Download Math 3329-Uniform Geometries — Lecture 11 1. The sum of three

Math 3329-Uniform Geometries — Lecture 11
1. The sum of three angles
This is where we are. In neutral geometry, we can show that any two angles of a triangle sum to less than
180◦.
Theorem 1. In neutral geometry, the sum of any two angles of a triangle sum to less than 180◦.
Recall that for a set of numbers S, if x is greater than or equal to every element of S, then we will say
that x is an upper bound for S. If y is the smallest possible upper bound for S, then y is said to be
a least upper bound (a.k.a. an infimum). For example, the number 7 is an upper bound for the set
[ 0, 1 ) = { x ∈ R | 0 ≤ x < 1 }, since it’s bigger than everything in [ 0, 1 ). The least upper bound is 1. Note
that this set has no largest element. Therefore, the least upper bound lies outside of the set. It only makes
sense to talk about a largest element, if the least upper bound lies inside the set (e.g., 1 is the least upper
bound for [ 0, 1 ]).
As an example, suppose that you have a triangle 4ABC, and you know that one of the angles measures 2◦.
Let’s say that ∠A = 2◦ . The other two angles must sum to less than 180◦ by Theorem 1. Therefore,
(1)
∠B + ∠C < 180◦,
and
(2)
∠A + ∠B + ∠C < 2◦ + 180◦ = 182◦ .
Similarly, if ∠A = ◦ , then ∠A + ∠B + ∠C < 180◦ + ◦ .
Apparently, if a triangle has one really tiny angle, then its angle sum can’t be much bigger than 180◦ . Our
plan of attack will be to show that every triangle has the same angle sum as a triangle with one really tiny
angle. We will use this observation to show that the angle sum of a triangle in neutral geometry can’t be
bigger than 180◦ at all.
Keep in mind that in neutral geometry, the angle sum of one triangle may be different from the angle sum
of another. We can’t show that every triangle has angle sum equal to 180◦ after all. Eventually, we’ll see
that if the area of the two triangles are the same, then the angle sums will be the same. We won’t assume
this fact, however.
2. Some triangles will have the same angle sum
We will start with some generic triangle 4ABC, and we want to show something about its angle sum. We
know a little something about triangles with one tiny angle, so we need to make some sort of connection.
We can find the midpoint of side BC, which is labeled E in Figure 1. We can then extend AE to a point F
so that AE ∼
= EF . The angles ∠4 and ∠40 are vertical angles, so they are congruent, and we have satisfied
the SAS criterion for triangles 4AEC and 4F EB. This is just like what we did with Figure ??.
We’re after the angle sum of 4ABC, which is
X
(3)
4ABC = ∠1a + ∠1b + ∠2 + ∠3.
We will show that this is the same as the angle sum for 4ABF , which is
X
(4)
4ABF = ∠1b + ∠2 + ∠30 + ∠5.
P
P
Since 4AEC and 4F EB are congruent triangles, ∠1a = ∠5, so
4ABC =
4F EB.
Furthermore, let’s suppose that ∠1 = ∠1a + ∠1b is the smallest angle of 4ABC. Then the smallest angle
of 4ABF is either ∠1b or ∠5. If ∠1b = ∠5, then they are both exactly half the measure of angle ∠1. If
∠1b 6= ∠5, then one of these angles must be less than half the measure of ∠1.
1
2
Figure 1. 4ABC has the same angle sum as 4AF B.
In any case, we know that the smallest angle of 4F EB is less than or equal to half the smallest angle of
4ABC.
3. Angle sums of triangles in neutral geometry cannot exceed 180◦
In the last section, we proved the following statement.
Theorem 2. Given any triangle T in neutral geometry, there is a triangle T 0 with the same angle sum
whose smallest angle measures less than or equal to half the measure of the smallest angle of T .
Suppose you have a triangle T0 in neutral geometry with angle sum 190◦ and smallest angle 40◦. Then
by Theorem 2 there is another triangle T1 that has the same angle sum, 190◦, and whose smallest angle
measures less than or equal to 20◦. Similarly, there is a triangle T2 with angle sum 190◦ and smallest angle
smaller than 10◦. There’s something wrong, however. The two larger angles of T2 sum to less than 180◦.
