Classification
Named by 9 essays across 6 fields — each of them below, with the objects they name alongside it.
Thirteen more when one word is dropped
The list of regular solids stops at five because the definition asks for two things at once. Ask for only the second — every corner alike — and thirteen more appear, each of them cut off a Platonic solid at a depth found rather than chosen.
Six in four dimensions, and three forever after
The count of regular solids goes five in three dimensions, six in four, and then three in every dimension above — for good. Four dimensions is the last place anything unusual happens, and it happens twice.
A disc sewn to a Möbius band
The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.
Every fifth one divides
p(4) is 5, p(9) is 30, p(14) is 135, and every partition count at a number leaving four on division by five is divisible by five. Ramanujan read it off a table; the explanation is a way of splitting those partitions into five equal heaps.
Sixteen polygons with one dot inside
Fix one of Pick's two counts at one and ask what is left. The answer is a finite list, the list has exactly sixteen entries, each one is its own kind of object with a dual that is another entry, and the whole classification is a search a page can carry out.
Every power of x that draws a hyperoval
In a plane of order 2^h, the graph of x^k plus two points at infinity is sometimes a hyperoval — as many points as a plane allows with no three in line. Searching every exponent in every plane from order 4 to 4096 finds hundreds that work, and once six symmetries of the problem are applied they fall into exactly the families already known: the conic, the translation curves, Segre's x⁶ and Glynn's two. Whether that list is complete in every order is open.
Closed sets obey the continuum hypothesis
Whether some set of reals is bigger than the whole numbers and smaller than the line is a question the axioms cannot answer. For closed sets it has an answer, found in 1883: strip away the isolated points, again and again, and what is left is either nothing or a perfect set, which is as large as the line — so every closed set is countable or of the line's size. The same answer holds for Borel and analytic sets, and fails to be provable exactly one step further up.
Two copies of the polygon, cross-matched
The two-sided surface that lies over a one-sided one can be built from the gluing word alone: take two copies of the polygon, one read each way round, glue a letter within the copies when it is used once each way, and across them when it is used twice the same way. The recipe works for every surface at once, doubles every count, and shows that a sphere with k cross-caps is covered by the surface with k − 1 handles — and that among all its connected double covers, exactly one is two-sided.
One block of quantifiers, read off an algebra
Which properties of words can be stated with existential quantifiers alone — 'there are positions, in this order, carrying these letters' — and Boolean combinations of such statements? Imre Simon answered in 1975: exactly those whose syntactic monoid is J-trivial. Of the 57,068 languages that automata of up to four states recognise, 848 pass the test, and every one of them is decided by which words of length three or less it contains as scattered subwords.
Named alongside it
The objects these essays reach for when they reach for this one.
DualityEuler characteristicExhaustive searchConjectureCross-capEquivalenceGluing diagramOrientabilityPlatonic solidsPolyhedronProjective planeVertex configuration