Wikipedia

Cube

For other uses, seeCube (disambiguation).

Ingeometry,acubeorregular hexahedronis athree-dimensionalsolid object bounded by six congruentsquarefaces, a type ofpolyhedron.It has twelve congruent edges and eight vertices. It is a type ofparallelepiped,with pairs of parallel opposite faces, and more specifically arhombohedron,with congruent edges, and arectangular cuboid,withright anglesbetween pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra:Platonic solid,regular polyhedron,parallelohedron,zonohedron,andplesiohedron.Thedual polyhedronof a cube is theregular octahedron.

Cube
TypePlatonic solid
Regular polyhedron
Parallelohedron
Zonohedron
Plesiohedron
Hanner polytope
Faces6
Edges12
Vertices8
Symmetry groupoctahedral symmetry
Dihedral angle(degrees)90°
Dual polyhedronregular octahedron
Propertiesconvex,
face-transitive,
edge-transitive,
vertex-transitive,
non-composite

The cube is the three-dimensionalhypercube,a family ofpolytopesalso including the two-dimensional square and four-dimensionaltesseract.A cube withunitside length is the canonical unit ofvolumein three-dimensional space, relative to which other solid objects are measured.

The cube can be represented in many ways, one of which is the graph known as thecubical graph.It can be constructed by using theCartesian product of graphs.The cube was discovered in antiquity. It was associated with the nature ofearthbyPlato,the founder of Platonic solid. It was used as the part of theSolar System,proposed byJohannes Kepler.It can be derived differently to create more polyhedrons, and it has applications to construct a newpolyhedronby attaching others.

Properties

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A cube is a special case ofrectangular cuboidin which the edges are equal in length.[1]Like other cuboids, every face of a cube has four vertices, each of which connects with three congruent lines. These edges form square faces, making thedihedral angleof a cube between every two adjacent squares being theinterior angleof a square, 90°. Hence, the cube has six faces, twelve edges, and eight vertices.[2]Because of such properties, it is categorized as one of the fivePlatonic solids,apolyhedronin which all theregular polygonsarecongruentand the same number of faces meet at each vertex.[3]

Measurement and other metric properties

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A face diagonal in red and space diagonal in blue.
ThePrince Rupert's cube

Given that a cube with edge length.Theface diagonalof a cube is thediagonalof a square,and thespace diagonalof a cube is a line connecting two vertices that is not in the same face, formulated as.Both formulas can be determined by usingPythagorean theorem.The surface area of a cubeis six times the area of a square:[4] The volume of a cuboid is the product of length, width, and height. Because the edges of a cube are all equal in length, it is:[4]

One special case is theunit cube,so-named for measuring a singleunit of lengthalong each edge. It follows that each face is aunit squareand that the entire figure has a volume of 1 cubic unit.[5][6]Prince Rupert's cube,named afterPrince Rupert of the Rhine,is the largest cube that can pass through a hole cut into the unit cube, despite having sides approximately 6% longer.[7]A polyhedron that can pass through a copy of itself of the same size or smaller is said to have theRupert property.[8]

A unit cube and a cube with twice the volume

A geometric problem ofdoubling the cube—alternatively known as theDelian problem—requires the construction of a cube with a volume twice the original by using acompass and straightedgesolely. Ancient mathematicians could not solve this old problem until French mathematicianPierre Wantzelin 1837 proved it was impossible.[9]

Relation to the spheres

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With edge length,theinscribed sphereof a cube is the sphere tangent to the faces of a cube at their centroids, with radius.Themidsphereof a cube is the sphere tangent to the edges of a cube, with radius.Thecircumscribed sphereof a cube is the sphere tangent to the vertices of a cube, with radius.[10]

For a cube whose circumscribed sphere has radius,and for a given point in its three-dimensional space with distancesfrom the cube's eight vertices, it is:[11]

