In recreational mathematics, a **polyabolo** (also known as a **polytan**) is a polyform with an isosceles right triangle as the base form.^{[1]}

The name is a back formation from 'diabolo' although the shape formed by joining two triangles at just one vertex is not a proper polyabolo. On a false analogy as if di- in diabolo means twice, polyaboloes with from 1 to 10 triangles are called respectively monoboloes, diaboloes, triaboloes, tetraboloes, pentaboloes, hexaboloes, heptaboloes, octaboloes, enneaboloes, and decaboloes.

There are two ways in which a square in a polyabolo can consist of two isosceles right triangles, but polyaboloes are considered equivalent if they have the same boundaries. The number of nonequivalent polyaboloes composed of 1, 2, 3, … triangles is 1, 3, 4, 14, 30, 107, 318, 1116, 3743, … (sequence A006074 in OEIS).

Polyaboloes that are confined strictly to the plane and cannot be turned over may be termed one-sided. There are now four diaboloes (instead of three), six triaboloes (instead of four), 22 tetraboloes (instead of 14), and so on.

A non-simply connected polyabolo is one that has one or more holes in it. The smallest value of *n* for which an *n*-abolo is non-simply connected is 7.

## Tiling rectangles with copies of a single polyabolo

In 1968, David A. Klarner defined the *order* of a polyomino. Similarly, the order of a polyabolo P can be defined as the minimum number of congruent copies of P that can be assembled (allowing translation, rotation, and reflection) to form a rectangle.

A polyabolo has order 1 if and only if it is itself a rectangle. Polyaboloes of order 2 are also easily recognisable. Solomon W. Golomb found polyaboloes, including a triabolo, of order 8^{[2]}. Michael Reid found a heptabolo of order 6^{[3]}.

## References

- ↑ Gardner, Martin (June 1967). "The polyhex and the polyabolo, polygonal jigsaw puzzle pieces".
*Scientific American***216**(6): 124–132. - ↑ Golomb, Solomon W. (1994).
*Polyominoes (2nd ed.)*. Princeton University Press. pp. 101-101. ISBN 0-691-02444-8. - ↑ Goodman, Jacob E.; O'Rourke, Joseph (2004).
*Handbook of Discrete and Computational Geometry (2nd ed.)*. Chapman & Hall/CRC. pp. 349-349. ISBN 1-58488-301-4.

## External links

- Weisstein, Eric W., "Polyabolo" from MathWorld.