Explorations in Hyperbolic Geometry
Samuel Otten
Michigan State University
Spring 2008
SME 842
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Contents and Navigation:
A MODEL
PARALLELS
TRIANGLES
RECTANGLES
CONCLUSION
APPENDICES
INTRODUCTION
More than two thousand years ago Euclid of Alexandria collected, compiled, and
composed the thirteen volumes of geometry known as the Elements. This magnum
opus would become the quintessential model of the way in which mathematics is
structured, namely, the axiomatic method. Euclid began by defining his terms and
then laying forth his postulates and common notions, both of which can be viewed as
the assumptions he would work from as his did his geometry. He then set to work in a
proposition-proof format wherein each result was proved using only that which came
before it. Now, it should be noted that Euclid, though his work was masterful, wa s not
without error.1 He failed to recognize as we do now that it is logically futile to define
all terms and so there must be undefined terms; it has also been uncovered that right
from his first proof he made assumptions about things like betweenness and
continuity that were not listed in his postulates and common notions. Nevertheless,
Euclid’s Elements was a logical and mathematical tour de force that was the standardbearer of mathematical reasoning and certainty, the standard-bearer, that is, until it all
came crashing down.
The crash occurred when two mathematicians—János Bolyai of Hungary and
Nikolai Lobachevsky of Russia—independently discovered that Euclid’s famous fifth
postulate was independent of the others, leading to a consistent non-Euclidean
1
Remarkably, Euclid never made an error of commission, that is, none of his propositions turned out to be
false. The errors referred to here were errors of omission.
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geometry. So it was that mathematics’ surest foundation was shaken. To get a better
perspective on this historic event let us take a moment to consider Euclid’s postulates,
giving particular attention to his famous fifth.
Euclid’s postulates, as recorded in Book I of the Elements, are as follows:
1. A unique line segment exists between any two distinct points.
2. A line segment can be uniquely extended in a straight manner.
3. A circle exists given any center and radius.
4. A right angle is equal to any other right angle.
5. If a line falling on two other lines makes the interior angles on the same
side less than two right angles, then the two lines, if produced
indefinitely, meet on that side on which are the angles less than the two
right angles.
Many mathematicians felt (and it is hard to blame them!) that the fifth was too long
and complicated to be a postulate and believed that it could be derived from the first
four, all of which were intuitively clear and acceptable. The many attempts to prove
the fifth postulate, however, were unsuccessful. 2 Then in the early 19 th Century
Bolyai and Lobachevsky published their discoveries of hyperbolic geometry, the
former’s work based on replacing the fifth postulate with a parameter and the latter’s
based on the postulate’s negation. No longer was Euclidean geometry the sole study
of shape and space. Eventually it would be proved with the introduction of hyperbolic
models (embedded in Euclidean space) by Klein, Poincaré and Beltrami that the
consistencies of hyperbolic geometry and Euclidean geometry were logically
equivalent. Alas, the proof attempts of Euclid V were doomed from the start!
2
See Appendix A for an example of such a proof attempt.
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A close inspection of the fifth postulate reveals that two negations exist. One
negation is the statement that there exist two lines such that a transversal forms angles
on one side less than two right angles but, when produced indefinitely, the two lines
do not meet on either side;3 but to say that the two lines, if produced indefinitely, also
meet on the other side is another negation. The first negation leads to hyperbolic
geometry, which will be the environment of the explorations to come. The second
negation, on the other hand, leads to spherical geometry which is itself an intriguin g
world in which to do geometry but, unfortunately, does not satisfy Euclid’s first
postulate (there is more than one line segment between two distinct points) and will
not be discussed in the remainder of this paper, except for a few comparative
comments in passing.
As indicated above, what follows is a collection of explorations in the world of
hyperbolic geometry. The sections are written in an active voice, much like Euclid’s
own Elements (e.g., he would write “let AC be drawn through B” rather than “let AC
be the segment containing B”). As the reader, you should envision the paper as
documentation of a student’s investigative excursion into this non-Euclidean
landscape, complete with false starts and modifications.
Euclid’s first four postulates will be cited as axioms, as will a few of Hilbert’s
additional axioms, and there will be conjecturing and proving that takes place. All the
while, though, the geometry will appeal to intuition and be grounded on the models.
So that is where we begin.
3
To see how this assumption about two particular lines implies the Universal Hyperbolic Theorem, see
Appendix B.
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EXPLORATION 1: FINDING A MODEL
Euclidean geometry is the study of size, shape, distances, and so forth, in an
ambient space that is in some-sense flat. The most common manifestation of this is
the doing of geometry on a piece of paper on a desk. The ground on which we walk,
run, and generally live is also perceived to be flat. From such experiences it is natural
to assume several things because they seem to be intuitively true. First, between any
two points we can find a unique line. Second, if we have a segment of a line then we
can extend it in a straight manner. Third, we can construct a unique circle so long as
we know the center and the radius. And fourth, a right angle is a right angle is a right
