Cube

For other uses, see Cube (disambiguation).

Cube
Type	Hanner polytope, orthogonal polyhedron, parallelohedron, Platonic solid, plesiohedron, regular polyhedron, zonohedron
Faces	6
Edges	12
Vertices	8
Euler char.	2
Vertex configuration	${\displaystyle 8\times (4^{3})}$
Schläfli symbol	${\displaystyle \{4,3\}}$
Symmetry group	octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$
Volume	side3
Dihedral angle (degrees)	90°
Dual polyhedron	regular octahedron
Properties	convex, edge-transitive, face-transitive, non-composite, orthogonal faces, vertex-transitive

A cube is a three-dimensional solid object in geometry. A polyhedron, its eight vertices and twelve straight edges of the same length form six square faces of the same size. It is a type of parallelepiped, with pairs of parallel opposite faces with the same shape and size, and is also a rectangular cuboid with right angles between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra, such as Platonic solids, regular polyhedra, parallelohedra, zonohedra, and plesiohedra. The dual polyhedron of a cube is the regular octahedron.

The cube can be represented in many ways, such as the cubical graph, which can be constructed by using the Cartesian product of graphs. The cube is the three-dimensional hypercube, a family of polytopes also including the two-dimensional square and four-dimensional tesseract. A cube with unit side length is the canonical unit of volume in three-dimensional space, relative to which other solid objects are measured. Other related figures involve the construction of polyhedra, space-filling and honeycombs, and polycubes, as well as cubes in compounds, spherical, and topological space.

The cube was discovered in antiquity, and associated with the nature of earth by Plato, for whom the Platonic solids are named. It can be derived differently to create more polyhedra, and it has applications to construct a new polyhedron by attaching others. Other applications are found in toys and games, arts, optical illusions, architectural buildings, natural science, and technology.

Properties

3D model of a cube

A cuboid is a polyhedron that consists of six quadrilateral faces. When all of its interior angle (the angle formed inside) are right angles, 90°, the faces are transformed into rectangles, which is known as rectangular cuboid. A cube becomes a special case of a rectangular cuboid when all of the edges are equal in length. ^[1] Like a rectangular cuboid, every face of a cube has four vertices, each of which connects with three lines of the same length. These edges form square faces, so the dihedral angle of a cube—the angle between every two adjacent squares—is the interior angle of a square as well. Hence, the cube has six faces, twelve edges, and eight vertices. ^[2] As for all convex polyhedra, the cube has Euler characteristic of 2, according to the formula ${\displaystyle V-E+F=2}$; the three letters denote respectively the number of vertices, edges, and faces. ^[3]

Every three square faces surrounding a vertex are orthogonal to each other, meaning the planes are perpendicular, forming a right angle between two adjacent squares. Hence, the cube is classified as an orthogonal polyhedron. ^[4] Other special cases for a cube are a parallelepiped —a polyhedron with six parallelograms faces—because its pairs of opposite faces are congruent, ^[5] a rhombohedron —as a special case of a parallelepiped with six rhombi faces—because the interior angle of all of the faces is right, ^[6] and a trigonal trapezohedron —a polyhedron with congruent quadrilateral faces—since its square faces are the special cases of rhombi. ^[7]

Measurement and other metric properties

A face diagonal in red and space diagonal in blue

Given a cube with edge length ${\displaystyle a}$, the face diagonal of the cube is the diagonal of a square ${\displaystyle a{\sqrt {2}}}$, and the space diagonal of the cube is a line connecting two vertices that are not in the same face, formulated as ${\displaystyle a{\sqrt {3}}}$. Both formulas can be determined by using the Pythagorean theorem. The surface area of a cube ${\displaystyle A}$ is six times the area of a square: ^[8] $${\displaystyle A=6a^{2}.}$$ The volume of a cuboid is the product of its length, width, and height. Because all the edges of a cube are equal in length, the formula for the volume of a cube is the third power of its side length, leading to the use of the term cubic to mean raising any number to the third power: ^{[9] [8]} $${\displaystyle V=a^{3}.}$$

Prince Rupert's cube

One special case is the unit cube, so named for measuring a single unit of length along each edge. It follows that each face is a unit square and that the entire figure has a volume of 1 cubic unit. ^{[10] [11]} Prince Rupert of the Rhine, known for Prince Rupert's drop, questioned whether a cube could pass through a hole cut into the unit cube. Despite having sides approximately 6% longer, such a cube can pass through a copy of itself of the same size or smaller. ^{[12] [13]} An ancient problem of doubling the cube —alternatively known as the Delian problem—requires the construction of a cube with a volume twice the original by using only a compass and straightedge. This was concluded by French mathematician Pierre Wantzel in 1837, proving that it is impossible to implement since a cube with twice the volume of the original—the cube root of 2, ${\displaystyle {\sqrt[{3}]{2}}}$—is not constructible. ^[14]

