Mathematical curiosities
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This is the beginning of a collection of "strange but true" facts in mathematics. As usual, the list will grow as my time permits.

The hereditary base b representation of a number is obtained by first writing the number as a sum of powers of b, then writing each of the exponents as a sum of powers of b, etc., until you can't go on any more. Now pick any natural number. Construct a second one by expressing the first number in hereditary base 2 and subtracting 1. Construct a third one by expressing the second one in hereditary base 3 and subtracting 1. If you continue like that indefinitely, you will produce a Goodstein sequence. Such sequences have a remarkable property. Taking the first number to be greater than or equal to 4, the sequence will appear to grow rapidly, seemingly without bound. Taking 19 as the first number, the second number is already 7625597484990, and the sequence will seemingly keep growing even faster. And yet, Goodstein's theorem states that any Goodstein sequence will eventually decrease again and reach zero! Even for the Goodstein sequence starting with 4 it takes an enormous number of steps for this to happen: the numbers start to decrease only after about 2.6 x 10^60605351 steps, and it takes approximately 10^121210700 steps to reach zero. Equally interestingly, as Goodstein already suspected in 1944, the theorem can not be proven within ordinary Peano arithmetic. This was demonstrated by Paris and Kirby in 1982.

The Prime Number Spiral was allegedly discovered by physicist Stanislaw Ulam while doodling during a boring physics talk. When the natural numbers are arranged in a spiral on a grid, and the prime numbers are highlighted, an interesting pattern emerges: diagonal lines appear. The effect seems sufficiently obvious to preclude any sort of optical illusion, but thus far there is no explanation for the effect.

There is a precise and natural sense in which some irrational numbers are more irrational than others. The best rational approximations to a number are obtained by constructing its continued fraction expansion, which defines a sequence of rational numbers that are successively better approximations of the number under consideration. To determine how irrational a number is, one can then look at how slowly its continued fraction expansion converges. The "most irrational number" is the one with the most slowly converging continued fraction expansion. This is the number corresponding to the continued fraction 1+1/(1+1/(1+..., which is none other than the golden ratio.

Here's another thing numerologists will like. The "area" of a unit sphere in 2-dimensional Euclidean space (a circle of radius 1) is 2Pi, for a unit sphere in 3D space the area is 4Pi, for a unit sphere in 4D it is 2Pi^2, etc. Naively one might think that the area of a unit hypersphere keeps growing with the dimension of the Euclidean space it is embedded in, but this is not the case. A maximum area is reached in 7 dimensions, beyond which the area asymptotically goes to zero as the dimension is increased.

There exist curves that fill the plane without holes. It is easier to show the existence of curves that fill a square without holes, but it is no less surprising that this can be done. Guiseppe Peano was the first to come up with a proof. Usually what one does is to define a sequence of curves that do not fill the square themselves, but then prove that the sequence has a limit and that this limit curve does fill the square. Some constructions use sequences of curves that are not self-intersecting. By looking at examples of proofs, once you are convinced that square-filling curves exist, you might get the impression that there are square-filling curves that don't self-intersect. However, as shown by Luitzen Brouwer, this is not the case.

Click here to see the large online collection of mathematical curiosities (ranging from almost trivial to very difficult) maintained by the mathematics department of Harvey Mudd College.



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