Matpic
mathematics, in pictures
Concept

Crossing number

The fewest pairs of edges that must cross in any drawing of a graph in the plane.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

the trefoil

Three moves, and what they cannot undo

A knot is a closed loop of string, and two knots are the same if one can be wiggled into the other. Reidemeister reduced all possible wiggling to three local pictures — which is what makes it possible to prove that a knot is knotted.

topology · knots 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.

discrete · planarity

Named alongside it

The objects these essays reach for when they reach for this one.

Topological invariantComplete graphCounting argumentEquivalenceEuler formulaGraphGraph colouringInvariantKnotOrientationPlanar graphPlanarity

All concepts