The cube has three types of closed geodesics, or paths on a cube'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. One type lies in a plane parallel to any face of the cube, forming a square congruent to a face, four times the length of each edge. Another type lies in a plane perpendicular to the long diagonal, forming a regular hexagon; its length is ${\displaystyle 3{\sqrt {2}}}$ times that of an edge. The third type is a non-planar hexagon. ^[15]

Insphere, midsphere, circumsphere

With edge length ${\displaystyle a}$, the inscribed sphere of a cube is the sphere tangent to the faces of a cube at their centroids, with radius ${\textstyle {\frac {1}{2}}a}$. The midsphere of a cube is the sphere tangent to the edges of a cube, with radius ${\textstyle {\frac {\sqrt {2}}{2}}a}$. The circumscribed sphere of a cube is the sphere tangent to the vertices of a cube, with radius ${\textstyle {\frac {\sqrt {3}}{2}}a}$. ^[16]

For a cube whose circumscribed sphere has radius ${\displaystyle R}$, and for a given point in its three-dimensional space with distances ${\displaystyle d_{i}}$ from the cube's eight vertices, it is: ^[17] $${\displaystyle {\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{4}+{\frac {16R^{4}}{9}}={\biggl (}{\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{2}+{\frac {2R^{2}}{3}}{\biggr )}^{2}.}$$

Symmetry

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

The cube has octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$ of order 48. In other words, the cube has 48 isometries, each of which transforms the cube to itself. These transformations include nine reflection symmetries (where two halves cut by a plane are identical): three cut the cube at the midpoints of its edges, and six cut diagonally. The cube also has thirteen axes of rotational symmetry (whereby rotation around the axis results in an identical appearance): three axes pass through the centroids of the cube's opposite faces, six through the midpoints of the cube's opposite edges, and four through the cube's opposite vertices; these axes are 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°). ^{[18] [19] [20] [21]}

The dual polyhedron can be obtained from each of the polyhedra's vertices tangent to a plane by a process known as polar reciprocation. ^[22] One property of dual polyhedra is that the polyhedron and its dual share their three-dimensional symmetry point group. In this case, the dual polyhedron of a cube is the regular octahedron, and both of these polyhedra have the same octahedral symmetry. ^[23]

The cube is face-transitive, meaning its two squares are alike and can be mapped by rotation and reflection. ^[24] It is vertex-transitive, meaning all of its vertices are equivalent and can be mapped isometrically under its symmetry. ^[25] It is also edge-transitive, meaning the same kind of faces surround each of its vertices in the same or reverse order, and each pair of adjacent faces has the same dihedral angle. Therefore, the cube is a regular polyhedron. ^[26] Each vertex is surrounded by three squares, so the cube is ${\displaystyle 4.4.4}$ by vertex configuration or ${\displaystyle \{4,3\}}$ in a Schläfli symbol. ^[27]

Appearances

A six-sided die

A completed Skewb

The Alamo sculpture

Cubes have appeared in many roles in popular culture. It is the most common form of dice. ^[24] Puzzle toys such as pieces of a Soma cube, ^[28] Rubik's Cube, and Skewb are built of cubes. ^[29] Minecraft is an example of a sandbox video game of cubic blocks. ^[30] The outdoor sculpture Alamo (1967) is a cube that spins around its vertical axis. ^[31] Optical illusions such as the impossible cube and Necker cube have been explored by artists such as M. C. Escher. ^[32] Salvador Dalí's painting Corpus Hypercubus (1954) contains a tesseract unfolding into a six-armed cross; a similar construction is central to Robert A. Heinlein's short story "And He Built a Crooked House" (1940). ^{[33] [34]} The cube was applied in Alberti's treatise on Renaissance architecture, De re aedificatoria (1450). ^[35] Cube houses in the Netherlands are a set of cubical houses whose hexagonal space diagonals become the main floor. ^[36]

Simple cubic crystal structure

Table salt cubic crystals

Ball-and-stick model of cubane

Cubes are also found in natural science and technology. It is applied to the unit cell of a crystal known as a cubic crystal system. ^[37] Table salt is an example of a mineral with a commonly cubic shape. ^[38] Pyrite has the same shape as well, although there are many variations. ^[39] The radiolarian Lithocubus geometricus, discovered by Ernst Haeckel, has a cubic shape. ^[40] A historical attempt to unify three physics ideas of relativity, gravitation, and quantum mechanics used the framework of a cube known as a cGh cube. ^[41] Cubane is a synthetic hydrocarbon consisting of eight carbon atoms arranged at the corners of a cube, with one hydrogen atom attached to each carbon atom. ^[42]

