Cube

For other uses, see Cube (disambiguation).

Cube
Type	Platonic solid Regular polyhedron Parallelohedron Zonohedron Plesiohedron Hanner polytope
Faces	6
Edges	12
Vertices	8
Symmetry group	octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$
Dihedral angle (degrees)	90°
Dual polyhedron	regular octahedron
Properties	convex, face-transitive, edge-transitive, vertex-transitive

In geometry, a cube is a three-dimensional solid object bounded by six square faces. It has twelve edges and eight vertices. It can be represented as the rectangular cuboid with six faces are all squares, and parallelepiped with the edges are all equal. It is an example of many type of solids: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron.

The cube can be represented in many ways, one of which is the graph known as the cubical graph. It can be constructed by using the Cartesian product of graphs. The cube was discovered in antiquity. It was associated with the nature of earth by Plato, the founder of Platonic solid. It was used as the part of the Solar System, proposed by Johannes Kepler. It can be derived differently to create more polyhedrons, and it has applications to construct a new polyhedron by attaching others. It can be generalized as tesseract in four-dimensional space.

Properties

A cube is a special case of rectangular cuboid in 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 the dihedral angle of a cube between every two adjacent squares being the interior angle of 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 five Platonic solids, a polyhedron in which all the regular polygons are congruent and the same number of faces meet at each vertex. ^[3]

Measurement and other metric properties

A face diagonal in red and space diagonal in blue.

The Prince Rupert's cube

Given that a cube with edge length ${\displaystyle a}$. The face diagonal of a cube is the diagonal of a square ${\displaystyle a{\sqrt {2}}}$, and the space diagonal of a cube is a line connecting two vertices that is not in the same face, formulated as ${\displaystyle a{\sqrt {3}}}$. Both formulas can be determined by using Pythagorean theorem. The surface area of a cube ${\displaystyle A}$ is six times the area of a square: ^[4] $${\displaystyle A=6a^{2}.}$$ 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] $${\displaystyle V=a^{3}.}$$

A unit cube is a special case where each cube's edge is 1 unit length. The surface area and the volume of a unit cube is 1. ^{[5] [6]} It has Rupert property, meaning a polyhedron with the same or larger size can pass through into the hole of the unit cube. The Prince Rupert's cube, named after Prince Rupert of the Rhine, is the largest cube that can fit inside, with the size being 6% larger. ^[7]

A unit cube and a cube with twice the volume

A geometric 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 a compass and straightedge solely. Ancient mathematicians could not solve this old problem until French mathematician Pierre Wantzel in 1837 proved it was impossible. ^[8]

Relation to the spheres

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 ${\textstyle {\frac {1}{\sqrt {2}}}a}$. The circumscribed sphere of a cube is the sphere tangent to the vertices of a cube, with radius ${\textstyle {\frac {\sqrt {3}}{2}}a}$. ^[9]

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: ^[10] $${\displaystyle {\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{4}+{\frac {16R^{4}}{9}}=\left({\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{2}+{\frac {2R^{2}}{3}}\right)^{2}.}$$

Symmetry

The cube has octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$. It is composed of reflection 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 of rotational symmetry, a symmetry by rotating it around the axis, from which the appearance is interchangeable. It has octahedral rotation symmetry ${\displaystyle \mathrm {O} }$: 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°). ^{[11] [12] [13]}

The dual polyhedron of a cube is the regular octahedron

The dual polyhedron can be obtained from each of the polyhedron's vertices tangent to a plane by the process known as polar reciprocation. ^[14] One property of dual polyhedrons generally 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 polyhedron has the same symmetry, the octahedral symmetry. ^[15]

The cube is face-transitive, meaning its two squares are alike and can be mapped by rotation and reflection. ^[16] It is vertex-transitive, meaning all of its vertices are equivalent and can be mapped isometrically under its symmetry. ^[17] It is also edge-transitive, meaning the same kind of faces surround each of its vertices in the same or reverse order, all two adjacent faces have the same dihedral angle. Therefore, the cube is regular polyhedron because it requires those properties. ^[18]

