Heptagon
Encyclopedia
In geometry
, a heptagon is a polygon
with seven sides and seven angle
s. In a regular
heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5π/7 radian
s, 128.5714286 degree
s. Its Schläfli symbol is {7}. The area (A) of a regular heptagon of side length a is given by
The heptagon is also occasionally referred to as the septagon, using "sept-" (an elision of septua-, a Latin
-derived numerical prefix
, rather than hepta-, a Greek
-derived numerical prefix together with the Greek suffix -agon meaning angle).
with compass and straightedge
but is constructible with a marked ruler
and compass. This type of construction is called a Neusis construction
. It is also constructible with compass, straightedge and angle
trisector. The impossibility of straightedge and compass construction follows from the observation that 2cos(2π/7) ≈ 1.247 is a zero of the irreducible
cubic
x3 + x2 - 2x - 1. Consequently this polynomial is the minimal polynomial
of 2cos(2π/7), whereas the degree of the minimal polynomial for a constructible number
must be a power of 2.
is shown in the drawing. Let 
lie on the circumference of the circumcircle. Draw arc 
. Then 
gives an approximation for the edge of the heptagon.
can be inscribed in a circle of radius 
with an error of less than 0.00013%.
This follows from a rational approximation of A
.
can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor
being the interval of connection.
Blue, {7/2} and green {7/3} star heptagons inside a red heptagon.
currently (2010) has two heptagonal coin
s, the 50p and 20p pieces, and the Barbados
Dollar is also heptagonal. The 20 eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a Reuleaux heptagon, a curvilinear heptagon to make them curves of constant width
: the sides are curved outwards so that the coin will roll smoothly in vending machine
s. Botswana pula
coins in the denominations of 2 Pula, 1 Pula, 50 Thebe and 5 Thebe are also shaped as equilateral-curve heptagons. Coins in the shape of Reuleaux heptagons are in circulation in Mauritius, U.A.E.,Tanzania, Samoa, Papua New Guinea, São Tomé and Príncipe, Haiti, Jamaica, Liberia, Haiti, Ghana, the Gambia, Jordan, Jersey, Guernsey, Isle of Man, Gibraltar, Guyana, Solomon Islands, Falkland Islands and Saint Helena.The 1000 Kwacha coin of Zambia is a true heptagon.
The Brazil
ian 25 cents coin has a heptagon inscribed in the coin's disk. Some old versions of Coat of arms of Georgia (country)
including Soviet days had used {7/2} heptagram as an element.
Apart from the heptagonal prism
and heptagonal antiprism
, no polyhedron made entirely out of regular polygons contains a heptagon as a face.
Regular heptagons can tile the hyperbolic plane
, as shown in this Poincaré disk model
projection:
is often drawn as a regular heptagon with all 21 edges connected. This graph also represents an orthographic projection
of the 7 vertices and 21 edges of the 6-simplex. The 21 and 35 vertices of the rectified and birectified 6-simplex also orthogonally project into regular heptagons.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a heptagon is a polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
with seven sides and seven angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s. In a regular
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5π/7 radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s, 128.5714286 degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
s. Its Schläfli symbol is {7}. The area (A) of a regular heptagon of side length a is given by
The heptagon is also occasionally referred to as the septagon, using "sept-" (an elision of septua-, a Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
-derived numerical prefix
Numerical prefix
Number prefixes are prefixes derived from numbers or numerals. In English and other European languages, they are used to coin numerous series of words, such as unicycle – bicycle – tricycle, dyad – triad – decade, biped – quadruped, September – October – November – December, decimal – hexadecimal,...
, rather than hepta-, a Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
-derived numerical prefix together with the Greek suffix -agon meaning angle).
Construction
A regular heptagon is not constructibleConstructible polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not....
with compass and straightedge
Compass and straightedge
Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
but is constructible with a marked ruler
Ruler
A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
and compass. This type of construction is called a Neusis construction
Neusis construction
The neusis is a geometric construction method that was used in antiquity by Greek mathematicians.- Geometric construction :The neusis construction consists of fitting a line element of given length in between two given lines , in such a way that the line element, or its...
. It is also constructible with compass, straightedge and angle
trisector. The impossibility of straightedge and compass construction follows from the observation that 2cos(2π/7) ≈ 1.247 is a zero of the irreducible
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
cubic
Cubic function
In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
x3 + x2 - 2x - 1. Consequently this polynomial is the minimal polynomial
Minimal polynomial (field theory)
In field theory, given a field extension E / F and an element α of E that is an algebraic element over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p = 0...
of 2cos(2π/7), whereas the degree of the minimal polynomial for a constructible number
Constructible number
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...
must be a power of 2.
