7 problems

Let $\mathcal M_n$ denote the space of integral curves with $n$ edges, and let $\mathcal M_4$ denote the corresponding space of unit rhombi. Finite-rhombi conjecture. For every int…

Let $\mathcal M_n$ denote the space of integral curves with $n$ edges, and let $\mathcal M_4$ denote the corresponding space of unit rhombi. Rhombus reduction conjecture. For every…

Let $\Delta_1,\Delta_2\subset\mathbb R^3$ be unit triangles, and let $\Upsilon=\Delta_1\cup\Delta_2$. Two-triangle non-domability conjecture. There exist unit triangles $\Delta_1$…

Let $\mathcal R^3$ be the unit distance graph of $\mathbb R^3$, and let $\rho_\lozenge=[uvwx]\subset\mathbb R^3$ be the unit rhombus defined in the source. A coloring is a map…

Let $\mathcal A$ be the set of $a\geq 0$ for which the planar rhombus $\rho(a,\sqrt{4-a^2})$ can be domed, and write $\rho_\lozenge:=\rho(1/2,\sqrt{15}/2)$. The $\rho_\lozenge$ con…

Let $\mathcal A$ be the set of all $a\geq 0$ such that the planar rhombus $\rho(a,\sqrt{4-a^2})$ can be domed. Countability conjecture for $\mathcal A$. The set $\mathcal A$ is cou…

Let $\Delta$ be an isosceles triangle with side lengths $(2,2,1)$. The $(2,2,1)$ triangle conjecture. The triangle $\Delta$ cannot be domed. This conjecture is a central obstructio…