 The cube is the Platonic solid  (also called
the regular hexahedron). It is composed of six square
faces that meet each other at right
angles and has eight vertices and 12 edges. It is also the uniform polyhedron  and Wenninger
model  . It is described by the Schläfli symbol  and Wythoff symbol  .
The cube is illustrated above, together with a wireframe version and a net (top figures). The bottom figures show three symmetric
projections of the cube.
There are a total of 11 distinct nets for the cube (Turney 1984-85, Buekenhout and Parker 1998, Malkevitch), illustrated above,
the same number as the octahedron.
Questions of polyhedron coloring
of the cube can be addressed using the Pólya
enumeration theorem.
A cube with unit edge lengths is called a unit
cube.
The surface area and volume of a cube with edge length  are
Because the volume of a cube of edge length  is given by  , a number of the form  is called a
cubic number (or sometimes simply
"a cube"). Similarly, the operation of taking a number to the third power is called cubing.
A unit cube has inradius, midradius,
and circumradius of
The cube has a dihedral angle
of
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(6)
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In terms of the inradius  of a cube, its
surface area  and volume  are given by
so the volume, inradius, and surface area are related by
 |
(9)
|
where  is the harmonic
parameter (Dorff and Hall 2003, Fjelstad and Ginchev 2003).
The illustration above shows an origami
cube constructed from a single sheet of paper (Kasahara and Takahama 1987, pp. 58-59).
Sodium chloride (NaCl; common table salt) naturally forms cubic crystals.
The world's largest cube is the Atomium, a structure built for the 1958 Brussels World's Fair, illustrated above (© 2006 Art Creation (ASBL); Artists Rights
Society (ARS), New York; SABAM, Belgium). The Atomium is 334.6 feet high, and the
nine spheres at the vertices and center
have diameters of 59.0 feet. The distance between the spheres along the edge of the
cube is 95.1 feet, and the diameter of the tubes connecting the spheres is 9.8 feet.
The dual polyhedron of a unit cube is an octahedron with edge lengths  .
The cube has the octahedral group  of symmetries, and is an equilateral zonohedron and a rhombohedron.
It has 13 axes of symmetry:  (axes joining midpoints of opposite
edges),  (space diagonals), and  (axes joining
opposite face centroids).
The connectivity of the vertices of the cube is given by the cubical graph.
Using so-called "wallet hinges," a ring of six cubes can be rotated continuously (Wells 1975; Wells 1991, pp. 218-219).
The illustrations above show the cross sections obtained by cutting a unit cube centered at the origin with various planes. The following
table summarizes the metrical properties of these slices.
As shown above, a plane passing through the midpoints of opposite edges (perpendicular
to a  axis) cuts the cube in a regular hexagonal cross
section (Gardner 1960; Steinhaus 1999, p. 170; Kasahara 1988, p. 118;
Cundy and Rollett 1989, p. 157; Holden 1991, pp. 22-23). Since there are
four such axes, there are four possible hexagonal cross sections. If the vertices of
the cube are  , then the vertices of
the inscribed hexagon are  ,  ,  ,  ,  , and  . A hexagon
is also obtained when the cube is viewed from above a corner along the extension
of a space diagonal (Steinhaus 1999, p. 170).
The maximal cross sectional area that can be obtained by cutting a unit cube with a plane passing through its center is  , corresponding
to a rectangular section intersecting the cube in two diagonally opposite edges and
along two opposite face diagonals. The area obtained as a function of normal to the
plane  is illustrated above (Hidekazu).
A hyperboloid of one sheet is obtained as the envelope of a cube rotated about a space diagonal (Steinhaus 1999, pp. 171-172;
Kabai 2002, p. 11). The resulting volume for a cube with edge length  is
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(10)
|
(Cardot and Wolinski 2004).
More generally, consider the solid of revolution obtained for revolution axis passing through the center and the
point  , several examples of which are shown above.
As shown by Hidekazu, the solid with maximum volume is obtained for parameters of approximately  . This corresponds
to the rightmost plot above.
The centers of the faces of an octahedron form a cube, and the centers of the faces of a cube form an octahedron (Steinhaus 1999, pp. 194-195). The largest
square which will fit inside a cube
of edge length  has each corner a distance 1/4 from a
corner of a cube. The resulting square
has edge length  , and the cube containing that
edge is called Prince Rupert's
cube.
