Hexagonal prism

Properties

A hexagonal prism has twelve vertices, eighteen edges, and eight faces. Every prism has two faces known as its bases, and the bases of a hexagonal prism are hexagons. The hexagons has six vertices, each of which pairs with another hexagon's vertex, forming six edges. These edges form three parallelograms as other faces. A prism is said to be right if the edges are of the same length and perpendicular to the base.

If faces are all regular, the hexagonal prism is a semiregular polyhedron—more generally, a uniform polyhedron—and the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t{2,6}. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product {6}×{}. The symmetry group of a right hexagonal prism is prismatic symmetry D_{6 \mathrm{h}} of order 24, consisting of rotation around an axis passing through the regular hexagon bases' center, and reflection across a horizontal plane.{{citation

As in most prisms, the volume is found by taking the area of the base, with a side length of a , and multiplying it by the height h, giving the formula: V = \frac{3 \sqrt{3}}{2}a^2h, and its surface area is by summing the area of two regular hexagonal bases and the lateral faces of six squares: S = 3a(\sqrt{3}a+2h).

Honeycombs

The hexagonal prism is one of the parallelohedra, a polyhedral class that can be translated without rotations in Euclidean space, producing honeycombs; this class was discovered by Evgraf Fedorov in accordance with his studies of crystallography systems. The hexagonal prism is generated from four line segments, three of them parallel to a common plane and the fourth not. Its most symmetric form is the right prism over a regular hexagon, forming the hexagonal prismatic honeycomb.{{citation

The hexagonal prism also exists as cells of four prismatic uniform convex honeycombs in 3 dimensions:

It also exists as cells of a number of four-dimensional uniform 4-polytopes, including:

References

References

1. Pugh, Anthony. (1976). "Polyhedra: A Visual Approach". University of California Press.
2. Wheater, Carolyn C.. (2007). "Geometry". Career Press.
3. Alexandrov, A. D.. (2005). "Convex Polyhedra". Springer.