13 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…

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…

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.