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What a system cannot say — page 3

Rules asked a question about themselves, and an answer that is provably not available from inside — which is a different kind of limit from not knowing yet.

Hippias's quadratrix, traced by two uniform motions. A unit square with a quarter circle, several positions of a turning radius and a falling horizontal line, their crossings, and the curve through them ending on the base at 2/π.

Computation

A curve that divides any angle

Let a radius turn at a steady rate while a horizontal line falls at a steady rate, both finishing together, and mark where they cross. The curve they trace turns heights into angles, so dividing a height — which a ruler and compass can always do — divides the angle in the same ratio. The same curve meets its base at 2/π of the side, a length from which a square with the area of a circle follows. It reaches what no marked ruler or conic can, and the ancient objection to it is exact.

A rectangle of the parity table, nearly balanced. A 32 by 32 grid of +1 and −1 entries — the parity of the inner product of row and column — with a 15 by 14 rectangle of chosen rows and columns highlighted; its entries sum to 8.

Computation

Two weak sources make one fair bit

No fixed rule can turn every weakly random source into fair bits: for any rule, some source with almost full unpredictability makes it constant. Two independent sources are different. Multiply their bits in pairs, add, and keep the parity — and if the two together carry more unpredictability than the length of one, the result is nearly fair, whatever else the sources do. The reason is that the table of those parities is balanced on every large rectangle.

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