متعدد سطوح بالإنگليزية: polyhedron أومتعدد أوجه هومجسّم مضلع في فضاء ثلاثي الأبعاد ويستخدم حالياً في فضاءات ذات أبعادٍ أكثر. لمتعدد الأوجه أوجه مستوية وحواف مستقيمة.

أساسيات التعريف

الخصائص

Polyhedral surface

A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.

Edges

Edges have two important characteristics (unless the polyhedron is complex):

• An edge joins just two vertices.
• An edge joins just two faces.

These two characteristics are dual to each other.

محددة اويلر

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

${\displaystyle \chi =V-E+F.\$

For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected and whose boundary is a manifold, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos.

Orientability

Some polyhedra, such as all convex polyhedra, have two distinct sides to their surface, for example one side can consistently be coloured black and the other white. We say that the figure is orientable.

But for some polyhedra this is not possible, and the figure is said to be non-orientable. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even χ < 2 may or may not be orientable.

Vertex figure

For every vertex one can define a vertex figure, which describes the local structure of the figure around the vertex. If the vertex figure is a regular polygon, then the vertex itself is said to be regular.

Duality

For every polyhedron there exists a dual polyhedron having:

• faces in place of the original's vertices and vice versa,
• the same number of edges
• the same Euler characteristic and orientability

The dual of a convex polyhedron can be obtained by the process of polar reciprocation.

الحجم

الأشكال التقليدية لمتعدد السطوح

متعددات السطوح متماثلة

Many of the most studied polyhedra are highly symmetrical.

Uniform polyhedra and their duals

منطق رئيسي: Uniform polyhedron

Noble polyhedra

Symmetry groups

The polyhedral symmetry groups are all point groups and include:

• T - chiral tetrahedral symmetry; the rotation group for a regular tetrahedron; order 12.
• T_d - full tetrahedral symmetry; the symmetry group for a regular tetrahedron; order 24.
• T_h - pyritohedral symmetry; order 24. The symmetry of a pyritohedron.
• O - chiral octahedral symmetry;the rotation group of the cube and octahedron; order 24.
• O_h - full octahedral symmetry; the symmetry group of the cube and octahedron; order 48.
• I - chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron; order 60.
• I_h - full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron; order 120.
• C_nv - -fold pyramidal symmetry
• D_nh - -fold prismatic symmetry
• D_nv - -fold antiprismatic symmetry

Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. The snub Archimedean polyhedra have this property.

Other polyhedra with regular faces

Equal regular faces

A few families of polyhedra, where every face is the same kind of polygon:

• Deltahedra have equilateral triangles for faces.

• With regard to polyhedra whose faces are all squares: if coplanar faces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (a,b,c), with integers a,b,c of which at most one is even.

• There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron.

There exists no polyhedron whose faces are all identical and are regular polygons with six or more sides because the vertex of three regular hexagons defines a plane. (See infinite skew polyhedron for exceptions with zig-zagging vertex figures.)

Deltahedra

A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:

• 3 regular convex polyhedra (3 of the Platonic solids)
  • Tetrahedron
  • Octahedron
  • Icosahedron
• 5 non-uniform convex polyhedra (5 of the Johnson solids)
  • Triangular dipyramid
  • Pentagonal dipyramid
  • Snub disphenoid
  • Triaugmented triangular prism
  • Gyroelongated square dipyramid

Johnson solids

Other important families of polyhedra

Pyramids

Pyramids include some of the most time-honoured and famous of all polyhedra.

Stellations and facettings

منطق رئيسي: Stellation

Zonohedra

Compounds

Orthogonal polyhedra

Generalisations of polyhedra

Apeirohedra

Complex polyhedra

Curved polyhedra

Spherical polyhedra

Curved spacefilling polyhedra

General polyhedra

Hollow faced or skeletal polyhedra

Tessellations or tilings

Non-geometric polyhedra

Topological polyhedra

Abstract polyhedra

Polyhedra as graphs

History

Prehistory

Greeks

Muslims and Chinese

Renaissance

متعدد السطوح النجمي

انظر أيضا:

• Regular polyhedron: History
• Regular polytope: History of discovery.

