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Measuring Discrete Symmetry

Dennis Collins
Organization: University of Puerto Rico

Wolfram Technology Conference 2013
Conference location

Champaign, Illinois, USA

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.

*Wolfram Technology

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