Geometry Explorer Help
Exploring in a Hyperbolic World:
Historical Background
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Euclid:
Hyperbolic Geometry belongs to a class of geometries called
"Non-Euclidean."
To understand what a non-Euclidean geometry is, we need to understand
exactly
what is meant by Euclidean geometry. Euclidean geometry is based
on an axiomatic system of thought that was first put into place about
2000
years ago. An axiomatic system is based on a set of axioms (or
self-evident
truths) that are accepted as being true without having to be
proven.
Such a set of axioms is needed in a logical system to prevent circular
reasoning and the infinite regression of reasoning. In other words, one
must accept a certain number of statements as facts, and then proceed
to
state other results in terms of such axiomatic facts.
Euclid, in about 300BC, set down five axioms to base his geometry
upon.
These can be stated as follows:
- To draw a straight line from any point to any other.
- To produce a finite straight line continuously in a straight
line.
- To describe a circle with any center and distance.
- That all right angles are equal to each other.
- That, if a straight line falling on two straight lines make the
interior
angles on the same side less than two right angles, the given straight
line will, if produced indefinitely, meet on that side on which the
angles
are less than the two right angles.
The fifth axiom (or postulate) has become known as the "Parallel"
postulate.
This is because it is logically equivalent to an axiom developed by
James
Playfair in about 1790. His axiom states
Given a line and a point not on the line, it is possible to draw exactly
one
line through the given point parallel to the line.
It is clear that the first four axioms are essential to doing geometry.
One needs to be able to draw lines from a point to another point.
Lines should be extendible. One must be able to construct
circles.
All right angles should be congruent. The right angle axiom basically
says
that the space we construct things in is uniform - a right angle in one
place means the same thing as a right angle in another place.
Euclid wanted to create a set of axioms that was the bare minimum so
that all of geometry could be based on these axioms. These axioms
were to be abstractions of doing geometry with a straight-edge and
compass,
i.e. the geometry of lines and circles. It would be hard to imagine an
axiomatic base for geometry that did not have the first four. But what
about axiom 5? It does not seem to fit with the other four. In fact,
Euclid
himself was not that satisfied with having to include this fifth axiom
and delayed using it in his classic geometry text, The Elements,
until Proposition 28 of Book I.
Gauss, Bolyai, and Lobachevsky:
Over the years since Euclid, many mathematicians were just as bothered
with the fifth axiom as Euclid was. One way out of using the
fifth
axiom would be to prove that it is a statement that can be proven using
the other four axioms. Then, the fifth axiom would become a proposition
or theorem, and Euclidean geometry could be based solely on the first
four
axioms. Many mathematicians attempted the proof of the fifth
axiom,
using the first four axioms, but none could succeed in the effort.
Then, in the 1800's Gauss, Bolyai, and Lobachevsky decided to try a
different approach to the problem of the fifth postulate. Before
these three it was generally assumed that the universe must be
Euclidean,
that it must satisfy the five Euclidean axioms. In fact, the
great
philosopher Kant asserted that statements in Euclidean geometry had an a
priori nature. That is, Euclidean geometry existed
independent
of our observations about the real world. A corollary of this would be
that Euclidean geometry was the "inevitable" geometry that one
"discovers"
under deductive reasoning. Gauss, Bolyai, and Lobachevsky
questioned
this view. In particular, they wondered what would happen if you
replaced
the fifth axiom with other axioms. Thinking of Playfair's axiom, one
replacement
would be to suppose that:
Given a line and a point not on the line, it is possible to draw more
than one line through the given point parallel to the line.
Using this as a new fifth axiom, Gauss and the others discovered that
a new type of geometry would be possible, and that this geometry seemed
to be just as rigorous and logically sound as Euclidean geometry was.
This
obviously disturbed the Kantians a great deal! This new geometry
is called hyperbolic geometry.
Later in the 1800's Riemann, who at one time worked with Gauss, showed
that you could replace the parallel postulate with the axiom:
Given a line and a point not on the line, there are no
parallels
to the line.
This produces a third type of geometry called elliptic geometry.
This revolution in the understanding of geometry prepared the
groundwork
for the revolution in physics about a century later when Einstein
proposed
the idea that the universe was curved (i.e. non-Euclidean) and
not
flat (i.e. Euclidean).
For a very nice discussion of the history surrounding the
discovery
of Non-Euclidean Geometry look here.