متعدد السطوح في الطبيعة

المصادر

• Coxeter, H.S.M.; Regular complex Polytopes, CUP (1974).
• Cromwell, P.;Polyhedra, CUP hbk (1997), pbk. (1999).
• 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.
• Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461–488. (pdf)
• Pearce, P.; Structure in nature is a strategy for design, MIT (1978)

خط عن متعدد السطوح

خط للقراء، وللاستعمال المدرسي

• Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
• Cundy, H.M. & Rollett, A.P.; Mathematical models, 1st Edn. hbk OUP (1951), 2nd Edn. hbk OUP (1961), 3rd Edn. pbk Tarquin (1981).
• Holden; Shapes, space and symmetry, (1971), Dover pbk (1991).
• Pearce, P and Pearce, S: Polyhedra primer, Van Nost. Reinhold (May 1979), ISBN 0442264968, ISBN 978-0442264963.
• Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
• Senechal, M. & Fleck, G.; Shaping Space a Polyhedral Approch, Birhauser (1988), ISBN 0817633510
• Tarquin publications: books of cut-out and make card models.
• Wenninger, Magnus; Polyhedron models for the classroom, pbk (1974)
• Wenninger, M.; Polyhedron models, CUP hbk (1971), pbk (1974).
• Wenninger, M.; Spherical models, CUP.
• Wenninger, M.; Dual models, CUP.

خط جامعية

• Coxeter, H.S.M. DuVal, Flather & Petrie; The fifty-nine icosahedra, 3rd Edn. Tarquin.
• Coxeter, H.S.M. Twelve geometric essays. Republished as The beauty of geometry, Dover.
• Thompson, Sir D'A. W. On growth and form, (1943). (not sure if this is the right category for this one, I haven't read it).

التصميم والعمارة

• Critchlow, K.; Order in space.
• Pearce, P.; Structure in nature is a strategy for design, MIT (1978)
• Williams, R.; The geometrical foundation of natural structure, Dover (1979).

نصوص رياضية متقدمة

• Coxeter, H.S.M.; Regular Polytopes 3rd Edn. Dover (1973).
• Coxeter, H.S.M.; Regular complex polytopes, CUP (1974).
• Lakatos, Imre; Proofs and Refutations, Cambridge University Press (1976) - discussion of proof of Euler characteristic
• Several more to add here.

خط تاريخية

• Brückner, M. (1900). . Leipzig: B.G. Treubner. ISBN 978-1418165901. (بالألمانية)

WorldCat English: Polygons and Polyhedra: Theory and History.

• Fejes Toth, L.;
• Kepler, J.; De harmonices Mundi (Latin. Available in English translation).
• Pacioli, L.;

انظر أيضا

وصلات خارجية

النظرية العامة

• Eric W. Weisstein, Polyhedron at MathWorld.
• Polyhedra Pages
• uniform solution for uniform polyhedra by Dr. Zvi Har'El
• Symmetry, Crystals and Polyhedra
• Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons

قوائم قواعد البيانات عن متعدد السطوح

• The Uniform Polyhedra
• Virtual Reality Polyhedra - The Encyclopedia of Polyhedra
  • Polyhedra and Pyramids
• Interactive 3D polyhedra in Java
• Electronic Geometry Models - Contains a peer reviewed selection of polyhedra with unusual properties.
• Origami Polyhedra - Models made with Modular Origami
• Polyhedra Collection - Various virtual and physical polyhedra models.
• Polyhedra plaited with paper strips - Polyhedra models constructed without use of glue.
• Rotatable polyhedron models - that work right in your web browser
• Rotatable non convex self intersecting polyhedra these ones also work right in your browser

برمجيات

• A Plethora of Polyhedra An interactive and free collection of polyhedra in JAVA. Features includes nets, planar sections, duals, truncations and stellations of more than 300 polyhedra.
• Stella: Polyhedron Navigator - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
• World of Polyhedra - Comprehensive polyhedra in flash applet, showing vertices and edges (but not shaded faces)
• Hyperspace Star Polytope Slicer - Explorer java applet, includes a variety of 3d viewer options.

مصادر لعمل نماذج ، نماذج لبيع

• Making Polyhedra
• Paper Models of Polyhedra Free nets of polyhedra
• Paper Models of Uniform (and other) Polyhedra
• Polyhedra software, die-cast models, & posters
• Simple instructions for building over 30 paper polyhedra

متفرقات

• PictureSpice - A site that lets you make polyhedra with your own uploaded pictures.
• uniform polyhedra in 3d

نطقب:Polyhedra نطقب:Polyhedron navigator