Augmented hexagonal prism

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Augmented hexagonal prism
Augmented hexagonal prism.png
Type Johnson
J53J54J55
Faces 4 triangles
5 squares
2 hexagons
Edges 22
Vertices 13
Vertex configuration 2x4(42.6)
1(34)
4(32.4.6)
Symmetry group C2v
Dual polyhedron monolaterotruncated hexagonal bipyramid
Properties convex
Net
Johnson solid 54 net.png

In geometry, the augmented hexagonal prism is one of the Johnson solids (J54). As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid (J1) to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism (J55), a metabiaugmented hexagonal prism (J56), or a triaugmented hexagonal prism (J57).

Contents

Construction

The augmented hexagonal prism is constructed by attaching one equilateral square pyramid onto the square face of a hexagonal prism, a process known as augmentation. [1] This construction involves the removal of the prism square face and replacing it with the square pyramid, so that there are eleven faces: four equilateral triangles, five squares, and two regular hexagons. [2] A convex polyhedron in which all of the faces are regular is a Johnson solid, and the augmented hexagonal prism is among them, enumerated as . [3] Relatedly, two or three equilateral square pyramids attaching onto more square faces of the prism give more different Johnson solids; these are the parabiaugmented hexagonal prism , the metabiaugmented hexagonal prism , and the triaugmented hexagonal prism . [1]

Properties

An augmented hexagonal prism with edge length has surface area [2] the sum of two hexagons, four equilateral triangles, and five squares area. Its volume [2] can be obtained by slicing into one equilateral square pyramid and one hexagonal prism, and adding their volume up. [2]

It has an axis of symmetry passing through the apex of a square pyramid and the centroid of a prism square face, rotated in a half and full-turn angle. Its dihedral angle can be obtained by calculating the angle of a square pyramid and a hexagonal prism in the following: [4]

Related Research Articles

In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians Norman Johnson and Victor Zalgaller.

<span class="mw-page-title-main">Gyroelongated square bipyramid</span> 17th Johnson solid

In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.

<span class="mw-page-title-main">Triaugmented triangular prism</span> Convex polyhedron with 14 triangle faces

The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron, composite polyhedron, and Johnson solid.

<span class="mw-page-title-main">Gyroelongated square pyramid</span> 10th Johnson solid (13 faces)

In geometry, the gyroelongated square pyramid is the Johnson solid that can be constructed by attaching an equilateral square pyramid to a square antiprism. It occurs in chemistry; for example, the square antiprismatic molecular geometry.

<span class="mw-page-title-main">Square pyramid</span> Pyramid with a square base

In geometry, a square pyramid is a pyramid with a square base, having a total of five faces. If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral. It is called an equilateral square pyramid, an example of a Johnson solid.

<span class="mw-page-title-main">Gyroelongated square bicupola</span> 45th Johnson solid

In geometry, the gyroelongated square bicupola is the Johnson solid constructed by attaching two square cupolae on each base of octagonal antiprism. It has the property of chirality.

<span class="mw-page-title-main">Elongated triangular pyramid</span> Polyhedron constructed with tetrahedra and a triangular prism

In geometry, the elongated triangular pyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically self-dual.

<span class="mw-page-title-main">Elongated square pyramid</span> Polyhedron with cube and square pyramid

In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid.

<span class="mw-page-title-main">Elongated triangular bipyramid</span> 14th Johnson solid; triangular prism capped with tetrahedra

In geometry, the elongated triangular bipyramid or triakis triangular prism a polyhedron constructed from a triangular prism by attaching two tetrahedrons to its bases. It is an example of Johnson solid.

<span class="mw-page-title-main">Elongated square bipyramid</span> Cube capped by two square pyramids

In geometry, the elongated square bipyramid is the polyhedron constructed by attaching two equilateral square pyramids onto a cube's faces that are opposite each other. It can also be seen as 4 lunes linked together with squares to squares and triangles to triangles. It is also been named the pencil cube or 12-faced pencil cube due to its shape.

<span class="mw-page-title-main">Elongated pentagonal bipyramid</span> 16th Johnson solid; pentagonal prism capped by pyramids

In geometry, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two pentagonal pyramids onto the base of a pentagonal prism. It is an example of Johnson solid.

<span class="mw-page-title-main">Augmented triangular prism</span> 49th Johnson solid

In geometry, the augmented triangular prism is a polyhedron constructed by attaching an equilateral square pyramid onto the square face of a triangular prism. As a result, it is an example of Johnson solid. It can be visualized as the chemical compound, known as capped trigonal prismatic molecular geometry.

<span class="mw-page-title-main">Biaugmented triangular prism</span> 50th Johnson solid

In geometry, the biaugmented triangular prism is a polyhedron constructed from a triangular prism by attaching two equilateral square pyramids onto two of its square faces. It is an example of Johnson solid. It can be found in stereochemistry in bicapped trigonal prismatic molecular geometry.

<span class="mw-page-title-main">Augmented pentagonal prism</span> 52nd Johnson solid

In geometry, the augmented pentagonal prism is a polyhedron that can be constructed by attaching an equilateral square pyramid onto the square face of pentagonal prism. It is an example of Johnson solid.

<span class="mw-page-title-main">Biaugmented pentagonal prism</span> 53rd Johnson solid

In geometry, the biaugmented pentagonal prism is a polyhedron constructed from a pentagonal prism by attaching two equilateral square pyramids onto each of its square faces. It is an example of Johnson solid.

<span class="mw-page-title-main">Elongated triangular cupola</span> Polyhedron with triangular cupola and hexagonal prism

In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.

<span class="mw-page-title-main">Elongated triangular orthobicupola</span> Johnson solid with 20 faces

In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.

<span class="mw-page-title-main">Elongated triangular gyrobicupola</span> 36th Johnson solid

In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in . It is an example of Johnson solid.

<span class="mw-page-title-main">Augmented truncated tetrahedron</span> 65th Johnson solid

In geometry, the augmented truncated tetrahedron is a polyhedron constructed by attaching a triangular cupola onto an truncated tetrahedron. It is an example of a Johnson solid.

<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

References

  1. 1 2 Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 8489. doi:10.1007/978-93-86279-06-4. ISBN   978-93-86279-06-4.
  2. 1 2 3 4 Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR   0290245.
  3. Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
  4. Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics . 18: 169–200. doi: 10.4153/cjm-1966-021-8 . MR   0185507. S2CID   122006114. Zbl   0132.14603.