39 problems

In the setting of Theorem, concerning robustness classes of convex solids and platonic solids, let downward external robustness and downward full robustness be the corresponding do…

Let $\pp$ be a compact, oriented, irreducible, atoroidal $3$-manifold with boundary equipped with an angled block structure, and let $\tilde{\pp}$ denote its universal cover. The a…

Milnor's continuity and vanishing conjecture. The volume function $V$ admits a continuous extension to $\overline{\Omega}$. Furthermore, the points on $\partial\Omega$ where $V$ va…

$4_n$ existence and zigzag conjecture. (i) A z-knotted $4_n$ exists if and only if $n\geq 30$ and $n\equiv 2\pmod 4$. (ii) A tight $4_n$ exists if and only if $n\geq 8$ and…

Symmetry-group conjecture. (i) $3_n(D_{2h})$ exists if and only if $\frac{n}{4}$ is even and $n\geq 16$; there are no tight $3_n(D_{2h})$. (ii) $3_n(D_{2d})$ exists if and only if…

Curvature-graph conjecture. The graph of curvatures of any tight graph of type $3_n$ is the graph in the first of those three cases.

Let $\mathcal{P}_a$ be the stellated tetrahedron with vertices … … … … where edges join all $p_i,p_j$ and $p_i,q_j$ for $i\ne j$, and $a\in(0.5,0.57)$. A polyhedron is Rupert if it…

Let $\mathcal{P}$ be a polyhedron. It is Rupert if there exist $R_1,R_2\in\operatorname{SO}(3)$ and $t\in\mathbb{R}^2$ such that … where $P:\mathbb{R}^3\to\mathbb{R}^2$ drops the…

Let $\mathcal{P}$ be a convex polyhedron, meaning the convex hull of a finite set of points in $\mathbb{R}^3$ in convex position. A polyhedron is Rupert if there exist…

Domokos–Horváth–Goriely–Regős conjecture. Every polyhedral tiling can be completely softened.

Finite-subcomplex relaxation conjecture. Condition (3) can be relaxed so that it is required to hold only outside a finite sub-complex of $\mathcal{D}$.

Let $P$ be any “polytope” with a full-dimensional affine span in $\mathbb R^d$. Let $\mathbf p_{n+1}$ be placed anywhere in $\mathbb R^d$. Let $\Omega$ be any stress for the bar fr…

Soft polyhedric realization conjecture. There exists a combinatorially equivalent, soft polyhedric tiling $M'$.

Let $K$ be a regular $n$-sided polygon with $n\geq 3$, circumscribed by a circle of unit radius. For a positive even integer $p$, let $L_{22}^{(p)}$ denote the corresponding…

A geometric polyhedron here may be nonconvex and self-intersecting; its faces, dihedral angles, edge lengths, and partially-flat vertices are understood in the sense of the paper.…

A hyperbolic strictly-convex polyhedron is a strictly-convex polyhedron in hyperbolic space, with dihedral angles measured along its edges. Stoker's conjecture. Every hyperbolic st…

Asymptotic count conjecture. As $v \to \infty$, the counts $\alpha(v)$ and $\beta(v)$ are respectively asymptotic to

Zero-condition conjecture. The conditions from the Clean Condition are satisfied only when $F(i)=0$.

Let $P_1$ and $P_2$ be polyhedra with the same $1$-skeleton $\Gamma$. Suppose that, for every edge $e$ of $\Gamma$, the dihedral angles of $P_1$ and $P_2$ at the corresponding edge…

Let $Q$ be a polyhedron and let $D$ be its dual. A polyhedron is locally Rupert if it has a local Rupert passage, and reverse locally Rupert if it has the corresponding reverse pas…

A polyhedron is locally Rupert if it admits a Rupert passage after an arbitrarily small alteration of a suitable orientation, and locally reverse Rupert is the analogous reverse-pa…

Polyhedron-bisecting curve conjecture. For each polyhedron $P$ in $\mathbb{R}^3$ there exists a smooth simple closed curve $\mathcal{C}$ such that

The Rhombicosidodecahedron is the point-symmetric Archimedean solid considered here, and Rupert's property means that a congruent copy can pass through a hole cut into the original…

A convex polyhedron is a polyhedron in $\operatorname{\mathbb{R}}^3$ whose faces are convex polygons. Such a polyhedron has Rupert's property if a hole can be cut into it through w…

Irreducible non-realizability conjecture. There is an irreducible SOD satisfying the axioms that is not the visibility map $S_\mathcal P$ of any polyhedron $\mathcal P$ with respec…