angle. These assumptions, or axioms, are based on the familiar “flat” geometry, but
also hold on other surfaces such as surfaces with constant positive curvature (e.g., a
sphere) and surfaces with constant negative curvature (e.g., a hyperboloid). Let us see
what happens if we delve into the latter case, known as hyperbolic geometry.
Our first order of business is to make sure that
we understand what the axioms are saying in a
negatively curved environment. We will take words
like “between,” “on”, “point,” “line” and
“congruent” to be undefined terms. This does not
mean we are without guidance with regard to their
Figure 1: A hyperboloid
meaning because intuition plays an important role. For instance, we can think of two
figures as being congruent if we can rigidly move one precisely onto the other, and a
line can be conceptualized as the path marking the shortest distance between its
points.
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What is the shortest path between two points A and B in hyperbolic geometry? 4
Using our model, we can stretch a string tautly along the surface of the hyperboloid.
Based on investigations of this sort we see that a “straight line” on our model is the
intersection of the hyperboloid with a plane through the central point.5 The result of
such an intersection can be a hyperbola (of which we would only use half), an ellipse
or a circle. In the latter two cases we run into a problem because two points can
determine more than one line. Specifically, if A and B are antipodal points of a circle
or ellipse, such as the one shown in figure 2, then either arc of the circle or ellipse is a
line segment between the two points. This is a clear violation of Axiom 1.
A
B
Figure 2: The “line” on the right is problematic
Perhaps we will not be able to proceed in a way similar to the explorations of
spherical geometry. Perhaps it is not easy to find a negatively curved surface on
which to physically conduct hyperbolic business. 6 Hence we must return and
contemplate what it is we are trying to accomplish.
4
It is important to note that the term geodesic is being avoided due to the fact that, as Greenberg points out,
it is not equivalent to the notion of shortest path. If a shortest path exists, then it is an arc of a geodesic, but
the converse is not necessarily true. For instance, take a major arc of a great circle on the sphere.
This is strikingly similar to a “straight line” on the sphere which is the intersection of the sphere with a
plane through its center, namely, a great circle.
5
6
The pseudosphere is a possibility, but it only captures a portion of the hyperbolic plane.
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We have four axioms in hand and want to explore a geometry in which the
ambient space is not necessarily “flat.” Another way to think about this is that the
lines in the geometry are not necessarily “straight.” These two ideas are related
because a perceived curvature of lines could really be just a symptom of the curvature
of the underlying space, but rather than try to identify that space we can just accept
the fact that lines appear to be curved. Of course line segments would stil l be the
shortest path between two points because it could be the case that what looks like a
straight path actually rises or dips through the ambient curvature, making it longer
than it seems. So how can we model this geometry containing “curved” lines?
Let A and B be two distinct points. We want to define a line l through A and B,
E●
but it has to be unique to satisfy Axiom 1.
B●
Thus it cannot be simply any curve containing
l
the points because there are many of those. A
D●
A●
third point C that was non-collinear with A
●
and B would determine a unique circle, and
we could define l to be the minor arc between
A and B of that circle. Assuming C is fixed,
F●
C
●
G
for points D and E that are collinear with C we
could define the line segment between them to
be the normal straight line segment. However,
Figure 3: An attempt at
curved lines
as soon as we fix C there are points, say F and G, which lie diametrically opposed to
each other with respect to their circle formed with C. In this case there is not a unique
line segment and Axiom 1 is violated. (Axiom 2 also fails—the “lines” are compact.)
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Again, let A and B be distinct points. Instead of fixing a point we can fix a line l
below A and B. If m is the perpendicular bisector of the Euclidean segment AB, then
m either intersects l at a point C or is parallel to l (again, in the Euclidean sense). In
the first case, Axiom 3 gives us a unique circle Γ through A and B with C as the
center. In the second case, we have a ray n emanating perpendicularly from l and
containing A and B. In either case, we have a way to define line segments for all
points lying in the half plane above line l.
m
B●
●B
A●
A●
Γ
l
Figure 5: The half-plane model for hyperbolic geometry
Let us quickly check the four axioms. Per the paragraph above, we know that a
unique line segment exists for any two points above line l because we can choose the
arc of the circle that lies above l (or else we have a case of the vertical ray which also
presents a unique line segment). If we use the open half-plane above l then any line
segment has an open neighborhood around it, and thus we can extend the line segment
to include a bit more of the hemisphere. (This suggests, however, that distances grow
exponentially as you get nearer to l.) We can define a hyperbolic circle as the set of
all points a fixed distance away from a fixed center, which satisfies the third axiom by
design. Finally, we can define hyperbolic angle measures to be the same as the
Euclidean angle measures between the tangent lines of the intersecting arcs; ergo, the
fourth axiom in Euclidean geometry implies the fourth axiom in our model.
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Thankfully, we seem to have found a workable model for the geometry that we
wish to investigate (indeed, in finding the model we have already been investigating
quite intensely). A summary seems appropriate.