Symmetry

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The cube hasoctahedral symmetry.It is composed ofreflection symmetry,a symmetry by cutting into two halves by a plane. There are nine reflection symmetries: the five are cut the cube from the midpoints of its edges, and the four are cut diagonally. It is also composed ofrotational symmetry,a symmetry by rotating it around the axis, from which the appearance is interchangeable. It has octahedral rotation symmetry:three axes pass through the cube's opposite faces centroid, six through the cube's opposite edges midpoints, and four through the cube's opposite vertices; each of these axes is respectively four-fold rotational symmetry (0°, 90°, 180°, and 270°), two-fold rotational symmetry (0° and 180°), and three-fold rotational symmetry (0°, 120°, and 240°).[12][13][14]

The dual polyhedron of a cube is the regular octahedron

Thedual polyhedroncan be obtained from each of the polyhedron's vertices tangent to a plane by the process known aspolar reciprocation.[15]One property of dual polyhedrons generally is that the polyhedron and its dual share theirthree-dimensional symmetry point group.In this case, the dual polyhedron of a cube is theregular octahedron,and both of these polyhedron has the same symmetry, the octahedral symmetry.[16]

The cube isface-transitive,meaning its two squares are alike and can be mapped by rotation and reflection.[17]It isvertex-transitive,meaning all of its vertices are equivalent and can be mappedisometricallyunder its symmetry.[18]It is alsoedge-transitive,meaning the same kind of faces surround each of its vertices in the same or reverse order, all two adjacent faces have the samedihedral angle.Therefore, the cube isregular polyhedronbecause it requires those properties.[19]

Classifications

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3D model of a cube

The cube is a special case among everycuboids.As mentioned above, the cube can be represented as therectangular cuboidwith edges equal in length and all of its faces are all squares.[1]The cube may be considered as theparallelepipedin which all of its edges are equal edges.[20]

The cube is aplesiohedron,a special kind of space-filling polyhedron that can be defined as theVoronoi cellof a symmetricDelone set.[21]The plesiohedra include theparallelohedrons,which can betranslatedwithout rotating to fill a space—calledhoneycomb—in which each face of any of its copies is attached to a like face of another copy. There are five kinds of parallelohedra, one of which is the cuboid.[22]Every three-dimensional parallelohedron iszonohedron,acentrally symmetricpolyhedron whose faces arecentrally symmetric polygons,[23]

Construction

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Nets of a cube

An elementary way to construct a cube is using itsnet,an arrangement of edge-joining polygons constructing a polyhedron by connecting along the edges of those polygons. Eleven nets for the cube are shown here.[24]

Inanalytic geometry,a cube may be constructed using theCartesian coordinate systems.For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, theCartesian coordinatesof the vertices are.[25]Its interior consists of all pointswithfor all.A cube's surface with centerand edge length ofis thelocusof all pointssuch that

The cube isHanner polytope,because it can be constructed by usingCartesian productof three line segments. Its dual polyhedron, the regular octahedron, is constructed bydirect sumof three line segments.[26]

Representation

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As a graph

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Main article:Hypercube graph
The graph of a cube, and its construction

According toSteinitz's theorem,thegraphcan be represented as theskeletonof a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties. It isplanar,meaning the edges of a graph are connected to every vertex without crossing other edges. It is also a3-connected graph,meaning that, whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected.[27][28]The skeleton of a cube can be represented as the graph, and it is called thecubical graph,aPlatonic graph.It has the same number of vertices and edges as the cube, twelve vertices and eight edges.[29]

The cubical graph is a special case ofhypercube graphor-cube—denoted as—because it can be constructed by using the operation known as theCartesian product of graphs.To put it in a plain, its construction involves two graphs connecting the pair of vertices with an edge to form a new graph.[30]In the case of the cubical graph, it is the product of two;roughly speaking, it is a graph resembling a square. In other words, the cubical graph is constructed by connecting each vertex of two squares with an edge. Notationally, the cubical graph can be denoted as.[31]As a part of the hypercube graph, it is also an example of aunit distance graph.[32]

Like other graphs of cuboids, the cubical graph is also classified as aprism graph.[33]

