A set of generic 6 points on the parabola is [[t[1], t[1]^2], [t[2], t[2]^2], [t[3], t[3]^2], [t[4], t[4]^2], [t[5], t[5]^2], [t[6], t[6]^2]] The Pascal configuation leads to diffent circuits [[1], [2], [3], [4], [5], [6], {1, 5}, {1, 6}, {2, 4}, {3, 4}, {7, 9}, {8, 10}, {11, 12}], [[1], [2], [3], [4], [5], [6], {1, 5}, {2, 4}, {2, 6}, {3, 5}, {7, 8}, {9, 10}, {11, 12}] yet thanks to Pascal, they begat the same line! Here are the detailed (different) genealogies that lead to the same (Pascal) line The Genealogy of the line (woman) Ms. , 13 who happens to be , - (t[1] t[2] t[4] - t[1] t[2] t[5] - t[1] t[3] t[4] + t[1] t[3] t[6] - t[1] t[4] t[5] + t[1] t[4] t[6] + t[2] t[3] t[5] - t[2] t[3] t[6] + t[2] t[4] t[5] - t[2] t[5] t[6] - t[3] t[4] t[6] + t[3] t[5] t[6]) x/(t[1] t[2] t[4] t[6] - t[1] t[2] t[5] t[6] - t[1] t[3] t[4] t[5] + t[1] t[3] t[5] t[6] + t[2] t[3] t[4] t[5] - t[2] t[3] t[4] t[6]) - (t[1] t[5] - t[1] t[6] - t[2] t[4] + t[2] t[6] + t[3] t[4] - t[3] t[5]) y/( t[1] t[2] t[4] t[6] - t[1] t[2] t[5] t[6] - t[1] t[3] t[4] t[5] + t[1] t[3] t[5] t[6] + t[2] t[3] t[4] t[5] - t[2] t[3] t[4] t[6]) + 1 = 0 By Shalosh B. Ekhad At the Sixth day of Creation, God created, 6, men (alias points) , whose names, and identity are as follows. 2 Adam[1], alias, Mr., 1, who happens to be the point, [t[1], t[1] ] 2 Adam[2], alias, Mr., 2, who happens to be the point, [t[2], t[2] ] 2 Adam[3], alias, Mr., 3, who happens to be the point, [t[3], t[3] ] 2 Adam[4], alias, Mr., 4, who happens to be the point, [t[4], t[4] ] 2 Adam[5], alias, Mr., 5, who happens to be the point, [t[5], t[5] ] 2 Adam[6], alias, Mr., 6, who happens to be the point, [t[6], t[6] ] and it came to pass that Mr., 1, knew Mr. , 5, and together they begat Ms., 7 (t[1] + t[5]) x y who happens to be the line, - --------------- + --------- + 1 t[1] t[5] t[1] t[5] and it came to pass that Mr., 1, knew Mr. , 6, and together they begat Ms., 8 (t[1] + t[6]) x y who happens to be the line, - --------------- + --------- + 1 t[1] t[6] t[1] t[6] and it came to pass that Mr., 2, knew Mr. , 4, and together they begat Ms., 9 (t[2] + t[4]) x y who happens to be the line, - --------------- + --------- + 1 t[2] t[4] t[2] t[4] and it came to pass that Mr., 3, knew Mr. , 4, and together they begat Ms., 10 (t[3] + t[4]) x y who happens to be the line, - --------------- + --------- + 1 t[3] t[4] t[3] t[4] and it came to pass that Ms., 7, knew Ms. , 9, and together they begat Mr., 11 t[1] t[5] - t[2] t[4] who happens to be the point, [-------------------------, t[1] + t[5] - t[2] - t[4] t[1] t[2] t[4] - t[1] t[2] t[5] - t[1] t[4] t[5] + t[2] t[4] t[5] - -----------------------------------------------------------------] t[1] + t[5] - t[2] - t[4] and it came to pass that Ms., 8, knew Ms. , 10, and together they begat Mr., 12 t[1] t[6] - t[3] t[4] who happens to be the point, [-------------------------, t[1] + t[6] - t[3] - t[4] t[1] t[3] t[4] - t[1] t[3] t[6] - t[1] t[4] t[6] + t[3] t[4] t[6] - -----------------------------------------------------------------] t[1] + t[6] - t[3] - t[4] and FINALLY, it came to pass that Mr., 11, knew Mr. , 12, and together they begat our hero, Ms., 13 who happens to be the line, - (t[1] t[2] t[4] - t[1] t[2] t[5] - t[1] t[3] t[4] + t[1] t[3] t[6] - t[1] t[4] t[5] + t[1] t[4] t[6] + t[2] t[3] t[5] - t[2] t[3] t[6] + t[2] t[4] t[5] - t[2] t[5] t[6] - t[3] t[4] t[6] + t[3] t[5] t[6]) x/(t[1] t[2] t[4] t[6] - t[1] t[2] t[5] t[6] - t[1] t[3] t[4] t[5] + t[1] t[3] t[5] t[6] + t[2] t[3] t[4] t[5] - t[2] t[3] t[4] t[6]) - (t[1] t[5] - t[1] t[6] - t[2] t[4] + t[2] t[6] + t[3] t[4] - t[3] t[5]) y/( t[1] t[2] t[4] t[6] - t[1] t[2] t[5] t[6] - t[1] t[3] t[4] t[5] + t[1] t[3] t[5] t[6] + t[2] t[3] t[4] t[5] - t[2] t[3] t[4] t[6]) + 