Euclidean tilings by convex regular polygons refer to a type of tiling (or tessellation) of the plane in which the entire plane is covered using one or more types of convex regular polygons without overlaps and without leaving any gaps. A convex regular polygon is a polygon that is both convex (all interior angles are less than 180 degrees) and regular (all sides and angles are equal).

Geometrography is a term that isn't widely recognized in established academic or scientific literature, which may lead to variations in interpretation. It seems to combine elements of geometry and geography, possibly referring to the study or representation of geometric aspects within geographical contexts, such as mapping spatial relationships, analyzing geographical data through geometric frameworks, or exploring the geometric properties of landforms and geographical features.

"Napoleon's problem" typically refers to a well-known geometrical problem in mathematics, specifically in the context of triangle geometry.

Pascal's theorem, also known as Pascal's Mystic Hexagram, is a theorem in projective geometry that deals with a hexagon inscribed in a conic section (such as a circle, ellipse, parabola, or hyperbola).

Polygon is a protocol and framework for building and connecting Ethereum-compatible blockchain networks. It seeks to address some of the scalability issues faced by the Ethereum network by enabling the creation of Layer 2 scaling solutions. Originally known as Matic Network, it rebranded to Polygon in early 2021.

A special right triangle is a type of right triangle that has specific, well-defined angle measures and side lengths that can be derived from simple ratios. There are two primary types of special right triangles: 1. **45-45-90 Triangle**: - This triangle has two angles measuring 45 degrees and one right angle (90 degrees). - The sides opposite the 45-degree angles are of equal length.

The term "pendent" can refer to different concepts depending on the context. Here are a couple of common meanings: 1. **In Architecture**: A "pendent" often refers to a decorative feature that is suspended from a structure, such as a pendant light. It can also describe a type of architectural element that protrudes or hangs down from a surface, like a pendant in a domed ceiling.

The British Flag Theorem is a geometric theorem that relates to specific points in a rectangular configuration. It states that for any rectangle \( ABCD \) and any point \( P \) in the plane, the sum of the squared distances from point \( P \) to two opposite corners of the rectangle is equal to the sum of the squared distances from \( P \) to the other two opposite corners.

The distance from a point to a plane in three-dimensional space can be calculated using a specific formula.

The Equal Incircles Theorem is a result in geometry that addresses the relationship between certain triangles and their incircles (the circle inscribed within a triangle that is tangent to all three sides). The theorem states that if two triangles are similar and have the same inradius, then their incircles are equal in size. To clarify in more detail: 1. **Inradius**: The radius of the incircle of a triangle is referred to as its inradius.

In geometry, "expansion" can refer to multiple concepts depending on the context. Here are a few interpretations: 1. **Geometric Expansion**: This often refers to increasing the size of a shape while maintaining its proportions. For example, if you expand a square by a certain factor, you multiply the lengths of its sides by that factor, which increases the area of the square.

It seems there might be a slight confusion in your question. You might be referring to "Haruki Murakami," who is a renowned Japanese author known for his works that blend elements of magical realism, surrealism, and themes of loneliness and existentialism. Some of his most famous novels include "Norwegian Wood," "Kafka on the Shore," and "The Wind-Up Bird Chronicle.

The measurement of a circle involves several key concepts and formulas that describe its dimensions. The primary measurements of a circle include: 1. **Radius (r)**: The distance from the center of the circle to any point on its circumference. 2. **Diameter (d)**: The distance across the circle, passing through the center. The diameter is twice the radius: \[ d = 2r \] 3.

"On Spirals" is a work by the philosopher and cultural critic J.J. (John James) Merrell, exploring the nature of spirals in various contexts, particularly in philosophy, science, art, and architecture. The book delves into how spirals symbolize growth, evolution, and the interconnectedness of different systems or ideas. The concept of spirals can also be metaphorical, representing nonlinear progress or the complexity of experiences in life and thought.

In mathematics and physics, a "root system" refers to a specific structure that arises in the study of Lie algebras, algebraic groups, and other areas such as representation theory and geometry. A root system generally consists of: 1. **Set of Roots**: A root system is a finite set of vectors (called roots) in a Euclidean space that satisfy certain symmetric properties. Each root typically corresponds to some symmetry in a Lie algebra.

A simplicial polytope is a specific type of polytope that is defined in terms of its vertices and faces. More formally, a simplicial polytope is a convex polytope where every face is a simplex. ### Key Characteristics: 1. **Vertices**: A simplicial polytope is described by its vertices. The vertices are points in a multidimensional space (typically in \( \mathbb{R}^n \)).

A two-point tensor, often referred to as a second-order tensor, is a mathematical object that can be represented as a rectangular array of numbers arranged in a 2-dimensional grid. In the context of physics and engineering, tensors are used to describe physical quantities that have multiple components and can occur in various coordinate systems. A two-point tensor typically has two indices, which can be thought of as pairs of values that represent how the tensor transforms under changes in coordinate systems.

Varignon's theorem is a principle in the geometry of polygons that applies specifically to quadrilaterals. It states that the area of a quadrilateral can be determined by considering the midpoints of its sides.

In algebraic topology, the cohomology ring is an important algebraic structure associated with a topological space. It is formed from the cohomology groups of the space, which provide algebraic invariants that help in understanding the topological properties of spaces.

The Eilenberg-Moore spectral sequence is a mathematical construct used in the field of algebraic topology and homological algebra. It arises in the context of homotopical algebra, particularly when dealing with fibred categories and the associated homotopy theoretic situations.