Other technological cubes include the spacecraft device CubeSat, ^[43] and thermal radiation demonstration device Leslie cube. ^[44] Cubical grids are usual in three-dimensional Cartesian coordinate systems. ^[45] In computer graphics, an algorithm divides the input volume into a discrete set of cubes known as the unit on isosurface, ^[46] and the faces of a cube can be used for mapping a shape. ^[47]

Sketch of a cube by Johannes Kepler

Kepler's Platonic solid model of the Solar System

The Platonic solids are five polyhedra known since antiquity. The set is named for Plato who, in his dialogue Timaeus, attributed these solids to nature. One of them, the cube, represented the classical element of earth because of its stability. ^[48] Euclid's Elements defined the Platonic solids, including the cube, and showed how to find the ratio of the circumscribed sphere's diameter to the edge length. ^[49] Following Plato's use of the regular polyhedra as symbols of nature, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids; he decorated the cube's side with a tree. ^[48] In his Mysterium Cosmographicum , Kepler also proposed that the ratios between sizes of the orbits of the planets are the ratios between the sizes of the inscribed and circumscribed spheres of the Platonic solids. That is, if the orbits are great circles on spheres, the sphere of Mercury is tangent to a regular octahedron, whose vertices lie on the sphere of Venus, which is in turn tangent to a regular icosahedron, within the sphere of Earth, within a regular dodecahedron, within the sphere of Mars, within a regular tetrahedron, within the sphere of Jupiter, within a cube, within the sphere of Saturn. In fact, the orbits are not circles but ellipses (as Kepler himself later showed), and these relations are only approximate. ^[50]

Construction

The eleven nets of a cube

An elementary way to construct a cube is using its net, an arrangement of edge-joined polygons, by connecting the free edges of those polygons. Eleven nets for the cube are possible. ^{[51] [52]}

In analytic geometry, a cube may be constructed using the Cartesian coordinate systems. For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are ${\displaystyle (\pm 1,\pm 1,\pm 1)}$. ^[53] Its interior consists of all points ${\displaystyle (x_{0},x_{1},x_{2})}$ with ${\displaystyle -1<x_{i}<1}$ for all ${\displaystyle i}$. A cube's surface with center ${\displaystyle (x_{0},y_{0},z_{0})}$ and edge length of ${\displaystyle 2a}$ is the locus of all points ${\displaystyle (x,y,z)}$ such that $${\displaystyle \max\{|x-x_{0}|,|y-y_{0}|,|z-z_{0}|\}=a.}$$

The cube is a Hanner polytope, because it can be constructed by using the Cartesian product of three line segments. Its dual polyhedron, the regular octahedron, is constructed by the direct sum of three line segments. ^[54]

Representation

As a graph

Main article: Hypercube graph

The graph of a cube

According to Steinitz's theorem, a graph can be represented as the vertex-edge graph of a polyhedron. Such a graph has two properties: planar (the edges of a graph are connected to every vertex without crossing other edges), and 3-connected (whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected). ^{[55] [56]} The skeleton of a cube, represented as the graph, is called the cubical graph, a Platonic graph. It has the same number of vertices and edges as the cube, twelve vertices and eight edges. ^[57] The cubical graph is also classified as a prism graph, resembling the skeleton of a cuboid. ^[58]

The cubical graph is a special case of hypercube graph or ${\displaystyle n}$-cube—denoted as ${\displaystyle Q_{n}}$—because it can be constructed by using the Cartesian product of graphs: two graphs connecting the pair of vertices with an edge to form a new graph. ^[59] In the case of the cubical graph, it is the product of ${\displaystyle Q_{2}\mathbin {\Box } Q_{1}}$. In other words, the cubical graph is constructed by connecting each vertex of two squares with an edge. Notationally, the cubical graph is ${\displaystyle Q_{3}}$. ^[60] Like any hypercube graph, it has a cycle which visits every vertex exactly once, ^[61] and it is also an example of a unit distance graph. ^[62]

The cubical graph is bipartite, meaning every independent set of four vertices can be disjoint and the edges connected in those sets. ^[63] However, every vertex in one set cannot connect all vertices in the second, so this bipartite graph is not complete. ^[64] It is an example of both a crown graph and a bipartite Kneser graph. ^{[65] [63]}

In orthogonal projection

An object illuminated by parallel rays of light casts a shadow on a plane perpendicular to those rays, called an orthogonal projection. A polyhedron is considered equiprojective if, 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 a regular hexagon. ^[66]

As a configuration matrix

The cube can be represented as a configuration matrix, a matrix in which the rows and columns correspond to the elements of a polyhedron as the vertices, edges, and faces. The diagonal of 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. The cube's eight vertices, twelve edges, and six faces are denoted by each element in a matrix's diagonal (8, 12, and 6). The first column of the middle row indicates that there are two vertices on 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: ^[67] $${\displaystyle {\begin{bmatrix}{\begin{matrix}8&3&3\\2&12&2\\4&4&6\end{matrix}}\end{bmatrix}}}$$