Therefore, the angle sum of T2 must be less than 190◦. This contradicts what we just found. Therefore, T0
can’t have an angle sum of 190◦!
As you can see, anytime we assume that the angle sum of a triangle in neutral geometry is greater than
180◦, we can use Theorems 1 and 2 to arrive at a contradiction. Let’s have you do the proof in general.
Theorem 3. The angle sum of a triangle in neutral geometry is less than or equal to 180◦.
Note that the number 180 is arbitrary. The number would be π, if we used that for the measure of a straight
angle. Without numbers, we might say that the angle sum is less than or equal to the measure of a straight
angle.
Here is the proof of the theorem.
Let T0 be a triangle in neutral geometry, and assume that its angle sum is greater than 180◦. In particular,
there exists some number > 0 such that the angle sum of T0 is 180◦ + ◦ . The triangle T0 has a smallest
angle, and let’s say that it measures α◦ .
By Theorem 1, we know that α < 90, because it plus one of the other angles sums to less than 180◦ , and
since α is the measure of the smallest angle, that other angle must be bigger than α.
By Theorem 2, there exists triangle T1 with angle sum 180◦ + ◦ and smallest angle measuring less than
There is also a triangle T2 with angle sum 180◦ + ◦ and smallest angle measuring
α◦
2 .
α◦
.
4
We can continue this indefinitely, and in general, triangle Tn has angle sum 180◦ + ◦ and the smallest angle
will go to zero.
3
In particular, there exists some n such that the smallest angle measures less than ◦ . The triangle Tn ,
therefore, will have smallest angle smaller than ◦ , while the other two angles sum to less than 180◦ . The
total angle sum, therefore, must be less than 180◦ + ◦ . That’s our contradiction.
Note that we can only prove that the angle sum is less than or equal to 180◦ .
4. Summary
Before Gauss, Bolyai, and Lobachevski, everyone was trying to show that neutral geometry and Euclidean
geometry were the same. That is, they wanted to prove Euclid’s Fifth Postulate as a theorem rather than
assume it as an axiom. If you could prove in neutral geometry that the angle sum of a triangle was exactly
the same as the measure of a straight angle, you could prove Euclid’s Fifth Postulate. The best you can do,
however, is to show that the angle sum is 180◦ or smaller.
Gauss, Bolyai, and Lobachevski, for the most part independently and about the same time, had the
obvious thought, “Hey, maybe you can’t prove Euclid’s Fifth Postulate.” But how could you prove that
doing this was impossible? One way is to come up with some odd geometry that satisfies all of the axioms
of neutral geometry, but in this geometry, Euclid’s Fifth Postulate is false. If you could do that, then
any neutral geometry proof would work in this geometry, and you certainly couldn’t prove Euclid’s Fifth
Postulate.
Two questions arise immediately. Does such a geometry exist? and if so, How real is that geometry?
If we take all of Hilbert’s axioms, including Euclid’s Axiom, we (supposedly) have an axiom system for
Euclidean geometry. If we replace Euclid’s Axiom by the following axiom, we get an axiom system for
a geometry called hyperbolic geometry. I’ve also seen references like Lobachevski’s and/or Bolyai’s
geometry.
The Hyperbolic Axiom. Given a line l and a point P not on l, there are at least two lines through P that
are parallel to l.
Next, we’ll explore a model of hyperbolic geometry. In Bonola’s Non-Euclidean Geometry, a description of
such a model starts on page 164. I don’t see this mentioned in the text, but this model is generally associated
with the mathematician Eugenio Beltrami. I plan on using a different model named after Henri Poincaré.
4.1. Quiz.
–1– Assuming that hyperbolic geometry exists (it does), and keeping in mind the things you’ve proven,
is it possible that the angle sums of all triangles in hyperbolic geometry is 180◦?
–2– Can there be any triangles in hyperbolic geometry that have angle sums greater than 180◦ ?
