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Decagon

From Wikipedia, the free encyclopedia

Shape with ten sides

Regular decagon
A regular decagon
Type	Regular polygon
Edges andvertices	10
Schläfli symbol	{10}, t{5}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D₁₀), order 2×10
Internal angle (degrees)	144°
Properties	Convex,cyclic,equilateral,isogonal,isotoxal
Dual polygon	Self

Ingeometry, adecagon (from the Greek δέκαdéka and γωνίαgonía, "ten angles") is a ten-sidedpolygon or10-gon.^[1] The total sum of theinterior angles of asimple decagon is 1440°.

Regular decagon

Aregular decagon has all sides of equal length and each internal angle will always be equal to 144°.^[1] ItsSchläfli symbol is {10}^[2] and can also be constructed as atruncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.

Decagons often appear in tilings with (partial) 5-fold symmetry. The images show anIslamic geometric pattern (15th century), an illustration in Kepler'sHarmonices Mundi (1619) and aPenrose tiling.

Side length

The picture shows a regular decagon with side length $a {\displaystyle a}$ and radius $R {\displaystyle R}$ of thecircumscribed circle.

The triangle $E_{10}E_{1}M$ has two equally long legs with length $R {\displaystyle R}$ and a base with length $a {\displaystyle a}$
The circle around $E_{1}$ with radius $a {\displaystyle a}$ intersects $]M\,E_{10}[$ in a point $P {\displaystyle P}$ (not designated in the picture).
Now the triangle ${E_{10}E_{1}P}\;$ is anisosceles triangle with vertex $E_{1}$ and with base angles $m\angle E_{1}E_{10}P=m\angle E_{10}PE_{1}=72^{\circ }\;$ .
Therefore $m\angle PE_{1}E_{10}=180^{\circ }-2\cdot 72^{\circ }=36^{\circ }\;$ . So $\;m\angle ME_{1}P=72^{\circ }-36^{\circ }=36^{\circ }\;$ and hence $\;E_{1}MP\;$ is also an isosceles triangle with vertex $P {\displaystyle P}$ . The length of its legs is $a {\displaystyle a}$ , so the length of $[P\,E_{10}]$ is $R-a$ .
The isosceles triangles $E_{10}E_{1}M\;$ and $PE_{10}E_{1}\;$ have equal angles of 36° at the vertex, and so they aresimilar, hence: $\;{\frac {a}{R}}={\frac {R-a}{a}}$
Multiplication with the denominators $R,a>0$ leads to the quadratic equation: $\;a^{2}=R^{2}-aR\;$
This equation for the side length $a\,$ has one positive solution: $\;a={\frac {R}{2}}(-1+{\sqrt {5}})$

So the regular decagon can be constructed withruler and compass.

Further conclusions

$\;R={\frac {2a}{{\sqrt {5}}-1}}={\frac {a}{2}}({\sqrt {5}}+1)\;$ and the base height of $\Delta \,E_{10}E_{1}M\,$ (i.e. the length of $[M\,D]$ ) is $h={\sqrt {R^{2}-(a/2)^{2}}}={\frac {a}{2}}{\sqrt {5+2{\sqrt {5}}}}\;$ and the triangle has the area: $A_{\Delta }={\frac {a}{2}}\cdot h={\frac {a^{2}}{4}}{\sqrt {5+2{\sqrt {5}}}}$ .

Area

Thearea of a regular decagon of side lengtha is given by:^[3]

A={\frac {5}{2}}a^{2}\cot \left({\frac {\pi }{10}}\right)={\frac {5}{2}}a^{2}{\sqrt {5+2{\sqrt {5}}}}\simeq 7.694208843\,a^{2}

In terms of theapothemr (see alsoinscribed figure), the area is:

A=10\tan \left({\frac {\pi }{10}}\right)r^{2}=2r^{2}{\sqrt {5\left(5-2{\sqrt {5}}\right)}}\simeq 3.249196962\,r^{2}

In terms of thecircumradiusR, the area is:

A=5\sin \left({\frac {\pi }{5}}\right)R^{2}={\frac {5}{2}}R^{2}{\sqrt {\frac {5-{\sqrt {5}}}{2}}}\simeq 2.938926261\,R^{2}

An alternative formula is $A=2.5da$ whered is the distance between parallel sides, or the height when the decagon stands on one side as base, or thediameter of the decagon'sinscribed circle.By simpletrigonometry,

d=2a\left(\cos {\tfrac {3\pi }{10}}+\cos {\tfrac {\pi }{10}}\right),

and it can be writtenalgebraically as

d=a{\sqrt {5+2{\sqrt {5}}}}.

Construction

As 10 = 2 × 5, apower of two times aFermat prime, it follows that a regular decagon isconstructible usingcompass and straightedge, or by an edge-bisection of a regularpentagon.^[4]

Construction of decagon

Construction of pentagon

An alternative (but similar) method is as follows:

Construct a pentagon in a circle by one of the methods shown inconstructing a pentagon.
Extend a line from each vertex of the pentagon through the center of thecircle to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon. In other words, the image of a regular pentagon under apoint reflection with respect of its center is aconcentriccongruent pentagon, and the two pentagons have in total the vertices of a concentricregular decagon.
The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.

The golden ratio in decagon

Both in the construction with given circumcircle^[5] as well as with given side length is thegolden ratio dividing a line segment by exterior division thedetermining construction element.

In the construction with given circumcircle the circular arc around G with radiusGE₃ produces the segmentAH, whose division corresponds to the golden ratio.

