Akshar Varma<p>My favorite ways to frame some of the most interesting (counterintuitive) facts about high <a href="https://mathstodon.xyz/tags/dimensional" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>dimensional</span></a> <a href="https://mathstodon.xyz/tags/geometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>geometry</span></a>.</p><p>1. Cubes are Caltrops. (yes, <a href="https://mathstodon.xyz/tags/dnd" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>dnd</span></a> reference)<br>2. Spheres are tops. (the tops you spin, thin on both ends, fat in the middle)<br>3. Mangoes > Oranges (but only in high dimensions, in 3D I prefer oranges)<br>4. Orthogonality (I don't have a catchy way of saying this. Upstanding vectors?)<br>5. Orthogonal everywhere (when dimensions are <a href="https://mathstodon.xyz/tags/infinite" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>infinite</span></a>)<br>6. Unit sphere vectors are tiny (their coordinates are mostly very small; yes I know the statement as is, is wrong)</p><p>Caltrops is simply a consequence of my (everyone's?) favorite theorem about right angled triangles: <a href="https://mathstodon.xyz/tags/Pythagoras" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Pythagoras</span></a> theorem. </p><p>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. </p><p>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. </p><p>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.</p><p>The remaining facts are consequences of exponential decay, which I'll go through in the next post in this thread.</p>