Regular octahedron

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Short description: Solid with eight equal triangular faces


Regular octahedron
Typeantiprism,
bipyramid,
cross-polytope,
deltahedron,
Hanner polytope,
octahedron,
Platonic solid,
regular polyhedron,
simplicial
Faces8
Edges12
Vertices6
Schläfli symbol{3,4}
Symmetry groupoctahedral symmetry
Dihedral angle (degrees)109.47°
Dual polyhedroncube
Propertiescomposite,
convex,
isohedral,
isogonal,
isotoxal
Net

In geometry, a regular octahedron is an eight-sided polyhedron with equilateral triangles as its faces. Known for its highly symmetrical form, the regular octahedron is a Platonic solid, and more generally, a regular polyhedron. If the faces are isosceles triangles, the regular octahedron becomes a square bipyramid. The regular octahedron is an example of many classifications as deltahedron and simplicial polyhedron.

Regular octahedra occur in nature and science, such as the crystal structures and in stereochemistry as a resemblance of a chemical molecule known as octahedral molecular geometry. Other appearances are popular culture and music theory. It can be the core of polyhedra construction, and it can tile with different polyhedra to create a honeycomb.

The vertices and edges of a regular octahedron give rise to a graph, a discrete structure drawn in a plane. It is an example of a four-connected simplicial well-covered graph. It is also one of the six connected graphs in which the neighborhood of every vertex is a cycle of length four or five. Within this structure, the graph forms a topological surface called a Whitney triangulation.

Description

The regular octahedron is a polyhedron with eight equilateral triangles, where each vertex is the meet of four edges and four faces. It is one of the Platonic solids, a set of convex polyhedra whose faces are congruent regular polygons.[1] Platonic solids are the ancient set of five polyhedra named after Plato, relating them to classical elements in his Timaeus dialogue. The regular octahedron represents wind.[2] Following his attribution with nature, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids.[2] In his Mysterium Cosmographicum, Kepler also proposed the Solar System by using the Platonic solids, setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: regular octahedron, regular icosahedron, regular dodecahedron, regular tetrahedron, and cube.[3]

Sketch of a regular octahedron by Johannes Kepler
Kepler's Platonic solid model of the Solar System

A regular octahedron is the cross-polytope in 3-dimensional space. It can be oriented and scaled so that its axes align with Cartesian coordinate axes and its vertices have coordinates (±1,0,0), (0,±1,0), and (0,0,±1). Such an octahedron has edge length 2.[4]

Symmetry and duality

The dual of a regular octahedron is a cube. Both have the same symmetry.

The regular octahedron has three-dimensional symmetry groups, namely the octahedral symmetry. The regular octahedron has thirteen axes rotatonal symmetry: three axes of four-fold rotational symmetry (0°, 90°, 180°, and 270°) passing through a pair of vertices oppositing each other, four axes of three-fold rotational symmetry (0°, 120°, and 240°) passing through the center of opposite triangular faces, and six axes of two-fold rotational symmetry (0° and 180°) passing through the pair of opposite edges at their midpoints.[5] Additionally, the regular octahedron has nine reflectional planes.

The dual polyhedron can be obtained from each of the polyhedra's vertices tangent to a plane by a process known as polar reciprocation. One property of dual polyhedra is that the polyhedron and its dual share their three-dimensional symmetry point group. In the case of a regular octahedron, its dual polyhedron is the cube, and they have the same three-dimensional symmetry groups.[6] Like its dual, the regular octahedron has three properties: any two faces, two vertices, and two edges are transformed by rotation and reflection under the symmetry orbit, such that the appearance remains unchanged; these are isohedral, isogonal, and isotoxal respectively. Hence, it is considered a regular polyhedron. Four triangles surround each vertex, so the regular octahedron is 3.3.3.3 by vertex configuration or {3,4} by Schläfli symbol.[7]

