A small hexagonal hexecontahedron is a polyhedron that is classified as a member of the family of convex polyhedra. Specifically, it is a type of Archimedean solid. The term "hexecontahedron" indicates that it has 60 faces. In the case of the small hexagonal hexecontahedron, these faces include hexagons and other polygons.

The snub icosidodecadodecahedron is a fascinating geometric shape that belongs to the category of Archimedean solids. It is a complex polyhedron characterized by its unique combination of faces, vertices, and edges. ### Key Features: - **Faces**: The snub icosidodecadodecahedron has 62 faces, 12 of which are regular pentagons and 50 are equilateral triangles.

The snub square antiprism is a type of Archimedean solid, which is a convex polyhedron that has identical vertices and faces that are regular polygons. Specifically, the snub square antiprism can be described as a modification of the square antiprism. It has the following characteristics: - **Faces**: The snub square antiprism has 38 faces in total, consisting of 8 triangles and 30 squares.

A square orthobicupola is a type of polyhedron that belongs to the category of Archimedean solids. Specifically, it is formed by the combination of two square cupolas and has a unique geometric configuration. ### Features of the Square Orthobicupola: 1. **Faces**: The square orthobicupola has a total of 24 faces. These consist of: - 8 square faces - 16 triangular faces 2.

A triangular bifrustum is a three-dimensional geometric shape that is essentially formed by truncating the top and bottom of a triangular prism. Specifically, it consists of two parallel triangular bases—one larger than the other—and three rectangular lateral faces that connect the corresponding sides of the two triangular bases.

A triangular hebesphenorotunda is a type of convex polyhedron, which belongs to a specific category of Archimedean solids. To understand it better, it can be described as a truncated version of a triangular prism combined with the properties of other geometric shapes. Here's a breakdown of the name: - **Triangular:** This refers to the shape of the base, specifically that it is a triangle.

The triaugmented dodecahedron is a geometric shape that is categorized as an Archimedean solid. It is formed by augmenting a regular dodecahedron (which has 12 faces, each a regular pentagon) with three additional pyramidal structures.

A trigonal trapezohedron is a type of polyhedron that has specific characteristics and belongs to the category of trapezohedra. It has 6 faces, each of which is a kite shape. The vertices of a trigonal trapezohedron correspond to the faces of a triangular bipyramid. The trigonal trapezohedron can be thought of as a convex polyhedron that has: - **Faces**: 6 faces, all of which are congruent kites.

The truncated square antiprism is a type of convex polyhedron that belongs to the family of Archimedean solids. It can be described as a modification of the square antiprism, which is an 8-faced solid formed by two square bases that are connected by eight triangular lateral faces. In the truncated version, each of the vertices of the square antiprism is truncated (or cut off), resulting in additional faces.

A truncated trapezohedron is a type of Archimedean solid derived from the trapezohedron, which itself is a 3D shape with trapezoidal faces. Specifically, a truncated trapezohedron results from truncating (cutting off) the vertices of the original trapezohedron. The geometry of a truncated trapezohedron features a combination of polygons as its faces—specifically, in this case, it will include hexagonal and quadrilateral faces.

Fenchel's theorem, often referred to in the context of convex analysis, deals with the correspondence between the convex functions and their subgradients. Specifically, it provides a characterization of convex functions through their conjugate functions.

The Appell–Humbert theorem is a result in the theory of complex numbers and multidimensional analysis. It relates to the behavior of certain classes of functions, particularly those that are harmonic or analytic. The theorem states conditions for when a function can be expressed as a series of its values on a certain domain.

Hurwitz's automorphisms theorem is a result in the field of group theory and topology, particularly in the study of Riemann surfaces and algebraic curves. It deals with the automorphisms of compact Riemann surfaces and their relationship to the structure of these surfaces.

The Midpoint Theorem in the context of conics, specifically concerning ellipses, refers to a property related to the midpoints of line segments connecting points on the ellipse. While the term "Midpoint Theorem" can also be associated with other geometrical contexts, such as triangles, in the realm of conics, it is often used to describe certain relationships and properties referring to the midpoints of chords.

Hesse's theorem is a result in geometry that deals with the properties of projective spaces. Specifically, it states that if you have a configuration of points in a projective plane, under certain conditions, the points will lie on a conic (a curve defined by a quadratic polynomial). In a more precise sense, the theorem can be framed in terms of the collinearity of points and the conditions under which these points create a conic.

The Veblen–Young theorem is a result in set theory and topology that pertains to the structure of certain well-ordered sets and their properties. It is primarily focused on the relationship between well-ordered sets and their representations as ordinals, specifically in the context of a well-ordered set being isomorphic to an ordinal if it exhibits certain properties.

The Indiana Pi Bill, formally known as House Bill 246, was a piece of legislation introduced in the Indiana General Assembly in 1897. It is famously associated with an attempt to define the mathematical constant π (pi) in a way that was not consistent with its actual mathematical properties. The bill proposed to establish an incorrect value of pi as 3.2, among other erroneous definitions related to geometry.