Polyabolo
Encyclopedia
In recreational mathematics
, a polyabolo (also known as a polytan) is a polyform
with an isosceles right triangle as the base form.
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, … .
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 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.
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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. Michael Reid found a heptabolo of order 6.
Higher orders are possible.
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
, a polyabolo (also known as a polytan) is a polyform
Polyform
In recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. The basic polygon is often a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as...
with an isosceles right triangle as the base form.
The name is a back formation from 'diabolo
Diabolo
The diabolo is a juggling prop consisting of a spool which is whirled and tossed on a string tied to two sticks held one in each hand. A huge variety of tricks are possible using the sticks, string, and various body parts...
' 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, … .
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
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 with a connected interior....
, 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 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 rectangleRectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square 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
Solomon W. Golomb
Solomon Wolf Golomb is an American mathematician and engineer and a professor of electrical engineering at the University of Southern California, best known to the general public and fans of mathematical games as the inventor of polyominoes, the inspiration for the computer game Tetris...
found polyaboloes, including a triabolo, of order 8. Michael Reid found a heptabolo of order 6.
Higher orders are possible.