5. More Summary
OK. In general terms, we can prove theorems of the form < stuff > implies l k m in neutral geometry. We
need to add Euclid’s axiom, however, to go the other way, and only in Euclidean geometry can we prove
theorems of the form l k m
P implies < stuff◦ >. Similarly, in neutral geometry we can only prove that the
angle sum of a triangle is
4ABC ≤ 180 .
Let’s go back to the triangle angle sum picture we saw before. Given 4ABC, we can construct lines l and
m, as in Figure 2, so that ∠1 = ∠10 and ∠2 = ∠20 . In neutral geometry, both l and m must be parallel to
the line through A and B. The problem is, of course, there may be more than one parallel through C. We
can break neutral geometry into two specific cases now: One is Euclidean, and one is hyperbolic. Let’s look
at the two cases.
4
Figure 2. The lines l and m are parallel to the base of the triangle AB.
In Euclidean, we have Euclid’s Axiom, which says that there can only be one parallel through C. Therefore,
l and m must be the same line. In this case, ∠10 , ∠3, and ∠20 form a straight angle. Therefore,
∠1 + ∠2 + ∠3 = ∠10 + ∠20 + ∠3 = 180◦
(5)
in Euclidean geometry.
OK. So we know that Euclid’s Axiom will allow us to prove that angle sums of triangles are always 180◦.
Does this go the other way? That is, if we knew that triangle angles sums were always 180◦, then could we
prove Euclid’s Axiom? If so, then we would say that the two statements are equivalent in neutral geometry.
Figure 3. Assuming triangle angle sums are always 180◦, 12 ∠AB1 P = ∠AB2 P .
It’s probably a bit more convenient to show that triangle angle sums are always 180◦ implies Euclid’s Fifth
Postulate, which we already know is equivalent to Euclid’s Axiom.
Here’s the proof. We are assuming that all triangles have angle sums of exactly 180◦. We will do a proof
by contradiction, and assume that Euclid’s Fifth Postulate is false. That is, there are lines l and m, as in
Figure 3, such that l k m, and ∠P AB1 + AP Q < 180◦ . The point B1 can be any point on l to the right of
A. We can construct the point B2 so that P B1 = B1 B2 . Since triangle 4P B1 B2 is isosceles, we can easily
prove that those two angles marked β 0 are congruent. We know, therefore, that
(6)
2β 0 + α0 = 180◦
β + α0 = 180◦
Therefore, β 0 = 12 β. By repeating this construction, we can get a sequence of points, B3 , B4 , etc. so that
∠ABn P → 0. Since triangle angle sums are always 180◦, it follows that the sum α + ∠AP Bn → 180◦ . Since
all of the lines P Bn lie below m, it follows that ∠AP Q 6=< 180◦, which is our contradiction.
5
Here is what we have. In neutral geometry, Euclid’s Axiom, Euclid’s Fifth Postulate, Parallel implies
alternate interior angles (et al) are congruent, and all triangles have angle sums of 180◦ are all equivalent.
In other words, neutral plus any of these as an axiom would give you Euclidean geometry.
In hyperbolic geometry, we have the Hyperbolic Axiom, which is the negation of Euclid’s Axiom (i.e., it is
the statement that says Euclid’s Axiom is false). We will see that all triangles in hyperbolic geometry will
have angle sums smaller than 180◦ . In fact, if you look at Figure 1, the two triangles 4ABC and 4ABF
have the same angle sum. This is true in neutral geometry, and therefore, is also true in hyperbolic. Note
that the two triangles have the same base and the same height, which would suggest that they have the same
area. We’ll see that in hyperbolic geometry, the angle sum of a triangle will depend on the area.
6. A model for hyperbolic geometry
When you build an axiom system for something like Euclidean geometry, you try to gather a basic
set of statements that, if assumed to be true, define the object you’re trying to axiomatize. In the case of
Euclidean geometry, one such axiom system is Hilbert’s. The hope is that if you take the set of theorems
you can prove from Hilbert’s axioms, they are exactly the same as the theorems you would expect to be true
in your vision of Euclidean geometry.
