crust on each slice is the same. (Assume that
the slices are made parallel, each perpendicular
to a fixed diameter of the loaf.)
The region of a sphere bounded between two parallel planes is called a spherical segment or a zone. If
the radius of the sphere is r and the distance between
the two planes is h, then the surface area of the zone is
2πrh. (Notice that this formula is indeed independent
of where the two planes slice the sphere.) The solid
obtained by rotating a sector of a circle about the
diameter of the circle is called a spherical sector, and a
spherical wedge is any segment of a sphere bounded
between two great circles.
A SPHERICAL TRIANGLE is any region on the surface
of a sphere bounded by arcs of three great circles, and
spherical trigonometry is the study of spherical triangles and the angles formed by intersecting great circles.
The study of SPHERICAL GEOMETRY shows that the
three angles in any spherical triangle sum to more than
180°. A spherical polygon is the figure formed on the
surface of a sphere by three or more arcs of great circles. Mathematicians have shown that the angles in an
n-sided spherical polygon sum to value greater than
180(n – 2)°, but no more than 180n°.
The famous hairy ball theorem asserts that it is
impossible to construct a continuous, nowhere zero,
VECTOR FIELD on the surface of a sphere.
A single sphere divides three-dimensional space
into two regions: an inside and an outside. Two intersecting spheres divide space into four regions (the
region interior to both spheres, two regions each interior to one sphere but not the other, and the outer
external region), three intersecting spheres divide space
into eight regions, four spheres divide space into 16
regions, and five spheres into 30 regions. In general, N
mutually intersecting spheres divide space into
(N
2 – 3N + 8)
distinct regions. (Thus, for instance, 10 spheres can be
arranged in space to produce 260 separate regions.)
In n-dimensional space, the maximal number of
nonintersecting spheres (or hyperspheres) of a fixed
radius r that can be arranged about a central sphere with
each touching or “kissing” the central sphere is called
the n-dimensional kissing number. (It is also called the
contact number, coordination number, ligancy, or the
Newton number.) As it is possible to arrange two line
segments at either end of a given line segment, the onedimensional kissing number is 2. One can arrange six
(but not seven) circles about a given circle, and so the
two-dimensional kissing number is 6. SIR ISAAC NEWTON (1642–1727) correctly believed that the kissing
number in three dimensions is 12 (although this fact was
not properly proved until late in the 19th century).
In three-dimensional space there are just three periodic (that is, self-repeating) patterns for stacking an infinite collection of identical spheres. Each arrangement is
called a packing. The “face-centered cubic lattice”
arranges spheres in layers, with each sphere surrounded
by just four spheres within that layer. The “cubic and
hexagonal lattices” arrange spheres in a manner such
that each sphere in a layer is surrounded by six other
spheres within that layer. (The two packings differ in
how a third layer of spheres is placed in relation to a first
layer.) The density of a packing is the fraction of space
occupied by the spheres. The cubic and hexagonal
packings each have a density of
≈ 74.05%.
In 1611 German mathematician and astronomer
JOHANNES KEPLER (1571–1630) conjectured that the
cubic and hexagonal packings are the densest possible. It
was not until 1998 that this was finally proved to be the
case.
The famous FOUR-COLOR THEOREM asserts that
just four colors are needed to paint any map on the
surface of a sphere. EULER’S THEOREM shows that if v
is the number of vertices in any such map, e the number of edges between countries, and r is the number of
regions defined by the map, then v – e + r = 2.
As soap film is elastic and wants to “pull in on
itself,” any soap bubble containing a fixed volume of
air adopts a shape that minimizes surface area. That
soap bubbles are spherical provides a physical demonstration of the following three-dimensional variation of
the ISOPERIMETRIC PROBLEM:
Of all figures in three-dimensional space of a
fixed volume, the sphere has the least surface
area.
See also MERCATOR’S PROJECTION; OBLATE/PROLATE; SOAP BUBBLES; SOLID ANGLE; SPHERICAL COORDINATES; TORUS.
spherical coordinates (spherical polar coordinates) In
three-dimensional space, the location of a point P can
π
––
3√
–
2
N –
3
472 spherical coordinates
the slices are made parallel, each perpendicular
to a fixed diameter of the loaf.)