Other properties

Measurements

File:Octahedron.stl The surface area A of a regular octahedron can be ascertained by summing the area of all its eight equilateral triangles, whereas its volume V is twice the volume of a square pyramid; if the edge length is a,[8] A=23a2≈3.464a2,V=132a3≈0.471a3. The radius of a circumscribed sphere ru (one that touches the octahedron at all vertices), the radius of an inscribed sphere ri (one that tangent to each of the octahedron's faces), and the radius of a midsphere rm (one that touches the middle of each edge), are:[9] ru=22a≈0.707a,ri=66a≈0.408a,rm=12a=0.5a. The dihedral angle of a regular octahedron between two adjacent triangular faces is 2arctan⁡2, which is about 109.47°. This can be obtained from the dihedral angle of an equilateral square pyramid: its dihedral angle between two adjacent triangular faces is the dihedral angle of an equilateral square pyramid between two adjacent triangular faces, and its dihedral angle between two adjacent triangular faces on the edge in which two equilateral square pyramids are attached is twice the dihedral angle of an equilateral square pyramid between its triangular face and its square base.[10]

The regular octahedron has two types of closed geodesics. The closed geodesics are the paths on a regular octahedron's surface that are locally straight. In other words, they avoid the vertices, follow line segments across the faces that they cross, and form complementary angles on the two incident faces of each edge that they cross. These geodesics have the length of 3 and 23≈3.464.[11]

Combinatorial structure

The graph of a regular octahedron. It is a complete tripartite graph, meaning the vertices can be grouped into three different colors, as shown in the illustration.

The skeleton of a regular octahedron is the (undirected) graph formed by its vertices and edges. Steinitz's theorem guarantees that the this graph can be drawn with no edge crossing another, which is 3-connected. Being k-connected means a graph remains connected whenever k−1 vertices are removed.[12][13] Its graph called the octahedral graph, a Platonic graph.[1]

The octahedral graph is a complete tripartite graph K2,2,2. It means that the octahedral graph is partitioned into three independent sets, each consisting of two opposite vertices, and there exists an edge between every pair of vertices from different independent sets.[14] It is an example of a Turán graph T6,3.

As a 4-connected simplicial, the octahedral graph is one of only four well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size (i.e., the same number of edges). The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.[15]

The octahedral graph is one of only six connected graphs in which the neighborhood of every vertex is a cycle of length four or five, the others being the Fritsch graph, the icosahedral graph, and the edge graphs of the pentagonal bipyramid, snub disphenoid and gyroelongated square bipyramid. More generally, when every vertex in a graph has a cycle of length at least four as its neighborhood, the triangles of the graph automatically link up to form a topological surface called a Whitney triangulation. These six graphs come from the six Whitney triangulations that, when their triangles are equilateral, have positive angular defect at every vertex. This makes them a combinatorial analogue of the positively curved smooth surfaces. They come from six of the eight deltahedra—excluding the two that have a vertex with a triangular neighborhood.[16]

Other appearances

Beyond its existence as a Platonic solid, the regular octahedron appears in many fields, such as nature and science, popular culture, and music theory.

In nature and science

Fluorite with octahedral structure

The natural crystals with octahedral structures are commonly discovered in diamond,[17] alum,[18] and fluorite. The plates of kamacite alloy in octahedrite-structural meteorites are arranged parallel to the eight faces of an octahedron. Many metal ions coordinate six ligands in an octahedral or distorted octahedral configuration. Widmanstätten patterns in nickel-iron crystals.

Octahedral molecular geometry is a chemical molecule resembling a regular octahedron in stereochemistry. This structure has a main-group element without an active lone pair, which can be described by a model that predicts the geometry of molecules known as VSEPR theory.[19]

The regular octahedron is the known solution of a six-electron case in Thomson problem, concerning the minimum-energy configuration of charged particles on a sphere. The solution is done by placing the vertices of a regular octahedron inscribed in a sphere.[20]

If each edge of an octahedron is replaced by a one-ohm resistor, the resistance between opposite vertices is 12 ohm, and that between adjacent vertices 512 ohm.[21]

Two identically formed Rubik's Snakes can approximate an octahedron.