Rather than construct an explicit surface on which to do hyperbolic geometry,
we have changed our visual image of “line” and relegated the ambient
curvature to the background.

The set of points for our hyperbolic plane model is the open upper half-plane
as determined by a line l.

The line segment between two points is either the arc of the circle with center
on l containing the two points, or is the segment of the ray perpendicular to l
containing the two points.

As you move closer and closer to l the underlying space curves more and
more, that is to say, the hyperbolic distances do not match the Euclidean
distances present in our model. 7
7
For a brief discussion of two other models, see Appendix C.
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EXPLORATION 2: PARALLEL LINES
With a model of hyperbolic geometry at our disposal we can now examine the
nature of lines and line segments in this new world. From past experience we know
that parallel lines in Euclidean geometry are everywhere equidistant in a certain
sense, and in spherical geometry parallel lines do not exist. One illuminating way to
formulate this distinction is by choosing a line m and a point P not on m. The question
is: how many lines parallel to m contain P?8 The Euclidean answer is one, and the
spherical answer is zero. Let us seek the hyperbolic answer. 9
To proceed, it is necessary to make explicit
what we mean by “parallel.”
Definition. Two lines are parallel if they have no
points in common.
Furthermore, it is important to note that in
Euclidean geometry two distinct circles can meet
in 0 points, 1 point, or 2 points, and the single
point situation occurs if and only if the circles
meet tangentially at that point. We will use this
because the hyperbolic lines of our model can
Figure 6: Circle Meetings
also be thought of as circles in the traditional Euclidean sense.
Now, let m be a line in the hyperbolic plane and let P be a point not on m. Label
the boundary points of m as A and B. We can construct the Euclidean line segment PA
and then bisect it perpendicularly. If this perpendicular bisector intersects line l then
8
The Euclidean situation, a unique parallel, is Playfair’s postulate and is equivalent to the fifth postulate.
9
The answer is found deductively in Appendix B, but here we will proceed more empirically.
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we can use this intersection point as the center of a circle and construct the hyperbolic
line n that passes through P and A (though A is not actually in the hyperbolic plane,
this is important!). The Euclidean circles m and n meet at the point A, and there they
are both orthogonal to line l which means that they meet tangentially. This means that
P●
A is the only point at which they meet.
n
m
A
But A is technically off the hyperbolic
B
plane, so necessarily m and n do not
meet in the hyperbolic plane. Thus, by
m
definition, they are parallel hyperbolic
●
P
A
P
B
lines. If the perpendicular bisector of PA
does not intersect line l then we can
●
construct the ray from A to P. This is a
m
hyperbolic line that meets m only at the
A
B
Figure 7: Construction of a Parallel
(three cases)
point A, and so is also parallel to m in
hyperbolic geometry.
The paragraph above has proven the following result in hyperbolic geometry.
Proposition 1. If m is a line and P is a point not on m, then there exists a
line through P parallel to m.
So we see that hyperbolic geometry is inherently different than spherical geometry.
Moreover, it is inherently different than Euclidean geometry because we can repeat
the argument above using the point B in place of A, and this will give us another line
through P parallel to m!
Proposition 1 (updated). There exist at least two lines through P parallel to m.
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A bit more examination uncovers infinitely many parallels to m through P (see
figure 8).10 However, we are seeing that the difference between parallels in hyperbolic
geometry and Euclidean geometry is more than just a matter of multitude, there is a
qualitative difference as well. In hyperbolic geometry we have some parallel lines
(like m and n in figure 7) that diverge in one direction but converge in the other, and
we have other parallel lines that diverge in both directions.
Definition. Parallel lines are ultraparallel if they diverge in both directions, and are
asymptotically parallel if they converge in one direction.
●
P
m
Figure 8: Many lines through P parallel to m
The asymptotically parallel lines (of which there are two, based on our proof of
