geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a hexagonal pyramid is a pyramid with a

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...

al base upon which are erected six

triangular A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...

faces that meet at a point (the apex). Like any pyramid, it is self- dual.

Properties

A hexagonal pyramid has seven vertices, twelve edges, and seven faces. One of its faces is

, a base of the pyramid; six others are triangles. Six of the edges make up the pentagon by connecting its six vertices, and the other six edges are known as the ''lateral edges'' of the pyramid, meeting at the seventh vertex called the

apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics) A-Bomb Abomination Absorbing Man Abraxas Abyss Abyss is the name of two characters appearing in Ameri ...

. Like other right pyramids with a regular polygon as a base, this pyramid has

pyramidal symmetry In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object. They are ...

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

C_

: the pyramid is left invariant by rotations of one, two, three, four, and five in six of a full turn around its

axis of symmetry An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...

, 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 In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with vertices can also be defined as the 1-skeleton of an pyramid. Some authors write to denote a wheel graph ...

W_6

; more generally, a wheel graph

W_n

is the representation of the

skeleton A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...

of a sided pyramid. It is self-dual, meaning its

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

is the hexagonal pyramid itself.

References

External links

*{{MathWorld , urlname=HexagonalPyramid , title=Hexagonal Pyramid
Virtual Reality Polyhedra
www.georgehart.com: The Encyclopedia of Polyhedra

Try: "Y6"

Hexagonal pyramid - Polytope Wiki Pyramids (geometry) Self-dual polyhedra Prismatoid polyhedra