recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

, a polyabolo (also known as a polytan) is a shape formed by gluing isosceles right triangles edge-to-edge, making a

polyform In recreational mathematics, a polyform is a plane (mathematics), plane figure or solid compound constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex polygon, convex plane-filling pol ...

with the isosceles right triangle as the base form. Polyaboloes were introduced by

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...

in his June 1967 "

Mathematical Games column Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, through June 1986, Gardner wrote 9 more columns, ...

" in ''

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...

''.

Nomenclature

The name ''polyabolo'' is a

back formation In etymology, back-formation is the process or result of creating a new word via inflection, typically by removing or substituting actual or supposed affixes from a lexical item, in a way that expands the number of lexemes associated with the ...

from the juggling object '

diabolo The diabolo ( ; commonly misspelled ''diablo'') is a juggling or circus prop consisting of an axle () and two cups (hourglass/egg timer shaped) or discs derived from the Chinese yo-yo. This object is spun using a string attached to two hand ...

', although the shape formed by joining two triangles at just one vertex is not a proper polyabolo. By false analogy, treating the di- in diabolo as meaning "two", polyaboloes with from 1 to 10 cells are called respectively monaboloes, diaboloes, triaboloes, tetraboloes, pentaboloes, hexaboloes, heptaboloes, octaboloes, enneaboloes, and decaboloes. The name ''polytan'' is derived from Henri Picciotto's name ''tetratan'' and alludes to the ancient Chinese amusement of

tangram The tangram () is a dissection puzzle consisting of seven flat polygons, called ''tans'', which are put together to form shapes. The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pie ...

Combinatorial enumeration

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, … . Polyaboloes that are confined strictly to the plane and cannot be turned over may be termed one-sided. The number of one-sided polyaboloes composed of 1, 2, 3, … triangles is 1, 4, 6, 22, 56, 198, 624, 2182, 7448, … . As for a

polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in pop ...

, a polyabolo that can be neither turned over nor rotated may be termed fixed. A polyabolo with no symmetries (rotation or reflection) corresponds to 8 distinct fixed polyaboloes. 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 David Anthony Klarner (October 10, 1940March 20, 1999) was an American mathematician, author, and educator. He is known for his work in combinatorial enumeration, polyominoes, and box-packing.rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...

. 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 Solomon Wolf Golomb (; May 30, 1932 – May 1, 2016) was an American mathematician, engineer, and professor of electrical engineering at the University of Southern California, best known for his works on mathematical games. Most notably, he inve ...

found polyaboloes, including a triabolo, of order 8. Michael Reid found a heptabolo of order 6. Higher orders are possible. There are interesting tessellations of the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

involving polyaboloes. One such is the

tetrakis square tiling In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed b ...

, a monohedral tessellation that fills the entire Euclidean plane with 45–45–90 triangles.

Tiling a common figure with various polyaboloes

The ''Compatibility Problem'' is to take two or more polyaboloes and find a figure that can be tiled with each. This problem has been studied far less than the Compatibility Problem for polyominoes. Systematic results first appeared in 2004 at Erich Friedman's website Math Magic.

References

External links

* * {{Polyforms Polyforms