Related figures

Truncation

The chamfered cube

Each of the cube's vertices can be truncated, and the resulting polyhedron is the Archimedean solid, the truncated cube. ^[68] When its edges are truncated, it is a rhombicuboctahedron. ^[69] 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". The same figure can be derived in the same way from the cube's dual, the regular octahedron. ^[70]

The chamfered cube is constructed from a cube by a truncating operator called chamfer. The resulting polyhedron has twelve hexagonal and six square centrally symmetric faces, a zonohedron. ^{[71] [72]}

Other polyhedra's construction

Some of the derived cubes, the stellated octahedron and tetrakis hexahedron

In addition to truncation, the cube can be applied in many constructions of a polyhedron. Some of its types can be derived differently in the following:

• When faceting a cube, meaning removing part of the polygonal faces without creating new vertices of a cube, the resulting polyhedron is the stellated octahedron. ^[73]
• The cube is a non-composite polyhedron, meaning there is no plane intersecting its surface only along edges, by which it could be cut into two or more polyhedra.
• New convex polyhedra can be constructed by attaching less-regular polyhedra to a cube's faces. ^[74] The cube is thus a component of two Johnson solids, the elongated square pyramid and elongated square bipyramid, the latter being a cube with square pyramids on opposite faces. ^[75]
• Attaching a low pyramid to each face of a cube produces its Kleetope, the tetrakis hexahedron, ^[76] dual to the truncated octahedron.
• The barycentric subdivision of a cube (or its dual, the regular octahedron) is the disdyakis dodecahedron, a Catalan solid. ^[77]
• The corner region of a cube can also be truncated by a plane (e.g., spanned by the three neighboring vertices), resulting in a trirectangular tetrahedron. ^[78]
• The snub cube is an Archimedean solid that can be constructed by separating the cube's faces, and filling the gaps with twisted angle equilateral triangles, a process known as a snub. ^[79]
• Three mutually perpendicular golden rectangles can be constructed from a pair of vertices located on the midpoints of the opposite edges on a cube's surface, drawing a segment line between those two, and dividing that segment line in a golden ratio from its midpoint. The corners of these rectangles are the vertices of a regular icosahedron with twenty equilateral triangles. ^[80]

The cube can be constructed with six square pyramids, tiling space by attaching their apices. In some cases, this produces the rhombic dodecahedron circumscribing a cube. ^{[81] [82]}

Polycubes

Main article: Polycube

Dalí cross, one of 261 nets of a tesseract

The polycube is a polyhedron in which the faces of many cubes are attached. Analogously, it can be interpreted as the polyominoes in three-dimensional space. ^[83]

When four cubes are stacked vertically, and four others are attached to the second-from-top cube of the stack, the resulting polycube is the Dalí cross, named after Salvador Dalí. The Dalí cross can be folded in a fourth dimension to enclose a tesseract. It can also tile three-dimensional space. ^{[84] [85]} Both cube and tesseract are known as three-dimensional and four-dimensional hypercubes, respectively. ^[86]

Space-filling and honeycombs

Hilbert's third problem asks whether every two equal-volume polyhedra can always be dissected into polyhedral pieces and reassembled into each other. If yes, then the volume of any polyhedron could be defined axiomatically as the volume of an equivalent cube into which it could be reassembled. Max Dehn solved this problem by inventing the Dehn invariant, answering that not all polyhedra can be reassembled into a cube. ^[87] It showed that two equal volume polyhedra should have the same Dehn invariant, except for the two tetrahedra whose Dehn invariants were different. ^[88]

Cubic honeycomb

The cube has a Dehn invariant of zero, meaning that cubes can achieve a honeycomb. It is also a space-filling tile in three-dimensional space in which the construction begins by attaching a polyhedron onto its faces without leaving a gap. ^[89] The cube is a plesiohedron, a special kind of space-filling polyhedron that can be defined as the Voronoi cell of a symmetric Delone set. ^[90] The plesiohedra include the parallelohedra, which can be translated without rotating to fill a space 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 parallelepiped. ^[91] Every three-dimensional parallelohedron is a zonohedron, a centrally symmetric polyhedron whose faces are centrally symmetric polygons. ^[92] In the case of the cube, it can be represented as a cell. Some honeycombs have cubes as the only cells; one example is the cubic honeycomb, the only regular honeycomb in Euclidean three-dimensional space, which has four cubes around each edge. ^{[93] [94]}

Miscellaneous

Enumeration according to Skilling (1976): compound of six cubes with rotational freedom ${\displaystyle \mathrm {UC} _{7}}$, three cubes ${\displaystyle \mathrm {UC} _{8}}$, and five cubes ${\displaystyle \mathrm {UC} _{9}}$

