22 problems

Let $G$ be a $3$-graph on $n$ vertices, and let $\delta_2(G)$ denote its minimum codegree. Define the $4$-graph $H$ on $V(G)$ by making a $4$-set an edge whenever it spans a tetrah…

Let the reciprocal and twinning operations be the two involutions on tetrahedra defined in the surrounding discussion. Their compositions in either order are not themselves involut…

Let $\alpha,\beta,\gamma,\delta,\varsigma\in\mathbb R$ satisfy … and let $\varrho_+(\alpha,\beta,\gamma,\delta)$ denote the unique positive root of the discriminant quadratic descr…

A quadruple of points in the special affine plane determines triangle and Varignon-parallelogram areas. Consider the corresponding quadruple of Euclidean-plane points for which the…

There is a finite packing in space consisting of twelve tetrahedra around a rhombic dodecahedron, with the property that no single member can move rigidly without disturbing the ot…

Let $T$ be a regular tetrahedron in three-dimensional space, and let $\vartheta_L(T)$ denote its lattice covering density. Forcade–Lamoreaux conjecture. … The value is supported by…

Let a tetrahedron have corresponding face centers, and consider the cevians joining the vertices to those face centers. Equal-cevian-length conjecture. If the cevians to the corres…

Let a reference tetrahedron and its associated central tetrahedron be given, and suppose the two tetrahedra are similar, meaning that corresponding edge lengths are proportional. L…

Let a reference tetrahedron and its associated central tetrahedron be given. A tetrahedron is regular when all its edges are equal. Regular central-tetrahedron conjecture. If the c…

Let a reference tetrahedron and its associated central tetrahedron be given. A tetrahedron is isosceles when it has the corresponding symmetry/equality of edge lengths intended in…

Let a tetrahedron be isodynamic, and let a center function assign a point on each face. A center function is hyperbolic when the cevians to the corresponding face centers form a hy…

Sommerville–Goldberg classification conjecture. The Sommerville and Goldberg lists of face-to-face tetrahedral tiles are complete. In particular, every tetrahedral tile has at leas…

Tetrahedral tile classification conjecture. The list of tetrahedral tiles identified in previous literature is exhaustive.

Extension conjecture. The authors conjecture that Theorem is an extension of these results.

Non-right-dihedral gentile tetrahedra conjecture. There are no gentile tetrahedra that do not have a dihedral angle of exactly $pi/2$.

Gentile acute tetrahedra conjecture. There are no gentile acute tetrahedra.

Ten-triangle conjecture. Any dissection of an acute spherical diangle into acute spherical triangles requires at least ten triangles.

Hill tetrahedra uniqueness conjecture. The Hill tetrahedra are the only reptile tetrahedra.

Uniqueness conjecture. Up to scaling, this is the unique such tetrahedron with only one rational face area. The source gives this as a conjecture; no resolution is supplied here.

Let the packing density of a packing of congruent tetrahedra be the fraction of space covered by the tetrahedra, and let $\delta$ denote the fraction of empty space. Optimal packin…

Let auxiliary tetrahedra be scissor-congruent when they are related by the scissor-congruence symmetry identified in the semiclassical description of the volume spectrum. The assoc…

Let $\mathcal{T}_0$ be a non-isosceles tetrahedron, with associated dynamical system of tetrahedra $(\mathcal{T}_n)$, centroids $(G_n)$, and circumsphere center $O$. Suppose the li…