My favorite ways to frame some of the most interesting (counterintuitive) facts about high #dimensional #geometry.
1. Cubes are Caltrops. (yes, #dnd reference)
2. Spheres are tops. (the tops you spin, thin on both ends, fat in the middle)
3. Mangoes > Oranges (but only in high dimensions, in 3D I prefer oranges)
4. Orthogonality (I don't have a catchy way of saying this. Upstanding vectors?)
5. Orthogonal everywhere (when dimensions are #infinite)
6. Unit sphere vectors are tiny (their coordinates are mostly very small; yes I know the statement as is, is wrong)
Caltrops is simply a consequence of my (everyone's?) favorite theorem about right angled triangles: #Pythagoras theorem.
Consider the unit cube in 𝑛 dimensional space. The circumradius, via repeated application of Pythagoras, is equal to √𝑛. On the other hand, the inradius is fixed at 1/2.
This gives rise to the pointy cubes visual for high dimensions. The vertices/corners of the unit cube will shoot out compared to the unit circle, despite the faces of the cube staying close.
I've also heard this referred to as "cubes in high dimensions are sea urchins." I like both names, but caltrop is just a fun word to say.
The remaining facts are consequences of exponential decay, which I'll go through in the next post in this thread.