In roleplaying games, this solid is known as a "d8", one of the more common polyhedral dice.[22]

In music theory

The hexany is the octahedron's orthogonal projection. Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad.[23]

As other special cases

Square bipyramid
The dual of a square bipyramid, the square prism

A regular octahedron is one of the eight convex deltahedra, polyhedra whose faces are all equilateral triangles.[24] It is a composite polyhedron constructed by attaching two equilateral square pyramids base-to-base.[25][6] When the square pyramids are a right pyramid, the regular octahedron becomes a square bipyramid, wherein its faces are all isosceles triangles.[26] In the case of a square bipyramid, its dual is a square prism.[27] Regardless of the different types of triangles, both a regular octahedron and a square bipyramid are examples of a simplicial polyhedron.

The regular octahedron is a type of trigonal antiprism, formed by taking a trigonal prism with equilateral triangle bases and rectangular lateral faces, and replacing the rectangles by alternating isosceles triangles. In the case of the regular octahedron, all of the resulting faces are congruent equilateral triangles.[28] The regular octahedron can also be considered a rectified tetrahedron, sometimes called a tetratetrahedron (by analogy to the cuboctahedron and icosidodecahedron); if alternate faces are considered to have different types (e.g. different colors or orientations), the octahedron can be considered a type of quasiregular polyhedron, a polyhedron in which two different types of polygonal faces alternate around each vertex.[29] It exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are Wythoff constructions within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.[30][31]

A regular octahedron is the three-dimensional case of the more general concept of a cross-polytope.[32]

In construction of polyhedra

The regular octahedron represents the central intersection of two tetrahedra
The truncated octahedron by removing the vertices of a regular octahedron
The triakis octahedron by attaching triangular pyramids on each face

The interior of the compound of two dual tetrahedra is an octahedron, and this compound—called the stella octangula—is its first and only stellation.[33] Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense, it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids.

Several constructions of polyhedra commence from the regular octahedron. The truncated octahedron is an Archimedean solid, constructed by removing all of the regular octahedron's vertices, resulting in six squares and eight hexagons, leaving out six square pyramids.[34] The triakis octahedron is a Catalan solid, the Kleetope of a regular octahedron, by attaching triangular pyramids onto its faces, topologically similar to the stellated octahedron.[35] The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement. It has four of the triangular faces and three central squares.[36]

One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. Five octahedra define any given icosahedron in this fashion, and together they define a regular compound. A regular icosahedron produced this way is called a "snub octahedron".[37]

Honeycomb

Tetrahedral-octahedral honeycomb by regular octahedra and tetrahedra
Rectified cubic honeycomb by regular octahedra and cuboctahedra

The Dehn invariant of a regular octahedron can be defined as a tensor product of the edge length and the dihedral angle of a regular octahedron, 12a⊗arccos⁡(−13), which is non-zero. Every polyhedron with a Dehn invariant of zero can tile a space with its copy by attaching its faces to another, forming a honeycomb. The regular octahedron, nevertheless, cannot tile a space. Instead, two different polyhedra that join to tile the space can have the Dehn invariant of zero.[38] In the case of a regular octahedron, it can tile alternately to regular tetrahedra to form a vertex, edge, and face-uniform tessellation of space, which is named tetrahedral-octahedral honeycomb.[39] R. Buckminster Fuller, in the 1950s, applied these alternating polyhedra as a space frame, which developed the strongest building structure for resisting cantilever stresses.[40] Another honeycomb is tesselating the regular octahedra alternately with cuboctahedra, named the rectified cubic honeycomb.[41]

Miscellanea

Spherical octahedron

The spherical octahedron represents a regular octahedron projected to a sphere, a part of spherical polyhedron. There are eight spherical triangles,[42] formed by the arc of great circles. Fuller identified that there are 25 great circles.

The regular octahedron is a Hanner polytope, because it can be constructed by using the direct sum of three line segments. Its dual polyhedron, the cube, is constructed by the Cartesian product of three line segments.[43]

A regular octahedron is a 3-ball in the Manhattan (ℓ1) metric.

Characteristic orthoscheme

Like all regular convex polytopes, the octahedron can be dissected into an integral number of disjoint orthoschemes, all of the same shape characteristic of the polytope. A polytope's characteristic orthoscheme is a fundamental property because the polytope is generated by reflections in the facets of its orthoscheme. The orthoscheme occurs in two chiral forms which are mirror images of each other. The characteristic orthoscheme of a regular polyhedron is a quadrirectangular irregular tetrahedron.

