Pentagonal pyramid explained

Type:Pyramid
Johnson
Edges:10
Vertices:6
Symmetry:

C5v

Vertex Config:

5 x (32 x 5)+1 x 35

Angle:As a Johnson solid:
Properties:convex,
elementary (Johnson solid)
Net:frameless

In geometry, a pentagonal pyramid is a pyramid with a pentagon base and five triangular faces, having a total of six faces. It is categorized as a Johnson solid if all of the edges are equal in length, forming equilateral triangular faces and a regular pentagonal base. Pentagonal pyramids occur as pieces and tools in the construction of many polyhedra. They also appear in stereochemistry, specifically in pentagonal pyramidal molecular geometry.

Properties

A pentagonal pyramid has six vertices, ten edges, and six faces. One of its faces is pentagon, a base of the pyramid; five others are triangles. Five of the edges make up the pentagon by connecting its five vertices, and the other five edges are known as the lateral edges of the pyramid, meeting at the sixth vertex called the apex. A pentagonal pyramid is said to be regular if its base is circumscribed in a circle that forms a regular pentagon, and it is said to be right if its altitude is erected perpendicularly to the base's center.

C5v

: the pyramid is left invariant by rotations of one, two, three, and four in five of a full turn around its axis of symmetry, the line connecting the apex to the center of the base. It is also mirror symmetric relative to any perpendicular plane passing through a bisector of the base. It can be represented as the wheel graph

W5

, meaning its skeleton can be interpreted as a pentagon in which its five vertices connects a vertex in the center called the universal vertex. It is self-dual, meaning its dual polyhedron is the pentagonal pyramid itself.

J2

. The dihedral angle between two adjacent triangular faces is approximately 138.19° and that between the triangular face and the base is 37.37°. It is an elementary polyhedron, meaning that it cannot be separated by a plane to create two small convex polyhedrons with regular faces. A polyhedron's surface area is the sum of the areas of its faces. Therefore, the surface area of a pentagonal pyramid is the sum of the areas of the four triangles and the one pentagon. The volume of every pyramid equals one-third of the area of its base multiplied by its height. So, the volume of a pentagonal pyramid is one-third of the product of the height and a pentagonal pyramid's area. In the case of Johnson solid with edge length

a

, its surface area

A

and volume

V

are: \begin A &= \frac\sqrt \approx 3.88554a^2, \\ V &= \frac a^3 \approx 0.30150a^3.\end

Applications

J9

, a gyroelongated pentagonal pyramid

J11

, a pentagonal bipyramid

J13

, an elongated pentagonal bipyramid

J16

, an augmented dodecahedron

J58

, a parabiaugmented dodecahedron

J59

, a metabiaugmented dodecahedron

J60

, and a triaugmented dodecahedron

J61

.[1] Relatedly, the removal of a pentagonal pyramid from polyhedra is an example of a technique known as diminishment; the metabidiminished icosahedron

J62

and tridiminished icosahedron

J63

are the examples in which their constructions begin by removing pentagonal pyramids from a regular icosahedron.

In stereochemistry, an atom cluster can have a pentagonal pyramidal geometry. This molecule has a main-group element with one active lone pair of electrons, which can be described by a model that predicts the geometry of molecules known as VSEPR theory. An example of a molecule with this structure include nido-cage carbonate CB5H9.

A study of applied the pentagonal (as well as the hexagonal pyramids) to demonstrate the shell assembly in the underlying potential energy surfaces. To measure such energy, the pyramids represent the building blocks for an icosahedral-shaped shell virus capsid by corresponding to the protein subunits assembled into the symmetries. The subsequent step is taking the interaction of two pyramids determined by the distance between the pyramids' apices, each involving the Lennard - Jones potential. Summarizing the conclusion, the assembly shell is possible through the changes of Lennard - Jones parameterization and the high energy consisting of pyramids.

References

Works cited

External links

Notes and References

  1. , pp. 84 - 88. See Table 12.3, where

    Pn

    denotes the prism and

    An

    denotes the antiprism.

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