Testwiki:Reference desk/Archives/Mathematics/2024 July 26

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July 26

I joined the X and Y axes together. What Wikipedia page already mentions the concept?

I had / have this fairly rather long blabbersome brilliant idea. My question is since there is no way that I, with a rather low IQ, could come up with something "new", then my idea certainly must be merely a rehash of some ideas already mentioned on several Wikipedia pages, in fact probably just a sentence on just one page. But which one(s)? Thanks. Jidanni (talk) 04:56, 26 July 2024 (UTC)

Your page is cumbersome to follow, but if I'm correct in interpreting it, you are essentially proposing the use of a pairing function or Hilbert curve. It is not possible to continuously reduce dimension in this manner (more precisely, 1D and 2D space are not homeomorphic). It would help if you would more rigorously formulate the function you are proposing rather than merely using examples.--Jasper Deng (talk) 06:22, 26 July 2024 (UTC)

Wikipedia defines "pairing function" only for natural numbers, but below I use the term for (not necessarily unique) functions wrapping up two values (not necessarily integers) into a single one of the same type.

Letting $I$ stand for the unit interval $[0,1],$ the Hilbert curve can be described as a function $h:I\to I\times I.$ Input one number $z,$ output $h(z)=(x,y):$ a pair of numbers. This function is surjective, which implies that there exists an inverse pairing function $h^{-1}:I\times I\to I.$ Being an inverse means that when $h(z)=(x,y),$ we have $h^{-1}(x,y)=z.$ Function $h$ is also continuous, but it is not injective. Many output pairs are reached several times; for example, $h(\frac{1}{6})=h(\frac{1}{2})=h(\frac{5}{6})=(\frac{1}{2},\frac{1}{2}).$ So the inverse is not unique.

Numbers in $I$ can be written in base 2; for example, $0=.000000..{.}_2,$ $\frac{3}{7}=.011011..{.}_2,$ and $1=.111111..{.}_2,$ This expansion is not unique: $.0111111..{.}_2=.100000..{.}_2=\frac{1}{2}.$ We can view these binary expansions as infinite sequences $a={(a_n)}_{n=1}^{\mathrm{\infty }}\in 2^{\omega }={\mathbb{N}}^{+}\to \{0,1\},$ The function $r:2^{\omega }\to I$ given by $r(a)={\sum }_{n=1}^{\mathrm{\infty }}{a}_n{2}^{-n}$ interprets a binary expansion as a real number. This function $r$ is continuous and surjective, just like function $h$ before, so it has an inverse. But, as before, function $r$ is not injective, so the inverse is not unique. However, by convention, it has a "canonical" inverse: other than $r^{-1}(1)=111111...,$ the only possible expansion of $1$ in the domain of $r,$ avoid sequences ending in an infinite stream of $1$s.

Now, using $r,$ we can define a bicontinuous pairing function $\pi :2^{\omega }\times 2^{\omega }\to 2^{\omega }$ such that

$h(r(\pi (u,v)))=(r(u),r(v)).$

This means that we can give a "canonical" definition for $h^{-1}$ by using $r$ and its canonical inverse:

$h^{-1}(x,y)=r(\pi (r^{-1}(x),r^{-1}(y))).$

The function $\pi$ can be described in the form of a 4-state finite-state automaton that gobbles up two streams of bits and produces a single stream of bits. It takes two bits at a time, one from each of the two input streams, and outputs two bits on the output stream.

I suspect that the "brilliant idea" is akin to this way of pairing $x$ and $y.$ I expect the idea is well known, but perhaps only as folklore, and I doubt that it is described or even hinted at in Wikipedia mainspace.  --Lambiam, edited 10:51, 28 July 2024 (UTC) (originall 11:11, 26 July 2024 (UTC))