The polyhedral compounds, in which the cubes share the same centre, are uniform polyhedron compounds, meaning they are polyhedral compounds whose constituents are identical—although possibly enantiomorphous — uniform polyhedra, in an arrangement that is also uniform. Respectively, the list of compounds enumerated by Skilling (1976) in the seventh to ninth uniform compounds for the compound of six cubes with rotational freedom, three cubes, and five cubes. ^[95] Two compounds, consisting of two and three cubes were found in Escher's wood engraving print Stars and Max Brückner's book Vielecke und Vielflache. ^[96]

Spherical cube

A view in three-dimensional torus

The spherical cube represents the spherical polyhedron, which can be modeled by the arc of great circles, creating bounds as the edges of a spherical square. ^[97] Hence, the spherical cube consists of six spherical squares with 120° interior angles on each vertex. It has vector equilibrium, meaning that the distance from the centroid and each vertex is the same as the distance from that and each edge. ^{[98] [99]} Its dual is the spherical octahedron. ^[97]

The topological object three-dimensional torus is a topological space defined to be homeomorphic to the Cartesian product of three circles. It can be represented as a three-dimensional model of the cube shape. ^[100]

See also

• Bhargava cube, a configuration to study the law of binary quadratic form and other such forms, of which the cube's vertices represent the integer.
• Chazelle polyhedron, a notched opposite faces of a cube.
• Cubism, an art movement of revolutionized painting and the visual arts.
• Hemicube, an abstract polyhedron produced by identifying opposite faces of a cube
• Schläfli double six, a configuration of 30 points and 12 lines in three-dimensional Euclidean space
• Squaring the square's three-dimensional analogue, cubing the cube
• Superellipsoid, a solid whose horizontal sections are of the same squareness