In orthogonal projection

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An object illuminated by parallel rays of light casts a shadow on a plane perpendicular to those rays, called anorthogonal projection.A polyhedron is consideredequiprojectiveif, for some position of the light, its orthogonal projection is a regular polygon. The cube is equiprojective because, if the light is parallel to one of the four lines joining a vertex to the opposite vertex, its projection is aregular hexagon.Conventionally, the cube is 6-equiprojective.[34]

As a configuration matrix

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The cube can be represented asconfiguration matrix.A configuration matrix is amatrixin which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. Thediagonalof a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element. As mentioned above, the cube has eight vertices, twelve edges, and six faces; each element in a matrix's diagonal is denoted as 8, 12, and 6. The first column of the middle row indicates that there are two vertices in (i.e., at the extremes of) each edge, denoted as 2; the middle column of the first row indicates that three edges meet at each vertex, denoted as 3. The following matrix is:[35]

Appearances

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In antiquity

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Sketch of a cube by Johannes Kepler
Kepler'sPlatonic solid model of theSolar System

ThePlatonic solidis a set of polyhedrons known since antiquity. It was named afterPlatoin hisTimaeusdialogue, who attributed these solids with nature. One of them, the cube, represented theclassical elementofearthbecause of its stability.[36]Euclid'sElementsdefined the Platonic solids, including the cube, and using these solids with the problem involving to find the ratio of the circumscribed sphere's diameter to the edge length.[37]

Following its attribution with nature by Plato,Johannes Keplerin hisHarmonices Mundisketched each of the Platonic solids, one of them is a cube in which Kepler decorated a tree on it.[36]In hisMysterium Cosmographicum,Kepler also proposed theSolar Systemby 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.[38]

Polyhedron, honeycombs, and polytopes

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Some of the derived cube, thestellated octahedronandtetrakis hexahedron.

The cube can appear in the construction of a polyhedron, and some of its types can be derived differently in the following:

  • Whenfacetinga cube, meaning removing part of the polygonal faces without creating new vertices of a cube, the resulting polyhedron is thestellated octahedron.[39]
  • The cube isnon-composite polyhedron,meaning it is a convex polyhedron that cannot be separated into two or more regular polyhedrons. The cube can be applied to construct a new convex polyhedron by attaching another.[40]Attaching asquare pyramidto each square face of a cube produces itsKleetope,a polyhedron known as thetetrakis hexahedron.[41]Suppose one and two equilateral square pyramids are attached to their square faces. In that case, they are the construction of anelongated square pyramidandelongated square bipyramidrespectively, theJohnson solid's examples.[42]
  • Each of the cube's vertices can betruncated,and the resulting polyhedron is theArchimedean solid,thetruncated cube.[43]When its edges are truncated, it is arhombicuboctahedron.[44]Relatedly, the rhombicuboctahedron can also be constructed by separating the cube's faces and then spreading away, after which adding other triangular and square faces between them; this is known as the "expanded cube". It also can be constructed similarly by the cube's dual, the regular octahedron.[45]
  • The corner region of a cube can also be truncated by a plane (e.g., spanned by the three neighboring vertices), resulting in atrirectangular tetrahedron.
  • Thesnub cubeis an Archimedean solid that can be constructed by separating away the cube square's face, and filling their gaps with twisted angle equilateral triangles;a process known assnub.[46]

Thehoneycombis the space-filling ortessellationin three-dimensional space, meaning it is an object in which the construction begins by attaching any polyhedrons onto their faces without leaving a gap. The cube can be represented as thecell,and examples of a honeycomb arecubic honeycomb,order-5 cubic honeycomb,order-6 cubic honeycomb,andorder-7 cubic honeycomb.[47]The cube can be constructed with sixsquare pyramids,tiling space by attaching their apices.[48]

