# Heptagon

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Regular heptagon
A regular heptagon
Type Regular polygon
Edges and vertices 7
Schläfli symbol {7}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D7), order 2×7
Internal angle (degrees)≈128.571°
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.

## Contents

The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua- , a Latin-derived numerical prefix, rather than hepta- , a Greek-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 angles of 5π/7 radians (12847 degrees). Its Schläfli symbol is {7}.

### Area

The area (A) of a regular heptagon of side length a is given by:

${\displaystyle A={\frac {7}{4}}a^{2}\cot {\frac {\pi }{7}}\simeq 3.634a^{2}.}$

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 as the common side. The apothem is half the cotangent of ${\displaystyle \pi /7,}$ and the area of each of the 14 small triangles is one-fourth of the apothem.

The area of a regular heptagon inscribed in a circle of radius R is ${\displaystyle {\tfrac {7R^{2}}{2}}\sin {\tfrac {2\pi }{7}},}$ while the area of the circle itself is ${\displaystyle \pi R^{2};}$ thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.

### Construction

As 7 is a Pierpont prime but not a Fermat prime, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. It is the smallest regular polygon with this property. 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 ${\displaystyle \scriptstyle {2\cos {\tfrac {2\pi }{7}}\approx 1.247}}$ is a zero of the irreducible cubic x3 + x2 − 2x − 1. Consequently, this polynomial is the minimal polynomial of 2cos(7), whereas the degree of the minimal polynomial for a constructible number must be a power of 2.

 A neusis construction of the interior angle in a regular heptagon. An animation from a neusis construction with radius of circumcircle ${\displaystyle {\overline {OA}}=6}$, according to Andrew M. Gleason [1] based on the angle trisection by means of the tomahawk. This construction relies on the fact that ${\displaystyle 6\cos \left({\frac {2\pi }{7}}\right)=2{\sqrt {7}}\cos \left({\frac {1}{3}}\arctan \left(3{\sqrt {3}}\right)\right)-1.}$

### Approximation

An approximation for practical use with an error of about 0.2% is shown in the drawing. It is attributed to Albrecht Dürer. [2] Let A lie on the circumference of the circumcircle. Draw arc BOC. Then ${\displaystyle \scriptstyle {BD={1 \over 2}BC}}$ gives an approximation for the edge of the heptagon.

This approximation uses ${\displaystyle \scriptstyle {{\sqrt {3}} \over 2}\approx 0.86603}$ for the side of the heptagon inscribed in the unit circle while the exact value is ${\displaystyle \scriptstyle 2\sin {\pi \over 7}\approx 0.86777}$.

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), 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. [4]

### Diagonals and heptagonal triangle

The regular heptagon's side a, shorter diagonal b, and longer diagonal c, with a<b<c, satisfy [5] :Lemma 1

${\displaystyle a^{2}=c(c-b),}$
${\displaystyle b^{2}=a(c+a),}$
${\displaystyle c^{2}=b(a+b),}$
${\displaystyle {\frac {1}{a}}={\frac {1}{b}}+{\frac {1}{c}}}$ (the optic equation)

and hence

${\displaystyle ab+ac=bc,}$

and [5] :Coro. 2

${\displaystyle b^{3}+2b^{2}c-bc^{2}-c^{3}=0,}$
${\displaystyle c^{3}-2c^{2}a-ca^{2}+a^{3}=0,}$
${\displaystyle a^{3}-2a^{2}b-ab^{2}+b^{3}=0,}$

Thus –b/c, c/a, and a/b all satisfy the cubic equation ${\displaystyle t^{3}-2t^{2}-t+1=0.}$ However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis.

The approximate lengths of the diagonals in terms of the side of the regular heptagon are given by

${\displaystyle b\approx 1.80193\cdot a,\qquad c\approx 2.24698\cdot a.}$

We also have [6]

${\displaystyle b^{2}-a^{2}=ac,}$
${\displaystyle c^{2}-b^{2}=ab,}$
${\displaystyle a^{2}-c^{2}=-bc,}$

and

${\displaystyle {\frac {b^{2}}{a^{2}}}+{\frac {c^{2}}{b^{2}}}+{\frac {a^{2}}{c^{2}}}=5.}$

A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles ${\displaystyle \pi /7,2\pi /7,}$ and ${\displaystyle 4\pi /7.}$ Thus its sides coincide with one side and two particular diagonals of the regular heptagon. [5]

### In polyhedra

Apart from the heptagonal prism and heptagonal antiprism, no convex polyhedron made entirely out of regular polygons contains a heptagon as a face.

## Star heptagons

Two kinds of star heptagons (heptagrams) 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.

## Tiling and packing

Triangle, heptagon, and 42-gon vertex
Hyperbolic heptagon tiling

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, tilings by regular heptagons are possible.

The regular heptagon has a double lattice 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. [7]

## Empirical examples

The United Kingdom, as of 2022, has two heptagonal coins, the 50p and 20p pieces, and the Barbados Dollar 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. 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 Brazilian 25-cent coin has a heptagon inscribed in the coin's disk. Some old versions of the coat of arms of Georgia, including in Soviet days, used a {7/2} heptagram as an element.

In architecture, heptagonal floor plans are very rare. A remarkable example is the Mausoleum of Prince Ernst in Stadthagen, Germany.

Many police badges in the US have a {7/2} heptagram outline.

## Related Research Articles

In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition, subtraction, multiplication, division, and square roots.

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. The bounded plane region, the bounding circuit, or the two together, may be called a polygon.

In geometry, a hexagon or sexagon is a six-sided polygon or 6-gon creating the outline of a cube. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.

In geometry, an octagon is an eight-sided polygon or 8-gon.

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

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 διαγώνιος diagonios, "from angle to angle" ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus.

In geometry, a dodecagon or 12-gon is any twelve-sided polygon.

Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.

In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.

In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.

The apothem 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. The word "apothem" can also refer to the length of that line segment and come from the ancient Greek ἀπόθεμα, made of ἀπό and θέμα, indicating a generic line written down. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent.

A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:

In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself. In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons.

## References

1. Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon p. 186 (Fig.1) –187" (PDF). The American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. Archived from the original (PDF) on 19 December 2015.
2. G.H. Hughes, "The Polygons of Albrecht Dürer-1525, The Regular Heptagon", Fig. 11 the side of the Heptagon (7) Fig. 15, image on the left side, retrieved on 4 December 2015
3. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN   978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
4. Salthouse, J.A; Ware, M.J. (1972). Point group character tables and related data. Cambridge: Cambridge University Press. ISBN   0 521 08139 4.
5. Abdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", Forum Geometricorum 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf
6. Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", Mathematics Magazine 46 (1), January 1973, 7–19.
7. Kallus, Yoav (2015). "Pessimal packing shapes". Geometry & Topology. 19 (1): 343–363. arXiv:. doi:10.2140/gt.2015.19.343. MR   3318753.