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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
.


Regular tridecagon

A ''
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
tridecagon'' is represented by
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
. The measure of each internal angle of a
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
tridecagon is approximately 152.308 degrees, and the area with side length ''a'' is given by :A = \fraca^2 \cot \frac \simeq 13.1858\,a^2.


Construction

As 13 is a
Pierpont prime In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...
but not a
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...
, the regular tridecagon cannot be constructed using a
compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
. However, it is constructible using
neusis In geometry, the neusis (; ; plural: ) is a geometric construction method that was used in antiquity by Greek mathematics, Greek mathematicians. Geometric construction The neusis construction consists of fitting a line element of given length ...
, or an angle trisector. The following is an animation from a ''neusis construction'' of a regular tridecagon with radius of circumcircle \overline = 12, according to
Andrew M. Gleason Andrew Mattei Gleason (19212008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in teaching a ...
, based on the
angle trisection Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
by means of the
Tomahawk A tomahawk is a type of single-handed axe used by the many Native Americans in the United States, Indian peoples and nations of North America, traditionally resembles a hatchet with a straight shaft. Etymology The name comes from Powhatan langu ...
(light blue). An approximate construction of a regular tridecagon using
straightedge A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler. Straightedges are used in the automotive service and ma ...
and
compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with No ...
is shown here. Another possible animation of an approximate construction, also possible with using straightedge and compass.


Based on the unit circle r = 1

nit of length NiT, NIT, or Nits may refer to: Education * Namgyal Institute of Tibetology, Sikkim, India * Narula Institute of Technology, West Bengal, India * National Institutes of Technology, India * Naval Institute of Technology, Biliran, Philippines * Nip ...
/h2>

* Constructed side length in
GeoGebra GeoGebra (a portmanteau of ''geometry'' and ''algebra'') is an interactive geometry, algebra, statistics and calculus application, intended for learning and teaching mathematics and science from primary school to university level. GeoGebra is a ...
a = 0.478631328575115\;
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
/math> * Side length of the tridecagon a_ = r \cdot 2 \cdot \sin\left(\frac \right) = 0.478631328575115\ldots\;
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
/math> * Absolute error of the constructed side length: : Up to the maximum precision of 15 decimal places, the absolute error is F_a = a - a_ = 0.0\;
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
/math> * Constructed central angle of the tridecagon in GeoGebra (display significant 13 decimal places, rounded) \mu = 27.6923076923077^\circ * Central angle of tridecagon \mu_ = \left( \frac\right) = 27.^\circ * Absolute angular error of the constructed central angle: : Up to 13 decimal places, the absolute error is F_\mu = \mu - \mu_ = 0.0^\circ


Example to illustrate the error

At a circumscribed circle of radius r = 1 billion km (a distance which would take light approximately 55 minutes to travel), the absolute error on the side length constructed would be less than 1 mm.


Symmetry

The ''regular tridecagon'' has Dih13 symmetry, order 26. Since 13 is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
there is one subgroup with dihedral symmetry: Dih1, and 2
cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
symmetries: Z13, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon.
John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many br ...
labels these by a letter and group order. Full symmetry of the regular form is r26 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g13 subgroup has no degrees of freedom but can be seen as directed edges.


Numismatic use

The regular tridecagon is used as the shape of the Czech 20 korun coin.Colin R. Bruce, II, George Cuhaj, and Thomas Michael, ''2007 Standard Catalog of World Coins'', Krause Publications, 2006, , p. 81. :


Related polygons

A tridecagram is a 13-sided
star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...
. There are 5 regular forms given by
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
s: , , , , and . Since 13 is prime, none of the tridecagrams are compound figures. Culturally, this shape is a symbol for
immortality Immortality is the concept of eternal life. Some species possess "biological immortality" due to an apparent lack of the Hayflick limit. From at least the time of the Ancient Mesopotamian religion, ancient Mesopotamians, there has been a con ...
.


Petrie polygons

The regular tridecagon is the
Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...
12-simplex:


References


External links

* {{Polygons Polygons by the number of sides