Polycubeis a polyhedron in which the faces of many cubes are attached. Analogously, it can be interpreted as thepolyominoesin three-dimensional space.[49]When four cubes are stacked vertically, and the other four are attached to the second-from-top cube of the stack, the resulting polycube isDali cross,afterSalvador Dali.The Dali cross is a tile space polyhedron,[50][51]which can be represented as the net of atesseract.A tesseract is a cube analogous'four-dimensional spacebounded by twenty-four squares, and it is bounded by the eight cubes known as itscells.[52]

References

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  1. ^abMills, Steve; Kolf, Hillary (1999).Maths Dictionary.Heinemann. p. 16.ISBN978-0-435-02474-1.
  2. ^Johnson, Norman W.(1966)."Convex polyhedra with regular faces".Canadian Journal of Mathematics.18:169–200.doi:10.4153/cjm-1966-021-8.MR0185507.S2CID122006114.Zbl0132.14603.See table II, line 3.
  3. ^Herrmann, Diane L.; Sally, Paul J. (2013).Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory.Taylor & Francis. p. 252.ISBN978-1-4665-5464-1.
  4. ^abKhattar, Dinesh (2008).Guide to Objective Arithmetic(2nd ed.).Pearson Education.p. 377.ISBN978-81-317-1682-3.
  5. ^Ball, Keith (2010). "High-dimensional geometry and its probabilistic analogues". InGowers, Timothy(ed.).The Princeton Companion to Mathematics.Princeton University Press. p.671.ISBN9781400830398.
  6. ^Geometry: Reteaching Masters.Holt Rinehart & Winston. 2001. p. 74.ISBN9780030543289.
  7. ^Sriraman, Bharath (2009). "Mathematics and literature (the sequel): imagination as a pathway to advanced mathematical ideas and philosophy". In Sriraman, Bharath; Freiman, Viktor; Lirette-Pitre, Nicole (eds.).Interdisciplinarity, Creativity, and Learning: Mathematics With Literature, Paradoxes, History, Technology, and Modeling.The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education. Vol. 7. Information Age Publishing, Inc. pp. 41–54.ISBN9781607521013.
  8. ^Jerrard, Richard P.; Wetzel, John E.; Yuan, Liping (April 2017). "Platonic passages".Mathematics Magazine.90(2). Washington, DC:Mathematical Association of America:87–98.doi:10.4169/math.mag.90.2.87.S2CID218542147.
  9. ^Lützen, Jesper (2010)."The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle".Centaurus.52(1): 4–37.doi:10.1111/j.1600-0498.2009.00160.x.
  10. ^Coxeter (1973)Table I(i), pp. 292–293. See the columns labeled,,and,Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter usesas the edge length (see p. 2).
  11. ^Poo-Sung, Park, Poo-Sung (2016)."Regular polytope distances"(PDF).Forum Geometricorum.16:227–232.{{cite journal}}:CS1 maint: multiple names: authors list (link)
  12. ^French, Doug (1988). "Reflections on a Cube".Mathematics in School.17(4): 30–33.JSTOR30214515.
  13. ^Cromwell, Peter R. (1997).Polyhedra.Cambridge University Press. p. 309.ISBN978-0-521-55432-9.
  14. ^Cunningham, Gabe; Pellicer, Daniel (2024)."Finite 3-orbit polyhedra in ordinary space, II".Boletín de la Sociedad Matemática Mexicana.30(32).doi:10.1007/s40590-024-00600-z.See p. 276.
  15. ^Cundy, H. Martyn;Rollett, A.P. (1961). "3.2 Duality".Mathematical models(2nd ed.). Oxford: Clarendon Press. pp. 78–79.MR0124167.
  16. ^Erickson, Martin (2011).Beautiful Mathematics.Mathematical Association of America.p. 62.ISBN978-1-61444-509-8.
