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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a heptagon or septagon is a seven-sided
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 to ...
or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''
septua- Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cy ...
'', a
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
-derived
numerical prefix Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cyc ...
, rather than ''
hepta- Numeral or number prefixes are prefixes derived from Numeral (linguistics), numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1 ...
'', a
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.


Regular heptagon

A regular heptagon, in which all sides and all angles are equal, has
internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
s of 5π/7
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s (128
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
s). Its Schläfli symbol is .


Area

The area (''A'') of a regular heptagon of side length ''a'' is given by: :A = \fraca^2 \cot \frac \simeq 3.634 a^2. This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then halving each triangle using the
apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...
as the common side. The apothem is half the cotangent of \pi/7, and the area of each of the 14 small triangles is one-fourth of the apothem. The area of a regular heptagon
inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
in a circle of
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
''R'' is \tfrac\sin\tfrac, while the area of the circle itself is \pi R^2; thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.


Construction

As 7 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 use ...
but not a
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first 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, 17, 257, 65537, 429496 ...
, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked
ruler A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...
and compass. It is the smallest regular polygon with this property. This type of construction is called a
neusis construction In geometry, the neusis (; ; plural: grc, νεύσεις, neuseis, label=none) is a geometric construction method that was used in antiquity by Greek mathematicians. Geometric construction The neusis construction consists of fitting a line e ...
. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge and compass construction follows from the observation that \scriptstyle is a zero of the irreducible cubic . Consequently, this polynomial is the minimal polynomial of whereas the degree of the minimal polynomial for a
constructible number In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...
must be a power of 2.


Approximation

An approximation for practical use with an error of about 0.2% is shown in the drawing. It is attributed to Albrecht Dürer. Let ''A'' lie on the circumference of the circumcircle. Draw arc ''BOC''. Then \scriptstyle gives an approximation for the edge of the heptagon. This approximation uses \scriptstyle \approx 0.86603 for the side of the heptagon inscribed in the unit circle while the exact value is \scriptstyle 2\sin \approx 0.86777. ''Example to illustrate the error:
At a circumscribed circle radius ''r = 1 m'', the absolute error of the 1st side would be ''approximately -1.7 mm''


Symmetry

The ''regular heptagon'' belongs to the D7h point group (
Schoenflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the ...
), order 28. The symmetry elements are: a 7-fold proper rotation axis C7, a 7-fold improper rotation axis, S7, 7 vertical mirror planes, σv, 7 2-fold rotation axes, C2, in the plane of the heptagon and a horizontal mirror plane, σh, also in the heptagon's plane.


Diagonals and heptagonal triangle

The regular heptagon's side ''a'', shorter
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
''b'', and longer diagonal ''c'', with ''a''<''b''<''c'', satisfyAbdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", ''
Forum Geometricorum ''Forum Geometricorum: A Journal on Classical Euclidean Geometry'' is a peer-reviewed open-access academic journal that specializes in mathematical research papers on Euclidean geometry. It was founded in 2001, is published by Florida Atlantic U ...
'' 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf
:a^2=c(c-b), :b^2 =a(c+a), :c^2 =b(a+b), :\frac=\frac+\frac (the
optic equation In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers ''a'' and ''b'' to equal the reciprocal of a third positive integer ''c'':Dickson, L. E., ''History of the Theory of Numbers, V ...
) and hence : ab+ac=bc, and :b^3+2b^2c-bc^2-c^3=0, :c^3-2c^2a-ca^2+a^3=0, :a^3-2a^2b-ab^2+b^3=0, Thus –''b''/''c'', ''c''/''a'', and ''a''/''b'' all satisfy the
cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
t^3-2t^2-t + 1=0. However, no
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations ( addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
s with purely real terms exist for the solutions of this equation, because it is an example of
casus irreducibilis In algebra, ''casus irreducibilis'' (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining roots th ...
. The approximate lengths of the diagonals in terms of the side of the regular heptagon are given by :b\approx 1.80193\cdot a, \qquad c\approx 2.24698\cdot a. We also have :b^2-a^2=ac, :c^2-b^2=ab, :a^2-c^2=-bc, and :\frac+\frac+\frac=5. A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles \pi/7, 2\pi/7, and 4\pi/7. Thus its sides coincide with one side and two particular
diagonals In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
of the regular heptagon.


