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Whilst this doesn't answer the original question, I shall show that it's possible to partition $\mathbb{R}^3 \setminus \{ 0 \}$ into pairwise-linked disjoint circles. I'll begin with a couple of obse …

We can express $\mathbb{R}^3$ as a disjoint union of circles. There are some constructive ways of doing this, although it's easier to construct them sequentially by transfinite induction, applying the …

Contrary to your claim, the overlapping number $N$ can exceed the kissing number $K$. Moreover, there exists some fixed value of $K$ for which $N$ can be arbitrarily large. This is a sketch of a proo …

Your symmetric distance is fine, as long as we're particularly careful in the definition of a 'shape'. I believe the following will suffice: A shape is an equivalence class of Borel sets, where two B …

Firstly, if you just have pairwise distances rather than coordinates of points in $\mathbb{R}^3$, then you can attempt to embed the mesh into $\mathbb{R}^3$ using Isomap. Under certain conditions, suc …

In Sphere Packings, Lattices and Groups, Conway and Sloane explore laminated lattices. If we let $X_d$ be the set of $d$-dimensional Euclidean lattices where every pair of points are separated by dist …

Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a function that maps lines to lines, and suppose that there are three non-collinear points in the image of $f$. Lemma 1: If $f[\ell_1]$ and $f[\ell_2 …

A bounded-length straightedge can emulate an arbitrarily large straightedge (even without requiring any compass), so the rusty compass theorem is sufficient. Note that, in particular, it suffices to s …

If $r < \frac{1}{3}$, then the graph cannot possibly be connected. Indeed, the largest possible connected components are of four points clustered around one of the lattice points. Proof: Suppose we h …

Let $M_n \subseteq SO(2n)$ be the set of real $2n \times 2n$ matrices $J$ satisfying $J + J^{T} = 0$ and $J J^T = I$. Equivalently, these are the linear transformations such that, for all $x \in \math …

Suppose we have a shape bounded by a simple closed curve $\gamma \subset \mathbb{C}$, with points $A,B,C,D$ in cyclic order on the curve. If we randomly color the interior of that shape in half red a …

Does there exist an irreducible polynomial $f \in \mathbb{R}[x, y, z]$ such that: $$ V := \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) \leq 0 \} $$ is a solid of constant width with a finite symmetry gr …

The number $65520$ arises in two very different scenarios: It occurs in the formula for the theta series of the Leech lattice: $$ \Theta_{\Lambda_{24}}(q) = 1 + \sum\limits_{m=1}^{\infty} \dfrac{655 …

Yes, we can. Consider the usual drawing of the Fano plane with 7 vertices, 6 lines, and a circle. Replace the circle with a line through two of the three vertices. Now we have 7 lines with 6 triple i …

If the triangle is equilateral, then all well-defined triangle centres coincide, and it's impossible to determine the size of the equilateral triangle. Otherwise, three triangle centres suffice: the i …