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A golygon, or more generally a serial isogon of 90°, is any
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
with all
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s (a
rectilinear polygon A rectilinear polygon is a polygon all of whose sides meet at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic polygons. In many cases another definition is ...
) whose sides are consecutive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
lengths. Golygons were invented and named by
Lee Sallows Lee Cecil Fletcher Sallows (born April 30, 1944) is a British electronics engineer known for his contributions to recreational mathematics. He is particularly noted as the inventor of golygons, self-enumerating sentences, and geomagic squares. ...
, and popularized by
A.K. Dewdney Alexander Keewatin Dewdney (born August 5, 1941) is a Canadians, Canadian mathematician, computer scientist, author, filmmaker, and conspiracy theorist. Dewdney is the son of Canadian artist and author Selwyn Dewdney, and brother of poet Christopher ...
in a 1990 ''
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 ...
'' column (Smith). Variations on the definition of golygons involve allowing edges to cross, using sequences of edge lengths other than the consecutive integers, and considering turn
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
s other than 90°.


Properties

In any golygon, all horizontal edges have the same parity as each other, as do all vertical edges. Therefore, the number ''n'' of sides must allow the solution of the system of equations :\pm 1 \pm 3 \pm \cdots \pm (n-1) = 0 :\pm 2 \pm 4 \pm \cdots \pm n = 0. It follows from this that ''n'' must be a multiple of 8. For example, in the figure we have -1 + 3 + 5 - 7 = 0 and 2 - 4 - 6 + 8 = 0. The number of golygons for a given permissible value of ''n'' may be computed efficiently using
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
s . The number of golygons for permissible values of ''n'' is 4, 112, 8432, 909288, etc. Finding the number of solutions that correspond to non-crossing golygons seems to be significantly more difficult. There is a unique eight-sided golygon (shown in the figure); it can
tile Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or o ...
the plane by 180-degree rotation using the
Conway criterion In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements:Will It Tile? Try ...
.


Examples

Golygon 16-sides.svg, 16-sided golygon.
Spirolateral In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M ...
1690°1,3,6,8,11 golygon 32-sides.svg, 32-sided golygon. Spirolateral 3290°1,3,5,7,11,12,14,17,19,21,23,26,29,31


Generalizations

A serial-sided isogon of order ''n'' is a closed polygon with a constant angle at each vertex and having consecutive sides of length 1, 2, ..., ''n'' units. The polygon may be self-crossing. Golygons are a
special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is ...
of serial-sided isogons. A
spirolateral In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M ...
is similar construction, notationally ''n''θ''i''1,''i''2,...,''i''''k'' which sequences lengths 1,2,3,...,''n'' with internal angles θ, with option of repeating until it returns to close with the original vertex. The ''i''1,''i''2,...,''i''''k'' superscripts list edges that follow opposite turn directions. serial isogon_9_120.svg, A serial-sided isogon order 9, internal angle 60°.
Spirolateral In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M ...
60°91,4,7. Serial_isogon_11_60.svg, A serial-sided isogon order 11, internal angle 60°.
Spirolateral In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M ...
60°114,5,7,8. serial isogon 12 60.svg, A serial-sided isogon order 12, internal angle 120°.
Spirolateral 120°121,4,8. Serial isogon 5 60 120.svg, A serial-sided isogon order 5, internal angles 60° and 120°.


Golyhedron

The three-dimensional generalization of a golygon is called a golyhedron – a closed simply-connected solid figure confined to the faces of a cubical lattice and having face areas in the sequence 1, 2, ..., ''n'', for some integer ''n'', first introduced in a MathOverflow question. Golyhedrons have been found with values of ''n'' equal to 32, 15, 12, and 11 (the minimum possible).Golyhedron update
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References

{{Reflist


External links


Golygons
at the
On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
Types of polygons