In polyhedra

Apart from the heptagonal prism and
heptagonal antiprism In geometry, the heptagonal antiprism is the fifth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each ot ...
, no convex polyhedron made entirely out of regular polygons contains a heptagon as a face.


Star heptagons

Two kinds of star heptagons (
heptagram A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek suffix ''-gram''. The ''-gram'' suffix derives from ''γρ ...
s) can be constructed from regular heptagons, labeled by Schläfli symbols , and , with the
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
being the interval of connection.
Blue, and green star heptagons inside a red heptagon.


Tiling and packing

A regular triangle, heptagon, and 42-gon can completely fill a plane vertex. However, there is no tiling of the plane with only these polygons, because there is no way to fit one of them onto the third side of the triangle without leaving a gap or creating an overlap. In the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
, tilings by regular heptagons are possible. The regular heptagon has a
double lattice In mathematics, especially in geometry, a double lattice in is a discrete subgroup of the group of Euclidean motions that consists only of translations and point reflections and such that the subgroup of translations is a lattice. The orbit of an ...
packing of the Euclidean plane of packing density approximately 0.89269. This has been conjectured to be the lowest density possible for the optimal double lattice packing density of any convex set, and more generally for the optimal packing density of any convex set.


Empirical examples

The United Kingdom, , has two heptagonal
coin A coin is a small, flat (usually depending on the country or value), round piece of metal or plastic used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order t ...
s, the 50p and 20p pieces, and the
Barbados Dollar The dollar has been the currency of Barbados since 1935. Globally its currency has the ISO 4217 code ''BBD'', however, unofficially in Barbados the International vehicle registration code code BDS is also commonly used, a currency code that is ...
are also heptagonal. The 20- eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a Reuleaux heptagon, a curvilinear heptagon which has curves of constant width; the sides are curved outwards to allow the coins to roll smoothly when they are inserted into a
vending machine A vending machine is an automated machine that provides items such as snacks, beverages, cigarettes, and lottery tickets to consumers after cash, a credit card, or other forms of payment are inserted into the machine or otherwise made. The ...
. 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 also in circulation in Mauritius, U.A.E., Tanzania, Samoa, Papua New Guinea, São Tomé and Príncipe, Haiti, Jamaica, Liberia, 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 ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...
ian 25-cent coin has a heptagon inscribed in the coin's disk. Some old versions of the
coat of arms of Georgia The coat of arms of Georgia is one of the national symbols of the republic. It is partially based on the medieval arms of the Georgian royal house and features Saint George, the traditional patron saint of Georgia. In addition to St. George, t ...
, including in Soviet days, used a heptagram as an element. In architecture, heptagonal floor plans are very rare. A remarkable example is the
Mausoleum of Prince Ernst The Mausoleum of Prince Ernst in Stadthagen, Lower Saxony, is a mausoleum erected by Ernst of Schaumburg (d. 1622) and his widow Hedwig of Hesse-Kassel in the years 1620–1627. Its unusual architecture and the resurrection monument by Adrian de V ...
in
Stadthagen Stadthagen () is the capital of the district of Schaumburg, in Lower Saxony, Germany. It is situated approximately 20 km east of Minden and 40 km west of Hanover. The city consists of the districts Brandenburg, Enzen-Hobbensen, Hörkamp-L ...
,
Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...
. Many police badges in the US have a heptagram outline.


See also

*
Heptagram A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek suffix ''-gram''. The ''-gram'' suffix derives from ''γρ ...
*
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 to ...


References


External links


Definition and properties of a heptagon
With interactive animation
Heptagon according Johnson

Another approximate construction method



Recently discovered and highly accurate approximation for the construction of a regular heptagon.
* Heptagon, an approximating construction as an animation * A heptagon with a given side, an approximating construction as an animation {{Polygons Polygons by the number of sides 7 (number) Elementary shapes