Geometry from Euclid to Knots

Preface to the Dover Edition

It gives me particular pleasure to have this book republished by Dover Publications, Inc. After all, they do list Euclid's The Elements, the most influential mathematical book of all time, in their catalog, and it is hard not to derive some satisfaction from this association. Especially so, as I have had so many occasions to actually consult this source in the preparation of various lectures. The mathematical world owes Dover a debt of gratitude for making this text so readily available.

I must confess that when I opened Euclid’s book for the first timed during my undergraduate years, I felt rather disappointed. The questionable definitions, the lack of symbolic notation, and, most damning, the seemingly nitpicking proofs of obvious facts, all combined to dull my interest. It was not until I was assigned to teach a course in modern geometries that I returned to Euclid’s opus. This time I knew more about non-Euclidean geometry, as well as mathematics in general, and so could appreciate it better. It is for this reason that the first chapter of this book is devoted to the concrete and plausible description of several alternative, non-Euclidean, geometries. Familiarity with other geometries shows the students that the obvious need not be true. This demonstrates to them the need for logical proofs. That this can be accomplished by introducing the students to the geometry of the upper halfplane, a mathematical tool that played an essential role in the recent resolutions of Fermat’s and Poincar’s conjectures, as well as in the classification of the finite simple groups, comes as a bonus.

The readers who know of my junior/senior level textbooks in algebra and real analysis might wonder why in writing this book I chose to depart from the historical, or evolutionary, approach to pedagogy. The reason is that since I myself found it impossible to conceive of a hyperbolic geometry without a model, I did not deem it realistic to expect my students to do so. On the other hand, perhaps it could be said that my approach in this book is also historical in the sense that it acknowledges the enormous difficulties the mathematical community experienced in trying to resolve the question of the status of Euclid's fifth postulate. It took two thousand years to do so, and not for lack of trying.

Saul Stahl        

December 2009

Preface

Organization

The main purpose of this book is to provide prospective high school mathematics teachers with the geometric background they need. Its core, consisting of Chapters 2 to 5, is therefore devoted to a fairly formal (that is, axiomatic) development of Euclidean geometry. Chapters 6, 7, and 8 complement this with an exposition of transformation geometry. The first chapter, which introduces teachers-to-be to non-Euclidean geometries, provides them with a perspective meant to enhance their appreciation of axiomatic systems. The much more informal Chapters 9 through 12 are meant to give students a taste of some more recent geometric discoveries.

The development of synthetic Euclidean geometry begins by following Euclid’s Elements very closely. This has the advantage of convincing students that they are learning the real thing. It also happens to be an excellent organization of the subject matter. Witness the well-known fact that the first 28 propositions are all neutral. These subtleties might be lost on the typical high school student, but familiarity with Euclid’s classic text must surely add to the teacher’s confidence and effectiveness in the classroom. I am also in complete agreement with the sentiments Todhunter displayed in the previous excerpt: No other system of teaching geometry is better than Euclid’s, provided, of course that his list of propositions is supplemented with a sufficient number of exercises. Occasionally, though, because some things have changed over two millennia, or else because of errors in the Elements, it was found advantageous either to expound both the modern and ancient versions in parallel or else to part ways with Euclid.

In order to convince prospective teachers of the need to prove obvious propositions, the axiomatic development of Euclidean geometry is preceded by the informal description of both spherical and hyperbolic geometry. The trigonometric formulas of these geometries are included in order to lend numerical substance to these alternate and unfamiliar systems. The part of the course dedicated to synthetic geometry covers the standard material about triangles, parallelism, circles, ratios, and similarity; it concludes with the classic theorems that lead to projective geometry. These lead naturally to a discussion of ideal points and lines in the extended plane. Experience indicates that the nonoptional portions of the first five chapters can be completed in about three quarters of a one semester course. During that time the typical weekly homework assignment called for about a dozen proofs.

Chapters 6 and 7 are concerned mostly with transformation geometry and symmetry. The planar rigid motions are completely and rigorously classified. This is followed by an informal discussion of frieze patterns and wallpaper designs. Inversions are developed formally and their utility for both Euclidean and hyperbolic geometry is explained.

The exposition in Chapters 8 through 12 is informal in the sense that few proofs are either offered or required. Their purpose is to acquaint students with some of the geometry that was developed in the last two centuries. Care has been taken to supply a great number of calculational exercises that will provide students with hands-on experience in these advanced topics.

