ثماني الأضلاع

Regular octagon
Regular octagon
	A regular octagon
النوع	مضلع منتظم
الأضلاع والرؤوس	{{{p8 جانب}}}
رمز شلفلي	{{{p8-شلفلي}}}
مخططات كوكستر-دنكن	;
مجموعة التماثل	Dihedral (D8), order 2×8
الزاوية الداخلية (الدرجات)	135°
الخصائص	Convex, cyclic, equilateral, isogonal, isotoxal

في الهندسة الرياضية، ثماني الأضلاع هو مضلع له 8 أضلاع.

A regular octagon has Schläfli symbol {8} ^[1] and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, {16}. A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square.Note: We can make Octagon by compass & by protector. There are 2 ways of making an Octagon. They are; 1^st= On the basis of side & 2^nd= On the basis of radious(Circle)

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خصائص المثمّن العام

The diagonals of the green quadrilateral are equal in length and at right angles to each other

The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°.

If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).^[2]^{:Prop. 9}

The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.^[2]^{:Prop. 10}

ثماني منتظم (مثمن)

مثمن .

المثمن هو ثماني أضلاع أضلاعه متساوية وزواياه متساوية.
قياس الزاوية الداخلية يساوي 135°.
مجموع قياسات زواياه الداخلية 1080°.
مساحة المثمن المنتظم الذي طول ضلعه a تعطى بالعلاقة:

A=2\cot {\frac {\pi }{8}}a^{2}=2(1+{\sqrt {2}})a^{2}\simeq 4.828427\,a^{2}.

وتعطى المساحة بدلالة R نصف قطر الدائرة المحيطة به بالعلاقة:

A=4\sin {\frac {\pi }{4}}R^{2}=2{\sqrt {2}}R^{2}\simeq 2.828427\,R^{2}.

وتعطى المساحة بدلالة r نصف قطر الدائرة المحاطة داخله بالعلاقة:

A=8\tan {\frac {\pi }{8}}r^{2}=8({\sqrt {2}}-1)r^{2}\simeq 3.3137085\,r^{2}.

الأقطار

The regular octagon, in terms of the side length a, has three different types of diagonals:

القطر القصير;
Medium diagonal (also called span or height), which is twice the length of the inradius;
Long diagonal, which is twice the length of the circumradius.

The formula for each of them follows from the basic principles of geometry. Here are the formulas for their length:^{[بحاجة لمصدر]}

القطر القصير: $a{\sqrt {2+{\sqrt {2}}}}$ ;
Medium diagonal: $(1+{\sqrt {2}})a$ ; (silver ratio times a)
Long diagonal: $a{\sqrt {4+2{\sqrt {2}}}}$ .

الإنشاء والخصائص الابتدائية

building a regular octagon by folding a sheet of paper

A regular octagon at a given circumcircle may be constructed as follows:

Draw a circle and a diameter AOE, where O is the center and A, E are points on the circumcircle.
Draw another diameter GOC, perpendicular to AOE.
(Note in passing that A,C,E,G are vertices of a square).
Draw the bisectors of the right angles GOA and EOG, making two more diameters HOD and FOB.
A,B,C,D,E,F,G,H are the vertices of the octagon.

Octagon at a given circumcircle

Octagon at a given side length, animation
(The construction is very similar to that of hexadecagon at a given side length.)

A regular octagon can be constructed using a straightedge and a compass, as 8 = 2³, a power of two:

Meccano octagon construction.

The regular octagon can be constructed with meccano bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required.

Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices. Its area can thus be computed as the sum of 8 isosceles triangles, leading to the result:

{\text{Area}}=2a^{2}({\sqrt {2}}+1)

for an octagon of side a.

الإحداثيات القياسية

The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:

(±1, ±(1+√2))
(±(1+√2), ±1).

تشريح

إسقاط 8-cube	تشريح 24 معيّن
	منتظم	Isotoxal

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.^[3] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octagon, m=4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in a Petrie polygon projection plane of the tesseract. The list (المتتالية A006245 في OEIS) defines the number of solutions as 8, by the 8 orientations of this one dissection. هذه المربعات والمعينات تُستخدم في تبليطات أمان-بينكر.