References

1. Mills, Steve; Kolf, Hillary (1999). Maths Dictionary. Heinemann. p. 16. ISBN   978-0-435-02474-1.
2. Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics . 18: 169–200. doi: 10.4153/cjm-1966-021-8 . MR   0185507. S2CID   122006114. Zbl   0132.14603. See table II, line 3.
3. Richeson, D. S. (2008). Euler's Gem: The polyhedron formula and the birth of topology. Princeton University Press. pp. 1–2.
4. Jessen, Børge (1967). "Orthogonal icosahedra". Nordisk Matematisk Tidskrift. 15 (2): 90–96. JSTOR   24524998. MR   0226494.
5. Calter, Paul; Calter, Michael (2011). Technical Mathematics. John Wiley & Sons. p. 197. ISBN   978-0-470-53492-2.
6. Hoffmann (2020), p.  83.
7. Chilton, B. L.; Coxeter, H. S. M. (1963). "Polar zonohedra". The American Mathematical Monthly . 70 (9): 946–951. doi:10.1080/00029890.1963.11992147. JSTOR   2313051. MR   0157282.
8. Khattar, Dinesh (2008). Guide to Objective Arithmetic (2nd ed.). Pearson Education. p. 377. ISBN   978-81-317-1682-3.
9. Thomson, James (1845). An Elementary Treatise on Algebra: Theoretical and Practical. London: Longman, Brown, Green, and Longmans. p. 4.
10. Ball, Keith (2010). "High-dimensional geometry and its probabilistic analogues". In Gowers, Timothy (ed.). The Princeton Companion to Mathematics. Princeton University Press. p.  671. ISBN   9781400830398.
11. Geometry: Reteaching Masters. Holt Rinehart & Winston. 2001. p. 74. ISBN   9780030543289.
12. 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. ISBN   9781607521013.
13. 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. S2CID   218542147.
14. 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.
15. 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 (inactive 1 July 2025).
16. Coxeter (1973) Table I(i), pp. 292–293. See the columns labeled ${\displaystyle {}_{0}\!\mathrm {R} /\ell }$, ${\displaystyle {}_{1}\!\mathrm {R} /\ell }$, and ${\displaystyle {}_{2}\!\mathrm {R} /\ell }$, Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses ${\displaystyle 2\ell }$ as the edge length (see p. 2).
17. Poo-Sung, Park, Poo-Sung (2016). "Regular polytope distances" (PDF). Forum Geometricorum . 16: 227–232. Archived from the original (PDF) on 2016-10-10. Retrieved 2016-05-24.
18. French, Doug (1988). "Reflections on a Cube". Mathematics in School. 17 (4): 30–33. JSTOR   30214515.
19. Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 309. ISBN   978-0-521-55432-9.
20. Cunningham, Gabe; Pellicer, Daniel (2024). "Finite 3-orbit polyhedra in ordinary space, II". Boletín de la Sociedad Matemática Mexicana. 30 (32) 32. doi: 10.1007/s40590-024-00600-z . See p. 276.
21. Kane, Richard (2001). Reflection Groups and Invariant Theory. Springer. p. 16. ISBN   978-0-387-98979-2.
22. Cundy, H. Martyn; Rollett, A.P. (1961). "3.2 Duality". Mathematical models (2nd ed.). Oxford: Clarendon Press. pp. 78–79. MR   0124167.
23. Erickson, Martin (2011). Beautiful Mathematics. Mathematical Association of America. p. 62. ISBN   978-1-61444-509-8.
24. McLean, K. Robin (1990). "Dungeons, dragons, and dice". The Mathematical Gazette . 74 (469): 243–256. doi:10.2307/3619822. JSTOR   3619822. S2CID   195047512. See p. 247.
25. Grünbaum, Branko (1997). "Isogonal Prismatoids". Discrete & Computational Geometry. 18 (1): 13–52. doi:10.1007/PL00009307.
26. Senechal, Marjorie (1989). "A Brief Introduction to Tilings". In Jarić, Marko (ed.). Introduction to the Mathematics of Quasicrystals. Academic Press. p. 12.
27. Walter, Steurer; Deloudi, Sofia (2009). Crystallography of Quasicrystals: Concepts, Methods and Structures. Springer Series in Materials Science. Vol. 126. p. 50. doi:10.1007/978-3-642-01899-2. ISBN   978-3-642-01898-5.
28. Masalski, William J. (1977). "Polycubes". The Mathematics Teacher. 70 (1): 46–50. doi:10.5951/MT.70.1.0046. JSTOR   27960702.
29. Joyner, David (2008). Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys (2nd ed.). The Johns Hopkins University Press. p. 76. ISBN   978-0-8018-9012-3.
30. Moore, Kimberly (2018). "Minecraft Comes to Math Class". Mathematics Teaching in the Middle School. 23 (6): 334–341. doi:10.5951/mathteacmiddscho.23.6.0334. JSTOR   10.5951/mathteacmiddscho.23.6.0334.
31. Reaven, Marci; Zeilten, Steve (2006). Hidden New York: A Guide to Places that Matter. Rutgers University Press. p. 77. ISBN   978-0-8135-3890-7.
32. Barrow, John D. (1999). Impossibility: The Limits of Science and the Science of Limits. Oxford University Press. p. 14. ISBN   9780195130829.
33. Kemp, Martin (1 January 1998). "Dali's dimensions". Nature . 391 (27): 27. Bibcode:1998Natur.391...27K. doi: 10.1038/34063 .
34. Fowler, David (2010). "Mathematics in Science Fiction: Mathematics as Science Fiction". World Literature Today. 84 (3): 48–52. doi:10.1353/wlt.2010.0188. JSTOR   27871086. S2CID   115769478. Robert Heinlein's "And He Built a Crooked House," published in 1940, and Martin Gardner's "The No-Sided Professor," published in 1946, are among the first in science fiction to introduce readers to the Moebius band, the Klein bottle, and the hypercube (tesseract).