Proposition 1) seem to be the bounds of a region that contains m, and any hyperbolic
line through P contained in that region will necessarily intersect m. Conversely, any
hyperbolic line through P outside of that region will be ultraparallel.
We have defined parallel as non-intersecting. There is another notion, however,
related to parallelism that is worth consideration—the parallel transport.
Definition. Two lines are parallel transports of one another if there exists a
transversal that creates equal corresponding angles.
10
For a proof of this result, see Appendix B.
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In Euclidean geometry two lines are parallel transports if and only if they are
parallel.11 Does such a result hold in hyperbolic geometry?
Proposition 2. If two lines are ultraparallel, then they are parallel transports.
Let m and n be ultraparallels. Our task is to find a third line p that creates equal
angles in corresponding positions with regard to m and n. Recall that the hyperbolic
angles in our model are conformal to the Euclidean angles.
small angle
small angle
p
n
large angle
m
large
angle
Figure 9: Parallels and “Extreme” Transversals
Intuitively, if we think of a very small (in the sense of Euclidean circles)
transversal, this will create an angle with respect to m that is nearly zero and a
corresponding angle with respect to n that is nearly two right angles (see figure 9).
Now, we let the radius of the transversal circle (i.e., the hyperbolic line) grow until it
is nearly the largest transversal possible. In this case, the angle in the same position as
before is nearly two right angles with respect to m and is nearly zero with respect to n.
They have switched the inequality! Since this process of growth was continuous, 12 by
the intermediate value theorem, there exists some transversal p that creates equal
corresponding angles. Thus m and n are parallel transports along p.
11
12
In fact, every transversal creates equal corresponding angles. This is equivalent to Euclid V.
Hence we are using a continuity axiom.
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It is important to note that the argument for Proposition 2 fails for asymptotically
parallel lines. This is because the angles in the “smallest” case are nearly a right angle
and so do not pass all the way from zero to two right angles nor vice versa, thus we
can not apply the intermediate value theorem in the same way.
Conjecture 1. If two lines are parallel transports, then they are parallel.
Let m and n be lines such that a third line p exists creating equal corresponding
angles. Since angles are conformal, supplementary angles work as expected in our
hyperbolic model. So we know
that alternate interior angles are
p
m
n
A
B
equal and same-side interior
X
angles are supplementary. Now,
label the intersection points of l
Figure 10: Parallel Transports
with m and n as A and B, respectively. Suppose to the contrary that m and n meet at a
point X on one side of l. Then ABX forms what seems to be a hyperbolic triangle. We
noted above that ABX and BAX sum to two right angles. The addition of AXB
gives us an angle sum for ABX of at least two right angles. So if we knew that the
sum of the interior angles of hyperbolic triangles was always less than two right
angles we would have a contradiction. In particular, such a theorem about the interior
angles of triangles would make Conjecture 1 true.
Therefore, let us turn our attention to hyperbolic triangles.
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EXPLORATION 3: HYPERBOLIC TRIANGLES
We have a fairly good grasp of lines and line segments to this point, so there is
nothing preventing us from defining polygons.
Definition. A polygon is the union of a finite collection of points (or vertices) and
line segments (or sides) such that each vertex is incident with exactly two sides and
the sides do not intersect except at a vertex.
A polygon with three vertices (and thus three
sides) is a triangle. We can construct some in our
model. Empirically, it seems that the angles of the
triangles are “scrunched up” or, in other words,
smaller than the angles of triangles in the
Figure 11: Triangles in the
Hyperbolic Plane
Euclidean plane. Is this always the case?
Conjecture 2. The sum of the interior angles of a hyperbolic triangle is
less than two right angles (i.e., π).
Proving this conjecture is the goal of the remainder of this exploration. 13
Let ABC be a hyperbolic triangle with interior angles  ,  and  . Let AB, BC
and CA be extended so that we have exterior angles (    ,    and    ,
respectively). Label the boundary intersections P, Q and R (see figure 12).
A



