# Space diagonal

In a rectangular box or a magic cube, the four **space diagonals** are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner. These lines are also called *triagonals* or *volume diagonals*.

For the cube to be considered magic, these four lines must sum correctly.

The word triagonal is derived from the fact that as a variable point travels down the line, three coordinates change. The equivalent in a square is diagonal, because two coordinates change. In a tesseract it is quadragonal because 4 coordinates change, etc.

The **space diagonal** of a cube with side length is .

*r*-agonals

This section applies particularly to Magic hypercubes.

The magic hypercube community has started to recognize an abbreviated expression for these *space diagonals*. By using *r* as a variable to describe the various
*agonals*, a concise notation is possible.

If *r* =

- 2 then we have a diagonal. 2 coordinates change.
- 3 = a
*triagonal*. 3 coordinates change - 4 = a
*quadragonal*. 4 coordinates change *n*= the dimension of the hypercube, the*2*agonals are required to sum correctly for the hypercube to be considered magic.^{n-1}

...
By extension, if *r* =

- 1, the line is parallel to a face. Only 1 coordinate changes. A 1-agonal may be called a monagonal, in keeping with a diagonal, a triagonal, etc. Lines parallel to the faces of the hypercube have, in the past, also been referred to as i-rows.

Because the prefix *pan* indicates *all*, we can concisely state the characteristics or a magic hypercube.

For example;

- If pan-
*r*-agonals sum correctly for*r*= 1 and 2, we know the square is pandiagonal magic. - If pan-
*r*-agonals sum correctly for*r*= 1 and 3, we have a pantriagonal magic cube (the equivalent of a pandiagonal magic square). - If the
*r*-agonals sum correctly for*r*= 1 and*n*, then the magic hypercube is simple magic regardless of what dimension it is.

The length of an *r*-agonal of a hypercube with side length *a* is .

## See also

## 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