نجمة ثمانية

(تم التحويل من Octagram)
Regular octagram
Regular star polygon 8-3.svg
octagram منتظم
النوعمضلع نجمي منتظم
الأضلاع والرؤوس{{{p8/3-جوانب}}}
رمز شلفلي{{{{p8/3-شلفلي}}}}
مخططات كوكستر-دنكنCDel node 1.pngCDel 8.pngCDel rat.pngCDel d3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel rat.pngCDel d3.pngCDel node 1.png
مجموعة التماثلثنائي الأسطح (D{{{p8/3-جوانب}}})
الزاوية الداخلية (الدرجات){{{زاوية p8/3}}}°
الخصائصنجمي، دائري، متساوي الأضلاع، متوازي، متجانس الأضلاع

في الهندسة، النجمة الثمانية octagram، هي مضلع ثماني الرؤوس.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

تفاصيل

A regular octagram with each side length equal to 1

عموماً، an octagram is any self-intersecting octagon (8-sided polygon).

The regular octagram is labeled by the Schläfli symbol {8/3}, which means an 8-sided star, connected by every third point.


تنويعات

These variations have a lower dihedral, Dih4, symmetry:

Regular truncation 4 1.5.svg
Narrow
Regular truncation 4 2.svg
Wide
(45 degree rotation)
Isotoxal octagram.png
Octagram-in-square.svg
Isotoxal
Ancient mapuche flag.svg
An old Flag of Chile contained this octagonal star geometry with edges removed (the Guñelve).
KRG1891.svg
The regular octagonal star is very popular as a symbol of rowing clubs in the Cologne Lowland, as seen on the club flag of the Cologne Rowing Association.
Star Guñelve.svg
The geometry can be adjusted so 3 edges cross at a single point, like the Auseklis symbol
Compass rose en 08p.svg
An 8-point compass rose can be seen as an octagonal star, with 4 primary points, and 4 secondary points.

The symbol Rub el Hizb is a Unicode glyph ۞  at U+06DE.

As a quasitruncated square

Deeper truncations of the square can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated square is an octagon, t{4}={8}. A quasitruncated square, inverted as {4/3}, is an octagram, t{4/3}={8/3}.[1]

The uniform star polyhedron stellated truncated hexahedron, t'{4,3}=t{4/3,3} has octagram faces constructed from the cube in this way. It may be considered for this reason as a three-dimensional analogue of the octagram.

Isogonal truncations of square and cube
Regular Quasiregular Isogonal Quasiregular
Regular quadrilateral.svg
{4}
Regular polygon truncation 4 1.svg
t{4}={8}
Regular polygon truncation 4 2.svg Regular polygon truncation 4 3.svg
t'{4}=t{4/3}={8/3}
Regular Uniform Isogonal Uniform
Cube truncation 0.00.png
{4,3}
Cube truncation 0.50.png
t{4,3}
Cube truncation 3.50.png Cube truncation 2.50.png
t'{4,3}=t{4/3,3}

Another three-dimensional version of the octagram is the nonconvex great rhombicuboctahedron (quasirhombicuboctahedron), which can be thought of as a quasicantellated (quasiexpanded) cube, t0,2{4/3,3}.

مركبات مضلعات نجمية

There are two regular octagrammic star figures (compounds) of the form {8/k}, the first constructed as two squares {8/2}=2{4}, and second as four degenerate digons, {8/4}=4{2}. There are other isogonal and isotoxal compounds including rectangular and rhombic forms.

Regular Isogonal Isotoxal
Regular star figure 2(4,1).svg
a{8}={8/2}=2{4}
Regular star figure 4(2,1).svg
{8/4}=4{2}
Octagram rectangle compound.png Octagram crossed-rectangle compound.png Octagram rhombic star.png

{8/2} or 2{4}, like Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.png + CDel node.pngCDel 4.pngCDel node 1.png, can be seen as the 2D equivalent of the 3D compound of cube and octahedron, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png + CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png, 4D compound of tesseract and 16-cell, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png + CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png and 5D compound of 5-cube and 5-orthoplex; that is, the compound of a n-cube and cross-polytope in their respective dual positions.


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

تمثيلات أخرى لنجم مضلع ثماني

An octagonal star can be seen as a concave hexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines.

star polygon Concave Central dissections
Squared octagonal-star3.svg
Compound 2{4}
Squared octagonal-star0.svg
|8/2|
Squared octagonal-star1.svg Squared octagonal-star4.svg Squared octagonal-star2.svg
Regular octagram star3.svg
Regular {8/3}
Regular octagram star0.svg
|8/3|
Regular octagram star1.svg Regular octagram star4.svg Regular octagram star2.svg
Auseklis star3.svg
Isogonal
Auseklis star0.svg Auseklis star1.svg Auseklis star4.svg Auseklis star2.svg
Square-compass-star3.svg
Isotoxal
Square-compass-star0.svg Square-compass-star1.svg Square-compass-star4.svg Square-compass-star2.svg

استخدامات أخرى

  • In Unicode, the "Eight Spoked Asterisk" symbol ✳ is U+2733.

انظر أيضا

الاستخدام
النجوم عموماً

المصادر

  1. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
  • Grünbaum, B. and G.C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1.
  • Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)

وصلات خارجية