35. March, Lionel (1996). "Renaissance mathematics and architectural proportion in Alberti's De re aedificatoria". Architectural Research Quarterly. 2 (1): 54–65. doi:10.1017/S135913550000110X. S2CID   110346888.
36. Alsina, Claudi; Nelsen, Roger B. (2015). A Mathematical Space Odyssey: Solid Geometry in the 21st Century. Vol. 50. Mathematical Association of America. p. 85. ISBN   978-1-61444-216-5.
37. Tisza, Miklós (2001). Physical Metallurgy for Engineers. Materials Park, Ohio: ASM International. p. 45. ISBN   978-1-61503-241-9.
38. Chieh, Chung (2013). "Polyhedra and Crystal Structures". In Senechal, Marjorie (ed.). Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination. Springer. p. 141. doi:10.1007/978-0-387-92714-5_10. ISBN   978-0-387-92713-8.
39. Hoffmann, Frank (2020). Introduction to Crystallography. Springer. p. 35. doi:10.1007/978-3-030-35110-6. ISBN   978-3-030-35110-6.
40. Haeckel, E. (1904). Kunstformen der Natur (in German). See here for an online book.
41. Padmanabhan, Thanu (2015). "The Grand Cube of Theoretical Physics". Sleeping Beauties in Theoretical Physics. Springer. pp. 1–8. ISBN   978-3319134420.
42. Biegasiewicz, Kyle; Griffiths, Justin; Savage, G. Paul; Tsanakstidis, John; Priefer, Ronny (2015). "Cubane: 50 years later". Chemical Reviews. 115 (14): 6719–6745. doi:10.1021/cr500523x. PMID   26102302.
43. Helvajian, Henry; Janson, Siegfried W., eds. (2008). Small Satellites: Past, Present, and Future. El Segundo, Calif.: Aerospace Press. p. 159. ISBN   978-1-884989-22-3.
44. Vollmer, Michael; Möllmann, Klaus-Peter (2011). Infrared Thermal Imaging: Fundamentals, Research and Applications. John Wiley & Sons. pp. 36–38. ISBN   9783527641550.
45. Kov´acs, Gergely; Nagy, Benedek Nagy; Stomfai, Gergely; Turgay, Nes¸et Deni̇z; Vizv´ari, B´ela (2021). "On Chamfer Distances on the Square and Body-Centered CubicGrids: An Operational Research Approach". Mathematical Problems in Engineering: 1–9. doi: 10.1155/2021/5582034 . hdl: 10831/111490 .
46. Chin, Daniel Jie Yuan Chin; Mohamed, Ahmad Sufril Azlan; Shariff, Khairul Anuar; Ishikawa, Kunio (23–25 November 2021). "GPU-Accelerated Enhanced Marching Cubes 33 for Fast 3D Reconstruction of Large Bone Defect CT Images". In Zaman, Halimah Badioze; Smeaton, Alan; Shih, Timothy; Velastin, Sergio; Terutoshi, Tada; Jørgensen, Bo Nørregaard; Aris, Hazleen Aris; Ibrahim, Nazrita Ibrahim (eds.). Advances in Visual Informatics. 7th International Visual Informatics Conference. Kajang, Malaysia. p. 376.
47. Greene, N (1986). "Environment mapping and other applications of world projections". IEEE Computer Graphics and Applications. 6 (11): 21–29. doi:10.1109/MCG.1986.276658. S2CID   11301955.
48. Cromwell (1997), p.  55.
49. Heath, Thomas L. (1908). The Thirteen Books of Euclid's Elements (3rd ed.). Cambridge University Press. pp. 262, 478, 480.
50. 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. ISBN   978-0-7679-0816-0.
51. 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. JSTOR   41199313.
52. Turney, Peter D. (1984–1985). "Unfolding the Tesseract". Journal of Recreational Mathematics . 17 (1).
53. Smith, James (2000). Methods of Geometry. John Wiley & Sons. p. 392. ISBN   978-1-118-03103-2.
54. 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.
55. Grünbaum, Branko (2003). "13.1 Steinitz's theorem". Convex Polytopes. Graduate Texts in Mathematics. Vol. 221 (2nd ed.). Springer-Verlag. pp. 235–244. ISBN   0-387-40409-0.
56. 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. ISBN   0-387-94365-X.
57. 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 July 2025). ISBN   9781316466919.
58. Pisanski, Tomaž; Servatius, Brigitte (2013). Configuration from a Graphical Viewpoint. Springer. p. 21. doi:10.1007/978-0-8176-8364-1. ISBN   978-0-8176-8363-4.
59. 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 .
60. Chartrand, Gary; Zhang, Ping (2012). A First Course in Graph Theory. Dover Publications. p. 25. ISBN   978-0-486-29730-9.
61. Gross, Jonathan L.; Yellen, Yellen (2006). Graph Theory and Its Applications, Second Edition. Taylor & Francis. p. 273. ISBN   978-1-58488-505-4.
62. Horvat, Boris; Pisanski, Tomaž (2010). "Products of unit distance graphs". Discrete Mathematics . 310 (12): 1783–1792. doi: 10.1016/j.disc.2009.11.035 . MR   2610282.
63. Berman, Leah (2014). "Geometric Constructions for Symmetric 6-Configurations". In Connelly, Robert; Weiss, Asia; Whiteley, Walter (eds.). Rigidity and Symmetry. Fields Institute Communications. Vol. 70. Springer. p. 84. doi:10.1007/978-1-4939-0781-6. ISBN   978-1-4939-0781-6.
64. Aldous, Joan; Wilson, Robin (2000). Graphs and Applications: An Introductory Approach. Springer. ISBN   978-1-85233-259-4.
65. Kitaev, Sergey; Lozin, Vadim (2015). Words and Graphs. p. 171. doi:10.1007/978-3-319-25859-1. ISBN   978-3-319-25859-1.
66. 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].
67. Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover Publications. pp.  122–123. See §1.8 Configurations.