The faces of the octahedron's characteristic tetrahedron lie in the octahedron's mirror planes of symmetry. The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess, among its mirror planes, some that do not pass through any of its faces. The octahedron's symmetry group is denoted B3. The octahedron and its dual polytope, the cube, have the same symmetry group but different characteristic tetrahedra.

The characteristic tetrahedron of the regular octahedron can be found by a canonical dissection[44] of the regular octahedron which subdivides it into 48 of these characteristic orthoschemes surrounding the octahedron's center. Three left-handed orthoschemes and three right-handed orthoschemes meet in each of the octahedron's eight faces, the six orthoschemes collectively forming a trirectangular tetrahedron: a triangular pyramid with the octahedron face as its equilateral base, and its cube-cornered apex at the center of the octahedron.[45]

Characteristics of the regular octahedron[46]
edge arc dihedral
𝒍 2 90° π2 109°28′ π−2ψ
𝟀 43≈1.155 54°44′8″ π2−κ 90° π2
𝝉[lower-alpha 1] 1 45° π4 60° π3
𝟁 13≈0.577 35°15′52″ κ 45° π4
0R/l 2≈1.414
1R/l 1
2R/l 23≈0.816
κ 35°15′52″ arc sec 32

If the octahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths 43, 1, 13 around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),[lower-alpha 1] plus 2, 1, 23 (edges that are the characteristic radii of the octahedron). The 3-edge path along orthogonal edges of the orthoscheme is 1, 13, 23, first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to the octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 90-60-30 triangle, which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a 45-90-45 triangle with edges 1, 2, 1, a right triangle with edges 13, 1, 23, and a right triangle with edges 43, 2, 23.

Notes

  1. ↑ 1.0 1.1 (Coxeter 1973) uses the Greek letter 𝝓 (phi) to represent one of the three characteristic angles 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the golden ratio constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions and use 𝝉 to represent the characteristic angle.

See also

  • Centered octahedral number, figurate number that counts the points of a three-dimensional integer lattice that lie inside a regular octahedron centered at the origin;
  • Compound of octahedra
  • Hexakis octahedron, another polyhedron's construction involving the regular octahedron's commencement;
  • Face Turning Octahedron
  • Octahedral number, figurate number that represents the number of spheres in a regular octahedron formed from close-packed spheres;
  • Octahedral prism, an example of four-dimensional polytope;
  • Octahedral sphere, spherical shape of a cross-polytope;
  • Skewb Diamond, an octahedral version of Rubik's cube
  • Superellipsoid, a solid whose horizontal sections are of the same squareness.

