Definitions:

Charged particles model: An imaginary model where a number of p particles or points in an n-dimensional Euclidian space are confined at an (n-1) dimensional surface of an n-dimensional sphere with radius r.
The particles have an equal imaginary charge and are creating an equal opposing force upon the other particles i.e. all the particles repell eachother.
The internal radii of all particles equal zero and they have no mass or other properties except the mentioned imaginary charge.
The force created by the charges is affected by distance in that sence that it is proportional to 1/d^(x-1) where d is mutual distance and x<=n, x>1
There is no specific propagation speed of the charge.
The particles are distributed at random or in a specific spatial way, no particle having equal n-dimensional coordinates.
All particles depending on the resultant force vector exercised by the other particles will try to find positions as far from each other as possible, remaining at fixed radius r, until they reach a final state where the force vectors exercised on each particle are pointing exactly in the opposite direction of the centre of the sphere.
Since the particles are confined, the sphere will show an opposite force vector resulting the final force vectors for each particle to be zero.
The sum of all mutual distances between particles or points, counted once, is an indicator for maximal distance.
Computational model: An approach of the charged particle model can be implemented on a computer where the particles or points can be distibuted at random or a specific spatial way whereby the imaginary force vectors are calculated in an iterative process, which will cause the particles to move apart. Iteration is stopped when all particles or points have a force vector 0.
During or after iteration the coordinates of the points forming the structure can be further processed by algorithms such as finding the closest distance between each particle or point and algorithms to find other relational properties used for classification.
A configuration [n][p]: Any group of p particles or points in n dimensions which can exist in the charged particle model.
A structure: A configuration which has unique shared characteristics in spatial distribution regardless of orientation and scale.
Stable structure: A structure of a certain configuration where the imaginary force vectors exercised on each particle or point are zero.
Structure with an unstable equilibrium: The structure has a state which is stable, but any small
disturbance will be amplified and destroy the state (like, a pencil standing on its point).
Structure with a stable equilibrium: The structure has a state which is stable, and will move back toward the stable state whenever given a small disturbance. "Small" is relative to how far away other equilibria are.
Stable or mono-stable configuration: For a certain configuration [n1][p] where p>n1 only one stable structure exists, which is not a stable structure of configuration [n2][p] where n2<n1.
Multi-stable or x-stable configuration: For a certain configuration [n1][p] where p>n1, two or more different stable structures exist, which is not a stable structure of configuration [n2][p] where n2<n1 and where x can be called bi (2), tri (3), quad (4) ec.
Distance: The sum of mutual distances between particles or points of a certain structure, based on r=1, all counted once.
Optimal distance: For a certain structure of a specific [n][p] configuration the distance is maximal.
Non optimal distance: For a certain structure of a specific [n][p] configuration the distance is not maximal.
Regular structure: Each particle or point has an equal number of nearest adjacent points higher than or equal to n, the distances of which are all equal.
Irregular structure: Each particle or point has an unequal number of nearest adjacent points or the distances of nearest adjacent points are not all equal or the number of adjecent points is not higher than or equal to n.
Euler's modified polyhedral theorem: where F(k) represents the number of n-dimensional faces and F(n) represents the n-dimensional body itself, counted as 1.

2003, Symen H. Hovinga

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