Classifications

3D model of a cube

The cube is a special case among every cuboids. As mentioned above, the cube can be represented as the rectangular cuboid with edges equal in length and all of its faces are all squares. ^[1] The cube may be considered as the parallelepiped in which all of its edges are equal edges. ^[19]

The cube is one of the types of parallelohedron, meaning it is a polyhedron that can be translated without rotating to fill a space—called honeycomb —in which all of its copies are being attached. ^[20] Every three-dimensional parallelohedron is zonohedron, a centrally symmetric polyhedron whose faces are centrally symmetric polygons, ^[21] and it is plesiohedron, a special kind of space-filling polyhedron that can be defined as the Voronoi cell of a symmetric Delone set. ^[22]

Construction

Nets of a cube

An elementary way to construct a cube is using the net. A net is an arrangement of edge-joining polygons constructing a polyhedron by connecting along the edges of those polygons. Here, there are eleven different cube's net. ^[23]

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 ${\displaystyle (\pm 1,\pm 1,\pm 1)}$. ^[24] 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 Hanner polytope, because it can be constructed by using Cartesian product of three line segments. Its dual polyhedron, the regular octahedron, is constructed by direct sum of three line segments. ^[25]

Representation

As a graph

Main article: Hypercube graph

The graph of a cube, and its construction

According to Steinitz's theorem, the graph can be represented as the skeleton of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties. It is planar, meaning the edges of a graph are connected to every vertex without crossing other edges. It is also 3-connected graph, meaning that, whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected. ^{[26] [27]} The skeleton of a cube can be represented as the graph, and it 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. ^[28]

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 operation known as the Cartesian 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. ^[29] In the case of the cubical graph, it is the product of two ${\displaystyle Q_{2}}$; 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 ${\displaystyle Q_{3}}$. ^[30] It is a unit distance graph. ^[31]

Like other graphs of cuboids, the cubical graph is also classified as the prism graph. ^[32]

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. Conventionally, the cube is 6-equiprojective. ^[33]

As a configuration matrix

The cube can be represented as configuration matrix. A configuration matrix is a matrix in which the rows and columns correspond to the elements of a polyhedron as in 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. 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: ^[34] $${\displaystyle {\begin{bmatrix}{\begin{matrix}8&3&3\\2&12&2\\4&4&6\end{matrix}}\end{bmatrix}}}$$

Appearances

In antiquity

Sketch of a cube by Johannes Kepler

Kepler's Platonic solid model of the Solar System

The Platonic solid is a set of polyhedrons known since antiquity. It was named after Plato in his Timaeus dialogue, who attributed these solids with nature. One of them, the cube, represented the classical element of earth because of its stability. ^[35] Euclid's Elements defined 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. ^[36]

Following its attribution with nature by Plato, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids, one of them is a cube in which Kepler decorated a tree on it. ^[35] 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. ^[37]

Polyhedron, honeycombs, and polytopes

Some of the derived cube, the stellated octahedron and tetrakis hexahedron.

The cube can appear in the construction of a polyhedron, and 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. ^[38]
• Attaching a square pyramid to each square face of a cube produces its Kleetope, a polyhedron known as the tetrakis hexahedron. ^[39] Suppose one and two equilateral square pyramids are attached to their square faces. In that case, they are the construction of an elongated square pyramid and elongated square bipyramid respectively, the Johnson solid's examples. ^[40]
• Each of the cube's vertices can be truncated, and the resulting polyhedron is the Archimedean solid, the truncated cube. ^[41] When its edges are truncated, it is a rhombicuboctahedron. ^[42] 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. ^[43]
• The snub cube is 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 as snub. ^[44]

The honeycomb is the space-filling or tessellation in 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 the cell, and examples of a honeycomb are cubic honeycomb, order-5 cubic honeycomb, order-6 cubic honeycomb, and order-7 cubic honeycomb. ^[45] The cube can be constructed with six square pyramids, tiling space by attaching their apices. ^[46]

