Enumerative Geometrical Genealogy (Or: The Sex Life of Points and Lines)

By
Shalosh B. Ekhad and Doron Zeilberger

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[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org]

First Written: June 19, 2014 (this page (but not article) updated July 2, 2014)

Joshua Cooper and Mark Walters initiated the subject that we call here Enumerative Geometric Genealogy, (and that lead to the hard-to-count sequence A140468) but they did not realize the broader impact of their research to the population explosion of a certain far-away planet, whose inhabitants are much smarter, and more civilized, than we are.

Sample Input and Output files for GeometryMiracles

• If you want to see the first miracle, concerning FOUR general points in the plane (taking place in the plane), as well as the the first miracle, concerning FIVE general points in the plane, as well as the Pascal miracle concerning SIX points on a conic

the input   yields the output

• If you want to see the beginning of the sequence enumerating the successive generations (alternating between points and lines), if they were FOUR initial points (i.e. four Adams), from the 0-th generation all the way to the 7-th

the input   yields the output This took less than tree seconds!

[Added July 2, 2014: but to get the next term, took much longer! (about 1.5*106 seconds), if you want also to see the number of creatures born in the 8th-generation (all males, of course)
the input   yields the output ]

• If you want to see the beginning of the sequence enumerating the successive generations (alternating between points and lines), if they were FIVE initial points (i.e. five Adams), from the 0-th generation all the way to the 4-th

the input   yields the output

• If you want to see the beginning of the sequence enumerating the successive generations (alternating between points and lines), if they were SIX initial points (i.e. six Adams), from the 0-th generation all the way to the 4-th

the input   yields the output

• If you want to see the beginning of the sequence enumerating the successive PURE (where inter-generational mating is forbidden) generations (alternating between points and lines), if they were FIVE initial points (i.e. five Adams), from the 0-th generation all the way to the 4-th

the input   yields the output

• If you want to see two detailed different genealogies of the same Pascal line for six points on a conic,

the input   yields the output

• If you want to see a detailed genealogy for a random 12-th generation object (that happens to be a line) staring with four initial ("Adams") random points

the input   yields the output

• If you want to see a picuture of a 4-generation points

the input   yields the output

• If you want to see 100 random 9-generation lines

the input   yields the output

Sample Input and Output files for RichardSerge

• If you want to see all the Schwartz-Tabachnikov miracles of the First Kind described in Theorem 4 of this beautiful article

the input   yields the output

• If you want to see all the Schwartz-Tabachnikov miracles of the Second Kind described in Theorems 1-3 of the above-mentioned article

the input   yields the output

Sample Input and Output files for GeneralizedMorley

If you want to see a 0.017-second proof of Euclid' result that the angle-bisectors or a triangle are concurrent, and a 0.152-second proof of Morley's famous Trisector Theorem, where instead of looking at the triangle formed by the points of intersection of trisetors closest to the edges, and less-trivially, the analogous theorem where one looks at the quadsectors (the lines that divide the angles into four equal parts), that took quite a bit longer (namely 71 seconds) to discover (and prove!),
the input   yields the output

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger