# Supplementary mathematics/Polyhedron

A polyhedron is a solid geometric object in three-dimensional space that has smooth and regular faces (each face in one plane) and sides or edges located on a straight line. So far, no single definition has been provided for it. A tetrahedron is a type of pyramid and a cube is an example of a hexagon. A polyhedron can be convex or non-convex. Polyhedrons such as pyramids and prisms can be made by extruding two-dimensional polygons. There can only be a finite number of convex polyhedra with regular faces and equiangular shapes, including Platonic solids and Archimedean solids. Some Archimedean solids can be made by cutting the top pyramid of Platonic solids. Due to the simplicity of construction, polyhedra are used in most architectural works such as geodesic domes and pyramids. Recently, due to the use of shapes, interest in multifaceted surfaces has increased. Some compact molecules and atoms, especially crystalline structures and Platonic hydrocarbons, as well as some radials have a shape similar to Platonic solids. Platonic solids are also used in making dice. Polyhedra have different characteristics and types and are placed in different symmetry groups. Other polyhedra can be created by operations on any polyhedra. Some of them have relationships with each other. Polyhedra have been of interest since the Stone Age. The sphere is also considered as a family of polyhedra. Cube, tetrahedron, parallelogram are geometric volumes that are also considered polyhedra.

## Definitions

Convex polyhedra are defined and convex polyhedra themselves are well-defined and can be calculated volume and area and can be used except for geometric volumes. But concave polyhedra are non-geometric volumes, and their definition is difficult and very difficult, and they do not have constant area and volume formulas. Geometric and non-geometric volumes are of the type of polyhedra, but their difference is in their concavity and convexity.
From these definitions, the following can be mentioned:
* A common and somewhat simple definition of a polyhedron is: a solid object whose outer surfaces can be covered with a large number of faces, or a solid formed by the union of convex polyhedra. A natural extension of this definition requires that the solid in question is bounded, its interior and possibly its boundary also connected. The faces of such a polyhedron can be defined as the connected space of the boundary parts inside each of the planes that cover it, and their sides and vertices as line segments and points where the faces meet. However, a polyhedron defined in this way does not include star-crossed polyhedrons whose faces may not form simple polygons and some of whose sides belong to more than two faces.
* Definitions based on the idea of a limiting surface rather than a solid are also common. For example, O'Rourke (1993) defines a polyhedron as a collection of convex polygons (its faces). These polygons are arranged in space in such a way that the intersection (or sharing) of both polygons is a common vertex or side or the null set, so that their union is a manifold. If a planar part of such a surface is not itself a convex polygon, O'Rourke stipulates that the part must be divided into pieces, each of which is a smaller convex polygon, such that the dihedral angles between them are flat. Somewhat more generally, Branko Grünbaum defines a polyhedron as a set of simple polygons that form an embedded manifold, with each vertex reached by at least three sides and any two faces only at vertices and sides common to each. They intersect. Cromwell's book on polyhedra gives a similar definition but without the restriction of at least three sides per vertex. Again, this type of definition does not include intersecting polyhedra. Similar concepts form the basis of topological definitions of polyhedra, as subsets of a topological manifold into topological disks (faces) whose binary intersections are points (vertices), topological arcs (sides). or the set is empty. However, there are topological polyhedra (even with perfect triangles) that cannot be understood as geometric polyhedra.
* A more modern definition based on the theory of single polyhedra is also prevalent. These polyhedra can be defined as sets with partial order, such that their elements are vertices, sides and faces of a polyhedron. When a vertex or side is smaller than a side or face, an element of the vertex or side is less than the element of the side or face (thus minor). Furthermore, there may be one special lower element of this partial order (denoting the null set) and one upper element representing the whole polyhedron. If the partial order segments between the elements of three faces apart (i.e., between each face and the bottom element and between the top element and each vertex) have the same structure as the abstract representation of a polygon, then these ordered sets contain exactly the same information as They carry topological polyhedra. However, these requirements are often permissive, instead requiring only that the cross-sections between elements of two faces from each other have the same structure as the abstract representation of a line segment. (This means that each side contains two vertices and belongs to two faces, and each vertex in a face belongs to two sides of that face.) A geometric polyhedron, defined in other ways, can be abstracted in this way. be described, but abstract polyhedra can also be used as a basis for defining geometric polyhedra. The realization of an abstract polyhedron is generally considered as a mapping of the vertices of the abstract polyhedron to geometric points, such that the vertices of each face are coplanar. So, a geometric polyhedron can be defined as the realization of an abstract polyhedron. Realizations that remove the flatness requirement, impose additional symmetry requirements, or map the vertices to higher dimensional spaces are also considered. The latter definition, in contrast to the definitions based on solids and processes, is quite suitable for star polyhedra. However, without additional restrictions, this definition allows the construction of polyhedra or unfaithful polyhedra (for example, by mapping all vertices to a single point) and the question: "How do we constrain the realization of some of these to avoid these polyhedra?" be prevented?" also remains unresolved.
* In all these definitions, a polytope can be understood as a three-dimensional instance of a more general polytope in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a tetrapolytope has a four-dimensional body and an additional set of three-dimensional "cells". However, some texts on higher-dimensional geometry use the term "polytope" to mean something else: not a three-dimensional polytope, but a shape that is somehow different from a polytope. For example, some sources define a convex polygon as the intersection of many half-spaces and a polytope as a bounded polytope. In this article, only 3D polyhedra are discussed.

### Angles

Flat angle: Each of the corner angles of the faces of polyhedral polygons is called a flat angle. Space angle: Each of the angles that a polyhedron covers on a vertex in three-dimensional space is called a space angle. Each of these angles is bounded by three or more than three right angle angles. Dihedral angle: Any angle between two polyhedral faces is called a dihedral angle.

### polyhedral surface

A "polyhedral surface" is the result of joining a limited number of flat polygonal faces and does not necessarily enclose space. Polyhedral surfaces can have boundary sides and boundary vertices (only when the polyhedral surface contains only one face).
Recently, due to the use of shapes, the interest in multifaceted surfaces in architecture has increased.

## Preliminaries

   Preliminaries are small things that we define simply.

### face

In Polyhedra, 'Face is any of Polygons having The area is"$^1$" which forms part of the boundary of a solid body. A three-dimensional solid bounded exclusively by faces is a polyhedron.
In more technical methods of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).

### vertex

"Vertex" (Arabic: "Ras") (English: vertex) in geometry is a point where two straight sides of an open or closed polygon meet. In other words, the vertex is the tip of the corners or the intersection of the lines of a geometric shape. A line is formed by connecting two vertices to each other, and a surface is formed by connecting three vertices to each other.
In 3D computer graphics models, vertices are usually used to define surfaces (usually triangles), and each vertex in these models is represented as a vector. In graph theory, vertices are also called nodes.
The number of vertices of any polygon in the plane is equal to the number of its sides.

### side

In geometry, a "side" or "line" or "edge" is a line segment that connects two adjacent vertices in a polygon; So in practice, a side is the interface for a one-dimensional line segment and two zero-dimensional objects.
Sides are the lines that make up each shape, and their number is often different in each shape compared to other shapes. For example, triangles have 3 sides and squares and rectangles have 4 sides.
A planar closed sequence of sides forms a polygon (and a face). In a polyhedron, exactly two faces touch each other on each side, while in higher dimensional polyhedra, three or more faces touch each other on each side.$^2$ sup>

### angles

A flat angle is the corner angle of a polygonal face.
A solid angle is an angle in three-dimensional space that covers a polyhedron on a vertex. This angle is enclosed by three or more than 3 solid angles.
Dihedral angle is the angle between two adjacent faces
$^{1.Thelateralfaceshaveasurfacearea}$
$^{2.Contactmeansbecausetheyhaveacommonvertex.}$

## FEATURES AND CHARACTERISTICS
### Number of multifaceted funds

Polyhedrons are classified and named based on the number of their faces and based on classical Greek; For example, tetrahedron means a polyhedron with four faces, pentahedron means a polyhedron with five faces, hexahedron means a polyhedron with six faces, and so on.

### interior angle of polyhedron

Polyhedra are made from regular polygons. Like polygons, polyhedra have internal and external angles.
The internal angle of polyhedra is obtained based on the number of faces and sides of the vertex.
For example, a tetrahedron has four equilateral triangles, and the interior angle of its face is 60 degrees, and the sum of its interior angles is 180 degrees. But a tetrahedron has a total interior angle of 720 degrees.
So the sum of the internal angle and the size of the internal angle are written accordingly.
$180n[(n'-2)]$
${\frac {180}{n'}}n[(n'-2)]$

### Shape and corners

For each vertex, a corner shape can be defined, which defines the shape of the polyhedron around the vertex. The exact definitions are variable, but a corner shape can be defined as a shape that is created by cutting the vertex of a polyhedron. If the polygon resulting from this process is regular, the vertex is considered regular.

#### vertex symbol

A vertex symbol or vertex configuration is a shorthand symbol for representing the corner shape of a polyhedron or tiling as a sequence of faces around a vertex. For uniform polyhedra there is only one type of corner shape and thus the vertex configuration completely defines the polyhedra.
The vertex symbol is represented as a sequence of numbers that represent the number of sides of the faces around the vertex. The notation "a.b.c" describes a vertex that has 3 faces around it, with sides a, b, and c.

#### Face configuration

Uniform binomials that are face symmetric can be represented by the same abbreviations as the vertex configuration, which is called the face configuration. These symbols are indicated by a V for difference. This symbol is defined as a consecutive count of the number of faces that are located at the vertices around the face. For example, the dodecahedron face configuration is V3,4,3,4 or $^2$(3,4)V.

### volume

Polyhedral solids have a specific value called volume, which measures the amount of space they occupy. Simple families of polyhedra may have simple formulas for their volumes. For example, the volume of pyramids, prisms, and parallelograms can be easily expressed in terms of side lengths or other specifications.
The volume of more complex polyhedra may not have simple formulas. By dividing the polyhedron into smaller parts, the volume of these polyhedrons is calculated. For example, the volume of a regular polyhedron can be calculated by dividing it into equal pyramids, such that each pyramid has one face of the polyhedron as its base and the center of the polyhedron as its vertex.
In general, it can be derived from the divergence theorem that the volume of a polyhedral solid is given by:
${\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {area} (F)\right|,$
where the sum over the faces of F is the polyhedron, is an arbitrary point on the face of F, is the unit vector perpendicular to F to the outside of the polyhedron, and is the point of multiplication of the inner product.

### area

The area of regular polyhedra has a surface area. The faces of regular polyhedra are regular polygons. Fixed polyhedra such as prisms, pyramids, and parallelograms have constant areas. It is regular, it is obtained based on the sum of the polygonal area and the lateral area of the prism (polygonal area x height).
Area of the polyhedron:$n[{\frac {1}{4}}n'a^{2}cot({\frac {\pi }{n'}})]$.
n here is the number of faces and 'n is the number of sides of the polygon, here the number π is in radians

### Ashlefly symbol

Schleffle notation is a notation for regular polytopes, including regular polytopes.
Regular polyhedra are denoted as {p,q}, where q is the Schleffle symbol of the corners and p is the Schleffle symbol of the polygon of each face.
The Schleffle symbol is a convex regular polygon of the form {p}, where p is the number of sides. The Schleffle symbol of a regular concave (star) polygon is in the form {p/q}, where p is the number of vertices and q is the number of sides between two vertices in connecting the vertices of the regular convex polygon to make it.
Polyhedra whose Schleffle symbols are similar are duals of each other.