References

  1. ↑ 1.0 1.1 Herrmann, Diane L.; Sally, Paul J. (2013). Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory. Taylor & Francis. p. 252. ISBN 978-1-4665-5464-1. https://books.google.com/books?id=b2fjR81h6yEC&pg=PA252. 
  2. ↑ 2.0 2.1 Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 55. ISBN 978-0-521-55432-9. https://archive.org/details/polyhedra0000crom/page/55. 
  3. ↑ The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. 2003. pp. 70–71. ISBN 0-7679-0816-3. https://books.google.com/books?id=bUARfgWRH14C. 
  4. ↑ Smith, James (2000). Methods of Geometry. John Wiley & Sons. p. 392. ISBN 978-1-118-03103-2. https://books.google.com/books?id=B0khWEZmOlwC&pg=PA392. 
  5. ↑ Cromwell (1997), p. 296.
  6. ↑ 6.0 6.1 Erickson, Martin (2011). Beautiful Mathematics. Mathematical Association of America. p. 62. ISBN 978-1-61444-509-8. https://books.google.com/books?id=LgeP62-ZxikC&pg=PA62. 
  7. ↑ Walter, Steurer; Deloudi, Sofia (2009). Crystallography of Quasicrystals: Concepts, Methods and Structures. Springer Series in Materials Science. 126. p. 50. doi:10.1007/978-3-642-01899-2. ISBN 978-3-642-01898-5. https://books.google.com/books?id=nVx-tu596twC&pg=PA50. 
  8. ↑ Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. 
  9. ↑ (Coxeter 1973) Table I(i), pp. 292–293. See the columns labeled 0R/ℓ, 1R/ℓ, and 2R/ℓ, Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses 2ℓ as the edge length (see p. 2).
  10. ↑ "Convex polyhedra with regular faces". Canadian Journal of Mathematics 18: 169–200. 1966. doi:10.4153/cjm-1966-021-8. 
  11. ↑ Fuchs, Dmitry; Fuchs, Ekaterina (2007). "Closed Geodesics on Regular Polyhedra". Moscow Mathematical Journal 7 (2): 265–279. doi:10.17323/1609-4514-2007-7-2-265-279. http://www.ams.org/distribution/mmj/vol7-2-2007/fuchs.pdf. Retrieved 2025-10-03. 
  12. ↑ "13.1 Steinitz's theorem", Convex Polytopes, Graduate Texts in Mathematics, 221 (2nd ed.), Springer-Verlag, 2003, pp. 235–244, ISBN 0-387-40409-0 
  13. ↑ "Chapter 4: Steinitz' Theorem for 3-Polytopes". Lectures on Polytopes. Graduate Texts in Mathematics. 152. Springer-Verlag. 1995. pp. 103–126. ISBN 0-387-94365-X. 
  14. ↑ Negami, S. (2016). "Faithful Embeddings of Planar Graphs on Orientable Closed Surfaces". Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, West Malvern, UK, July 2014. Springer Proceedings in Mathematics & Statistics. 159. Springer. p. 250. doi:10.1007/978-3-319-30451-9. ISBN 978-3-319-30451-9. https://books.google.com/books?id=HarWCwAAQBAJ&pg=PA250. 
  15. ↑ Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J. (2010). "On well-covered triangulations. III". Discrete Applied Mathematics 158 (8): 894–912. doi:10.1016/j.dam.2009.08.002. 
  16. ↑ Knill, Oliver (2019). "A simple sphere theorem for graphs". arXiv:1910.02708 [math.CO].
  17. ↑ Wells, Alexander Frank (1984). Structural Inorganic Chemistry (5th ed.). Oxford University Press. p. 127. ISBN 978-0-19-965763-6. https://books.google.com/books?id=uR77AAAAQBAJ&pg=PA127. 
  18. ↑ Glusker, Jenny P.; Lewis, Mitchell; Rossi, Miriam (1994). Crystal Structure Analysis for Chemists and Biologists. John Wiley & Sons. p. 45. ISBN 978-0-471-18543-7. https://books.google.com/books?id=O-wMN9CSeP0C&pg=PA45. 
  19. ↑ Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General Chemistry: Principles and Modern Applications. 1. Prentice Hall. pp. 413–414. ISBN 978-0-13-014329-7. https://books.google.com/books?id=EZEoAAAAYAAJ&pg=PA413.  See Table 11.1.
  20. ↑ "Minimal-energy clusters of hard spheres". Discrete & Computational Geometry 14 (3): 237–259. 1995. doi:10.1007/BF02570704. 
  21. ↑ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules". Croatica Chemica Acta 75 (2): 633–649. http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf. Retrieved 30 September 2006. 
  22. ↑ Slavicsek, Bill; Baker, Richard (2005). Dungeons & Dragons For Dummies. Wiley. ISBN 9780764599248. 
  23. ↑ Narushima, Terumi (2018). Microtonality and the Tuning Systems of Erv Wilson. Taylor & Francis. p. 151–155. ISBN 978-1-317-51343-8. https://books.google.com/books?id=VmkPEAAAQBAJ&pg=PA151. 