C
B
P
Q
R
Figure 12: A Triangle with Exterior Angles
13
The approach will be the same as with spherical geometry – find a formula for area in terms of angles.
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We connect P to Q, Q to R, and R to P. This forms a new type of triangle with
vertices on the boundary line (that is, with vertices at infinity).
Definition. A triangle with all of its vertices on the boundary line is an ideal triangle.
A triangle with two vertices on the boundary line is a 2/3 ideal triangle.
Note that angles formed on the boundary line necessarily have angle measure zero.
C
A
P
B
Q
R
Figure 13: An Ideal Triangle Comprising Four Other Triangles
Now, we see from figure 13 that PQR is ideal and it is composed of four other
triangles, three of which are 2/3 ideal and the fourth being ABC . That is,
PQR  PAQ  QBR  RCP  ABC .
(1)
Proposition 3. A 2/3 ideal triangle is completely determined by its non-ideal angle.
Consider two 2/3 ideal triangles with the same non-ideal angle. We can rigidly
move the first triangle so that a ray of the non-ideal angle coincides with a ray of the
non-ideal angle of the second triangle. Since the angles are equal we know that the
second rays can also be made to
coincide (using a reflection about the
first ray if necessary). Thus the two
triangles can be rigidly transformed
into a configuration like figure 14.
X
Y
Figure 14: 2/3 Ideal Triangles
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The last thing to check is that the sides XY coincide, but this is implied by the
uniqueness of the Euclidean circle with diameter XY. So the two triangles are
congruent.
Proposition 3 implies the existence of a (linear) area formula for 2/3 ideal
triangles that depends only on the non-ideal ideal, or equivalently, on the non-ideal
exterior angle. Let us use Area23 to denote the function that takes in the exterior
angle of a 2/3 ideal triangle and outputs its area. Since Area23 is linear we know that
it has the form Area23   k   for some constant k. We also know that if the nonideal angle is taken to the boundary of the hyperbolic plane (thus forming an ideal
triangle), its measure goes to zero. In this case the exterior angle becomes  and so
all ideal triangles have the same area, namely, Area23  . Let us denote this fixed
area of ideal triangles as I. Then the formula I  Area23   k   implies that
k  I  . So the area of a 2/3 ideal triangle with non-ideal angle  can be found via
the function
Area23  
I 

.
Now we return to Equation (1) from above. Based on our work (and the fact that
the exterior angles of the 2/3 ideal triangles are precisely the interior angles of
ABC ) this equation becomes
I
I 


I 


I 

 ABC .
We are actually interested in the area of ABC , so we rearrange the terms and find
ABC  I 
I


I


I
     
 I 1 
.




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But the area of ABC must be positive, so the quantity on the right must be positive
(since I is positive). This means
1
   
