Space diagonal

Last updated
AC' (shown in blue) is a space diagonal, while AC (shown in red) is a face diagonal. Cube diagonals.svg
AC' (shown in blue) is a space diagonal, while AC (shown in red) is a face diagonal.

In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals , which connect vertices on the same face (but not on the same edge) as each other. [1]

Contents

For example, a pyramid has no space diagonals, while a cube (shown at right) or more generally a parallelepiped has four space diagonals.

Axial diagonal

An axial diagonal is a space diagonal that passes through the center of a polyhedron.

For example, in a cube with edge length a, all four space diagonals are axial diagonals, of common length More generally, a cuboid with edge lengths a, b, and c has all four space diagonals axial, with common length

A regular octahedron has 3 axial diagonals, of length , with edge length a.

A regular icosahedron has 6 axial diagonals of length , where is the golden ratio . [2]

Space diagonals of magic cubes

A magic square is an arrangement of numbers in a square grid so that the sum of the numbers along every row, column, and diagonal is the same. Similarly, one may define a magic cube to be an arrangement of numbers in a cubical grid so that the sum of the numbers on the four space diagonals must be the same as the sum of the numbers in each row, each column, and each pillar.

See also

Related Research Articles

<span class="mw-page-title-main">Cube</span> Solid object with six equal square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.

<span class="mw-page-title-main">Golden ratio</span> Ratio between two quantities whose sum is at the same ratio to the larger one

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with ,

<span class="mw-page-title-main">Regular icosahedron</span> Polyhedron with 20 regular triangular faces

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

<span class="mw-page-title-main">Octahedron</span> Polyhedron with eight triangular faces

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

<span class="mw-page-title-main">Tetrahedron</span> Polyhedron with 4 faces

In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra.

<span class="mw-page-title-main">Hexagon</span> Shape with six sides

In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

<span class="mw-page-title-main">Diagonal</span> In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron

In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios, "from angle to angle" ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus.

In mathematics, a magic hypercube is the k-dimensional generalization of magic squares and magic cubes, that is, an n × n × n × ... × n array of integers such that the sums of the numbers on each pillar (along any axis) as well as on the main space diagonals are all the same. The common sum is called the magic constant of the hypercube, and is sometimes denoted Mk(n). If a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number

<span class="mw-page-title-main">5-cell</span> Four-dimensional analogue of the tetrahedron

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

<span class="mw-page-title-main">Rhombic triacontahedron</span> Catalan solid with 30 faces

In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

<span class="mw-page-title-main">Triakis icosahedron</span> Catalan solid with 60 faces

In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's Divina proportione, where it was named the icosahedron elevatum. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron.

<span class="mw-page-title-main">Bilunabirotunda</span> 91st Johnson solid (14 faces)

In geometry, the bilunabirotunda is one of the Johnson solids. A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra. They were named by Norman Johnson, who first listed these polyhedra in 1966.

Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics.

<span class="mw-page-title-main">Midsphere</span> Sphere tangent to every edge of a polyhedron

In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.

<span class="mw-page-title-main">Regular dodecahedron</span> Polyhedron with 12 regular pentagonal faces

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.

<span class="mw-page-title-main">Medial rhombic triacontahedron</span> Polyhedron with 30 faces

In geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.

<span class="mw-page-title-main">Bilinski dodecahedron</span> Polyhedron with 12 congruent golden rhombus faces

In geometry, the Bilinski dodecahedron is a convex polyhedron with twelve congruent golden rhombus faces. It has the same topology but a different geometry than the face-transitive rhombic dodecahedron. It is a parallelohedron.

References

  1. William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.116
  2. Sutton, Daud (2002), Platonic & Archimedean Solids, Wooden Books, Bloomsbury Publishing USA, p. 55, ISBN   9780802713865 .