Non-Euclidean Geometry, spring term 2017
Lecture notes
Faculty of Mathematics, NRU HSE, and Math in Moscow, IUM
1. Reading for Lecture 7
Last time we introduced Klein disk model of a hyperbolic plane. We now define explicitly the distance in the Klein model using the notion of cross-ratio and projective
transformations. We also study isometries of hyperbolic plane and tilings in hyperbolic,
Euclidean and spherical geometries [So, VS].
2. Axioms of a Hilbert plane
A Hilbert plane is a collection of points and lines together with incidence, betweenness and congruence relations that satisfy all axioms I − III listed below. The axioms
describe properties of undefined objects (points and lines) and relations between them
(incidence, betweenness, congruence, also described by verbs contain, lie between, and be
equal, respectively, and their synonyms).
Axioms of incidence geometry:
I1−2 For any two distinct points A and B, there exists a unique line containing A and B.
I3 Every line contains at least two points. There exist three noncollinear points (that is,
three points not all contained in a single line).
Axioms of order:
II1 If B is between A and C, then A, B and C are three distinct points on a line, and
also B is between C and A.
II2 For any two distinct points A and B, there exists a point C such that B is between
A and C.
II3 Given three distinct points on a line, at most one of them is between the other two.
II4 (Pasch) Let A, B, C be three noncollinear points, and let l be a line not containing
any of A, B, C. If l contains a point D lying between A and B, then it must also
contain either a point lying between A and C or a point lying between B and C.
Axioms of congruence: Congruence relation will be denoted by '. Following Hilbert,
by angle we mean the union of two rays originating at the same point.
III1 Given a line segment AB, and given a ray r originating at a point C, there exists a
unique point D on the ray r such that AB ' CD.
III2 If AB ' A0 B 0 and A0 B 0 ' A00 B 00 , then AB ' A00 B 00 (if two segments are each
congruent to a third segment, then they are congruent to each other).
III3 (Addition of segments) Let AB and BC be two segments on a line, and let A0 B 0 and
B 0 C 0 be two segments on another (or the same) line such that B is between A and C,
and B 0 is between A0 and C 0 If AB ' A0 B 0 and BC ' B 0 C 0 , then AC is congruent
to A0 C 0 .
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2
III4 Given an angle ∠BAC and given a ray DF originating at a point D, there exists a
unique ray DE on a given side of the line DF such that ∠EDF ' ∠BAC.
III5 (SAS) Given triangles ABC and A0 B 0 C 0 , if AB ' A0 B 0 , AC ' A0 C 0 , and ∠BAC '
∠B 0 A0 C 0 , then ∆ABC ' ∆A0 B 0 C 0 .
3. Necessary and discarded axioms
Last time we had a list of Hilbert’s axioms I − III for neutral (or absolute) geometry
(see Section 2). We now take a closer look at some of these axioms and their implications
mostly following [Ha05, Chapter 2]. We also discuss axioms IV − V that turn a Hilbert
plane into the usual Euclidean plane R2 [Hi, Sections 5 and 8] and their independence
from the rest of the axioms [Hi, Sections 10,12]. Note that our list of axioms is taken
directly from [Hi, Chapter 1] (up to possibly switching the groups III ↔ IV ) and is
slightly different from the one in [Ha05, Chapter 2].
There are 12 axioms total in groups I − III, however, the first edition of [Hi] included
two extra axioms, which were later discarded as unnecessary. Hilbert’s original goal was
to construct the system of axioms such that
(1) Any proposition of Euclid [Eu] can be rigourously proved using this system (without any extra explicit or implicit assumptions).
(2) No axiom can be deduced from the rest of the axioms (that is, the system is as
simple as possible).
Indeed, the system of axioms we had last time is minimal in the following sense. If we
omit any single axiom E, then it is possible to construct a model of a plane that satisfies
all axioms except for E and does not satisfy E. In other words, E is independent from
the other axioms. For instance, a usual Euclidean sphere in a 3-space yields a model for
a plane that satisfies axioms I1 (there is a line through any pair of points), I3 , II1−4 ,
III1−5 but does not satisfy axiom I2 (there is at most one line through any pair of points)
(see [Ha05, Exercise 34.13]). The usual Euclidean plane can be turned into a model that
satisfies all axioms except for III5 by modifying the definition of distance (see [Hi, Section
2.11]).
