The hypercube, also called an cube
to dimensions ,
where positive
integer . It is a regular polytope with mutually
perpendicular sides, and is therefore an orthotope .
It is denoted Schläfli symbol square
has symbol
The following table summarizes the names of
The number of coefficient of binomial coefficient . The number of
polytope vertices in the A000079 ), the
number of polytope edges is A001787 ),
the number of squares is A001788 ),
the number of cubes is A001789 ),
etc.
The numbers of distinct nets for the A091159 ; Turney 1984-85).
The above figure shows a projection of the tesseract in three-space. A tesseract has 16 polytope
vertices , 32 polytope edges , 24 squares ,
and eight cubes .
The dual of the tesseract is known as the 16-cell . For all dimensions, the dual of the hypercube is the cross
polytope (and vice versa).
An isometric projection of the 5-hypercube appears together with the great rhombic triacontahedron
on the cover of Coxeter's well-known book on polytopes (Coxeter 1973).
Wilker (1996) considers the point in the unit product of distances
to its polytope vertices (Trott 2004, p. 104).
The following table summarizes results for small
See also Cross Polytope ,
Cube ,
Cube-Connected Cycle Graph ,
Glome ,
Hamiltonian Graph ,
Hypercube
Graph ,
Hypercube Line Picking ,
Hypersphere ,
Orthotope ,
Parallelepiped ,
Polytope ,
Regular Polytope ,
Simplex ,
Tesseract Explore this topic in the MathWorld classroom
Explore with Wolfram|Alpha
References Born, M. Problems of Atomic Dynamics. Coxeter,
H. S. M. Regular
Polytopes, 3rd ed. Dewdney,
A. K. "Computer Recreations: A Program for Rotating Hypercubes Induces
Four-Dimensional Dementia." Sci. Amer. 254 , 14-23, Mar. 1986. Fischer,
G. (Ed.). Plates 3-4 in Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Gardner, M. "Hypercubes."
Ch. 4 in Mathematical
Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. Geometry Center. "The
Tesseract (or Hypercube)." http://www.geom.uiuc.edu/docs/outreach/4-cube/ . Pappas,
T. "How Many Dimensions Are There?" The
Joy of Mathematics. Skiena, S. "Hypercubes." §4.2.5 in Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Sloane, N. J. A.
Sequences A000079 /M1129, A001787 /M3444,
A001788 /M4161, A001789 /M4522,
and A091159 in "The On-Line Encyclopedia
of Integer Sequences." Trott, M. The
Mathematica GuideBook for Programming. https://www.mathematicaguidebooks.org/ . Trott,
M. "The Mathematica Guidebooks Additional Material: Hypercube Projections."
https://www.mathematicaguidebooks.org/additions.shtml#N_1_04 . Turney,
P. D. "Unfolding the Tesseract." J. Recr. Math. 17 ,
No. 1, 1-16, 1984-85. Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. Wilker, J. B. "An
Extremum Problem for Hypercubes." J. Geom. 55 , 174-181, 1996. Williams,
R. The
Geometrical Foundation of Natural Structure: A Source Book of Design. Referenced on Wolfram|Alpha Hypercube
Cite this as:
Weisstein, Eric W. "Hypercube." From MathWorld https://mathworld.wolfram.com/Hypercube.html
Subject classifications More... Less...
-cube or measure polytope, is a generalization of a 3-cube
to 
dimensions,
where 
is a positive
integer. It is a regular polytope with mutually
perpendicular sides, and is therefore an orthotope.
It is denoted 
and for 
has Schläfli symbol 
,
with 
entries equal to 3. The square
has symbol 
.
-dimensional hypercubes.
-cubes
contained in an 
-cube
can be found as the coefficient of 
in 
, namely 
,
where 
is a binomial coefficient. The number of
polytope vertices in the 
-hypercube is therefore 
(OEIS A000079), the
number of polytope edges is 
(OEIS A001787),
the number of squares is 
(OEIS A001788),
the number of cubes is 
(OEIS A001789),
etc.
-hypercube for 
, 3, ... are 1, 11, 261, ...
(OEIS A091159; Turney 1984-85).
-cube ![[0,1]^n](/images/equations/Hypercube/Inline25.svg)
that maximizes the product of distances
to its polytope vertices (Trott 2004, p. 104).
The following table summarizes results for small 
.