1 This ends this fascinating genealogy, that took, 0.014, seconds to generate. -------------------------------------------------------- The Genealogy of the line (woman) Ms. , 13 who happens to be , - (t[1] t[2] t[4] - t[1] t[2] t[5] - t[1] t[3] t[4] + t[1] t[3] t[6] - t[1] t[4] t[5] + t[1] t[4] t[6] + t[2] t[3] t[5] - t[2] t[3] t[6] + t[2] t[4] t[5] - t[2] t[5] t[6] - t[3] t[4] t[6] + t[3] t[5] t[6]) x/(t[1] t[2] t[4] t[6] - t[1] t[2] t[5] t[6] - t[1] t[3] t[4] t[5] + t[1] t[3] t[5] t[6] + t[2] t[3] t[4] t[5] - t[2] t[3] t[4] t[6]) - (t[1] t[5] - t[1] t[6] - t[2] t[4] + t[2] t[6] + t[3] t[4] - t[3] t[5]) y/( t[1] t[2] t[4] t[6] - t[1] t[2] t[5] t[6] - t[1] t[3] t[4] t[5] + t[1] t[3] t[5] t[6] + t[2] t[3] t[4] t[5] - t[2] t[3] t[4] t[6]) + 1 = 0 By Shalosh B. Ekhad At the Sixth day of Creation, God created, 6, men (alias points) , whose names, and identity are as follows. 2 Adam[1], alias, Mr., 1, who happens to be the point, [t[1], t[1] ] 2 Adam[2], alias, Mr., 2, who happens to be the point, [t[2], t[2] ] 2 Adam[3], alias, Mr., 3, who happens to be the point, [t[3], t[3] ] 2 Adam[4], alias, Mr., 4, who happens to be the point, [t[4], t[4] ] 2 Adam[5], alias, Mr., 5, who happens to be the point, [t[5], t[5] ] 2 Adam[6], alias, Mr., 6, who happens to be the point, [t[6], t[6] ] and it came to pass that Mr., 1, knew Mr. , 5, and together they begat Ms., 7 (t[1] + t[5]) x y who happens to be the line, - --------------- + --------- + 1 t[1] t[5] t[1] t[5] and it came to pass that Mr., 2, knew Mr. , 4, and together they begat Ms., 8 (t[2] + t[4]) x y who happens to be the line, - --------------- + --------- + 1 t[2] t[4] t[2] t[4] and it came to pass that Mr., 2, knew Mr. , 6, and together they begat Ms., 9 (t[2] + t[6]) x y who happens to be the line, - --------------- + --------- + 1 t[6] t[2] t[6] t[2] and it came to pass that Mr., 3, knew Mr. , 5, and together they begat Ms., 10 (t[3] + t[5]) x y who happens to be the line, - --------------- + --------- + 1 t[5] t[3] t[5] t[3] and it came to pass that Ms., 7, knew Ms. , 8, and together they begat Mr., 11 t[1] t[5] - t[2] t[4] who happens to be the point, [-------------------------, t[1] + t[5] - t[2] - t[4] t[1] t[2] t[4] - t[1] t[2] t[5] - t[1] t[4] t[5] + t[2] t[4] t[5] - -----------------------------------------------------------------] t[1] + t[5] - t[2] - t[4] and it came to pass that Ms., 9, knew Ms. , 10, and together they begat Mr., 12 t[2] t[6] - t[3] t[5] who happens to be the point, [-------------------------, t[2] + t[6] - t[3] - t[5] t[2] t[3] t[5] - t[2] t[3] t[6] - t[2] t[5] t[6] + t[3] t[5] t[6] - -----------------------------------------------------------------] t[2] + t[6] - t[3] - t[5] and FINALLY, it came to pass that Mr., 11, knew Mr. , 12, and together they begat our hero, Ms., 13 who happens to be the line, - (t[1] t[2] t[4] - t[1] t[2] t[5] - t[1] t[3] t[4] + t[1] t[3] t[6] - t[1] t[4] t[5] + t[1] t[4] t[6] + t[2] t[3] t[5] - t[2] t[3] t[6] + t[2] t[4] t[5] - t[2] t[5] t[6] - t[3] t[4] t[6] + t[3] t[5] t[6]) x/(t[1] t[2] t[4] t[6] - t[1] t[2] t[5] t[6] - t[1] t[3] t[4] t[5] + t[1] t[3] t[5] t[6] + t[2] t[3] t[4] t[5] - t[2] t[3] t[4] t[6]) - (t[1] t[5] - t[1] t[6] - t[2] t[4] + t[2] t[6] + t[3] t[4] - t[3] t[5]) y/( t[1] t[2] t[4] t[6] - t[1] t[2] t[5] t[6] - t[1] t[3] t[4] t[5] + t[1] t[3] t[5] t[6] + t[2] t[3] t[4] t[5] - t[2] t[3] t[4] t[6]) + 1 This ends this fascinating genealogy, that took, 0.007, seconds to generate.