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Even more is true for this theorem. Check out this drawing from Arseniy Akopyan wonderful book of Geometry in Figures (Second, extended edition, 2017). On page 65 we find Figure 4.7.29) In the fore …

In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of t …

Here is an idea. Consider the following parameterization, which is supposed to cover the configuration space in question. $$\mathcal{C}_7:=\left\{\pmatrix{x_k\\y_x\\z_k},\pmatrix{a_k\\b_k\\c_k}_{1\le …

Answer to Q1: All of the 261.  I looked at this question because of a video of Matt Parker and wrote an algorithm to find solutions. See here for an example of how a solution would look like. I dumped …

Looking at this old question again, I'm now fairly convinced that the easiest route of solving this problem is using similar ideas to the one suggested by David E Speyer in a comment, namely basically …

A (trivial) necessary condition is that the diagonal of the inner one is not longer than the diagonal of the outer one. So if $(a,b,c)$ is supposed to fit in $(x,y,z)$, then we should have $$a^2+b^2+ …

Not all simple polytopes are incribable, e.g. the dual of the cyclic polytope $C_4(8)$ is simple and not inscribable, as shown recently in Combinatorial Inscribability Obstructions for Higher-Dimensio …

While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ …

The answer seems to be $\frac{1}{2\pi}$, using a semi circle. See Moran, P. A. P. "On a problem of S. Ulam." Journal of the London Mathematical Society 1.3 (1946): 175-179.

Yashar Mermarian writes here that the answer is yes. And the argument he gives is pretty much the same as the one you already started. Taking $n=k$ and $\epsilon=\pi/2$ Gromov's waist inequality give …

I used sage to make a 3d animation of all 261 unfoldings. Here is a screenshot of the first few: The file cube-unfoldings.txt contains all the unfoldings, each line contains a list of 8 points. …

I am interested in the Hausdorff dimension of the Apollonian circle packing. There seem to be two numerical calculations of the value: 1.305686729(10) from P.B Thomas and D.Dhar, The Hausdorf[sic!] …

Let me answer at least some of your questions. I will only talk about your first definition of the cells, since these are somewhat nicer, as Igor Rivin pointed out. You consider the function $f(w_1)= …

Allow me look at one aspect, or special case, of your question, namely "finding the largest regular 3-dimensional tetrahedron inscribed in a d-dimensional unit cube". I. $4$-cube I can find the follow …

First, let's assume $a=1$; for other values we can scale a solution with $\sqrt{a}$. So we want to minimize $H=\sum_{i,j} (1-\|x_i-x_j\|^2)^2$. I globally optimized the problem numerically for $n=4 …