{\frac {\overline {AM}}{\overline {MH}}}={\frac {\overline {AH}}{\overline {AM}}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618{\text{.}}

In the construction with given side length^[6] the circular arc around D with radiusDA produces the segmentE₁₀F, whose division corresponds to thegolden ratio.

{\frac {\overline {E_{1}E_{10}}}{\overline {E_{1}F}}}={\frac {\overline {E_{10}F}}{\overline {E_{1}E_{10}}}}={\frac {R}{a}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618{\text{.}}

Decagon with given circumcircle,^[5] animation

Decagon with a given side length,^[6] animation

Symmetry

Symmetries of a regular decagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edges. Gyration orders are given in the center.

Theregular decagon hasDih₁₀ symmetry, order 20. There are 3 subgroup dihedral symmetries: Dih₅, Dih₂, and Dih₁, and 4cyclic group symmetries: Z₁₀, Z₅, Z₂, and Z₁.

These 8 symmetries can be seen in 10 distinct symmetries on the decagon, a larger number because the lines of reflections can either pass through vertices or edges.John Conway labels these by a letter and group order.^[7] Full symmetry of the regular form isr20 and no symmetry is labeleda1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), andi when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled asg for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only theg10 subgroup has no degrees of freedom but can be seen asdirected edges.

The highest symmetry irregular decagons ared10, anisogonal decagon constructed by five mirrors which can alternate long and short edges, andp10, anisotoxal decagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms areduals of each other and have half the symmetry order of the regular decagon.

Dissection

10-cube projection	40 rhomb dissection

Coxeter states that everyzonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected intom(m-1)/2 parallelograms.^[8]In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For theregular decagon,m=5, and it can be divided into 10 rhombs, with examples shown below. This decomposition can be seen as 10 of 80 faces in aPetrie polygon projection plane of the5-cube. A dissection is based on 10 of 30 faces of therhombic triacontahedron. The listOEIS: A006245 defines the number of solutions as 62, with 2 orientations for the first symmetric form, and 10 orientations for the other 6.

Regular decagon dissected into 10 rhombi
5-cube

Skew decagon

3 regular skew zig-zag decagons
{5}#{ }	{5/2}#{ }	{5/3}#{ }

A regular skew decagon is seen as zig-zagging edges of apentagonal antiprism, apentagrammic antiprism, and apentagrammic crossed-antiprism.

Askew decagon is askew polygon with 10 vertices and edges but not existing on the same plane. The interior of such a decagon is not generally defined. Askew zig-zag decagon has vertices alternating between two parallel planes.

Aregular skew decagon isvertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew decagon and can be seen in the vertices and side edges of apentagonal antiprism,pentagrammic antiprism, andpentagrammic crossed-antiprism with the same D_5d, [2⁺,10] symmetry, order 20.

These can also be seen in these four convex polyhedra withicosahedral symmetry. The polygons on the perimeter of these projections are regular skew decagons.

Orthogonal projections of polyhedra on 5-fold axes
Dodecahedron	Icosahedron	Icosidodecahedron	Rhombic triacontahedron

Petrie polygons

Theregular skew decagon is thePetrie polygon for many higher-dimensional polytopes, shown in theseorthogonal projections in variousCoxeter planes:^[9] The number of sides in the Petrie polygon is equal to theCoxeter number,h, for each symmetry family.

A₉	D₆		B₅
9-simplex	4₁₁	1₃₁	5-orthoplex	5-cube

See also

Decagonal number andcentered decagonal number,figurate numbers modeled on the decagon
Decagram, astar polygon with the same vertex positions as the regular decagon

References

^^a ^bSidebotham, Thomas H. (2003),The A to Z of Mathematics: A Basic Guide, John Wiley & Sons, p. 146,ISBN 9780471461630.
^Wenninger, Magnus J. (1974),Polyhedron Models, Cambridge University Press, p. 9,ISBN 9780521098595.
^The elements of plane and spherical trigonometry, Society for Promoting Christian Knowledge, 1850, p. 59. Note that this source usesa as the edge length and gives the argument of the cotangent as an angle in degrees rather than in radians.
^Ludlow, Henry H. (1904),Geometric Construction of the Regular Decagon and Pentagon Inscribed in a Circle, The Open Court Publishing Co..
^^a ^bGreen, Henry (1861),Euclid's Plane Geometry, Books III–VI, Practically Applied, or Gradations in Euclid, Part II, London: Simpkin, Marshall,& CO., p. 116. Retrieved 10 February 2016.
^^a ^bKöller, Jürgen (2005),Regelmäßiges Zehneck, → 3. Section "Formeln, Ist die Seite a gegeben ..." (in German). Retrieved 10 February 2016.
^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)
^Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
^Coxeter, Regular polytopes, 12.4 Petrie polygon, pp. 223-226.

External links

Weisstein, Eric W."Decagon".MathWorld.
Definition and properties of a decagon With interactive animation

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Polygons (List)

By number
of sides

1–10 sides	Monogon (1) Digon (2) Triangle (3) Quadrilateral (4) Pentagon (5) Hexagon (6) Heptagon (7) Octagon (8) Nonagon/Enneagon (9) Decagon (10)
11–20 sides	Hendecagon (11) Dodecagon (12) Tridecagon (13) Tetradecagon (14) Pentadecagon (15) Hexadecagon (16) Heptadecagon (17) Octadecagon (18) Icosagon (20)
>20 sides	Icositrigon (23) Icositetragon (24) Triacontagon (30) 257-gon Chiliagon (1000) Myriagon (10,000) 65537-gon Megagon (1,000,000) Apeirogon (∞)

Classes

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