 A Neusis construction of the interior angle in a regular heptagon. |
 A Neusis construction of the interior angle in a regular heptagon. (method by John Horton Conway John Horton Conway John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory... |
An animation of an approximate compass-and-straightedge construction of a regular heptagon. |
Approximation
A decent approximation for practical use with an accuracy of is shown in the drawing. Let lie on the circumference of the circumcircle. Draw arc . Then gives an approximation for the edge of the heptagon.A more exact approximation
A regular heptagon with sides can be inscribed in a circle of radius with an error of less than 0.00013%.This follows from a rational approximation of A
.Star Heptagons
Two kinds of star heptagonsHeptagram
A heptagram or septegram is a seven-pointed star drawn with seven straight strokes.- Geometry :In general, a heptagram is any self-intersecting heptagon ....
can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
being the interval of connection.
Blue, {7/2} and green {7/3} star heptagons inside a red heptagon.
Uses
The United KingdomUnited Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
currently (2010) has two heptagonal coin
Coin
A coin is a piece of hard material that is standardized in weight, is produced in large quantities in order to facilitate trade, and primarily can be used as a legal tender token for commerce in the designated country, region, or territory....
s, the 50p and 20p pieces, and the Barbados
Barbados
Barbados is an island country in the Lesser Antilles. It is in length and as much as in width, amounting to . It is situated in the western area of the North Atlantic and 100 kilometres east of the Windward Islands and the Caribbean Sea; therein, it is about east of the islands of Saint...
Dollar is also heptagonal. The 20 eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a Reuleaux heptagon, a curvilinear heptagon to make them curves of constant width
Curve of constant width
In geometry, a curve of constant width is a convex planar shape whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines.More generally, any compact...
: the sides are curved outwards so that the coin will roll smoothly in vending machine
Vending machine
A vending machine is a machine which dispenses items such as snacks, beverages, alcohol, cigarettes, lottery tickets, consumer products and even gold and gems to customers automatically, after the customer inserts currency or credit into the machine....
s. Botswana pula
Botswana pula
The pula is the currency of Botswana. It has the ISO 4217 code BWP and is subdivided into 100 thebe. Pula literally means "rain" in Setswana, because rain is very scarce in Botswana - home to much of the Kalahari Desert - and therefore valuable. Pula also means "blessing" as rain is considered a...
coins in the denominations of 2 Pula, 1 Pula, 50 Thebe and 5 Thebe are also shaped as equilateral-curve heptagons. Coins in the shape of Reuleaux heptagons are in circulation in Mauritius, U.A.E.,Tanzania, Samoa, Papua New Guinea, São Tomé and Príncipe, Haiti, Jamaica, Liberia, Haiti, Ghana, the Gambia, Jordan, Jersey, Guernsey, Isle of Man, Gibraltar, Guyana, Solomon Islands, Falkland Islands and Saint Helena.The 1000 Kwacha coin of Zambia is a true heptagon.
The Brazil
Brazil
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...
ian 25 cents coin has a heptagon inscribed in the coin's disk. Some old versions of Coat of arms of Georgia (country)
Coat of arms of Georgia (country)
The coat of arms of the Republic of Georgia was adopted on 1 October 2004. It is partially based on the medieval arms of the Georgian royal house of Bagrationi....
including Soviet days had used {7/2} heptagram as an element.
Apart from the heptagonal prism
Heptagonal prism
In geometry, the heptagonal prism is the fifth of the infinite set of convex prisms formed by square sides and two regular polygon caps, in this case two heptagons....
and heptagonal antiprism
Heptagonal antiprism
In geometry, the heptagonal antiprism is the fourth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms...
, no polyhedron made entirely out of regular polygons contains a heptagon as a face.
Regular heptagons can tile the hyperbolic plane
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, as shown in this Poincaré disk model
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the straight lines of the hyperbolic geometry are segments of circles contained in the disk...
projection:
Graphs
The K7 complete graphComplete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
is often drawn as a regular heptagon with all 21 edges connected. This graph also represents an orthographic projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...
of the 7 vertices and 21 edges of the 6-simplex. The 21 and 35 vertices of the rectified and birectified 6-simplex also orthogonally project into regular heptagons.
6-simplex (6D) |
Rectified 6-simplex Rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the... (6D) |
Birectified 6-simplex (6D) |