Beyond the issue of getting the correct set of theorems, there are three things you look for in an axiom
system, consistency, independence, and completeness. An axiom system is consistent, if it does not lead to
contradictions. If, in your axiom system, you could prove that the angle sum of a triangle is always 180◦,
and you could also prove that the angle sum is sometimes not 180◦ , then your axiom system would not be
consistent.
An axiom system is independent, if you cannot prove any of your axioms from the others. In the attempt
to prove Euclid’s Fifth Postulate from the other axioms, the underlying suspicion was that Euclid’s axiom
system was not independent. With independence, we are wanting to ensure that our set of axioms is minimal,
and that every axiom is actually needed.
Finally, an axiom system is complete, if we cannot add another axiom without losing consistency or independence. Hilbert’s axiom system has been shown to be incomplete several times since it’s original formulation,
and additional axioms were introduced to fix the various problems. We probably will never be absolutely
sure about the completeness of this axiom system, but it seems like the general assumption is that Hilbert’s
axiom system is complete.
None of these things is easy to prove, but the most important and easiest to verify is consistency (note the
use of verify, since prove would be too strong). In this assignment, we wish to investigate how the axioms
of hyperbolic geometry can be shown to be consistent, and at the same time, determine that the neutral
geometry axiom set is not complete.
6.1. Quiz.
–1– State Euclid’s Axiom.
–2– State the Hyperbolic Axiom.
–3– If you had both Euclid’s Axiom and the Hyperbolic Axiom in the same axiom system, would that
axiom system be consistent?
–4– Assuming that neutral geometry plus Euclid’s Axiom and neutral geometry plus the Hyperbolic
Axiom are both consistent, can neutral geometry be complete?
–5– From what we’ve been exploring the last couple of weeks, neutral geometry plus Euclid’s Axiom plus
angle sum of a triangle is 180◦ would definitely not be (consistent, independent, complete).
6
7. The Poincaré Disk
We can test the consistency of hyperbolic geometry by comparing it to something that we have confidence
in. The mathematics we work with today is built on axioms systems for logic and set theory. I don’t
want to get into it too much, but logic and set theory are simpler than geometry, and their axioms systems
were formulated to be axiom systems (Euclid was not an expert in axioms systems). As a result, we can
be more confident in these modern systems. I have great confidence in these axiom systems, but it is only
confidence. Mathematicians like to think that mathematics is absolute truth, but at best, we only have faith
in our understanding of logic. Mathematics is not a religion exactly, but at its foundations, we believe in
mathematics with fanatical conviction.
If we can build a mathematical object, a model, within our modern mathematical structure, and the axioms
of hyperbolic geometry are true for our model, then we can say that hyperbolic geometry, as an axiom system,
is as consistent as the rest of mathematics. That’s what we want to do here. The xy-plane is a model for
Euclidean geometry, and we saw that Euclid’s postulates, and therefore, some of Hilbert’s axioms, were true
in that model. I think it’s safe to assume that all of Hilbert’s axioms are true in our xy-plane model for
Euclidean geometry.
The model for hyperbolic geometry to be discussed in this assignment is called the Poincaré disk. Poincaré’s
model lives in the xy-plane (a.k.a, R2 ), and consists of all of the points inside a fixed circle. For convenience,
let’s nail down the circle. The points in this geometry are the points in the set P defined by
(7)
P = (x, y) ∈ R2 | x2 + y2 < 1 .
In words, P is the set of points inside the unit circle, but not including the circle itself. We might call this
an open disk. The lines in this geometry are parts of circles. In particular, we will consider any circle
that intersects the unit circle at right angles. The intersection of any such circle and P will be a line. It is
convenient to think of a straight line as a circle with infinite radius, and we will do that. Diameters of the
unit circle, therefore, are also lines (not including the endpoints). Figure 4 shows a few lines in the Poincaré
Disk. Viewed in terms of the xy-plane, these are straight lines or circles that meet the boundary at right
angles.
Figure 4. Some lines in the Poincaré Disk.
Note that near the center of Figure 4 there is a small triangle. Since one of the edges of this triangle is
curved slightly inward, two of the angles are slightly smaller than they would be if that edge were straight.
