3 ms·

Thanks for the link. Sometimes I wish Xanadu had actually happened. The moment I opened the article I saw something like |R^3. Honestly speaking I don't really

• card_zero 2y ago
  Blackboard bold R with a superscript 3? That's just space, like regular space that we inhabit. The 3 is the dimensions, the R is the real numbers, and this distinguishes it from other wacky kinds of space that mathematicians like to try things out in. Edit: we also have in there "the coefficients of the multipole expansion can be written as functions of the distance to the origin, r", so it sounds like r in italics is the origin - but that's usually (0, 0, 0), so maybe it's a distance to it. Anyway r in bold is probably a vector, vectors are usually in bold. And you're quite right about Xanadu, I wonder if there's some way to overhaul Wikipedia to put a key to notation in the sidebar of all pages with equations in.
  • 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/