Matpic
mathematics, in pictures
Ladder

Finite geometry — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 001010100110101011111point on line?0123456•••••••••••••••••••••L0L1L2L3L4L5L6seven points, seven lines, three points on every line and three lines through every pointthe drawing was checked against the algebra by searching all 5,040 relabellings — one of them carries GF(2)³ onto thispicture

    Seven points, seven lines

    A geometry with seven points, in which every two points lie on exactly one line and every two lines meet in exactly one point. There are no parallels, the whole thing is built out of the two-element field, and one of its lines has to be drawn as a circle.

    rung 1 · computation
  2. 01234567812 triples0 1 20 3 40 5 60 7 81 3 51 4 71 6 82 3 82 4 62 5 73 6 74 5 89 points, 12 triples, each point in 4 of them — and every one of the 36 pairs appears exactly oncefound by backtracking over the pairs, which decides existence rather than assuming it

    A schedule where every pair meets once

    Sort n people into groups of three so that every two of them share a group exactly once. Two divisions have to come out whole, that rules out most sizes — and at every size the divisions permit, a schedule exists.

    rung 2 · computation

All ladders