Since the straight lines only pass through the center of the disk, we will never have a triangle with three
straight sides. We will measure angles in this geometry as we see them (technically, we measure the angles
between tangent lines).
Basic Principle 1. Since the sides of triangles in the Poincaré disk curve inward, their angle sums will be
less than 180◦.
7
There is a horizontal diameter and a quarter circle in the lower left that intersect on the unit circle on the
left. Since the unit circle is not part of P, these two lines do not intersect. Since according to Euclid’s
definition of parallel lines (and ours), parallel lines are those pairs that do not intersect. Therefore, these
two lines are parallel.
Figure 5. The lines m and n contain P and are parallel to l.
In Figure 5, the lines m and n intersect l at the boundary, so in the model, they do not actually intersect.
Both lines, therefore, are parallel to l. This model satisfies the Hyperbolic Axiom (there are at least two
parallels through P ).
If you were to consider all of the lines through P in Figure 5 some are parallel to l and some intersect.
Imagine taking a single line t through P and rotating it about P . You will find that t intersects l for awhile,
and then it will be parallel. A little later, it will start intersecting again. During the change, there will
be one position such that everything before is not parallel and everything after is parallel. At this position
itself, is t parallel or does it intersect l? It has to be parallel, because if not, then there would be a point of
intersection. If this happens, then there would be other points of intersection on either side of it on l, and
therefore, there would be non-parallels on either side of it. In hyperbolic geometry, therefore, we have the
phenomenon of a last parallel.
8. Distance in the Poincaré model
One important aspect of this model must be different from how it looks. In hyperbolic geometry, lines must
be infinitely long. In order to satisfy the axioms of hyperbolic geometry, therefore, we must define distances
in a funny way. I won’t give you the general formula for the distance between any two points (this formula
would be called the metric). I’ll just tell you what the distance between any point and the point in the
center of the model. For any point P ∈ P, if the distance from the center is r, as we would measure it in the
xy-plane, then the hyperbolic distance will be
1+r
(8)
d(0, r) = ln
.
1−r
As a point is moved towards the boundary, r would approach 1. Its hyperbolic distance from the center
would therefore approach
1+r
(9)
lim d(0, r) = lim ln
= ∞.
r→1
r→1
1−r
The hyperbolic distance from the center out to the boundary, therefore, must be infinite.
8
8.1. Quiz.
–1– Suppose you were 2-dimensional, and the Poincaré disk was your universe. Suppose also that when
you stood on the x-axis at the center, the top of your head would be at the point (0, 0.1). What is
your height in hu’s (hu = hyperbolic unit)?
–2– If my height were 0.3 hu’s, and I stood on point (0, 0.8) with my head pointing up the y-axis, I
1.8
could
figure
out where my head was. My feet would be ln .2 = 2.197 hu. My head would be at
ln
1+r
1−r
hu. Therefore,
(10)
1+r
− 2.197
.3 = ln
1−r
1+r
2.497 = ln
1−r
1+r
e2.497 =
1−r
(1 − r)e2.497 = 1 + r
e2.497 − 1 = r + re2.497
e2.497−1 = r(1 + e2.497)
e2.497 − 1
=r
1 + e2.497
Therefore, r = 0.848, and my head would be at (0, 0.848). If you were standing on the point (0, 0.9)
with your head pointing up the y-axis, where would your head be?
9. What does the Poincaré disk tell us?
With the definitions I’ve given you, and a few more, you could show that all the axioms of neutral geometry
are true in this model, and in addition, the Hyperbolic Axiom is true in this model. It might take
you awhile, but you could show that all of the definitions are well-defined (i.e., don’t give contradictory
answers). This would mean that the Poincaré disk is consistent within our modern mathematical system,
and this would imply that the axioms of hyperbolic geometry are as consistent as the axioms for logic and set
theory. In other words, if the axioms of hyperbolic geometry are bad, then all of mathematics is in trouble.
That might be interesting, actually. As far as independence and completeness are concerned, the axioms of
hyperbolic geometry are probably both independent and complete. I’m not an expert on such matters, but
I don’t think either has been proven.
