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>>and this distinguishes it from other wacky kinds of space that mathematicians like to try things out in. What other space apart from 3 dimensional space exis

by kamaal 2y ago

>>and this distinguishes it from other wacky kinds of space that mathematicians like to try things out in. What other space apart from 3 dimensional space exists? And what does that even mean?

0_____0 2y ago
Math doesn't have the same limitations on dimensionality that our brains and the space that we interact with do. A line becomes a square becomes a cube becomes a hypercube, and so forth.
card_zero 2y ago
Well, the nicest ones are two dimensions and one dimension, which sort of (not really) exist as subsets of 3D. So a single variable is one-dimensional. Then you could have a gradient, say: https://en.wikipedia.org/wiki/Spatial_gradient https://en.wikipedia.org/wiki/Spatial_gradient And you could have a gradient (of colors if you like, or salinity or force of whatever) along a line, one dimension, or on a plane, two dimensions, or in space, three dimensions. By extension, just adding components to vectors and iterating over the components when you do stuff with them, you can apply the same logic about higher dimensions. This makes people thin and unhealthy and serves no purpose, but seems to have a compulsive appeal for a certain kind of mind. You can't always reliably extend some idea to different numbers of dimensions, because the interactions between components can get complicated, sometimes a thing that's valid for certain numbers of dimensions is not in fact valid for other numbers of dimensions, like maybe you have to have a odd number of dimensions to prevent positives and negatives cancelling out or something. Like many people, I knew this stuff for about a month once but forgot the specifics. But a tesseract is kind of fun to look at. https://en.wikipedia.org/wiki/Tesseract https://en.wikipedia.org/wiki/Tesseract This is like the next step in the sequence dot, line, square, cube. The animation is a "projection into 3 dimensions", like how the shadow of a cube on a piece of paper is in two dimensions. It can't be shown in four dimensions, due to four dimensions being complete fiction. (It's squirming like that because it's rotating, but the axis it's rotating around is at some 4D angle.)
Majromax 2y ago
> What other space apart from 3 dimensional space exists? And what does that even mean? Aside from the one and two-dimensional spaces mentioned in a sibling comment (ℝ¹ and ℝ²), there's also spaces with more interesting geometry. For example, S² is the two-sphere, more conventionally known as "the surface of a ball." On the sphere, the equivalent of straight lines are the great circles. S¹ would be the one-sphere (the circumference of a circle), and S³ would be an interesting, periodic three-dimensional space – kind of like what our universe would be if it were closed rather than (probably) open.
- card_zero 2y ago
  Here I will shoehorn in a mention of Hyperrogue, which is almost related. https://www.roguetemple.com/z/hyper/ https://www.roguetemple.com/z/hyper/

by kamaal 2y ago

>>and this distinguishes it from other wacky kinds of space that mathematicians like to try things out in. What other space apart from 3 dimensional space exists? And what does that even mean?

0_____0 2y ago

Math doesn't have the same limitations on dimensionality that our brains and the space that we interact with do. A line becomes a square becomes a cube becomes a hypercube, and so forth.

card_zero 2y ago

Well, the nicest ones are two dimensions and one dimension, which sort of (not really) exist as subsets of 3D. So a single variable is one-dimensional. Then you could have a gradient, say: https://en.wikipedia.org/wiki/Spatial_gradient https://en.wikipedia.org/wiki/Spatial_gradient And you could have a gradient (of colors if you like, or salinity or force of whatever) along a line, one dimension, or on a plane, two dimensions, or in space, three dimensions. By extension, just adding components to vectors and iterating over the components when you do stuff with them, you can apply the same logic about higher dimensions. This makes people thin and unhealthy and serves no purpose, but seems to have a compulsive appeal for a certain kind of mind. You can't always reliably extend some idea to different numbers of dimensions, because the interactions between components can get complicated, sometimes a thing that's valid for certain numbers of dimensions is not in fact valid for other numbers of dimensions, like maybe you have to have a odd number of dimensions to prevent positives and negatives cancelling out or something. Like many people, I knew this stuff for about a month once but forgot the specifics. But a tesseract is kind of fun to look at. https://en.wikipedia.org/wiki/Tesseract https://en.wikipedia.org/wiki/Tesseract This is like the next step in the sequence dot, line, square, cube. The animation is a "projection into 3 dimensions", like how the shadow of a cube on a piece of paper is in two dimensions. It can't be shown in four dimensions, due to four dimensions being complete fiction. (It's squirming like that because it's rotating, but the axis it's rotating around is at some 4D angle.)

Majromax 2y ago

> What other space apart from 3 dimensional space exists? And what does that even mean? Aside from the one and two-dimensional spaces mentioned in a sibling comment (ℝ¹ and ℝ²), there's also spaces with more interesting geometry. For example, S² is the two-sphere, more conventionally known as "the surface of a ball." On the sphere, the equivalent of straight lines are the great circles. S¹ would be the one-sphere (the circumference of a circle), and S³ would be an interesting, periodic three-dimensional space – kind of like what our universe would be if it were closed rather than (probably) open.

card_zero 2y ago
Here I will shoehorn in a mention of Hyperrogue, which is almost related. https://www.roguetemple.com/z/hyper/ https://www.roguetemple.com/z/hyper/