68. Cromwell (1997), pp.  81–82.
69. 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. ISBN   978-0-08-096529-1.
70. 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. ISBN   978-3-319-95587-2. See Fig. 6.
71. Gelişgen, Özcan; Yavuz, Serhat (2019). "Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces". Mathematical Sciences and Applications E-Notes. 7 (2): 174–182. doi:10.36753/mathenot.542272.
72. Deza, Antoinea; Michel, Deza; Grishukhin, Viatcheslav (1998). "Fullerenes and coordination polyhedra versus half-cube embeddings". Discrete Mathematics. 192 (1–3): 41–80. doi:10.1016/S0012-365X(98)00065-X.
73. Inchbald, Guy (2006). "Facetting Diagrams". The Mathematical Gazette . 90 (518): 253–261. doi:10.1017/S0025557200179653. JSTOR   40378613.
74. 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.
75. 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. ISBN   978-93-86279-06-4.
76. 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.
77. Langer, Joel C.; Singer, David A. (2010). "Reflections on the Lemniscate of Bernoulli: The Forty-Eight Faces of a Mathematical Gem". Milan Journal of Mathematics . 78 (2): 643–682. doi:10.1007/s00032-010-0124-5.
78. Coxeter (1973), p.  71.
79. Holme, A. (2010). Geometry: Our Cultural Heritage. Springer. doi:10.1007/978-3-642-14441-7. ISBN   978-3-642-14441-7.
80. Cromwell (1997), p.  70.
81. Barnes, John (2012). Gems of Geometry (2nd ed.). Springer. p. 82. doi:10.1007/978-3-642-30964-9. ISBN   978-3-642-30964-9.
82. Cundy, H. Martyn (1956). "2642. Unitary Construction of Certain Polyhedra". The Mathematical Gazette . 40 (234): 280–282. doi:10.2307/3609622. JSTOR   3609622.
83. 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. ISBN   978-1-48325-512-5.
84. Diaz, Giovanna; O'Rourke, Joseph (2015). "Hypercube unfoldings that tile ${\displaystyle \mathbb {R} ^{3}}$ and ${\displaystyle \mathbb {R} ^{2}}$". arXiv: 1512.02086 [cs.CG].
85. 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).
86. 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. JSTOR   2369565.
87. Gruber, Peter M. (2007). "Chapter 16: Volume of Polytopes and Hilbert's Third Problem". Convex and Discrete Geometry. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 336. Springer, Berlin. pp. 280–291. doi:10.1007/978-3-540-71133-9. ISBN   978-3-540-71132-2. MR   2335496.
88. Zeeman, E. C. (July 2002). "On Hilbert's third problem". The Mathematical Gazette . 86 (506): 241–247. doi:10.2307/3621846. JSTOR   3621846.
89. Lagarias, J. C.; Moews, D. (1995). "Polytopes that fill ${\displaystyle \mathbb {R} ^{n}}$ and scissors congruence". Discrete & Computational Geometry . 13 (3–4): 573–583. doi: 10.1007/BF02574064 . MR   1318797.
90. 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 . MR   1703597.. Voronoi conjectured that all tilings of higher-dimensional spaces by translates of a single convex polytope are combinatorially equivalent to Voronoi tilings, and Erdahl proves this in the special case of zonotopes. 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, see Grünbaum, 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 . MR   0585178.
91. Horváth, Ákos G. (1995). "On Dirichlet–Voronoi cell, part I: Classical problems". Periodica Polytechnica Mechanical Engineering. 39 (1): 25–42.
92. 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. MR   0365332.
93. Coxeter, H. S. M. (1968). The Beauty of Geometry: Twelve Essays. Dover Publications. p. 167. ISBN   978-0-486-40919-1. See table III.
94. Nelson, Roice; Segerman, Henry (2017). "Visualizing hyperbolic honeycombs". Journal of Mathematics and the Arts. 11 (1): 4–39. arXiv: 1511.02851 . doi:10.1080/17513472.2016.1263789.
95. Skilling, John (1976). "Uniform Compounds of Uniform Polyhedra". Mathematical Proceedings of the Cambridge Philosophical Society . 79 (3): 447–457. Bibcode:1976MPCPS..79..447S. doi:10.1017/S0305004100052440. MR   0397554.
96. Hart, George (16–20 July 2019). "Max Brücknerʼs Wunderkammer of Paper Polyhedra" (PDF). In Goldstein, Susan; McKenna, Douglas; Fenyvesi, Kristóf (eds.). Bridges 2019 Conference Proceedings. Linz, Austria: Tessellations Publishing, Phoenix, Arizona. ISBN   978-1-938664-30-4.
97. Yackel, Carolyn (26–30 July 2013). "Marking a Physical Sphere with a Projected Platonic Solid" (PDF). In Kaplan, Craig; Sarhangi, Reza (eds.). Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture. Banff, Alberta, Canada. pp. 123–130. ISBN   978-0-96652-020-0.
98. Popko, Edward S. (2012). Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere. CRC Press. pp. 100–101. ISBN   9781466504295.
99. Fuller, Buckimster (1975). Synergetics: Explorations in the Geometry of Thinking. MacMillan Publishing. p. 173. ISBN   978-0-02-065320-2.
100. Marat, Ton (2022). A Ludic Journey into Geometric Topology. Springer. p. 112. doi:10.1007/978-3-031-07442-4. ISBN   978-3-031-07442-4.

External links

• Weisstein, Eric W. "Cube". MathWorld .
• Cube: Interactive Polyhedron Model*
• Volume of a cube, with interactive animation
• Cube (Robert Webb's site)