mimic continuously changing input information, such as
the rate of flow of oil through a pipeline, for example.
See also BASE OF A NUMBER SYSTEM; ERROR.
dihedral Any geometric construct formed by the
intersection of two planes is called dihedral. For example, the line of intersection of two nonparallel planes is
called a dihedral line, and the angle between the two
planes is called a dihedral angle (or, sometimes, a dihedron). A dihedral angle can be computed by taking the
DOT PRODUCT of the two normal vectors to the planes.
The dihedral angle of a polyhedron is the angle
between two adjacent faces of the solid.
The word dihedral comes from the Greek prefix di-,
meaning “two,” and the word hedra, meaning “base,”
“seat,” or “surface.”
See also NORMAL TO A PLANE.
dilation See GEOMETRIC TRANSFORMATION; LINEAR
TRANSFORMATION.
dimension The number of coordinates needed to
specify the position of a particular point in space is
called the dimension of that space. Lines and curves are
considered one-dimensional, since the location of any
point on a curve can be specified by a single parameter,
namely, the distance along the curve at which it lies.
Points in two-dimensional space form a surface, and
points in three-dimensional space lie within a volume.
A physical theory is said to be multidimensional if it
describes the universe with a large number of parameters.
Since events in the world occur at specific locations and
specific times, four parameters are needed to describe
them: three spatial parameters x, y, and z and a fourth
parameter t for time. Because of this, “time” is often
cited as being the fourth dimension. However, a theory
that also considers the electric charge q of objects, say,
would be described as “five-dimensional,” and one that
also considers magnetic strength as “six-dimensional.”
Physicists have asserted that we live in a universe that is
as much as 18-dimensional. There is little meaning here
other than that scientists are asserting that all events in
the universe can be described fully through 18 different
variables. There is no specific reason for the fourth variable to always be interpreted as “time.”
To a mathematician, an n-dimensional object is one
that is fully described by n parameters. For example, a
circle sitting in two-dimensional space is described by
an equation with two variables: x
2
+y
2
=1. A SPHERE in
three-dimensional space is described by an equation
with three variables: x
2
+y
2
+z
2
=1. Adding more variables to the equation gives higher-dimensional spheres:
the equations x
2
+y
2
+z
2
+w
2
=1 and x
2
+y
2
+z
2
+w
2
+u
2 = 1,
for instance, might be said to describe “hyperspheres”
in four- and five-dimensional space. Although one cannot envision what these objects are, the mathematics of
these objects is little different from the mathematics of
ordinary circles and spheres.
In a geometric context, “dimension” can be
described through the notion of scaling. If we SCALE a
geometric object by a factor k, then its size changes
accordingly: any line of length a becomes a line of
length ka, any planar region of area A becomes a planar region of area k
2
A, and any solid of volume V is
replaced by a solid of volume k
3
V. An object can thus
be described as d-dimensional if its “size” scales
according to the rule:
new size=k
d
× old size
In this context, it is possible for a geometric object to
have fractional dimension. Such an object is called a
FRACTAL.
See also HYPERCUBE.
Diocles (ca. 240–180 B.C.E.) Greek Geometry Born
on Évvoia (Euboea), a Greek island (the exact birth
date is not known), Diocles is remembered for his work
on CONIC SECTIONS and his innovative solution to the
DUPLICATING THE CUBE problem via the invention of a
new curve called the cissoid curve.
Almost nothing is known of Diocles’ life, and
knowledge of his work (until recently) came to us
chiefly through references made by scholars after his
time. It is known that Diocles was a contemporary of
APOLLONIUS OF PERGA (ca. 260–190 B.C.E.) and may
have spent considerable time at Arcadia, the intellectual
center of Greek culture.
Diocles wrote one significant text, On Burning
Mirrors, which, although chiefly ignored by Greek
scholars, had considerable influence on Arab mathematicians. A translation of this work was only
136 dihedral
the rate of flow of oil through a pipeline, for example.
See also BASE OF A NUMBER SYSTEM; ERROR.
dihedral Any geometric construct formed by the
intersection of two planes is called dihedral. For example, the line of intersection of two nonparallel planes is
called a dihedral line, and the angle between the two
planes is called a dihedral angle (or, sometimes, a dihedron). A dihedral angle can be computed by taking the
DOT PRODUCT of the two normal vectors to the planes.
The dihedral angle of a polyhedron is the angle
between two adjacent faces of the solid.
The word dihedral comes from the Greek prefix di-,
meaning “two,” and the word hedra, meaning “base,”
“seat,” or “surface.”
See also NORMAL TO A PLANE.
dilation See GEOMETRIC TRANSFORMATION; LINEAR
TRANSFORMATION.
dimension The number of coordinates needed to
specify the position of a particular point in space is
called the dimension of that space. Lines and curves are
considered one-dimensional, since the location of any
point on a curve can be specified by a single parameter,
namely, the distance along the curve at which it lies.
Points in two-dimensional space form a surface, and
points in three-dimensional space lie within a volume.
A physical theory is said to be multidimensional if it
describes the universe with a large number of parameters.
Since events in the world occur at specific locations and
specific times, four parameters are needed to describe
them: three spatial parameters x, y, and z and a fourth
parameter t for time. Because of this, “time” is often
cited as being the fourth dimension. However, a theory
that also considers the electric charge q of objects, say,
would be described as “five-dimensional,” and one that
also considers magnetic strength as “six-dimensional.”
Physicists have asserted that we live in a universe that is
as much as 18-dimensional. There is little meaning here
other than that scientists are asserting that all events in
the universe can be described fully through 18 different
variables. There is no specific reason for the fourth variable to always be interpreted as “time.”
To a mathematician, an n-dimensional object is one
that is fully described by n parameters. For example, a
circle sitting in two-dimensional space is described by
an equation with two variables: x
2
+y
2
=1. A SPHERE in
three-dimensional space is described by an equation
with three variables: x
2
+y
2
+z
2
=1. Adding more variables to the equation gives higher-dimensional spheres:
the equations x
2
+y
2
+z
2
+w
2
=1 and x
2
+y
2
+z
2
+w
2
+u
2 = 1,
for instance, might be said to describe “hyperspheres”
in four- and five-dimensional space. Although one cannot envision what these objects are, the mathematics of
these objects is little different from the mathematics of
ordinary circles and spheres.
In a geometric context, “dimension” can be
described through the notion of scaling. If we SCALE a
geometric object by a factor k, then its size changes
accordingly: any line of length a becomes a line of
length ka, any planar region of area A becomes a planar region of area k
2
A, and any solid of volume V is
replaced by a solid of volume k
3
V. An object can thus
be described as d-dimensional if its “size” scales
according to the rule:
new size=k
d
× old size
In this context, it is possible for a geometric object to
have fractional dimension. Such an object is called a
FRACTAL.
See also HYPERCUBE.
Diocles (ca. 240–180 B.C.E.) Greek Geometry Born
on Évvoia (Euboea), a Greek island (the exact birth
date is not known), Diocles is remembered for his work
on CONIC SECTIONS and his innovative solution to the
DUPLICATING THE CUBE problem via the invention of a
new curve called the cissoid curve.
Almost nothing is known of Diocles’ life, and
knowledge of his work (until recently) came to us
chiefly through references made by scholars after his
time. It is known that Diocles was a contemporary of
APOLLONIUS OF PERGA (ca. 260–190 B.C.E.) and may
have spent considerable time at Arcadia, the intellectual
center of Greek culture.
Diocles wrote one significant text, On Burning
Mirrors, which, although chiefly ignored by Greek
scholars, had considerable influence on Arab mathematicians. A translation of this work was only
136 dihedral