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Hexahedron

In the article that follows, we will delve into the fascinating world of Hexahedron. From its impact on current society to its historical origins, through its multiple applications in different areas, we will immerse ourselves in an exhaustive analysis that will allow us to fully understand the importance of Hexahedron in our daily lives. With interviews with experts, relevant data and case studies, this article aims to shed light on a fundamental aspect that has marked a milestone in world history. Join us on this tour of Hexahedron and discover everything behind this phenomenon that is so relevant today.

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A hexahedron (pl.: hexahedra or hexahedrons) or sexahedron (pl.: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.

There are seven topologically distinct convex hexahedra,[1] one of which exists in two mirror image forms. Additional non-convex hexahedra exist, with their number depending on how polyhedra are defined. Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.

Convex

Cuboid

A hexahedron that is combinatorially equivalent to a cube may be called a cuboid, although this term is often used more specifically to mean a rectangular cuboid, a hexahedron with six rectangular sides. A cuboid possess 8 vertices, 6 faces and 12 edges. Different types of cuboids include the ones depicted and linked below.

Cuboids
Cube
(square) 		Rectangular cuboid
(three pairs of
rectangles) 		Trigonal trapezohedron
(congruent rhombi) 		Trigonal trapezohedron
(congruent quadrilaterals) 		Quadrilateral frustum
(apex-truncated
square pyramid) 		Parallelepiped
(three pairs of
parallelograms) 		Rhombohedron
(three pairs of
rhombi)

Others

There are seven topologically distinct convex hexahedra,[1] the cuboid and six others, which are depicted below. One of these is chiral, in the sense that it cannot be deformed into its mirror image.

Image
Name Triangular bipyramid Pentagonal pyramid Doubly truncated tetrahedron[2]
Features
  • 5 vertices
  • 9 edges
  • 6 triangles
  • 6 vertices
  • 10 edges
  • 4 triangles
  • 2 quadrilaterals
  • 6 vertices
  • 10 edges
  • 5 triangles
  • 1 pentagon
  • 7 vertices
  • 11 edges
  • 2 triangles
  • 4 quadrilaterals
  • 7 vertices
  • 11 edges
  • 3 triangles
  • 2 quadrilaterals
  • 1 pentagon
  • 8 vertices
  • 12 edges
  • 2 triangles
  • 2 quadrilaterals
  • 2 pentagons
Properties Simplicial
Dome

Concave

Three further topologically distinct hexahedra can only be realised as concave acoptic polyhedra. These are defined as the surfaces formed by non-crossing simple polygon faces, with each edge shared by exactly two faces and each vertex surrounded by a cycle of three or more faces.[3]

Concave
4.4.3.3.3.3 Faces
10 E, 6 V 		5.5.3.3.3.3 Faces
11 E, 7 V 		6.6.3.3.3.3 Faces
12 E, 8 V

These cannot be convex because they do not meet the conditions of Steinitz's theorem, which states that convex polyhedra have vertices and edges that form 3-vertex-connected graphs.[4] For other types of polyhedra that allow faces that are not simple polygons, such as the spherical polyhedra of Hong and Nagamochi, more possibilities exist.[5]

References

  1. ^ a b Dillencourt, Michael B. (1996), "Polyhedra of small order and their Hamiltonian properties", Journal of Combinatorial Theory, Series B, 66 (1): 87–122, doi:10.1006/jctb.1996.0008, MR 1368518
  2. ^ Kolpakov, Alexander; Murakami, Jun (2013), "Volume of a doubly truncated hyperbolic tetrahedron", Aequationes Mathematicae, 85 (3): 449–463, arXiv:1203.1061, doi:10.1007/s00010-012-0153-y, MR 3063880
  3. ^ Grünbaum, Branko (1999), "Acoptic polyhedra" (PDF), Advances in discrete and computational geometry (South Hadley, MA, 1996), Contemporary Mathematics, vol. 223, Providence, Rhode Island: American Mathematical Society, pp. 163–199, doi:10.1090/conm/223/03137, ISBN 978-0-8218-0674-6, MR 1661382; for the three non-convex acoptic hexahedra see p. 7 of the preprint version and Fig. 3, p. 30
  4. ^ Ziegler, Günter M. (1995), "Chapter 4: Steinitz' Theorem for 3-Polytopes", Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, pp. 103–126, ISBN 0-387-94365-X
  5. ^ Hong, Seok-Hee; Nagamochi, Hiroshi (2011), "Extending Steinitz's theorem to upward star-shaped polyhedra and spherical polyhedra", Algorithmica, 61 (4): 1022–1076, doi:10.1007/s00453-011-9570-x, MR 2852056


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