Editing Space diagonal

{{Short description|Line connecting two vertices that are not on the same face}}
[[Image:Cube diagonals.svg|thumb|right|250px|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 [[vertex (geometry)|vertices]] that are not on the same [[face (geometry)|face]]. Space diagonals contrast with ''[[face diagonal]]s'', which connect vertices on the same face (but not on the same [[edge (geometry)|edge]]) as each other.<ref>William F. Kern, James R Bland,''Solid Mensuration with proofs'', 1938, p.116</ref>

For example, a [[pyramid (geometry)|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 <math>a\sqrt {3}.</math> More generally, a [[cuboid]] with edge lengths ''a'', ''b'', and ''c'' has all four space diagonals axial, with common length <math>\sqrt{a^2+b^2+c^2}. </math>

A regular [[octahedron]] has 3 axial diagonals, of length <math>a\sqrt {2}</math>, with edge length ''a''. 

A [[regular icosahedron]] has 6 axial diagonals of length <math>a\sqrt {2+\varphi}</math>, where <math>\varphi</math> is the [[golden ratio]] <math>(1+\sqrt 5)/2</math>.<ref>{{citation|title=Platonic & Archimedean Solids|series=Wooden Books|first=Daud|last=Sutton|publisher=Bloomsbury Publishing USA|year=2002|isbn=9780802713865|page=55|url=https://books.google.com/books?id=vgo7bTxDmIsC&pg=PA55}}.</ref>

== Space diagonals of magic cubes==
{{main|Magic cube}}
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==
* [[Distance]]
* [[Face diagonal]]
* [[Magic cube classes]]
* [[Hypotenuse]]
* [[Spacetime interval]]

==References==
{{reflist}}
* John R. Hendricks, ''The Pan-3-Agonal Magic Cube'', Journal of Recreational Mathematics 5:1:1972, pp 51–54. First published mention of pan-3-agonals
* Hendricks, J. R., ''Magic Squares to Tesseracts by Computer'', 1998, 0-9684700-0-9, page 49
* Heinz & Hendricks, ''Magic Square Lexicon: Illustrated'', 2000, 0-9687985-0-0, pages 99,165
* Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 173, 1994.

== External links ==
* {{MathWorld |urlname=SpaceDiagonal |title=Space Diagonals}}
* [http://www.magichypercubes.com/Encyclopedia/index.html  de Winkel Magic Encyclopedia]
* [http://members.shaw.ca/hdhcubes/cube_basics.htm Heinz - Basic cube parts]
* [https://web.archive.org/web/20140207001658/http://members.shaw.ca/johnhendricksmath/ John Hendricks Hypercubes]

[[Category:Magic squares]]
[[Category:Elementary geometry]]