In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a space diagonal (also interior diagonal or body diagonal) of a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
is a line connecting two
vertices that are not on the same
face
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...
. Space diagonals contrast with ''
face diagonals'', which connect vertices on the same face (but not on the same
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed ...
) as each other.
For example, a
pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
has no space diagonals, while a
cube
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.
The cube is the only ...
(shown at right) or more generally a
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...
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
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.
The cube is the only ...
with edge length ''a'', all four space diagonals are axial diagonals, of common length
More generally, a
cuboid
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a c ...
with edge lengths ''a'', ''b'', and ''c'' has all four space diagonals axial, with common length
A regular
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) 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 ea ...
has 3 axial diagonals, of length
, with edge length ''a''.
A
regular icosahedron has 6 axial diagonals of length
, where
is the
golden ratio
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 a and b with a > b > 0,
where the Greek letter phi ( ...
.
[.]
Space diagonals of magic cubes
A
magic square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
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
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
*
Face diagonal
*
Magic cube classes
*
Hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...
*
Spacetime interval
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
References
* 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
de Winkel Magic EncyclopediaJohn Hendricks Hypercubes
Magic squares
Elementary geometry