The small dodecicosidodecahedron is one of the Archimedean solids and is classified as a polyhedron. More specifically, it is a convex polyhedral structure that consists of both regular and irregular faces.

The small icosihemidodecacron is a type of convex polyhedron that belongs to the family of Archimedean solids. Specifically, it is one of the deltahedra, which are polyhedra whose faces are all equilateral triangles. The small icosihemidodecacron has 20 faces, which are composed of equilateral triangles, along with 30 edges and 12 vertices.

A square bifrustum is a three-dimensional geometric shape, typically associated with the field of geometry, particularly in the study of polyhedra. It can be understood as a variation of a frustum, which is a portion of a solid (usually a cone or a pyramid) that lies between two parallel planes cutting through it.

The trapezo-rhombic dodecahedron is a type of convex polyhedron that belongs to the category of Archimedean solids. It is characterized by having 12 faces, which are a mix of trapezoids and rhombuses. Specifically, there are 6 trapezoidal faces and 6 rhombic faces.

The Triakis octahedron is a convex polyhedron that can be classified as a type of Archimedean solid. It is derived from the regular octahedron by adding a pyramid to each face of the octahedron, where each pyramid has a triangular base. This construction results in a solid that retains the overall symmetry of the octahedron but has additional vertices, edges, and faces.

The Tridyakis icosahedron is a type of convex polyhedron and a member of the family of Catalan solids. Specifically, it is associated with the dual of the icosahedron, which is a regular polyhedron with 20 triangular faces. The Tridyakis icosahedron itself has a unique structure characterized by its geometry.

A truncated cuboctahedral prism is a three-dimensional geometric shape derived from the cuboctahedral prism, which is itself formed by stacking two truncated octahedral shapes. To break it down further: 1. **Cuboctahedral Prism**: This is a prism whose bases are cuboctahedra.

The truncated great icosahedron is a type of Archimedean solid, which is a category of polyhedra that are highly symmetrical, convex, and composed of regular polygons. Specifically, the truncated great icosahedron can be understood as follows: - **Basic Definition**: It is formed by truncating (cutting off) the vertices of a great icosahedron.

The truncated triakis icosahedron is a convex Archimedean solid, a polyhedron that can be constructed by truncating (or slicing off the corners of) the triakis icosahedron. The triakis icosahedron itself is a non-convex polyhedron that can be thought of as an icosahedron where each triangular face has been replaced by three additional triangular pyramids.

Newton's theorem, often referred to as the "Newton's theorem on ovals," relates to the properties of an oval, particularly in the context of projective geometry and combinatorial geometry. The theorem essentially states that given a set of points in the plane, if these points are located on a smooth convex curve (an oval), then there exists a certain relationship concerning the tangents, secants, and other lines drawn from these points.

The Birkhoff–Grothendieck theorem is a fundamental result in the field of lattice theory and universal algebra. It characterizes the representability of certain types of categories, especially in the context of complete lattice structures. **Statement of the theorem:** The Birkhoff–Grothendieck theorem states that a distributive lattice can be represented as the lattice of open sets of some topological space if and only if it is generated by its finitely generated ideals.

Le Potier's vanishing theorem is a result in algebraic geometry concerning sheaf cohomology on certain types of varieties, specifically on smooth projective varieties. It is particularly concerned with the behavior of cohomology groups of coherent sheaves under the action of the derived category.

Circles are fundamental shapes in geometry, and several important theorems govern their properties and behaviors. Here are some key theorems about circles: 1. **Circumference Theorem**: The circumference \( C \) of a circle is given by the formula: \[ C = 2\pi r \] where \( r \) is the radius of the circle.

Hjelmslev's theorem is a result in the field of projective geometry that relates to the properties of conics (i.e., curves defined by quadratic equations) in projective spaces. Specifically, it addresses the conditions under which a conic in one projective plane can be transformed into an equivalent conic in another projective plane.

The Mohr–Mascheroni theorem is a result in geometry that states that it is possible to construct any length using only a compass, without the need for a straightedge. This theorem is named after the German mathematician Max Mohr and the Italian mathematician Giovanni Mascheroni, who independently proved this result. The theorem can be surprising because traditional geometric constructions often rely on both a compass and a straightedge.

The statement "five points determine a conic" refers to a fundamental result in projective geometry. It states that given any five points in a plane, no three of which are collinear, there exists a unique conic section (which can be an ellipse, parabola, hyperbola, or degenerate conic) that passes through all five points.

Cadaeic Cadenza is a unique and intriguing form of poetry known as a "nullptr" poem, which means the number of letters in each word corresponds to the digits of the number pi (π). Since pi is approximately equal to 3.

The Spiral of Theodorus, also known as the square root spiral or the spiral of square roots, is a mathematical construct that visually represents the square roots of natural numbers. It is named after the ancient Greek mathematician Theodorus of Cyrene, who is credited with demonstrating the irrationality of the square roots of non-square integers.

Spatiotemporal gene expression refers to the regulation of gene expression in both space (spatial) and time (temporal) within a biological system. This concept emphasizes that genes are not expressed uniformly throughout an organism or tissue at all times; rather, their expression can vary based on location and developmental stage. ### Key Aspects of Spatiotemporal Gene Expression: 1. **Spatial Variation**: - Gene expression can differ in various tissues, organs, or cellular compartments.

Grid cells are specialized types of neurons found in the entorhinal cortex of the brain, particularly involved in spatial navigation and the cognitive mapping of the environment. They play a crucial role in providing a metric for spatial navigation, helping to create a coordinate system that allows for the representation of space. Key characteristics of grid cells include: 1. **Hexagonal Grid Pattern**: The firing pattern of grid cells forms a hexagonal grid.