  17. ^McLean, K. Robin (1990). "Dungeons, dragons, and dice".The Mathematical Gazette.74(469): 243–256.doi:10.2307/3619822.JSTOR3619822.S2CID195047512.See p. 247.
  18. ^Grünbaum, Branko(1997). "Isogonal Prismatoids".Discrete & Computational Geometry.18(1): 13–52.doi:10.1007/PL00009307.
  19. ^Senechal, Marjorie (1989)."A Brief Introduction to Tilings".In Jarić, Marko (ed.).Introduction to the Mathematics of Quasicrystals.Academic Press.p. 12.
  20. ^Calter, Paul; Calter, Michael (2011).Technical Mathematics.John Wiley & Sons.p. 197.ISBN978-0-470-53492-2.
  21. ^Erdahl, R. M. (1999)."Zonotopes, dicings, and Voronoi's conjecture on parallelohedra".European Journal of Combinatorics.20(6): 527–549.doi:10.1006/eujc.1999.0294.MR1703597..Voronoi conjectured that all tilings of higher dimensional spaces by translates of a singleconvex polytopeare combinatorially equivalent to Voronoi tilings, and Erdahl proves this in the special case ofzonotopes.But as he writes (p. 429), Voronoi's conjecture for dimensions at most four was already proven by Delaunay. For the classification of three-dimensional parallelohedra into these five types, seeGrünbaum, Branko;Shephard, G. C.(1980)."Tilings with congruent tiles".Bulletin of the American Mathematical Society.New Series.3(3): 951–973.doi:10.1090/S0273-0979-1980-14827-2.MR0585178.
  22. ^Alexandrov, A. D.(2005)."8.1 Parallelohedra".Convex Polyhedra.Springer. pp. 349–359.
  23. ^In higher dimensions, however, there exist parallelopes that are not zonotopes. See e.g.Shephard, G. C. (1974). "Space-filling zonotopes".Mathematika.21(2): 261–269.doi:10.1112/S0025579300008652.MR0365332.
  24. ^Jeon, Kyungsoon (2009). "Mathematics Hiding in the Nets for a CUBE".Teaching Children Mathematics.15(7): 394–399.doi:10.5951/TCM.15.7.0394.JSTOR41199313.
  25. ^Smith, James (2000).Methods of Geometry.John Wiley & Sons.p. 392.ISBN978-1-118-03103-2.
  26. ^Kozachok, Marina (2012). "Perfect prismatoids and the conjecture concerning with face numbers of centrally symmetric polytopes".Yaroslavl International Conference "Discrete Geometry" dedicated to the centenary of A.D.Alexandrov (Yaroslavl, August 13-18, 2012)(PDF).P.G. Demidov Yaroslavl State University, International B.N. Delaunay Laboratory. pp. 46–49.
  27. ^Grünbaum, Branko(2003). "13.1 Steinitz's theorem".Convex Polytopes.Graduate Texts in Mathematics.Vol. 221 (2nd ed.). Springer-Verlag. pp. 235–244.ISBN0-387-40409-0.
  28. ^Ziegler, Günter M.(1995). "Chapter 4: Steinitz' Theorem for 3-Polytopes".Lectures on Polytopes.Graduate Texts in Mathematics.Vol. 152. Springer-Verlag. pp. 103–126.ISBN0-387-94365-X.
  29. ^Rudolph, Michael (2022).The Mathematics of Finite Networks: An Introduction to Operator Graph Theory.Cambridge University Press.p. 25.doi:10.1007/9781316466919(inactive 1 November 2024).ISBN9781316466919.{{cite book}}:CS1 maint: DOI inactive as of November 2024 (link)
  30. ^Harary, F.;Hayes, J. P.; Wu, H.-J. (1988). "A survey of the theory of hypercube graphs".Computers & Mathematics with Applications.15(4): 277–289.doi:10.1016/0898-1221(88)90213-1.hdl:2027.42/27522.
  31. ^Chartrand, Gary; Zhang, Ping (2012).A First Course in Graph Theory.Dover Publications.p. 25.ISBN978-0-486-29730-9.
