3 ms·

Slight nit-pick, but any number of dimensions can be reduced to one, using a pairing function https://en.wikipedia.org/wiki/Pairing_function https://en.wikipedi

Slight nit-pick, but any number of dimensions can be reduced to one, using a pairing function https://en.wikipedia.org/wiki/Pairing_function https://en.wikipedia.org/wiki/Pairing_function or a space-filling curve https://en.wikipedia.org/wiki/Space-filling_curve https://en.wikipedia.org/wiki/Space-filling_curve

chmod775 5y ago
Right. I don't immediately see how either of those would work for uncountable infinite sets, but it may be possible. Clearly some other limitations are required to make "dimensions" a useful metric in this context.
- chriswarbo 5y ago
  > I don't immediately see how either of those would work for uncountable infinite sets, but it may be possible. We can think of an N-dimensional space as a grid of (hyper)cube "cells": - Cantor's pairing function traces diagonal lines through the top-right quadrant of a (countable) 2D grid - We can extend Cantor pairing to the whole (countable) 2D plane by joining the diagonals of each quandrant (they meet at the axes). The resulting line spirals out from the origin, in a diamond shape. - We can extend the 2D plane to higher dimensions, again by joining diagonals when they meet at the axes. This results in a diamond-shaped "shell" spiralling out from the origin. The above gives us a 1D line which visits every cell one after another. Next we associate each unit-interval on our line with the cells we're traversing, i.e. the interval between 0 and 1 is associated with the 1st cell; the interval between 1 and 2 is associated with the 2nd cell; and so on. (We could also jump between positive and negative, like [0, 1), [-1, 0), [1, 2), [-2, -1), ... to include the negative half of our 1D line as well) Finally, we use a space-filling curve, like a Z-curve, to map each interval to the volume of its associated grid cell. This works for uncountable intervals and cell volumes. The result would be a line which starts at the origin, and gradually spirals outwards. From a distance, the line appears to form N-dimensional diamond-shaped 'shells'. If we look closer, we see each 'shell' has jagged edges, since it's formed of N-dimensional grid cells. If we look even closer, we see that the line traces a z-curve through each grid cell, filling its volume. The above works for any pairing function and space-filling curve, although I've written about cantor pairing at http://chriswarbo.net/projects/procedural/cantor.html http://chriswarbo.net/projects/procedural/cantor.html and z-curves at http://chriswarbo.net/projects/procedural/z.html http://chriswarbo.net/projects/procedural/z.html :)
- ClumsyPilot 5y ago
  I think once you are talking about "uncountable infinite sets" and different kinds of infinities, the definition has become very niche.

chmod775 5y ago

Right. I don't immediately see how either of those would work for uncountable infinite sets, but it may be possible. Clearly some other limitations are required to make "dimensions" a useful metric in this context.

chriswarbo 5y ago
> I don't immediately see how either of those would work for uncountable infinite sets, but it may be possible. We can think of an N-dimensional space as a grid of (hyper)cube "cells": - Cantor's pairing function traces diagonal lines through the top-right quadrant of a (countable) 2D grid - We can extend Cantor pairing to the whole (countable) 2D plane by joining the diagonals of each quandrant (they meet at the axes). The resulting line spirals out from the origin, in a diamond shape. - We can extend the 2D plane to higher dimensions, again by joining diagonals when they meet at the axes. This results in a diamond-shaped "shell" spiralling out from the origin. The above gives us a 1D line which visits every cell one after another. Next we associate each unit-interval on our line with the cells we're traversing, i.e. the interval between 0 and 1 is associated with the 1st cell; the interval between 1 and 2 is associated with the 2nd cell; and so on. (We could also jump between positive and negative, like [0, 1), [-1, 0), [1, 2), [-2, -1), ... to include the negative half of our 1D line as well) Finally, we use a space-filling curve, like a Z-curve, to map each interval to the volume of its associated grid cell. This works for uncountable intervals and cell volumes. The result would be a line which starts at the origin, and gradually spirals outwards. From a distance, the line appears to form N-dimensional diamond-shaped 'shells'. If we look closer, we see each 'shell' has jagged edges, since it's formed of N-dimensional grid cells. If we look even closer, we see that the line traces a z-curve through each grid cell, filling its volume. The above works for any pairing function and space-filling curve, although I've written about cantor pairing at http://chriswarbo.net/projects/procedural/cantor.html http://chriswarbo.net/projects/procedural/cantor.html and z-curves at http://chriswarbo.net/projects/procedural/z.html http://chriswarbo.net/projects/procedural/z.html :)
ClumsyPilot 5y ago
I think once you are talking about "uncountable infinite sets" and different kinds of infinities, the definition has become very niche.