polytope

See also: Polytope

English

A 3-dimensional Stasheff polytope

Etymology

Learned borrowing from German Polytop, originally coined by German mathematician Reinhold Hoppe in 1882, and first used in English by British mathematician Alicia Boole Stott in her 1910 paper Geometrical deduction of semiregular from regular polytopes and space fillings. By surface analysis, poly- (“many”) +‎ -tope (“surface”), from Ancient Greek τόπος (tópos, “region; area”).

Pronunciation

• (UK) IPA^(key): /ˈpɒlɪˌtəʊp/, /ˈpɒliːˌtəʊp/
• (US) IPA^(key): /ˈpɑliˌtoʊp/

• Audio (US):

  (file)

• Hyphenation: pol‧y‧tope
• Rhymes: -ɒlɪtəʊp, -ɒliːtəʊp

Noun

polytope (plural polytopes)

2. (geometry) A geometric shape (of any number of dimensions) which is fully enclosed and has flat sides, making it a member of the generalized class of shapes which includes the two-dimensional polygon and three-dimensional polyhedron; (formally) a finite region of n-dimensional space bounded by hyperplanes.
   • 1964, Victor Klee, On the Number of Vertices of a Convex Polytope: Canadian Journal of Mathematics, volume XVI, number 4, page 701:

     As is well known, the theory of linear inequalities is closely related to the study of convex polytopes.

   • 1998, F. Pierrot, M. Benoit, P. Dauchez, “SamoS: A Pythagorean Solution for Omnidirectional Underwater Vehicles”, in Jadran Lenar I, Manfred L. Husty, editors, Advances in Robot Kinematics: Analysis and Control, page 220:

     This polytope is mapped into a Cartesian force polytope (resp. torque polytope) in the Cartesian space. Such a polytope represents the exact force (resp. torque) that can be produced on the vehicle main body.

   • 2006, Rekha R. Thomas, Lectures in Geometric Combinatorics, page 27:

     Verify the Hirsch conjecture for the 3-cube, 4-cube and any other polytope that takes your fancy.
     The Steinitz theorem is a very satisfactory understanding of the graphs of three-dimensional polytopes.

Hyponyms

classes of a given dimension

• polygon (“2-dimensional shape”)
• polyhedron (“3-dimensional shape”)
• polychoron (“4-dimensional shape”)

• hypercube (“generalized cube of n-dimensions”)
• simplex (“generalized tetrahedron of n-dimensions”)
• tesseract (“4-dimensional equivalent to the cube”)

Derived terms

• 0-polytope
• 1-polytope
• 2-polytope
• 3-polytope
• 4-polytope
• 5-polytope
• abstract polytope
• Birkhoff polytope
• chiral polytope
• convex polytope
• cross-polytope
• cyclic polytope
• Hanner polytope
• integral polytope
• matroid polytope
• n-polytope
• parallelotope
• pentagonal polytope
• polytopal
• random polytope
• regular polytope
• uniform polytope
• zonotope

Related terms

• polytopic

Translations

geometric shape

• Catalan: polítop m
• Chinese:

  Mandarin: 多胞形 (zh) (duōbāoxíng)

• Esperanto: hiperpluredro
• French: polytope (fr) m
• German: Polytop m
• Greek: πολύτοπο n (polýtopo)
• Hungarian: politóp (hu)
• Italian: politopo (it) m
• Japanese: ポリトープ (poritōpu), 超多面体 (ちょうためんたい, chōtamentai)
• Korean: 다포체(多胞體) (dapoche)
• Persian: چندبر (fa)
• Polish: wielotop m, wielokomórka f, politop m
• Portuguese: politopo m
• Russian: полито́п (ru) m (politóp)
• Spanish: politopo m
• Swedish: polytop (sv) c
• Ukrainian: політо́п m (politóp)

French

Etymology

Borrowed from German Polytop. By surface analysis, poly- +‎ -tope, from Ancient Greek τόπος (tópos, “region; area”).

Pronunciation

• IPA^(key): /pɔ.li.tɔp/

• Audio (France (Vosges)):

  (file)

Noun

polytope m (plural polytopes)

1. (geometry) polytope