  32. ^Horvat, Boris;Pisanski, Tomaž(2010)."Products of unit distance graphs".Discrete Mathematics.310(12): 1783–1792.doi:10.1016/j.disc.2009.11.035.MR2610282.
  33. ^Pisanski, Tomaž; Servatius, Brigitte (2013).Configuration from a Graphical Viewpoint.Springer. p. 21.doi:10.1007/978-0-8176-8364-1.ISBN978-0-8176-8363-4.
  34. ^Hasan, Masud; Hossain, Mohammad M.; López-Ortiz, Alejandro; Nusrat, Sabrina; Quader, Saad A.; Rahman, Nabila (2010). "Some New Equiprojective Polyhedra".arXiv:1009.2252[cs.CG].
  35. ^Coxeter, H.S.M.(1973).Regular Polytopes(3rd ed.). New York:Dover Publications.pp.122–123.See §1.8 Configurations.
  36. ^abCromwell (1997),p.55.
  37. ^Heath, Thomas L.(1908).The Thirteen Books of Euclid's Elements(3rd ed.).Cambridge University Press.p. 262, 478, 480.
  38. ^Livio, Mario(2003) [2002].The Golden Ratio: The Story of Phi, the World's Most Astonishing Number(1st trade paperback ed.). New York City:Broadway Books.p. 147.ISBN978-0-7679-0816-0.
  39. ^Inchbald, Guy (2006). "Facetting Diagrams".The Mathematical Gazette.90(518): 253–261.doi:10.1017/S0025557200179653.JSTOR40378613.
  40. ^Timofeenko, A. V. (2010)."Junction of Non-composite Polyhedra"(PDF).St. Petersburg Mathematical Journal.21(3): 483–512.doi:10.1090/S1061-0022-10-01105-2.
  41. ^Slobodan, Mišić; Obradović, Marija; Ðukanović, Gordana (2015)."Composite Concave Cupolae as Geometric and Architectural Forms"(PDF).Journal for Geometry and Graphics.19(1): 79–91.
  42. ^Rajwade, A. R. (2001).Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem.Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89.doi:10.1007/978-93-86279-06-4.ISBN978-93-86279-06-4.
  43. ^Cromwell (1997),pp.81–82.
  44. ^Linti, G. (2013). "Catenated Compounds - Group 13 [Al, Ga, In, Tl]". In Reedijk, J.; Poeppelmmeier, K. (eds.).Comprehensive Inorganic Chemistry II: From Elements to Applications.Newnes. p. 41.ISBN978-0-08-096529-1.
  45. ^Viana, Vera; Xavier, João Pedro; Aires, Ana Paula; Campos, Helena (2019). "Interactive Expansion of Achiral Polyhedra". In Cocchiarella, Luigi (ed.).ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics 40th Anniversary - Milan, Italy, August 3-7, 2018.Advances in Intelligent Systems and Computing. Vol. 809. Springer. p. 1123.doi:10.1007/978-3-319-95588-9.ISBN978-3-319-95587-2.See Fig. 6.
  46. ^Holme, A. (2010).Geometry: Our Cultural Heritage.Springer.doi:10.1007/978-3-642-14441-7.ISBN978-3-642-14441-7.
  47. ^Coxeter, H. S. M.(1968).The Beauty of Geometry: Twelve Essays.Dover Publications.p. 167.ISBN978-0-486-40919-1.See table III.
  48. ^Barnes, John (2012).Gems of Geometry(2nd ed.). Springer. p. 82.doi:10.1007/978-3-642-30964-9.ISBN978-3-642-30964-9.
  49. ^Lunnon, W. F. (1972). "Symmetry of Cubical and General Polyominoes". In Read, Ronald C. (ed.).Graph Theory and Computing.New York:Academic Press.pp. 101–108.ISBN978-1-48325-512-5.
  50. ^Diaz, Giovanna;O'Rourke, Joseph(2015). "Hypercube unfoldings that tileand".arXiv:1512.02086[cs.CG].
  51. ^Langerman, Stefan;Winslow, Andrew (2016)."Polycube unfoldings satisfying Conway's criterion"(PDF).19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016).
  52. ^Hall, T. Proctor(1893). "The projection of fourfold figures on a three-flat".American Journal of Mathematics.15(2): 179–189.doi:10.2307/2369565.JSTOR2369565.
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