Matpic
mathematics, in pictures
Ladder

Planarity — the ladder

One essay so far against this idea. A ladder is the distinct arguments that stand against one idea, and this one has room to grow.
  1. K4no crossingsK51 crossing, and no drawing has noneK3,31 crossing, and no drawing has noneK4: 4 points, 6 edges, and at most 6 allowed; K5: 5 points, 10 edges, and at most 9 allowed; K3,3: 6 points, 9 edges,and at most 8 allowed6,000 layouts were tried for each; the best found is drawn, and the bound is what rules out anything better

    Two graphs that will not lie flat

    Five points, every pair joined: no matter how the points are placed or how the lines are drawn, two of the lines cross. The proof is not about drawing at all — it counts edges against faces and finds one edge too many.

    rung 1 · discrete

All ladders