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There are many proofs, and I'm not claiming that the following list is complete. New references are welcome. (First proof) Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs. Ak …

I consider a graph $G$ (possibly infinite, but locally finite) embedded in the Euclidean plane $\mathbb{E}^2 \cup \{\infty\}$ such that each local perturbation of the embedding "increases the total le …

The fractal dimension of the 3D Apollonian packing is computed in this paper. In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (Hausdo …

View the two examples, I think $P(n,k)$ is the $(n-k)$-rectified $n$-hypercube or the $(k-1)$-rectified $n$-cross-polytope (same thing). I believe the notion of rectification will be very helpful for …

The following statement has been largely improved from my original post thanks to discussions with @DmitriPanov ad well as the comment from @Wojowu. Let $f \colon \mathbb{S}^3 \to \mathbb{R}^2$ be …

Let $C_\pm$ be the two circles obtained by intersecting the cylinder $x^2+y^2=R^2$ with the planes $z=\pm 1$, on which we mark four points $A_\pm:(R,0,\pm 1)$ and $B_\pm:(-R,0,\pm 1)$. Assume that $R …

In a recent paper of Padrol and me, we studied several generalizations of this problem. http://arxiv.org/pdf/1508.03537v1.pdf Regarding Q1, Yoav already mentioned Schulte's work, and Gil mentioned t …

I know that there are a lot of TPMS with different symmetry groups. It seems like every space group is the symmetry group of some TPMS. But I can not find a reference that confirms this for all the …

While the answer to the question of OP is "no", I would like to mention a question raised by Grünbaum and Shephard in "Some problems on polyhedra.", J. Geom. 29 (1987), no. 2, 182–190, for the informa …

I recently showed that: The graph of a stacked $4$-polytope is $3$-ball packable if and only if it does not contain six $4$-cliques sharing a $3$-clique. While Eppstein, Kuperberg and Ziegler 20 …

On the wikipedia page of tetrahedron, there is a list of eight symmetry groups for a (possibly irregular) $3$-simplex (with unmarked faces). There is also a list on the page of 5-cell but doesn't see …

I have been working on minimal surfaces, only recently learnt about maximal surfaces and "maxfaces" in Lorentz spaces. I can clearly see the mathematical motivations. But I wonder if zero-mean-curvat …

For a subset $I$ of $[n]$, a hyperplane $H_I \subset \mathbb{R}^n$ is defined by $$\sum_{i \in I} x_i= \sum_{j \not\in I} x_j.$$ Have you seen the following hyperplane arrangements? Is there anything …

OK, I found the reference: For those who care, it's recently proved in a much stronger form by Oh and Shah in The asymptotic distribution of circles in the orbits of Kleinian groups. The paper is ab …

Finally, I find that the works of Ivanov and Tuzhilin seem to be very close to what I'm looking for.