Matpic
mathematics, in pictures
Concept

Euler formula

For a connected graph drawn in the plane, corners minus edges plus faces is two, counting the region outside as a face.

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

ABCDABABCBCDADCDABCD4 circles cut the plane into 14 pieces — Euler's count is 1414 of the 16 patterns appear; missing: A¬BC¬D, ¬AB¬CD

Four circles cannot do it

Three overlapping circles cut the plane into exactly the eight regions three sets need. Four circles cut it into fourteen, and sixteen are required — so the diagram everyone draws stops working at four, and the reason is a count.

logic · class diagrams 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.

ArrangementClosed curveComplete graphConvexityCounting argumentCrossing numberGraphGraph colouringMembership patternPlanar graphPlanarityRegion count

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