The region of a sphere bounded between two parallel planes is called a spherical segment or a zone. If
the radius of the sphere is r and the distance between
the two planes is h, then the surface area of the zone is
2πrh. (Notice that this formula is indeed independent
of where the two planes slice the sphere.) The solid
obtained by rotating a sector of a circle about the
diameter of the circle is called a spherical sector, and a
spherical wedge is any segment of a sphere bounded
between two great circles.
A SPHERICAL TRIANGLE is any region on the surface
of a sphere bounded by arcs of three great circles, and
spherical trigonometry is the study of spherical triangles and the angles formed by intersecting great circles.
The study of SPHERICAL GEOMETRY shows that the
three angles in any spherical triangle sum to more than
180°. A spherical polygon is the figure formed on the
surface of a sphere by three or more arcs of great circles. Mathematicians have shown that the angles in an
n-sided spherical polygon sum to value greater than
180(n – 2)°, but no more than 180n°.
The famous hairy ball theorem asserts that it is
impossible to construct a continuous, nowhere zero,
VECTOR FIELD on the surface of a sphere.
A single sphere divides three-dimensional space
into two regions: an inside and an outside. Two intersecting spheres divide space into four regions (the
region interior to both spheres, two regions each interior to one sphere but not the other, and the outer
external region), three intersecting spheres divide space
into eight regions, four spheres divide space into 16
regions, and five spheres into 30 regions. In general, N
mutually intersecting spheres divide space into
(N
2 – 3N + 8)
distinct regions. (Thus, for instance, 10 spheres can be
arranged in space to produce 260 separate regions.)
In n-dimensional space, the maximal number of
nonintersecting spheres (or hyperspheres) of a fixed
radius r that can be arranged about a central sphere with
each touching or “kissing” the central sphere is called
the n-dimensional kissing number. (It is also called the
contact number, coordination number, ligancy, or the
Newton number.) As it is possible to arrange two line
segments at either end of a given line segment, the onedimensional kissing number is 2. One can arrange six
(but not seven) circles about a given circle, and so the
two-dimensional kissing number is 6. SIR ISAAC NEWTON (1642–1727) correctly believed that the kissing
number in three dimensions is 12 (although this fact was
not properly proved until late in the 19th century).
In three-dimensional space there are just three periodic (that is, self-repeating) patterns for stacking an infinite collection of identical spheres. Each arrangement is
called a packing. The “face-centered cubic lattice”
arranges spheres in layers, with each sphere surrounded
by just four spheres within that layer. The “cubic and
hexagonal lattices” arrange spheres in a manner such
that each sphere in a layer is surrounded by six other
spheres within that layer. (The two packings differ in
how a third layer of spheres is placed in relation to a first
layer.) The density of a packing is the fraction of space
occupied by the spheres. The cubic and hexagonal
packings each have a density of
≈ 74.05%.
In 1611 German mathematician and astronomer
JOHANNES KEPLER (1571–1630) conjectured that the
cubic and hexagonal packings are the densest possible. It
was not until 1998 that this was finally proved to be the
case.
The famous FOUR-COLOR THEOREM asserts that
just four colors are needed to paint any map on the
surface of a sphere. EULER’S THEOREM shows that if v
is the number of vertices in any such map, e the number of edges between countries, and r is the number of
regions defined by the map, then v – e + r = 2.
As soap film is elastic and wants to “pull in on
itself,” any soap bubble containing a fixed volume of
air adopts a shape that minimizes surface area. That
soap bubbles are spherical provides a physical demonstration of the following three-dimensional variation of
the ISOPERIMETRIC PROBLEM:
Of all figures in three-dimensional space of a
fixed volume, the sphere has the least surface
area.
See also MERCATOR’S PROJECTION; OBLATE/PROLATE; SOAP BUBBLES; SOLID ANGLE; SPHERICAL COORDINATES; TORUS.
spherical coordinates (spherical polar coordinates) In
three-dimensional space, the location of a point P can
π
––
3√
–
2
N –
3
472 spherical coordinates