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. ^[47] When four cubes are stacked vertically, and the other four are attached to the second-from-top cube of the stack, the resulting polycube is Dali cross, after Salvador Dali. The Dali cross is a tile space polyhedron, ^{[48] [49]} which can be represented as the net of a tesseract. A tesseract is a cube analogous' four-dimensional space bounded by twenty-four squares, and it is bounded by the eight cubes known as its cells. ^[50]

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. 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.
4. Khattar, Dinesh (2008). Guide to Objective Arithmetic (2nd ed.). Pearson Education. p. 377. ISBN   978-81-317-1682-3.
5. 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.
6. Geometry: Reteaching Masters. Holt Rinehart & Winston. 2001. p. 74. ISBN   9780030543289.
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. ISBN   9781607521013.
8. 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.
9. 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).
10. Poo-Sung, Park, Poo-Sung (2016). "Regular polytope distances" (PDF). Forum Geometricorum . 16: 227–232.
11. French, Doug (1988). "Reflections on a Cube". Mathematics in School. 17 (4): 30–33. JSTOR   30214515.
12. Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 309. ISBN   978-0-521-55432-9.
13. 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.
14. Cundy, H. Martyn; Rollett, A.P. (1961). "3.2 Duality". Mathematical models (2nd ed.). Oxford: Clarendon Press. pp. 78–79. MR   0124167.
15. Erickson, Martin (2011). Beautiful Mathematics. Mathematical Association of America. p. 62. ISBN   978-1-61444-509-8.
16. 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.
17. Grünbaum, Branko (1997). "Isogonal Prismatoids". Discrete & Computational Geometry. 18: 13–52. doi:10.1007/PL00009307.
18. Senechal, Marjorie (1989). "A Brief Introduction to Tilings". In Jarić, Marko (ed.). Introduction to the Mathematics of Quasicrystals. Academic Press. p. 12.
19. Calter, Paul; Calter, Michael (2011). Technical Mathematics. John Wiley & Sons. p. 197. ISBN   978-0-470-53492-2.
20. Alexandrov, A. D. (2005). "8.1 Parallelohedra". Convex Polyhedra. Springer. pp. 349–359.
21. 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.
22. 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.
23. 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.
24. Smith, James (2000). Methods of Geometry. John Wiley & Sons. p. 392. ISBN   978-1-118-03103-2.
25. 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.
26. 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.
27. 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.
28. Rudolph, Michael (2022). The Mathematics of Finite Networks: An Introduction to Operator Graph Theory. Cambridge University Press. p. 25. doi:10.1007/9781316466919 (inactive 2024-07-17). ISBN   9781316466919.
29. 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 .
30. Chartrand, Gary; Zhang, Ping (2012). A First Course in Graph Theory. Dover Publications. p. 25.
31. 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.
32. 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.
33. 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].
34. Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover Publications. pp.  122–123. See §1.8 Configurations.
35. Cromwell (1997), p.  55.
36. Heath, Thomas L. (1908). The Thirteen Books of Euclid's Elements (3rd ed.). Cambridge University Press. p. 262, 478, 480.
37. 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.
38. Inchbald, Guy (2006). "Facetting Diagrams". The Mathematical Gazette . 90 (518): 253–261. doi:10.1017/S0025557200179653. JSTOR   40378613.
39. 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.
40. 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.
41. Cromwell (1997), pp.  81–82.
42. 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.
43. 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.
44. Holme, A. (2010). Geometry: Our Cultural Heritage. Springer. doi:10.1007/978-3-642-14441-7. ISBN   978-3-642-14441-7.
45. Coxeter, H. S. M. (1968). The Beauty of Geometry: Twelve Essays. Dover Publications. p. 167. ISBN   978-0-486-40919-1. See table III.
46. 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.
47. 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.
48. Diaz, Giovanna; O'Rourke, Joseph (2015). "Hypercube unfoldings that tile ${\displaystyle \mathbb {R} ^{3}}$ and ${\displaystyle \mathbb {R} ^{2}}$". arXiv: 1512.02086 [cs.CG].
49. 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).
50. 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.

External links

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