MEASURING DISCRETE SYMMETRY

Dennis Glenn Collins

UPR-Mayaguez (retired)

Introduction

Being able to measure things helps do science. This presentation presents some baby steps of the author’s patent on measuring symmetry of a data set.  Before the author’s patent, symmetry was typically measured relative to some points or lines or symmetry groups.  Here symmetry is calculated based only on the number SYM of pairs of equal distances between points.  SYM provides a scale for the horizontal axis on the Complexity Arc  Graph on p. 28 of the book Simplexity (2008) by Jeffrey Kluger.  For example if here are 5 points, the number of distinct distances between points is (combination of 5 things taken 2 at a time) =5*4/2= 10.  Then the possible number of pairs of equal distances is (combination of 10 things taken 2 at a time) = 10*9/2 = 45.  For example if all 5 points were at the same position, the distance between any two of them would be zero (0) and the number of pairs of equal distances, selecting any two of the ten 0’s would be 45.  Thus the SYM scale in this case goes from 0 if there are no pairs of equal distances to a maximum of 45.  A possible vertical scale for the Complexity Arc is then SYM*(MAXSYM - SYM).

However the important beginning thing is the histogram of number of equal distances versus distance.

Thanks for help by Mr. Glenn H. Collins.

Increasing Symmetry

Examples are given of some patterns obtainable with 5 points and their SYM measures.

0 equal pairs:  If all the points are “random” there will be no equal distances.

1 equal pair:  There could be 4 of the 5 points such that d(A,B) = d(C,D), which is not equal to any of the other distances.

2 equal pairs:  If 4 of the 5 points form a parallelogram with no other distances equal to the opposite-side distances of the parallelogram, there are 2 pairs of equal distances.

3 equal pairs:  If the parallelogram is a rectangle but the 5th point does not create any equal distances, there are three pairs of equal distances, namely the opposite sides and the diagonals of the rectangle.

7  equal pairs:  If the rectangle is a square, but the 5th point does not create any more equal distances, there are seven pairs of equal distances formed from the four sides taken 2 at a time (combination of 4 things taken 2 at a time = 4*3/2 = 6) plus the pair of diagonals of the square.

13 equal pairs:  If the 5th point is now put at the center of the square, there are 4 more equal distances from the center to the corners of the square, contributing 6 more pairs of equal distances,giving a total of 7+6= 13 pairs.

20 equal pairs:  If the 5 points form a pentagon, then there are 5 equal sides around the pentagon contributing 5*4/2 = 10 pairs plus 5 more equal distances creating a star shape, contributing another 10 pairs, giving a total of 20.

22 equal pairs:  At this point in the talk I usually ask if anyone can think of a shape with more symmetry, and no one can   However the equilateral triangle has powerful symmetry and 3 of them connected together in a “W” shape have 7 equal distances, contributing  7*6/2 = 21 equal pairs plus one pair of equal diagonals, for 22.

29 equal pairs:  If 4 of the points make a square base and the 5th point makes a 3-dimensional pyramid with 4 equilateral triangle sides, there are 8 equal distances contributing 8*7/2 = 28 equal pairs plus the diagonals on the base, totalling 29.  This was Sneferu’s attempted pyramid.

36 equal pairs:  At this point I again ask if anyone can think of how to get more symmetry, and no one can.  However by making an equilateral triangle base in 2 dimensions plus points above and below  in 3-dimensions to create equilateral triangles, there are 9 equal distances, contributing 9*8/2= 36 equal distances.  Out of the 10 distances only the top to bottom is different, but it is not paired.

45 equal pairs: With 5 dimensions the unit vectors (1,0,0,0,0), (0,1,0,0,0) , . . .,(0,0,0,0,1) are equal disances apart (SquareRoot of 2) contributing 45 pairs.  The same result can be achieved in 4 dimensions, which can be imagined as a tetrahedron that grows in time such that the space-time distance from the origin to each vertex is the same.  The auhor has calculated specific values in 4-dimensional space with a Mathematica program.

Possible “Simplexity” Measure

Taking as a first guess for he vertical axis of the Complxity Arc the product  SYM*(MAXSYM -SYM), by calculus the largest value of the “Symplexity” quadratic function would be at 45/2= 22.5.  The closest value to this out of the cases given would be the “W” shape with SYM = 22.  Surprisingly or not, this or  is attainable in 2 dimensions.  “Simplexity tends to a maximum.”  would perhaps tend to this pattern, with another close value being the pentagon shape with SYM = 20.   Interestingly Sneferu’s attempt to construct a large-scale pyramid with SYM = 29 resulted in the collapse/modification into the “bent pyramid.”

Row[{Plot[x*(45-x),{x,0,45}]},ImageSize->{400}]

Conclusion

The symmetry patent allows numerical discussion of quantities such as SYM and “Simplexity” which otherwise can be considered only in vague generalities.