However, when Hilbert published the first list of his axioms, other mathematicians
realized that some of his axioms are actually theorems.
Theorem 1 (Discarded axiom of order). Any four distinct points on the same line can
be labeled by A, B, C, D so that B is between A and C, and also between A and D, while
C is between B and D, and also between A and D.
In the first edition of [Hi] (published in 1899), this theorem is listed as an extra axiom
of order, but in 1902, American mathematicians E.H.Moore (the chair of the Department
of Mathematics at the University of Chicago) and R.L.Moore (an undergraduate student
of G.B.Halsted at the University of Texas) deduced this axiom from the other axioms of
order and incidence (they gave independently two different proofs [Mo, Hal]). In later
editions, Hilbert acknowledged their work and listed this axiom as a theorem.
Theorem 2 (Discarded axiom of congruence). For any three angles α, β and γ, if α = β
and α = γ, then β = γ.
This axiom was deduced from the other axioms by A. Rosenthal much later, in 1912
[Ro]. In the 7th edition, Hilbert converted this axiom to a theorem.
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4. Definition of Euclidean plane
The Euclidean plane is a collection of pairs (x, y) (called points) of real numbers. Lines
are triples (u : v : w) of real numbers (considered up to proportionality) such that u and v
are not simultaneously equal to 0. To define incidence relation, we say that a point (x, y)
belongs to a line (u : v : w) if ux + vy + w = 0. To define betweenness we use the order
on R. If A = (x, y), B = (x0 , y 0 ) and C = (x00 , y 00 ) are three points on a line, say that B
lies between A and C, if one of the following conditions holds:
• x < x0 < x00
• x > x0 > x00
• y < y 0 < y 00
• y > y 0 > y 00
To define congruence of segments first define the length |AB| of the segment with endpoints
A = (x, y), B = (x0 , y 0 ), as
p
(x − x0 )2 + (y − y 0 )2 .
Next, say that two segments are congruent if they have equal lengths.
To define congruence of angles use dot product of vectors and cosine of an angle. Consider an angle with vertex O = (x, y) and with two sides passing through points A = (x0 , y 0 )
and B = (x00 , y 00 ), respectively. Cosine of ∠AOB is defined as
(x − x0 )(x − x00 ) + (y − y 0 )(y − y 00 )
.
|OA||OB|
(The numerator of this formula is called dot product of vectors OA and OB.) Say that
two angles are congruent, if their cosines are equal. (These definitions of length and cosine
are motivated by Pythagorean theorem.)
Alternatively, Euclidean plane can be defined axiomatically by taking Hilbert axioms
I − III together with three extra axioms.
Axiom of parallels:
IV1 Let l be any line and A a point not on l. Then there exists at most one line, that
passes through A and does not intersect l.
Axioms of continuity:
V1 (Archimedes) For any two segments AB and CD, there exist finitely many points
A1 ,. . . , An on the line AB such that segments AA1 , A1 A2 ,. . . , An−1 An are congruent
to CD, and B lies between A and An .
V2 Points of a line can not be completed by new points so that axioms I1−2 , II, III1 ,
V1 are still true.
5. Non-Euclidean Hilbert planes
Last time we discussed additional axioms IV − V that turn a Hilbert plane into the
Euclidean plane R2 . We also recalled definition of a field. Using a field F we can define a
Cartesian plane F2 (named after a famous French mathematician and philosopher René
Descartes), also called a coordinate vector space of dimension two over F [Ha05, Section
3.14]. Geometry of a Cartesian plane is to some extent similar to geometry of a Hilbert
plane, however, it can be studied by purely algebraic methods [Ha05, Section 3.13].
We now briefly discuss relationship between Hilbert and Cartesian planes following
[Ha05, Chapter 3]. Note that a Cartesian plane by definition satisfies Hilbert axioms of
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incidence geometry (group I) [Ha05, Proposition 14.1] and the axiom of parallels (axiom
IV ). It satisfies axioms of order (group II) if and only if F is an ordered field such as a
subfield of R [Ha05, Proposition 15.3]. Axioms of congruence (group III) hold if and only
if F is Pythagorean [Ha05, Sections 3.16-3.17]. Finally, axioms of continuity (group V )
hold if and only if F = R [Ha05, Proposition 15.5]. There are non-Archimedean ordered
fields [Ha05, Section 3.18], in particular, their existence shows that the Archimedean
axiom V1 is independent from the rest of the axioms.
