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The cube, illustrated above together with a wireframe version and anet that can be used for its construction, is thePlatonic solid composed of sixsquare faces that meet each other atright angles and has eight vertices and 12 edges. It is also theuniform polyhedron with Maeder index 6 (Maeder 1997), Wenninger index 3 (Wenninger 1989), Coxeter index 18 (Coxeteret al.1954), and Har'El index 11 (Har'El 1993). It is described by theSchläfli symbol andWythoff symbol.

Three symmetric projections of the cube are illustrated above.

The cube is the unique regular convexhexahedron. The topologically distinctpentagonal wedge is the only other convexhexahedron that shares the same number of vertices, edges, and faces as the cube (though of course with different face shapes; the pentagonal wedge consists of triangles, 2 quadrilaterals, and 2 pentagons).

The cube is implemented in theWolfram Language asCube[] orUniformPolyhedron["Cube"]. Precomputed properties are available asPolyhedronData["Cube",prop].

The cube is aspace-filling polyhedronand therefore hasDehn invariant 0.

It is theconvex hull of theendododecahedronandstella octangula.

There are a total of 11 distinctnets for the cube (Turney 1984-85, Buekenhout and Parker 1998, Malkevitch), illustrated above, the same number as theoctahedron. Questions ofpolyhedron coloring of the cube can be addressed using thePólya enumeration theorem.

A cube with unit edge lengths is called aunit cube.

Thesurface area andvolume of a cube with edge length are

			(1)
			(2)

Because thevolume of a cube of edge length is given by, a numberof the form is called acubic number (or sometimes simply "a cube"). Similarly, the operation of taking a number to the thirdpower is calledcubing.

Aunit cube hasinradius,midradius, andcircumradius of

			(3)
			(4)
			(5)

The cube has adihedral angle of

(6)

In terms of theinradius of a cube, its surface area and volume are given by

			(7)
			(8)

so the volume, inradius, and surface area are related by

(9)

where is theharmonic parameter (Dorff and Hall 2003, Fjelstad and Ginchev 2003).

The illustration above shows anorigami cube constructedfrom 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 ninespheres 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.

Thedual polyhedron of aunit cube is anoctahedron with edge lengths.

The cube has theoctahedral group of symmetries, and is anequilateral zonohedron and arhombohedron. 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 thecubicalgraph.

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 aunit cube centered at the origin with various planes. The following table summarizes the metrical properties of these slices.

cutting plane	face shape	edge lengths	surface area	volume of pieces
	square	1	1	,
	rectangle	1,		,
	hexagon			,
	equilateral triangle			,

As shown above, aplane passing through themidpoints of opposite edges (perpendicular to a axis) cuts the cube in a regularhexagonalcross 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 possiblehexagonalcross sections. If the vertices of the cube are, then the vertices of the inscribedhexagon are,,,,, and. Ahexagon 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).

Ahyperboloid 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

(10)

(Cardot and Wolinski 2004).

More generally, consider thesolid 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 anoctahedron form a cube, and the centers of the faces of a cube form anoctahedron (Steinhaus 1999, pp. 194-195). The largestsquare which will fit inside a cube of edge length has each corner a distance 1/4 from a corner of a cube. The resultingsquare has edge length, and the cube containing that edge is calledPrince Rupert's cube.

The solid formed by the faces having the edges of thestella octangula (left figure) aspolygon diagonals is a cube (right figure; Ball and Coxeter 1987). Affixing asquare pyramid of height 1/2 on each face of a cube having unit edge length results in arhombic dodecahedron (Brückner 1900, p. 130; Steinhaus 1999, p. 185).

Since its eight faces are mutually perpendicular or parallel, the cube cannot bestellated.

The cube can be constructed byaugmentation of a unit edge-lengthtetrahedron by a pyramid with height. The following table gives polyhedra which can be constructed byaugmentation of acube by pyramids of given heights.

	result
	tetrakis hexahedron
	rhombic dodecahedron
	starequilateral 24-deltahedron

Thepolyhedron 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 thez-axis, the coordinates are (0, 0,), (0,,), (,,), (,,), (0,,), (,,), (,,), and the negatives of these vectors. Afaceted version is thegreat cubicuboctahedron.

See also

Explore with Wolfram|Alpha

More things to try:

• cube
• 7^3
• cbrt(3)

References

Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension."Disc. Math.186, 69-94, 1998.

Cundy, H. and Rollett, A. "Cube." and "Hexagonal Section of a Cube." §3.5.2 and 3.15.1 inMathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 85 and 157, 1989.

Dorff, M. and Hall, L. "Solids in Whose Area is the Derivative of the Volume."College Math. J.34, 350-358, 2003.

Hidekazu, T. "Research on a Cube."http://www.biwako.ne.jp/~hidekazu/materials/cubee.htm

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Cube." FromMathWorld--AWolfram Web Resource.https://mathworld.wolfram.com/Cube.html

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