of the 8th joint EPS-APS International Conference on Physics Computing '96, Kraków 17-21.9.1996, pp. 367-370

Some op-art creations look similar to these mappings ... Relation to platonic high-dimensional spaces are in the catalog for art exhibition "Komplex symplexu", by Jerzy Olek and Witold Szymański (Wrocław Technical University), 2001 (in Polish only, Sympleksy w wielu wymiarach.

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Last modification 1.09.2006,
Wlodzislaw Duch