Not all Hilbert planes are Cartesian. Last time, we discussed that a Hilbert plane is
Cartesian if and only if it satisfies the axiom of parallels IV . In this case, there are
two ways to associate a field with a Hilbert plane, that is, to define multiplication of
segments. One can use either the theory of proportions of Eudoxus–Euclid (relying on
the Archimedean axiom) [Eu, Book V] or the Pappus theorem (proved by Hilbert without
relying on the Archimedean axiom [Hi, Section 13], see also [Ha05, Chapter 4]).
Note that the addition of segments does not require the axiom of parallels, in particular,
one can measure lengths of segments on any Hilbert plane. In particular, if we impose
axioms V1−2 then the points of a line can be identified with real numbers. Still, such a
Hilbert plane might look quite different from the Euclidean plane. In order to see these
differences we have to look at the global picture, in particular, all proofs will require some
sort of limiting argument. Here is an example.
Theorem 3 (Saccheri–Legendre). If the Archimedean axiom (axiom V1 ) holds in a given
Hilbert plane, then the sum of angles of any triangle on this plane is less than or equal to
180◦ .
Legendre gave two different proofs of this theorem. We will discuss one of them in class,
for the other see [Ha05, Theorem 35.2].
Definition 1. We say that a Hilbert plane is non-Euclidean if there exists a triangle
whose sum of angles is not equal to 180◦ .
In a non-Euclidean Hilbert plane, there exists a remarkable additive function on triangles called defect [Ha05, Lemma 34.8]. It behaves very much like an area but is easier to
define.
Definition 2. For a triangle ABC, define the defect δ(ABC) by the formula
δ(ABC) = 180◦ − ∠A − ∠B − ∠C.
We will discuss properties and applications of the defect following [Ha05, Section 34].
6. Definition of Cartesian plane
If you remember the definition of a vector space over a field, then think of the Cartesian
plane F2 over a field F as of a two-dimensional vector space over F. Otherwise, replace R
with F in the definition of the Euclidean plane.
Namely, define F2 as the collection of pairs (x, y) (called points or vectors) such that
x, y ∈ F. Lines (or affine subspaces of dimension one) are triples (u : v : w) of elements
u, w, v ∈ F (considered up to proportionality) such that u and v are not simultaneously
equal to 0. To define incidence relation, we say that a point (x, y) belongs to a line
(u : v : w) if ux + vy + w = 0.
For many fields F, we can not define reasonable relations of betweenness and congruence. Nevertheless, there are very useful Cartesian planes even without such relations. In
particular, there are exotic Cartesian planes over finite fields, which have applications in
digital technologies. Here is the minimal example.
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Definition 3. The field of two elements F2 is a collection of elements O and I with
operations of addition and multiplication given by the following tables:
+
O
I
O I
O I ,
I O
·
O
I
O I
O O .
O I
Addition in F2 can be described by the power button rule: pressing the button twice
yields the same result as not pressing it at all.
7. Area: Euclidean and hyperbolic
Last time we discussed properties of non-Euclidean Hilbert planes such as the existence
of an additive function on triangles called defect. We now relate the defect to the hyperbolic area of a triangle following [Ha05, Section 36]. First, we review briefly the notion of
Euclidean area following [Ha05, Theorem 12.2] and [Ha05, Chapter 5], in particular, discuss the notion of equidecomposable (also called scissor-congruent) figures and the notion
of content.
We now live in hyperbolic plane.
Definition 4. By a hyperbolic plane we call a Hilbert plane that satisfies the following
axiom:
(L) Let l be any line and A a point not on l. Then there exist two rays AB and AC, not
lying on the same line, and not meeting l, such that any ray AE in the interior of the
angle BAC meets l.
We do not impose axioms V1−2 on a hyperbolic plane, however, you may use these axioms
if they make your solutions easier.
8. Hyperboloid model
Reading for this section is [CFKP, Sections 3, 4, 5, 10]. Last time we defined hyperbolic
plane. We now have to show that is exists. To do this we will construct a hyperboloid
model of hyperbolic plane using Minkowski quadratic form in R3 . Read your favorite linear
algebra textbook to review the notion of quadratic form, linear operator and multiplication
of matrices.