10. Is hyperbolic geometry real?
Most of us seem to be comfortable with the assumption that our universe is a 3-dimensional Euclidean
space, the xyz-space of calculus. In fact, Bertrand Russell filled a small book, I think it’s called An Essay
on the Foundations of Geometry, “proving” that this was indeed the case. This essay was written
around 1900. About 30 years later, Russell states as fact that our universe is a manifold. We’ll get to
manifolds later, but most manifolds are definitely not Euclidean. I’m working from memory here, but I
think the second reference is in something called Principles of Mathematics. I’m been trying to track these
down for awhile.
Now, the Poincaré disk lives in the Euclidean plane, so it definitely seems subordinate to Euclidean geometry.
We could, perhaps, put it on more equal footing by stretching it out so that distances could be measured
directly. Since things get smaller towards the boundary, we would have to stretch the Poincaré’s disk more
and more as we move away from the center. If we did this, we would get something called the hyperbolic
plane, and this plane will stretch out to infinity.
9
What would the hyperbolic plane look like? Well, if you took a small piece of the Poincaré disk, and did
the stretching, you would get something that looked like the picture on page 133 of Bonola’s Non-Euclidean
Geometry. Notice that it is really wrinkly. This is because hyperbolic geometry spreads out faster than
Euclidean geometry spreads out. For example, a circle in hyperbolic geometry has a larger circumference
than a circle in Euclidean geometry with the same radius. So in some sense, the hyperbolic plane is too big
to fit in our supposedly Euclidean space, so it has to be crammed into our space, and it gets wrinkled in the
process. This is not proof that our space is Euclidean, however. Later, we will describe how hyperbolic a
space is in terms of curvature. We’ll say that the hyperbolic plane has curvature K = −1. A hyperbolic
plane with curvature K = −.003 might look perfectly flat in our space, because if our universe is hyperbolic,
it would have very small curvature.
We’ll spend much of the rest of the semester trying to understand these ideas, but let me end with this. We
have direct experience with the geometry of our universe only on the surface of the earth, and the visible
universe has a radius of only 10-15 billion light-years. There are an infinite number of distinctly different
geometric spaces consistent with the information we have. One of the problems I have with Hilbert’s axioms
is that they lead to only two of these spaces.
References
[Bonola]
Roberto Bonola (1955). Non-Euclidean Geometry (H.S. Carslaw, Trans.). Dover Publications, New York. (Original
translation, 1912, and original work published in 1906.)
[Descartes] Rene Descartes (1954). The Geometry of Rene Descartes (D.E. Smith and M.L. Latham, Trans.). Dover Publications, New York. (Original translation, 1925, and original work published in 1637.)
[Euclid]
Euclid (1956). The Thirteen Books of Euclid’s Elements (2nd Ed., Vol. 1, T.L. Heath, Trans.). Dover Publications,
New York. (Original work published n.d.)
[Eves]
Howard Eves (1990). An Introduction to the History of Mathematics (6th Ed.). Harcourt Brace Jovanovich, Orlando, FL.
[Federico] P.J. Federico (1982). Descartes on Polyhdra: A study of the De Solidorum Elementis. Springer-Verlag, New York.
[Henderson] David W. Henderson (2001). Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces 2nd Ed.
Prentice Hall, Upper Saddle River, NJ.
[Henle]
Michael Henle (2001). Modern Geometries: Non-Euclidean, Projective, and Discrete 2nd Ed. Prentice Hall, Upper
Saddle River, NJ.
[Hilbert]
David Hilbert (1971). Foundations of Geometry (2nd Ed., L. Unger, Trans.). Open Court, La Salle, IL. (10th
German edition published in 1968.)
[Hilbert2]
D. Hilbert and S. Cohn-Vossen (1956). Geometry and the Imagination (P. Nemenyi, Trans.). Chelsea, New York.
(Original work, Anschauliche Geometrie, published in 1932.)
[Motz]
Lloyd Motz and Jefferson Hane Weaver (1993). The Story of Mathematics. Avon Books, New York.
[Weeks]
Jeffrey R. Weeks (1985). The Shape of Space. Marcel Dekker, New York.