  24. ↑ "An Infinite Class of Deltahedra". Mathematics Magazine 51 (1): 55–57. 1978. doi:10.1080/0025570X.1978.11976675. 
  25. ↑ Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra". St. Petersburg Mathematical Journal 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2. https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf. 
  26. ↑ O'Keeffe, Michael; Hyde, Bruce G. (2020). Crystal Structures: Patterns and Symmetry. Dover Publications. p. 141. ISBN 978-0-486-83654-6. https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA141. 
  27. ↑ Sibley, Thomas Q. (2015). Thinking Geometrically: A Survey of Geometries. Mathematical Association of American. p. 53. ISBN 978-1-939512-08-6. https://books.google.com/books?id=EUh2CgAAQBAJ&pg=PA53. 
  28. ↑ O'Keeffe & Hyde (2020), p. 140.
  29. ↑ Maekawa, Jun (2022). Art & Science of Geometric Origami: Create Spectacular Paper Polyhedra, Waves, Spirals, Fractals, and More!. Tuttle. p. 42. ISBN 978-1-4629-2398-4. https://books.google.com/books?id=Kq-kEAAAQBAJ&pg=PA42. 
  30. ↑ Regular Polytopes (3rd ed.). Dover Publications. 1973. Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction. ISBN 0-486-61480-8. 
  31. ↑ Huson, Daniel H. (September 1998). "Two Dimensional Symmetry Mutation". https://www.researchgate.net/publication/2422380. 
  32. ↑ "The Projection of Fourfold Figures on a Three-flat". American Journal of Mathematics 15 (2): 179–189. 1893. doi:10.2307/2369565. 
  33. ↑ Cromwell (1997), p. 171, 261.
  34. ↑ Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. 10. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2. https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39. 
  35. ↑ Brigaglia, Aldo; Palladino, Nicla; Vaccaro, Maria Alessandra (2018). "Historical notes on star geometry in mathematics, art and nature". in Emmer, Michele; Abate, Marco. Imagine Math 6: Between Culture and Mathematics. Springer International Publishing. pp. 197–211. doi:10.1007/978-3-319-93949-0_17. ISBN 978-3-319-93948-3. 
  36. ↑ Posamentier, Alfred S.; Maresch, Guenter; Thaller, Bernd; Spreitzer, Christian; Geretschlager, Robert; Stuhlpfarrer, David; Dorner, Christian (2022). Geometry In Our Three-dimensional World. World Scientific. pp. 267–268. ISBN 9789811237126. https://books.google.com/books?id=DGxYEAAAQBAJ&pg=PA268. 
  37. ↑ Kappraff, Jay (1991). Connections: The Geometric Bridge Between Art and Science (2nd ed.). World Scientific. p. 475. ISBN 978-981-281-139-4. https://books.google.com/books?id=tz76s0ZGFiQC&pg=PA475. 
  38. ↑ "Polytopes that fill ℝn and scissors congruence". Discrete & Computational Geometry 13 (3–4): 573–583. 1995. doi:10.1007/BF02574064.  See Equation 4.2 and the surrounding discussion.
  39. ↑ Posamentier et al. (2022), p. 232–234.
  40. ↑ Miura, Koryo; Pellegrino, Sergio (2020). Forms and Concepts for Lightweight Structures. Cambridge University Press. p. 28. doi:10.1017/9781139048569. ISBN 9781139048569. https://books.google.com/books?id=v9LQDwAAQBAJ&pg=PA28. 
  41. ↑ Posamentier et al. (2022), p. 234–235.
  42. ↑ Synergetics: Explorations in the Geometry of Thinking. Macmillan Publishing. 1982. p. 183. ISBN 978-0-02-065320-2. https://books.google.com/books?id=AKDgDQAAQBAJ&pg=PA183. 
  43. ↑ Kozachok, Marina (2012). "Yaroslavl International Conference "Discrete Geometry" Dedicated to the Centenary of A.D. Alexandrov (Yaroslavl, August 13-18, 2012)". P.G. Demidov Yaroslavl State University, International B.N. Delaunay Laboratory. pp. 46–49. https://www.dcglab.uniyar.ac.ru/sites/default/files/papers/Alexandrov2012Thesis.pdf#page=46. 
  44. ↑ Coxeter 1973, p. 130, §7.6 The symmetry group of the general regular polytope; "simplicial subdivision".
  45. ↑ Coxeter 1973, pp. 70–71, Characteristic tetrahedra; Fig. 4.7A.
  46. ↑ Coxeter 1973, pp. 292–293, Table I(i); "Octahedron, 𝛽3".
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds



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