Topology — page 4
Seven graphs that must link
Six points joined in every way cannot be placed in space without two disjoint triangles hooking together. Trade any triangle for a three-pointed star, or a star back for a triangle, and the property survives; doing it in every possible way from K₆ reaches exactly seven graphs, the Petersen graph among them, and stops. Every placement of every one of them links, by the same parity count. Remove any edge from any of them, and placements that link nothing appear at once. The seven are the whole answer: a graph must link exactly when it contains one of them.
Four equal quarters with two lines
Any flat shape, however lopsided, can be cut into four pieces of equal area by two perpendicular straight lines. The proof turns the pair of lines like the hands of a clock: the area in one quadrant, minus a quarter, reverses its sign every quarter-turn, so somewhere it is zero. The same kind of argument cuts a solid into eight equal pieces with three planes. It stops working in five dimensions, where some masses cannot be cut into thirty-two equal pieces by five hyperplanes — and in four, nobody knows.