0,

or equivalently,
      .
This means that we have verified Conjecture 2.
Proposition 4. The sum of the interior angles of a hyperbolic triangle
is less than  .14
Going back even farther to our parallel exploration, Proposition 4 gives us the
contradiction we needed to prove Conjecture 1.
Proposition 5. If two lines are parallel transports, then they are parallel.
So it is clear that Proposition 4 packs quite a punch! Indeed, it highlights an essential
difference between hyperbolic geometry and Euclidean geometry (and spherical
geometry). In the Euclidean world interior angles of triangles sum to  exactly (and
in the spherical world, the sum is greater than  ).
In this exploration we have uncovered an important characteristic of hyperbolic
triangles. Let us continue and see what we can discover about certain hyperbolic
quadrilaterals, specifically, rectangles.
The difference between π and the area of a triangle is called the triangle’s defect, and looking at the
formula for the area of a triangle at the bottom of page 17 reveals the fact that the only variable in the
equation is the angle sum. Because of this, a hyperbolic triangle’s area can be defined as its defect.
14
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EXPLORATION 4: HYPERBOLIC RECTANGLES
Given any convex hyperbolic quadrilateral, either diagonal creates two
hyperbolic triangles. Thus Proposition 4 immediately gives us the following.
Proposition 6. The sum of the interior angles of a convex quadrilateral is less than 2 .
So if we intend to find rectangles in the hyperbolic
plane it would be futile to use the definition wherein a
rectangle contains four right angles. But are there other
ways to conceptualize and define rectangles?
Figure 15: A Convex
Quadrilateral
One possibility is to start with a base
A
D
and onto it erect sides of equal length at
right angles to the base (this is called a
B
C
Saccheri quadrilateral). In Euclidean
Figure 16: A Saccheri Quadrilateral
geometry, the result is a rectangle. By
our note above the two summit angles cannot both be right in hyperbolic geometry.
But they can both be equal.
Conjecture 3. The summit angles of a Saccheri quadrilateral are equal.
In order to prove this we will use congruent triangles. We have from previous
explorations an intuitive notion of congruence, but let us now formalize this a bit
more. It is reasonable to assume that given two side lengths and the measure of the
included angle, there is a unique triangle (up to congruence) that can be constructed.
You simply lay off the sides at the given angle and there is nothing to do but close up
the third side. Thus, let us add side-angle-side (SAS) as axiom of congruence. 15
15
This is what Hilbert did when he made explicit the assumptions that Euclid had tacitly used.
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This added axiom will allow us to do quite a bit of work.
Definition. A triangle is isosceles if it has two congruent sides.
Proposition 7. The base angles of an isosceles triangle are congruent.
Let ABC be an isosceles triangle (AB = BC). Then BAC is congruent to
CAB (SAS). Therefore corresponding angles are congruent; in particular, B  C .
Definition. A line is a bisector of a segment if it cuts the segment into equal segments,
and is a perpendicular bisector if it is also perpendicular to the segment.
Proposition 8. The perpendicular bisector of the base of an isosceles
triangle passes through the summit vertex.
Let ABC be isosceles and let BC be bisected at M. Connect A to M. It will
suffice to show that AM meets BC perpendicularly. Since M is the midpoint we know
that BM = MC. By Proposition 7 B  C , and by assumption AB = AC. Thus SAS
assures us that AMB is congruent to AMC . This implies AMB  AMC , but
these angles form the straight line BC and so are both right. Hence the line through A
and M is the perpendicular bisector of BC, proving Proposition 8.
Proposition 9. Side-side-side (SSS) is a congruence condition for triangles. 16
Let ABC and DEF be triangles satisfying SSS.
A, D ●
We can rigidly move one so that AB coincides with DE
F
and (after a possible reflection) C and F are on the
B, E
same side of AB. Suppose to the contrary that, after this
Figure 17: Triangles
with SSS
motion, C and F are distinct points. Then CAF is
C
isosceles because AC = DF. Thus Proposition 8 tells us that the perpendicular bisector
16
Angle-angle-angle (AAA) is also a congruence condition in hyperbolic geometry – see Appendix D.
Otten 21
of CF contains A. Similarly, CBF is isosceles and so the perpendicular bisector of
CF contains B. But by Axiom 1 the only line containing A and B is line AB. So the
perpendicular bisector of CF is line AB. This is a contradiction, however, because C
and F are on the same side of AB.17
Now let us return to the Saccheri quadrilateral. By drawing in the diagonals we
see that ACB is congruent to DBC
A
D
by SAS. Hence AC = DB. This means
that ABD is congruent to DCA by
C
Figure 18: A Saccheri Quadrilateral
(again)
SSS (Proposition 9). Therefore
BAD  CDA because they
correspond. These are the summit angles of the Saccheri quadrilateral and so we have
our answer to Conjecture 3.
Proposition 10. The summit angles of a Saccheri quadrilateral are equal.