The purpose of Chapter 8 is threefold. First there is an exposition of some interesting facts, such as Euler’s equation and the Platonic and Archimedian solids. This is followed by a representation of the rotational symmetries of the regular solids by means of permutations, a discussion of their symmetry groups, and a visual definition of isomorphism. Both of these discussions aim to develop the prospective teacher’s ability to visualize three dimensional phenomena. Finally comes the tale of Monstrous Moonshine.

Chapter 9 consists of a short introduction to the notions of homeomorphism and isotopy. Chapter 10 acquaints students with some standard topics of graph theory: traversability, colorability, and planarity. The topology of surfaces, both of the closed and the bordered varieties, is the subject of Chapter 11. Algorithms are described for identification of the topological type of any bordered surface. Two knot theoretic invariants are described in Chapter 12, including the recently discovered Jones polynomial.

Exercises

In Chapters 2 to 5 exercises are listed following every two or three propositions. This helps the professor select appropriate homework assignments and eliminates some of the guesswork for students. The great majority of these exercises call for straightforward applications of the immediately preceding propositions. On the other hand, the chapter review exercises provide problems for which the determination of the applicable propositions does require thought. In the remaining and less formal chapters the exercises appear at the end of each section. Each chapter concludes with a list of review problems. Solutions and/or hints to selected exercises are provided at the end of the book.

The exercises that are interspersed with the propositions are of four types. There are relational and constructive propositions in whose answers the students should adhere to the same format that is used in the numbered propositions. The third type of exercises has to do with the alternate spherical, hyperbolic, and taxicab geometries; in these the appropriate response usually consists of one or two English sentences. The fourth, and last, type of exercises calls for the use of some computer program, and these are marked with a (C).

In the other chapters, namely Chapter 1 and Chapters 6 through 12, exercises are listed at the end of each section. The emphasis in these is on the algorithmic aspect of geometry. They mostly require the straightforward, albeit nontrivial, application of the methods expounded in the text.

Notation and Conventions

Chapters 2 to 5 of this text present most of the content of Book I and selected topics from Books, II, III, IV, and VI of The Elements. In addition to the conventional labeling of propositions by chapter, section, and number these propositions are also identified by a parenthesized roman numeral, number that pinpoints their appearance in Euclid's book. For example, the Theorem of Pythagoras is listed as Proposition 3.3.2(I.47). Exercises are identified in a similar manner: Exercise 5.3B.6 is the sixth exercise in group B of the third section in Chapter 5. The symbol (C) is used to distinguish exercises that call for the use of computer applications.

The justifications for the various steps of the construction and proof are stated, in abbreviated form, in brackets. The abbreviations used are DFN for definition, PT for postulate, CN for common notion, and PN for proposition. The symbol ∴ is used as an abbreviation for the word therefore. Optional sections or propositions are labeled with an asterisk. Difficult exercises are also designated by an asterisk. Propositions whose proof lies outside the scope of the text are followed by the symbol .

Acknowledgments

The author wishes to acknowledge the many valuable suggestions and corrections provided by Mark Hunacek as well as the technical help of his colleague Pawel Szeptycki. Jack Porter taught from an early draft and provided some corrections and useful suggestions. Dan Archdeacon and Marisa Debowsky helped improve portions of the manuscript. The final form of the book owes much to the suggestions of Prentice Hall's Acquisitions Editor George Lobell, Production Editor Bayani M. DeLeon, Arthur T. White, Larry W. Cusick, John Golden, and other reviewers who chose to remain anonymous. Larisa Martin converted my manuscript to LATEX. I owe a debt of gratitude to them all.

Saul Stahl

[email protected]

Credits

Illustrations

All the figures were produced by Adobe Illustrator®.

The cartoons on pp. xviii, 85, 267, and 331 are reprinted with the permission of their creator Sidney Harris.

The cartoon on p. 175 is reprinted with the gracious permission of its creator Susan H. Stahl.

Figure 6.26 is reprinted, by permission, from Owen Jones, The Complete "Chinese Ornament," p. 21. Copyright © 1990 by Dover Publications, Inc.

Figure 6.36 is reprinted, by permission, from W. and G. Audsley, Designs and Patterns from Historic Documents, Plates 19 and 20. Copyright © 1968 by Dover Publications, Inc.

The figures in Exercises 1–34 of Section 6.7 were generated with software written at the Geometry Center, University of Minnesota.

Figures 8.1, 8.4–8.6, 8.12–8.23 were imported from Mathematica®.

Figure 8.2 is reprinted from Ernst H. P. A. Haeckel, Report on the Scientific Results of the Voyage of H. M. S. Challenger, Vol. XVIII, Part 3, Pl. 117. H. M. S. O., 1887.