By analogy with the equation of a sphere x2 +y 2 +z 2 = R2 in R3 we define a two-sheeted
hyperboloid by the equation x2 + y 2 − z 2 = −c2 . Note that any non-degenerate quadratic
form on R3 can be reduced by the change of coordinates either to qs := x2 +y 2 +z 2 (positive
definite) or to qh := x2 + y 2 − z 2 (Minkowski or hyperbolic). So there is nothing special
about our choice of qh : it is the only (up to a linear change of coordinates) indefinite
non-degenerate quadratic form in R3 .
Denote by H the upper half of the hyperboloid, i.e., the surface {x2 + y 2 − z 2 =
−c2 , z > 0}. Our goal is to turn H into a model of hyperbolic plane. Again by analogy
with spherical geometry define (hyperbolic) lines in H as the intersections of H with
the planes passing through the origin. It is not hard to check that hyperbolic lines are
Euclidean hyperbolas (“great hyperbolas”).
There is an alternative more general definition of “lines” or geodesics on a surface. It
is based on the notion of metric.
Definition 5. Let S be a set. A real-valued function d(x, y) on pairs of points x, y ∈ S
is called the distance function (or metric) if the following properties hold:
(1) d(x, y) ≥ 0 for all x, y ∈ S (non-negativity);
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(2) d(x, y) = 0 if and only if x = y;
(3) d(x, y) = d(y, x);
(4) d(x, y) + d(y, z) ≥ (x, z) for all x, y, z ∈ S (triangle inequality).
One way to introduce a metric on a surface S ⊂ R3 is to use the standard Euclidean
metric on R3 and a bit of calculus (read your favorite analysis textbook to recall how to
measure the length of a smooth curve in R2 and R3 ). Suppose we can measure the length
of any smooth curve on a surface S. For any two points A, B ∈ S, we can define the
“segment” AB (usually called geodesic) as the shortest path between A and B (we only
consider paths that belong to S). It is not evident that this definition for a sphere indeed
yields great circles as geodesics. This can be proved both analytically and geometrically.
Last time we constructed the hyperboloid model of hyperbolic plane and defined hyperbolic lines. We now check that it satisfies axioms of hyperbolic plane. Our main tool
will be the group of isometries of hyperboloid with respect to Minkowski metric on R3 .
Read your favorite algebra textbook to recall the definition of a group.
Disclaimer: Minkowski metric is not a metric. It becomes a metric only after restriction to a suitable hyperboloid. the hyperboloid model of hyperbolic plane and defined
hyperbolic lines.
To introduce a hyperbolic metric on H p
we use Minkowski metric on R3 , namely, the
“length” of a vector (x, y, z) is defined as x2 + y 2 − z 2 . In general, this length is not
a nonnegative real number (or a real number at all). However, if the vector (x, y, z) is
tangent to a hyperbolic line l on H, then its length is positive. Hence, the length of any arc
AB of l with respect to the Minkowski metric is well-defined. It is not hard to show that
this length is equal to the area of the plane figure OAB (in the plane passing through
l and the origin O) bounded by the arc AB and the segments OA and OB (combine
homework problem 5.1 and calculation in problem 6 from problem solving session 5).
References
[Eu] Euclid, Elements, any edition
[Hi] David Hilbert, Foundations of Geometry, any edition
[CFKP] J.W.Cannon, W. J. Floyd, R. Kenyon, W. R. Parry, Hyperbolic Geometry, chapter from
Flavors of Geometry, ed. Silvio Levy, MSRI Publications 31, 1997
[Hal] G.B. Halsted, The Betweenness Assumption, Amer. Math. Monthly 9 (1902), 98–101
[Ha05] Robin Hartshorne, Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics,
Springer Science & Business Media, 2005
[Mo] E. H. Moore, On the Projective Axioms of Geometry, Transactions of the AMS, 3 (1902), no. 1,
142–158
[Ro] A. Rosenthal, Vereinfachungen des Hilbertschen Systems der Kongruenzaxiome, Math. Ann. 71
(1912), 257–274
[So] A.B. Sossinsky, Geometries, IUM, 2008
[VS] E. B. Vinberg, O. V. Schwarzman, Discrete groups of motions of spaces of constant curvature,
29 Encyclopaedia of Mathematical Sciences, Springer, 139-248