Unfortunately, this Saccheri quadrilateral, though it has some nice symmetry,
seems to fall short of what we think of as a rectangle. So our search continues for
something that feels like a rectangle in the hyperbolic plane.
Another possibility for a “hyperbolic rectangle” is to keep as many right angles as
possible by constructing a figure with three right angles, but then Proposition 6 would
force the fourth angle to be less than a right angle (this is called a Lambert
quadrilateral). Such a figure is not equiangular and hence we have lost an important
characteristic of a Euclidean rectangle. This is unacceptable, but perhaps we have
stumbled upon that which makes a rectangle a rectangle – equal angles.
17
There is an assumption about betweenness being made here, but a full treatment of this issue would take
us too far afield.
Otten 22
Definition. A hyperbolic rectangle is a quadrilateral with four equal angles.
This is an intuitive and acceptable definition for rectangle in our negatively
curved environment, but we should make sure that one exists. To construct a
hyperbolic rectangle we begin with an isosceles triangle ABC . Proposition 7 gives
B  C . Let  be the summit angle and let  be the base angle. If we move A
very close to BC then  gets very close to  while  gets very close to 0. On the
A
other hand, if we move A very far away from
BC then  gets very close to 0 while  is
B
C
larger than 0. Since this motion is continuous,
A'
the intermediate value theorem guarantees the
existence of an isosceles triangle built on BC
Figure 19: Construction of
a Hyperbolic Rectangle
with the angle relationship   2  . We reflect this triangle over BC and the result is a
hyperbolic rectangle ABA'C.
The rectangle just constructed has the additional property that it is equilateral
(since AB = BC by assumption and reflections preserve distance). Therefore, we have
just constructively proved the existence of an equiangular, equilateral quadrilateral in
hyperbolic geometry.
Definition. A hyperbolic square is a quadrilateral with four equal angles
and four equal sides.
Ironically, our quest for something resembling a rectangle has led us to
something even better – the existence of a hyperbolic square!
Otten 23
CONCLUSION
Hopefully, in reading this project you have found the world of hyperbolic geometry
to be as rewarding a mathematical playground as I have. It was especially helpful, from
an intuitive standpoint, to have a workable model as a guide for exploration. This
phenomenon of taking something as abstract as a non-Euclidean environment and
finding a tangible representation of it is one of the beautiful tendencies of mathematics.
Of course, the benefits of working with hyperbolic geometry are not restricted to
hyperbolic geometry itself. The thought processes, arguments, and careful
considerations that arose during the explorations also cultivated a deeper appreciation
for Euclidean geometry. For instance, we found that triangles in the hyperbolic plane
have an interior angle sum that may lie anywhere between zero and π, and we know
from elsewhere that spherical triangles have interior angles that sum to something
larger than π; Euclidean geometry is the remarkably special case where the angle sum is
always precisely π itself. Furthermore, Euclidean geometry is the case where precisely
one parallel line exists through a point off a given line, and it is only in Euclidean
geometry that shapes can be resized without distortion (since AAA is a congruence
condition in other geometries).
Work in a non-Euclidean environment exposes the formerly implicit assumptions
about space’s curvature (or lack thereof). Although its development was historically
seen as a devastating blow to the certainty of mathematics, I believe that with a new
perspective it can be seen as a demonstration of the power of mathematics to work
deductively from explicit axioms. Would that other disciplines, and mankind in general,
were so forthcoming about their presuppositions.
Otten 24
APPENDIX A: “PROOF” OF THE FIFTH POSTULATE
Farkas Bolyai, the father of János Bolyai, attempted to prove Playfair’s postulate
(which is equivalent to Euclid’s fifth) in the following way.
Let l be a line and let P be any point not on l. Drop a perpendicular from P to l, hitting l at
the point Q. Construct line m perpendicular to PQ at the point P. We wish to show that m is the
only line through P parallel to l (we know that it is parallel by Proposition 5 in our triangle
exploration). So let n be any line through P distinct from m. We must show that n intersects l.
Let A be any point between P and Q, and
let B be the unique point such that Q is on AB
C
and AQ = QB. Let R be the foot of the
P
perpendicular from A to n, and let C be the
m
R
unique point such that R is on AC and AR =
A
n
RC (see figure 20). We know that A, B and C
Q
are not collinear because then R and P would
coincide, contradicting the distinctness of n
from m. Hence there is a unique circle
B
Figure 20: Farkas’ Construction
containing A, B and C. Since l is the perpendicular bisector of chord AB and n is the perpendicular
bisector of chord AC, l and n necessarily meet at the center of the circle. Therefore m is the
unique parallel to l through P.