Figure 8.3 is reprinted, by permission, from David Wells, The Penguin Dictionary of Curious and Interesting Geometry, pp. 6–7. Copyright © 1991 by David Wells.

Quotations

The statements of Euclid's definitions, postulates, common notions, as well as many of the propositions in Chapters 2, 3, and 4 are reprinted, by permission, from Euclid, The Thirteen Books of the Elements (Sir Thomas L. Heath, translator), 2nd edition. Copyright © 1956 by Dover Publications, Inc.

The recognition chart for wallpaper symmetry groups is reprinted at the end of Chapter 6, by permission, from Doris Schattschneider's article The Plane Symmetry Groups: Their Recognition and Notation, Amer. Math. Monthly, 85(1978), p. 443. Copyright © 1978 by The American Mathematical Monthly.

Appendix C is modified from G. D. Birkhoff and R. Beatley, Basic Geometry, 3rd edition. Copyright © 1959 by Chelsea Publishing Company.

Appendix D is reprinted from UCSMP Geometry by Arthur Coxford, Zalman Usiskin, and Daniel Hirschorn. © 1991 by Scott, Foresman and Company. Published by Prentice Hall, Inc. Used by permission of Pearson Education, Inc.

Appendix E is modified, by permission, from David Hilbert, Foundations of Geometry (L. Unger, translator). Copyright © 1971 by Open Court Publishing Co.

Chapter 1

Other Geometries: A Computational Introduction

In order to provide a better perspective on Euclidean geometry, three alternative geometries are described. These are the geometry of the surface of the sphere, hyperbolic geometry, and taxicab geometry.

1.1Spherical Geometry

Due to its relationship with geography and astronomy, spherical geometry was studied extensively by the Greeks as early as 300 B.C. Menelaus (ca. 100) wrote the book Spherica on spherical trigonometry, which was greatly extended by Ptolemy (100–178) in his Almagest. Many later mathematicians, including Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855), made substantial contributions to this topic. Here it is proposed only to compare and contrast this geometry with that of the plane. Because the time to develop spherical geometry in the same manner as will be done with Euclidean geometry is not available, this discussion is necessarily informal and frequent appeals will be made to the readers' visual intuition.

Strictly speaking there are no straight lines on the surface of a sphere. Instead it is both customary and useful to focus on curves that share the shortest distance property with the Euclidean straight lines. The following thought experiment will prove instructive for this purpose. Imagine that two pins have been stuck in a smooth sphere in points that are not diametrically opposite and that a (frictionless) rubber band is held by the pins in a stretched state. Rotate this sphere until one of the two pins is directly above the other right in front of your mind’s eye. It is then hard to avoid the conclusion that the rubber band will be stretched out along the sphere in the plane formed by the two pins and the eye—the plane of the book’s page in Figure 1.1. The inherent symmetry of the sphere dictates that this plane should cut the sphere into two identical hemispheres; in other words, that this plane should pass through the center of the sphere. It is also clear that the tension of the stretched rubber band forces it to describe the shortest curve on the surface of the sphere that connects the two pins. The following may therefore be concluded.

Figure 1.1. A geodesic on the sphere

Proposition 1.1.1 (Spherical geodesics) If A and B are two points on a sphere that are not diametrically opposite, then the shortest curve joining A and B on the sphere is an arc of the circle that constitutes the intersection of the sphere with a plane that contains the sphere's center.

Such circles are called great circles and these arcs are called great arcs or geodesic segments. They are the spherical analogs of the Euclidean line segments.

Diametrically opposite points on the sphere present a dilemma. A stretched rubber band joining them will again lie along a great circle, but this circle is no longer uniquely determined since these points can clearly be joined by an infinite number of great semicircles. For example, assuming for the sake of argument that the earth is an exact sphere, each meridian is a great semicircle that joins the North and South Poles. Hence, the aforementioned analogy between the geodesic segments on the sphere and Euclidean line segments is not perfect. It is necessary either to exclude such meridians from the class of geodesic segments or else to accept that some points can be joined by many such segments. The first alternative is the one chosen in this text. Thus, by definition, the endpoints of geodesic segments on the sphere are never diametrically opposite.

Figure 1.2. The lune α

Next, the spherical analog of the angle is defined. Any two great semicircles that join two diametrically opposite points A and B, but are not contained in the same great circle, divide the sphere into two portions each of which is called a lune, or a spherical angle (Fig. 1.2). The measure of the spherical angle is defined to be the measure of the angle between their tangent lines at A (or at B). Alternately, this equals the measure of the angle formed by the radii from the center of the sphere to the midpoints of the bounding great semicircles. For example, each meridian forms a 90° angle with the equator at their point of intersection.

