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Things that cannot be done — page 6

Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.
sin(2πkx) for k up to 10, and the largest gap between every pair. Members of the sequence sin 2πkx drawn on one pair of axes, beside a table of the largest vertical gap between every two members, none of which is less than one. Analysis

The subsequence that has to exist

Every bounded list of numbers has a part that settles down. A bounded list of functions need not: the waves sin 2πkx never come within 1.76 of one another. One extra condition — that no member may change faster than a bound they all share — restores the guarantee, and it is the reason a differential equation with a continuous rule has a solution at all.

 Biproportional seats against their fair shares. A table of votes for 3 districts and 4 parties beside the seats the biproportional method gives, each with the fair share from the continuous fit, and the cell whose seats fall outside its quota marked. Applied

The table inside every quota

Give seats to districts and parties at once, and every cell of the table has a fair share it ought to round from. A table rounding every cell to its floor or its ceiling, with every total exact, always exists. The biproportional method does not always choose one: here it gives a party 2 seats where its fair share is 3.088.

 Sixteen halves in a three-by-three-by-three table of seats. A three-way table of fair shares drawn as three slices, one per group, with sixteen cells holding a half and every line total, along districts, parties and groups, equal to zero or one. Applied

Where the rounding runs out

In two dimensions a table of seats inside every fair share always exists. Add a third family of totals — every district and party split between groups — and it need not. Sixteen halves in a three-by-three-by-three table meet every total, and no whole table does it without a seat where the fair share is nothing, because the halves close a loop of seven.

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