the angles α and β satisfy the fundamental relationship
In general we have:
If the speed of travel through one medium is s
units per second, and through a second
medium is r units per second, then the optimal
route of travel across a boundary separating
the two media occurs when the angle of incidence α and the angle of refraction β satisfy
the relationship:
This principle is today called Snell’s law of refraction.
Snell observed that a beam of light moving from one
medium to another, say, from air to water, obeys this
law. Light of one color, that is, of one particular
wavelength, will travel through the same medium at a
speed different from that of a beam of a different
color, and so will have its own appropriate angle of
refraction. For this reason, white light—the composition of all colors—separates into a rainbow of colors
at the interchange of a new medium. Water droplets
suspended in the air after a rainstorm separate the colors of sunlight in this way to produce a large rainbow
visible from the ground.
soap bubbles The film of a soap solution is elastic
and has the property that it always “pulls in on itself”
to create a surface of least possible surface area. Noting
that a soap bubble, containing a fixed volume of air,
always adopts the shape of a SPHERE provides a physical demonstration of the following three-dimensional
variation of the ISOPERIMETRIC PROBLEM:
Of all figures with a fixed volume, the sphere
has the least surface area.
Polish mathematician Hermann Schwarz (1843–1921)
provided a mathematical proof of this observation
in 1882.
Many basic questions about arrangements of bubbles separating more than one chamber of air are unresolved. In 2000 mathematicians Frank Morgan, Michael
Hutchings, Manuel Ritoré, and Antonio Ros managed
to prove that the familiar double bubble seen in nature
(two portions of a sphere attached to a curved boundary
separating the two volumes) is indeed the shape of minimal total surface area for a configuration enclosing two
chambers of air. The corresponding triple-bubble problem remains open.
solid angle Just as an ANGLE in two-dimensional
space is the measure of the amount of length of a UNIT
CIRCLE “cut off” by two rays describing the angle, a
solid angle in three-dimensional space is the amount of
surface area of a unit SPHERE intercepted by a bundle of
rays emanating from the center of the sphere. A solid
angle need not have any particular shape; it is defined
by the nappe of a CONE, which can be of any shape.
The unit of measure of a solid angle is called a steradian. Because the surface area of a sphere is 4π(1)
2
, there
are a total of 4π steradians about a point in space. The
measure of a hemisphere is
· 4π = 2π steradians.
Solid angles are used in physics to measure the
fraction of the total emission of radiation from a
source in space. For example, the Earth, of radius
approximately 6370 km, exposes a total surface area of
· 4π(6370)
2 km
2 toward the Sun at any time. As the
distance of the Earth from the Sun is approximately
150 million km, we have that the Earth occupies the
fraction
of the surface
area of sphere about the Sun of radius 150 million km.
The solid angle of the Earth with respect to the Sun is
this fraction of the full 4π steradians of space about the
Sun. Notice that this fraction is exceedingly small. Life
on Earth is sustained by less than 1 billionth of the
total energy emitted by the Sun.
The polyhedral angle of a POLYHEDRON is the solid
angle of the region formed by projecting one face of the
polyhedron onto a unit sphere.
solid of revolution If the region under a curve y = f(x)
between x = a and x = b is rotated through one revolution about the x-axis, it generates a three-dimensional
figure called a solid of revolution. More generally, any
figure in the plane rotated about a line in the plane that
does not cut the figure produces a solid of revolution.
1
2
4 6370
4 150 10
1
1 110 000 000
2
6 2
⋅
×
≈
π
π
(
)
(
)
,
,
,
1
–
2
1
–
2
sin
sin
α
β
r
s
=
sin
sin
α
β
r
s
=
solid of revolution 469
In general we have:
If the speed of travel through one medium is s
units per second, and through a second
medium is r units per second, then the optimal
route of travel across a boundary separating
the two media occurs when the angle of incidence α and the angle of refraction β satisfy
the relationship:
This principle is today called Snell’s law of refraction.
Snell observed that a beam of light moving from one
medium to another, say, from air to water, obeys this
law. Light of one color, that is, of one particular
wavelength, will travel through the same medium at a
speed different from that of a beam of a different
color, and so will have its own appropriate angle of
refraction. For this reason, white light—the composition of all colors—separates into a rainbow of colors
at the interchange of a new medium. Water droplets
suspended in the air after a rainstorm separate the colors of sunlight in this way to produce a large rainbow
visible from the ground.
soap bubbles The film of a soap solution is elastic
and has the property that it always “pulls in on itself”
to create a surface of least possible surface area. Noting
that a soap bubble, containing a fixed volume of air,
always adopts the shape of a SPHERE provides a physical demonstration of the following three-dimensional
variation of the ISOPERIMETRIC PROBLEM:
Of all figures with a fixed volume, the sphere
has the least surface area.
Polish mathematician Hermann Schwarz (1843–1921)
provided a mathematical proof of this observation
in 1882.
Many basic questions about arrangements of bubbles separating more than one chamber of air are unresolved. In 2000 mathematicians Frank Morgan, Michael
Hutchings, Manuel Ritoré, and Antonio Ros managed
to prove that the familiar double bubble seen in nature
(two portions of a sphere attached to a curved boundary
separating the two volumes) is indeed the shape of minimal total surface area for a configuration enclosing two
chambers of air. The corresponding triple-bubble problem remains open.
solid angle Just as an ANGLE in two-dimensional
space is the measure of the amount of length of a UNIT
CIRCLE “cut off” by two rays describing the angle, a
solid angle in three-dimensional space is the amount of
surface area of a unit SPHERE intercepted by a bundle of
rays emanating from the center of the sphere. A solid
angle need not have any particular shape; it is defined
by the nappe of a CONE, which can be of any shape.
The unit of measure of a solid angle is called a steradian. Because the surface area of a sphere is 4π(1)
2
, there
are a total of 4π steradians about a point in space. The
measure of a hemisphere is
· 4π = 2π steradians.
Solid angles are used in physics to measure the
fraction of the total emission of radiation from a
source in space. For example, the Earth, of radius
approximately 6370 km, exposes a total surface area of
· 4π(6370)
2 km
2 toward the Sun at any time. As the
distance of the Earth from the Sun is approximately
150 million km, we have that the Earth occupies the
fraction
of the surface
area of sphere about the Sun of radius 150 million km.
The solid angle of the Earth with respect to the Sun is
this fraction of the full 4π steradians of space about the
Sun. Notice that this fraction is exceedingly small. Life
on Earth is sustained by less than 1 billionth of the
total energy emitted by the Sun.
The polyhedral angle of a POLYHEDRON is the solid
angle of the region formed by projecting one face of the
polyhedron onto a unit sphere.
solid of revolution If the region under a curve y = f(x)
between x = a and x = b is rotated through one revolution about the x-axis, it generates a three-dimensional
figure called a solid of revolution. More generally, any
figure in the plane rotated about a line in the plane that
does not cut the figure produces a solid of revolution.
1
2
4 6370
4 150 10
1
1 110 000 000
2
6 2
⋅
×
≈
π
π
(
)
(
)
,
,
,
1
–
2
1
–
2
sin
sin
α
β
r
s
=
sin
sin
α
β
r
s
=
solid of revolution 469