In the Euclidean plane the relationships between lengths of straight line segments and measures of angles are given by well-known trigonometric identities. Some fundamental theorems of spherical trigonometry are now stated without proof.

Any three points A, B, C on the sphere, no two of which are diametrically opposite, constitute the vertices of a spherical triangle denoted by ABC. The three sides of this triangle are the geodesic segments that join each pair. The sides opposite the vertices A, B, C (and their lengths) are denoted a, b, c respectively. The interior angle α at the vertex A is the lune between b and c. The interior angles β and γ at B and C are defined in a similar manner. The term spherical geometry refers to the geometry of the sphere of radius 1.

Proposition 1.1.2 (Spherical trigonometry) On a sphere of radius R = 1, let ABC be a spherical triangle with sides a, b, c and interior angles α, β, γ. Then

(i)  

(i′)   cos a = cos b cos c + cos α sin b sin c

(ii)  

(ii′)   cos α = cos a sin β sin γ − cos β cos γ

(iii)  

These are known as the first spherical law of cosines, the second spherical law of cosines, and the spherical law of sines. It should be noted that i and i′ are really the same equation as are ii and ii′, although, as will be demonstrated by the following examples, their uses are different. The solution of a triangle consists of the lengths of its sides and the measures of its interior angles. For any side a, radian units should be used in the evaluation of sin a and cos a.

Example 1.1.3 Solve the spherical triangle with sides a = 1, b = 2, and c = π/2.

It follows from the first spherical law of cosines that

so that

The angles β and γ are similarly shown to have measures 119.64° and 72.91°.

Example 1.1.4 On a sphere of radius 4000 miles, solve the triangle in which an interior angle of 50° lies between sides of lengths 7000 miles and 9000 miles, respectively.

Taking the radius as the unit, it follows that we may set

Figure 1.3. A thin spherical triangle

Hence, from the first law of cosines,

Now that all three sides of the triangle are known, the method of the previous example yields

Note that in both of the preceding examples the sum of the angles of the spherical triangle exceeds 180°. That is in fact true for all spherical triangles.

Proposition 1.1.5 The sum of the angles of every spherical triangle lies strictly between 180° and 540°.

A spherical triangle the sum of whose angles is close to 180° is formed by the equator together with two close meridians. Thus, the sum of the angles of the spherical ABC of Figure 1.3 is 90° + 90° + α = 180° + α. A spherical triangle A′B′C′ with angle sum near 540° is described in Figure 1.4, where A, B, C are points that are equally spaced along a great circle. As A′, B′, C′ approach A, B, C, respectively, the angles they form are flattened out and come arbitrarily close to 180° each. For example, since the spherical distance between any two of the points A, B, C is 2π/3, it might be assumed that the spherical distance between any two of the points A′, B′, C′ is a = 2π/3−0.00001, in which case each of the angles of A′B′C′ is

Figure 1.4. A nearly maximal spherical triangle

and their sum is 538.56°.

Since, by definition, each of the interior angles of the spherical triangle is less than 180°, it follows that the sum of these angles can never equal 540°. Similarly, as will be shown momentarily, the sum of these angle cannot equal the lower bound of 180° either.

The area of the spherical triangle is also of interest. An elegant proof of this formula is offered in Section 3.2.

Proposition 1.1.6 If a triangle on a sphere of radius R has angles with radian measures α, β, γ then it has area (α + β + γ − π)R².

For example, the spherical triangle formed by the equator, the Greenwich meridian and the 90° East meridian has all of its angles equal to π/2 and hence its area is

This answer is consistent with the fact that the said triangle constitutes one fourth of a hemisphere. Since the surface area of the sphere is 4πR², this triangle has area

which agrees with the previous calculation.

Figure 1.5. A spherical triangle

The quantity α + β + γ − π is called the excess of the spherical ABC. The preceding theorem in effect states that the area of a spherical triangle is proportional to its excess. This assertion is supported by the triangle of Figure 1.5, which has excess π/2 + π/2 + α − π = α and whose area is clearly proportional to α as long as A and B vary along the equator and C remains at the North Pole.

This area formula can be used to close a gap in the preceding discussion. Since every triangle has positive area, it follows that the sum of the angles of a spherical triangle never equals π or 180°, although, as was seen previously, it can come arbitrarily close to this lower bound.

