When the strands tilt, the surface of a hyperboloid of

one sheet appears.

See also

ELLIPSOID

;

PARABOLOID

.

hypercube The higher-dimensional analog of

SQUARE

in two dimensions and a

CUBE

in three dimensions is

called a hypercube.

A unit square in the plane has four vertices at loca-

tions (0,0),(1,0),(0,1) and (1,1). A unit cube in three-

dimensional space has eight vertices at locations

(0,0,0), (1,0,0), (0,1,0), (1,1,0), (0,0,1), (1,0,1), (0,1,1)

and (1,1,1). By analogy, a hypercube of side-length 1

sitting in four-dimensional space has 16 vertices given

by: (0,0,0,0), (1,0,0,0), (0,1,0,0),…,(1,1,1,1).

Some people find it helpful to interpret the fourth

coordinate as time. In this context, each particular

value of the fourth coordinate represents a given

instant in time, and the remaining three coordinates

describe an ordinary object in three-dimensional space.

A four-dimensional hypercube can thus be thought of

as an ordinary cube moving through time. (One can

also apply this interpretation to ordinary squares and

cubes of lower dimension. For example, a line segment

is the result of a point sliding horizontally over time, a

square is the result of a horizontal line segment sliding

in the direction of the positive y-axis over time, and a

cube is the result of a two-dimensional square sliding in

the direction of the positive z-axis over time.) A four-

dimensional hypercube is also called a tesseract.

A square in two-dimensional space has four ver-

tices and four edges. A cube in three-dimensional space

has eight vertices and six square faces. One can show

that a tesseract has eight vertices and six cubic “faces,”

and that, in general, an n-dimensional hypercube has

2nvertices and 2nfaces, each itself a hypercube of one

dimension less.

Four line segments hinged end-to-end fold to

make a square. Six squares arranged in a “cross” fold

to make a cube. Eight cubes arranged in an analogous

way—four in a row with the four remaining cubes

attached to the four exposed faces of the second

cube—fold in four-dimensional space to make a

hypercube. All in all, there are 261 different ways to

arrange eight ordinary cubes that fold in four-dimen-

sional space to make a hypercube. Each arrangement

is called a

NET

.

hypotenuse The longest side of a triangle that con-

tains a 90°angle, that is, a right triangle, is called the

hypotenuse of the triangle. P

YTHAGORAS

’

S THEOREM

shows that this longest side lies opposite the right

angle. (If the three side-lengths of a right triangle are a,

b, and c, with copposite the right angle, then c2= a2+

b2. This establishes that cis indeed larger than both a

and b.) The two remaining sides of a right triangle are

called the legs of the triangle.

The word hypotenuse is derived from the Greek

term hypoteinousa meaning “under tension” (the prefix

hypo-means “under” and teinein means “to stretch”).

Mathematicians have shown that an integer ccan

be the length of the hypotenuse of a right triangle with

integer side-lengths if, and only if, in the prime

FAC

-

TORIZATION

of c, no

PRIME

that appears an odd num-

ber of times is 3 more than a multiple of 4.

See also P

YTHAGOREAN TRIPLES

.

hypothesis testing See

STATISTICS

:

INFERENTIAL

.

256 hypercube

By rotating the top disc of this 1872 string model, one can trans-

form a cylinder of chords into the shape of a hyperboloid. (Photo

courtesy of the Science Museum, London/Topham-HIP/The

Image Works)