مثمن منتظم مشرّح
تسراكت	4 معيّنات و مربعان

مثمن منحرف

A regular skew octagon seen as edges of a square antiprism, symmetry D_4d, [2⁺,8], (2*4), order 16.

A skew octagon is a skew polygon with 8 vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zig-zag octagon has vertices alternating between two parallel planes.

A regular skew octagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew octagon and can be seen in the vertices and side edges of a square antiprism with the same D_4d, [2⁺,8] symmetry, order 16.

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مضلعات پتري

The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A₇, B₄, and D₅ Coxeter planes.

A₇	D₅	B₄
7-simplex	5-demicube	16-cell	Tesseract

تناظر المثمن

التناظر
	The 11 symmetries of a regular octagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position.

The regular octagon has Dih₈ symmetry, order 16. There are 3 dihedral subgroups: Dih₄, Dih₂, and Dih₁, and 4 cyclic subgroups: Z₈, Z₄, Z₂, and Z₁, the last implying no symmetry.

أمثلة مثمنات حسب التماثل
r16
d8	g8	p8
d4	g4	p4
d2	g2	p2
a1

On the regular octagon, there are 11 distinct symmetries. John Conway labels full symmetry as r16.^[4] The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r16 and no symmetry is labeled a1.

The most common high symmetry octagons are p8, an isogonal octagon constructed by four mirrors can alternate long and short edges, and d8, an isotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular octagon.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g8 subgroup has no degrees of freedom but can seen as directed edges.

استخدامات المثمنات

The octagonal floor plan, Dome of the Rock.

The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan. The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's Cathedral, Addis Ababa, Basilica of San Vitale (in Ravenna, Italia), Castel del Monte (Apulia, Italia), Florence Baptistery, Zum Friedefürsten Church (Germany) and a number of octagonal churches in Norway. The central space in the Aachen Cathedral, the Carolingian Palatine Chapel, has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral.

Architects such as John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, notably the Intelsat Headquarters in Washington D.C., Callam Offices in Canberra, and Octagon Offices in Parramatta, Australia.

استخدامات أخرى

Umbrellas often have an octagonal outline.
The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.
The street & block layout of Barcelona's Eixample district is based on non-regular octagons
Janggi uses octagonal pieces.
Japanese lottery machines often have octagonal shape.
Stop sign used in English-speaking countries, as well as in most European countries
An icon of a stop sign with a hand at the middle.
The trigrams of the Taoist bagua are often arranged octagonally
Famous octagonal gold cup from the Belitung shipwreck
Classes at Shimer College are traditionally held around octagonal tables
The Labyrinth of the Reims Cathedral with a quasi-octagonal shape.
The movement of the analog stick(s) of the Nintendo 64 controller, the GameCube controller, the Wii Nunchuk and the Classic Controller is restricted by a rotated octagonal area, allowing the stick to move in only eight different directions.

أشكال مشتقة

The truncated square tiling has 2 octagons around every vertex.
An octagonal prism contains two octagonal faces.
An octagonal antiprism contains two octagonal faces.
The truncated cuboctahedron contains 6 octagonal faces.
The omnitruncated cubic honeycomb

كثيرو الجوانب ذوو الصلة

The octagon, as a truncated square, is first in a sequence of truncated hypercubes: قالب:Truncated hypercube polytopes As an expanded square, it is also first in a sequence of expanded hypercubes: قالب:Expanded hypercube polytopes

انظر أيضاً

Bumper pool
Octagon house
Octagonal number
Octagram
Octahedron, 3D shape with eight faces.
Oktogon, a major intersection in Budapest, Hungary
Rub el Hizb (also known as Al Quds Star and as Octa Star)
Smoothed octagon

المراجع

^ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595, https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9 .
^ ^أ ^ب Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", Forum Geometricorum 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html
^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
^ 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)

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وصلات خارجية

Octagon Calculator
Definition and properties of an octagon With interactive animation

[1] Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595, https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9 .

[Oai-2] أ ^ب Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", Forum Geometricorum 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html

[3] Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

[4] 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)

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