Transcript Spherical Geometry

Spherical Geometry
The sole exception to this rule is one of the main characteristics of spherical geometry.
Two points which are a maximal distance apart, namely half the circumference, are said to
be at opposite poles, such as the North and South poles.
Every point P has its opposite pole, called its antipode P, obtained by drilling from P straight
through the center of the sphere and coming out the other side. “Digging through the
earth to China," so to speak.
Non-Euclidean geometry is an example
of a paradigm in the history of science.
The major difference between spherical
geometry and the other two branches,
Euclidean and hyperbolic, is that
distances between points on a sphere
cannot get arbitrarily large.
There is a maximum distance two
points can be apart. In our model of the
earth this maximal distance is half the
circumference of the earth, roughly
12,500 miles.
Distance between two points on a great
circle is always the shortest distance
between them.
The distance between Philadelphia and
New York is 120 miles, not the 24,880
miles it would take to go around the
other side of the world. The triangle
inequality holds in spherical
geometry.
In flat space
Find any two points and draw the STRAIGHT LINE they
define in space.
At A and B erect perpendicular, straight lines. Mark these
angles.
In Spherical Geometry
Find any two points A and B on the
equator and draw the STRAIGHT
LINE they define.
At A and B erect perpendicular,
straight lines. Mark these angles.
Find the midpoint of AB and call it Q.
Construct a perpendicular from AB at Q.
Project it until it eventually meets AO and BO.
(It must meet them since there are no parallels
in this geometry.)
Where will they meet?
repeat the construction.
Bisect AQ and from that point erect a
perpendicular that will pass through O.
Bisect QB and from that point erect a
perpendicular that will pass through O.
By repeating this process indefinitely, we can
divide the original interval AB into as many
equal sized parts as we like. Perpendiculars
raised from each of these points will all pass
through the point O.
As before, all these perpendiculars will have
the same length.
There is also a circle in the figure. While the line AGG'G''G''' is a straight line, it also has the
important property of being the circumference of a circle centered on O. Every point on
AGG'G''G''' is the same distance from O. That is the defining property of a circle.
And what an unusual circle it is. It has radius AO. That radius AO is equal in length to each of
the four segments AG, GG', G'G'', G''A that make up the circumference.
Radius = AO
Circumference = AG + GG' + G'G'' + G''A
AO = AG = GG' = G'G'' = G''A
That means that the circle AGG'G''G''' has the curious property that
Circumference = 4 x Radius
Contrast that with the properties familiar to us from circles in Euclidean geometry
Circumference = 2π x Radius
A longer analysis would tell us that the area of the circle AGG'G''G''' stands in an unexpected
relationship with the radius AO. Specifically
Area = (8/π) x Radius2
In Euclidean geometry, the area of a circle relates to its radius by Area = π x Radius2
Let us return to our starting point. Euclid's achievement appeared unshakeable to
the mathematicians and philosophers of the eighteenth century. The great
philosopher Immanuel Kant declared Euclid's geometry to be the repository of
synthetic, a priori truths, that is propositions that were both about the world but
could also be known true prior to any experience of the world. His ingenious means
of justifying their privileged status came from his view about how we interact with
what is really in the world. In our perceiving of the world, we impose an order and
structure on what we perceive; one manifestation of that is geometry.
The discovery of new geometries in the nineteenth century showed that we ought
not to be so certain that our geometry must be Euclidean. In the early twentieth
century Einstein showed that our actual geometry was not Euclidean. So what are
we to make of Kant's certainty? Einstein gave this diagnosis in his 1921 essay
"Geometry and Experience."