Hyperspheres - What are They? How does one visualize them?
Hyperspheres are spheres in a dimension N, greater than 3. In two dimensions we have a circle, in 3 dimensions we have our first sphere. It is very hard for us to visualize spheres in larger dimensions, since we leave in a 3D world.
Sphere packing is an interesting concept - filling N-D space with N-D spheres, which has led to exposed truths about 8 and 24 dimensions - truths which we don't even understand in 4 dimensions. We currently only know how to do sphere packing in dimensions 2, 3, 8 and 24.
Visualizing a hypersphere is very hard. An N-th dimensional space simply has N coordinates. For example a 4D dimensional space has 4 coordinates: (x, y, z, w). One way to visualize a 4D sphere, for example, is to use slicing. For example, if we move a 3D sphere through a 2D plane, we will obtain a 2D slice of our original sphere. Similarly, if we take a 4D sphere and pass it through a 3D cube, we will obtain 3D slices of our 4D sphere, which can be visualized.
Hyperspheres a very counterintuitive. For example, consider a circle inscribed in a square, so that it touches the squares walls. It covers 79% of the square's surface. If we consider the case in 3 dimensions, a sphere only occupies 52% of the area inside of a cube. If we keep going, in 30 dimensions the sphere will occupy only 10 to the -13 the size of the cube inside it. This is about as big as a grain of sand is to a sports arena, only that somehow the sand grain is touching each wall of the sports arena and is still round.
For more information and some nice visualizations please see the video below, by PBS Infinite Series:
Great post. But the biggest lie of them all is the world could possibly be flat!! Do you think the world is flat? I know this sounds absurd but it's all over net surely it should be looked into. Can you imagine how much money NASA has gone through and they could be full of shit.
Do you mind providing more information about those 'truths' in 8 and 24 dimensions? I am a bit curious here :)
PS: I can't watch the video at the moment.
Well, the sphere packing arrangements are not known for any other dimensions up from 3, except 8 and 24. You can find much more information about this in the following article: https://www.quantamagazine.org/sphere-packing-solved-in-higher-dimensions-20160330
The following excerpt from the article gives some further insight:
"Finding the best packing of equal-sized spheres in a high-dimensional space should be even more complicated than the three-dimensional case Hales solved, since each added dimension means more possible packings to consider. Yet mathematicians have long known that two dimensions are special: In dimensions eight and 24, there exist dazzlingly symmetric sphere packings called E8 and the Leech lattice, respectively, that pack spheres better than the best candidates known to mathematicians in other dimensions."
I got the main ideas. Thanks for providing the information, the link, and indirectly the references in the link.
Symmetries as always :)
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STOPMind = blown. Thanks for that👍