The error in this argument takes place when the existence of the circle is asserted. To
say that a circle can be constructed through any three non-collinear points is equivalent to
Euclid’s fifth postulate, so the reasoning above is circular. This becomes clearer when we
think about the construction of such a circle in Euclidean geometry. We find the intersection
of two perpendicular bisectors and use this as the center; but what if the perpendicular
bisectors never intersect? Farkas assumed they did intersect to prove that they intersected.
l
Otten 25
APPENDIX B: THE UNIVERSAL HYPERBOLIC THEOREM
We assume a negation of Playfair’s postulate, that is, there exists some line l and
some point P not on l such that at least two distinct lines parallel to l pass through P.
This can be generalized to the universal hyperbolic theorem which states that for all
lines l and for all points P not on l there exist infinitely many lines parallel to l that
pass through P.
Let l be any line and P any point not on l. Construct Q, the foot of the
perpendicular from P to l, and let m be the line through P perpendicular to PQ. Now,
let R be any point on l distinct from Q, and construct line t through R perpendicular to
l. Drop P perpendicularly to t and label the intersection point S (see figure 21).
t
P
m
S
R
Q
l
Figure 21: Construction of Parallels
By Proposition 5, m and PS are both parallel to l. We claim that they are also
distinct lines. Suppose to the contrary that S is on m. Then PQRS is a quadrilateral
with four right angles, but this contradicts Proposition 6 (which is where we use the
assumption). Thus m and line PS are distinct lines parallel to l. By varying R along l it
follows that infinitely many such lines exist.
Otten 26
APPENDIX C: OTHER HYPERBOLIC PLANE MODELS
Throughout our explorations we used the half-plane model for hyperbolic
geometry. This model includes a boundary line which represents infinity in the
hyperbolic plane. The one-point compactification of this half plane and its boundary
line results in the Poincaré disk model, wherein
hyperbolic lines are arcs of circles that intersect
the boundary circle orthogonally. Angles in this
model, as in the half-plane model, are
conformal.
The Beltrami-Klein disk model also utilizes
a boundary circle representing infinity;
however, hyperbolic lines are chords of the
boundary circle instead of arcs. This allows for
Figure 22: Poincaré and
Beltrami-Klein Models
an easier grasp of collinearity, but angles are distorted.
It can be shown via projective geometry that all of these models are isomorphic.
Otten 27
APPENDIX D: AAA CONGRUENCE
We will prove that similarity implies congruence in hyperbolic geometry; that is,
AAA is a congruence condition for hyperbolic triangles.
Suppose to the contrary that there exist triangles ABC and ABC which are
similar but not congruent. It must be the case that no sides are congruent, otherwise
the triangles would fall under ASA (which can be proven a valid congruence
condition using SAS). Then by the pigeon-hole principle, two of the sides of one
triangle must be larger than the two corresponding sides in the other triangle. Without
loss of generality, suppose AB  AB  and AC  AC . This means that there exist
points D and E on AB and AC, respectively, such that AD  AB  and AE  AC (see
figure 23).
A
We have ABC congruent to
A'
ADE by SAS. This gives B   ADE
D
B
E
and C  AED . We are assuming that
B'
C'
C
Figure 23: Similar Triangles
the corresponding angles of ABC and
ABC are also equal, specifically,
B  B   ADE and C  C  AED . Hence, by Proposition 5, line BC is
parallel to line DE. So quadrilateral BCED is convex. But B and BDE are
supplementary, as are C and CED , which means BCED has interior angles that
sum to 2 . This contradicts Proposition 6.
Therefore, no triangles with corresponding angles equal exist except ones that are
congruent. In other words, AAA assures congruence.
Otten 28
REFERENCES 18
Euclid. (1956). The Thirteen Books of the Elements (T. L. Heath, Trans. 2 ed.). New
York: Dover Publications.
Greenberg, M. J. (1993). Euclidean and Non-Euclidean Geometries: Development and
History. New York: W.H. Freeman Co.
Henderson, D. W. (2001). Experiencing Geometry: In Euclidean, spherical, and
hyperbolic spaces (2 ed.). Upper Saddle River, NJ: Prentice-Hall, Inc.
Hilbert, D., & Cohn-Vossen, S. (1952). Geometry and the Imagination (P. Nemenyi,
Trans.). New York: Chelsea Publishing Company.
Kline, M. (1982). Mathematics: The Loss of Certainty. Oxford: Oxford University Press.
Euclid’s text was used as a primary source for references to the Elements, and as a guide for the active
voice found throughout this paper. The primary contributions of Henderson’s book were ideas about
parallel transports and his sequence of problems that led me to the proof of the hyperbolic triangle sum
theorem found in Exploration 3. From Hilbert I harvested some important thoughts about axioms, and
Kline contributed the historical context regarding mathematical certainty and the non-Euclidean revolution.
The remainder of the material was taken from Greenberg (especially my appendices), though some of the
conjectures, propositions, and proofs are based exclusively on my own investigation and curiosity.
18