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Things that cannot be done — page 13

Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.
Hilbert's polygons, stage by stage. Hilbert's polygons at stages 1, 2, 3 and 6, each visiting the centres of 4, 16, 64 and 4,096 squares in order, from the bottom-left corner to the bottom-right. Analysis

A path that fills a square

A pen moving continuously for one unit of time can pass through every point of a square. Hilbert's polygons, each threading four times as many squares as the last, move no point more than 0.4507 × 2⁻ⁿ from where the previous one put it — and that one estimate, through the theorem that uniform limits stay continuous, is the whole proof. What the limit pays is exactness of a kind: it repeats some points four times, and its points can never be closer than the square root of their parameters allows.

 Squares meet four at a time, bricks three. A square covered by a 6 by 6 grid of closed squares, with the 25 inner corners where four meet, beside a brick wall of 21 closed bricks, where no point lies in more than three and 25 inner points lie in exactly three. Topology

Squares meet four at a time, bricks three

Tile a floor with square tiles and four meet at every corner; lay bricks in a staggered wall and only three meet anywhere. Can small closed pieces cover a square with only two meeting at any point? Lebesgue proved in 1911 that they cannot — three is forced — and that in every dimension the forced number is one more than the dimension. Counted at every corner and crossing: three for bricks, four for staggered bricks in space, three for random cells, and none of them can do better.

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