The solid formed by the faces having the edges of the stella octangula (left figure) as polygon diagonals is a cube (right figure; Ball and Coxeter
1987). Affixing a square pyramid
of height 1/2 on each face of a cube having unit edge length results in a rhombic dodecahedron (Brückner 1900, p. 130; Steinhaus
1999, p. 185).
Since its eight faces are mutually perpendicular or parallel, the cube cannot be
stellated.
The cube can be constructed by cumulation of a unit edge-length tetrahedron
by a pyramid with height  . The following table gives
polyhedra which can be constructed by cumulation
of a cube by pyramids of given heights  .
The polyhedron vertices of a cube of edge length 2 with face-centered axes are given by  .
If the cube is oriented with a space diagonal along the z-axis,
the coordinates are (0, 0,  ), (0,  ,  ), ( ,  ,  ), ( ,  ,  ), (0,  ,  ), ( ,  ,  ), ( ,  ,  ), and the negatives of these vectors. A faceted version is the great cubicuboctahedron.
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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York:
Dover, 1987.
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton,
FL: CRC Press, pp. 127 and 228, 1987.
Brückner, M. Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900.
Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension  ." Disc. Math. 186,
69-94, 1998.
Cardot C. and Wolinski F. "Récréations scientifiques." La
jaune et la rouge, No. 594, 41-46, 2004.
Cundy, H. and Rollett, A. "Cube.  " and "Hexagonal
Section of a Cube." §3.5.2 and 3.15.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin
Pub., pp. 85 and 157, 1989.
Davie, T. "The Cube (Hexahedron)." http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/cube.html.
Dorff, M. and Hall, L. "Solids in  Whose Area is
the Derivative of the Volume." College Math. J. 34, 350-358, 2003.
Eppstein, D. "Rectilinear Geometry." http://www.ics.uci.edu/~eppstein/junkyard/rect.html.
Fischer, G. (Ed.). Plate 2 in Mathematische Modelle aus den Sammlungen von Universitäten
und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 3, 1986.
Fjelstad, P. and Ginchev, I. "Volume, Surface Area, and the Harmonic Mean."
Math. Mag. 76, 126-129, 2003.
Gardner, M. "Mathematical Games: More about the Shapes That Can Be Made with
Complex Dominoes." Sci. Amer. 203, 186-198, Nov. 1960.
Geometry Technologies. "Cube." http://www.scienceu.com/geometry/facts/solids/cube.html.
Harris, J. W. and Stocker, H. "Cube" and "Cube (Hexahedron)." §4.2.4 and 4.4.3 in Handbook of Mathematics and Computational Science. New
York: Springer-Verlag, pp. 97-98 and 100, 1998.
Hidekazu, T. "Research on a Cube." http://www.biwako.ne.jp/~hidekazu/materials/cubee.htm.
Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.
Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica.
Püspökladány, Hungary: Uniconstant, p. 231, 2002.
Kasahara, K. "Cube A--Bisecting I," "Cube B--Bisecting II," "Cube C--Bisecting Horizontally," "Cube D--Bisecting on the Diagonal," "Cube
E--Bisecting III," "Making a Cube from a Cube with a Single Cut,"
and "Module Cube." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan
Publications, pp. 104-108, 112, and 118-120, and 208, 1988.
Kasahara, K. and Takahama, T. Origami for the Connoisseur. Tokyo: Japan Publications,
1987.
Kern, W. F. and Bland, J. R. "Cube." §9 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley,
pp. 19-20, 1948.
Malkevitch, J. "Nets: A Tool for Representing Polyhedra in Two Dimensions."
http://www.ams.org/new-in-math/cover/nets.html.
Malkevitch, J. "Unfolding Polyhedra." http://www.york.cuny.edu/~malk/unfolding.html.
Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.
Turney, P. D. "Unfolding the Tesseract." J. Recr. Math. 17,
No. 1, 1-16, 1984-85.
Wells, D. "Puzzle Page." Games and Puzzles. Sep. 1975.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, pp. 41-42 and 218-219, 1991.
Wenninger, M. J. "The Hexahedron (Cube)." Model 3 in Polyhedron Models. Cambridge, England: Cambridge University
Press, p. 16, 1989.
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