Exercises 1.1

1.   Let ABC be a spherical triangle with a right angle at C. Use the formulas of spherical trigonometry to prove the following:

(a)   sin a = sin α sin c

(b)   tan a = tan α sin b

(c)   tan a = cos β tan c

(d)   cos c = cos a cos b

(e)   cos α = sin β cos a

(f)   sin b = sin β sin c

(g)   tan b = tan β sin a

(h)   tan b = cos α tan c

(i)   cos c = cot α cot β

(j)   cos β = sin α cos b

2.   On a sphere of radius R, solve the spherical triangle with angles:

(a)   60°, 70°, 80°

(b)   70°, 70°, 70°

(c)   120°, 150°, 170°

(d)   θ, θ, θ where 60° < θ < 120°.

3.   On a sphere of radius R, solve the spherical triangle with sides:

(a)   R, R, R

(b)   R, 1.5R, 2R

(c)   πR/2, πR/2, πR/2

(d)   d, d, d, where 0 < d < 2πR

(e)   .2R, .3R, .4R

(f)   .02R, .03R, .04R

(g)   2R, 3R, 4R

4.   On a sphere of radius R, solve the spherical triangle with:

(a)   a = .5R, β = 60°, γ = 80°

(b)   b = R, α = 40°, γ = 100°

(c)   a = 2R, β = γ = 10°

(d)   a = 2R, β = γ = 170°

5.   On a sphere of radius R, solve the spherical triangle with:

(a)   a = 2R, b = R, γ = 100°

(b)   b = .5R, c = 1.2R, α = 100°

(c)   a = 2R, b = R, γ = 120°

(d)   b = .5R, c = 1.2R, α = 120°

6.   On a sphere of radius 75 cm, solve the spherical triangle with:

(a)   a = 100 cm, b = 125 cm, c = 140 cm

(b)   α = 100°, β = 125°, γ = 140°

(c)   α = 100°, b = 125 cm, c = 125 cm

(d)   a = 100 cm, β = 125°, γ = 125°

7.   Evaluate the limits of the angles of the spherical triangles below both as x → 0 and as x → π.

(a)   a = b = c = x

(b)   a = b = x, c = 2x

Figure 1.6. The shrinkage that defines the hyperbolic plane

(c)   a = x, b = c = 2x

8.   Which of the following congruence theorems hold for spherical triangles? Justify your answer.

(a)   SSS

(b)   SAS

(c)   ASA

(d)   SAA

(e)   AAA

1.2Hyperbolic Geometry

Imagine a two-dimensional universe, with a superimposed Cartesian coordinate system, in which the x-axis is infinitely cold. Imagine further that as the objects of this universe approach the x-axis, the drop in temperature causes them to contract (see Fig. 1.6). Thus, the inhabitants of this fictitious land will find that it takes them less time to walk along a horizontal line from A(0, 1) to B(1, 1) (Fig. 1.7) than it takes to walk along a horizontal line from C(0, .5) to D(1, .5). Since their rulers contract just as much as they do, this observation will not seem at all paradoxical to them. If it is assumed that the contraction is such that the outside observer sees the length of any object as being proportional to its distance from the x-axis, then the inhabitants will find that walking from C(0, .5) to D(1, .5) takes twice as long as walking from A(0, 1) to B(1, 1) and one-fifth of the time of walking from E(0, .1) to F(1, .1). To differentiate between the Euclidean length of such a segment and its length as experienced by these fictitious beings, it is customary to refer to the latter as the hyperbolic length of the segment. Accordingly, the hyperbolic lengths of the segments AB, CD, and EF of Figure 1.7 are 1, 2, and 10 respectively. It is customary to restrict this geometry to the upper half-plane [i.e., those points (x, y) for which y > 0]. In this half-plane, the hyperbolic length of a horizontal line segment at distance y from the x-axis is given by the formula

Figure 1.7. Paths of unequal hyperbolic lengths

Other curves also have a hyperbolic length and a method for computing it is given in Exercise 16.

Not surprisingly, perhaps, the Euclidean straight line segment joining two points does not constitute the curve of shortest hyperbolic length between them. When setting out from A(0, 1) to B(1, 1) the inhabitants of this strange land may find that if they bear a little to the north their journey will be somewhat shorter because, unbeknownst to them, their legs are longer on this route (Fig. 1.8). However, if they stray too far north the length of the detour will offset any advantages gained by the elongation of their stride and they will find the length of the tour to be excessive. They are therefore faced with a trade-off problem. Some deviation to the north will shorten the duration of the trip from A to B, but too much will extend it. Which path, then, is it that makes the trip as short as possible?

Figure 1.8. Which path has shorter hyperbolic length?

The answer to this question is surprisingly easy to describe, though not to justify. The path of shortest hyperbolic length that connects A(0, 1) to B(1, 1) is the arc of the circle that is centered at (.5, 0) and contains A and B (Fig. 1.9). Its hyperbolic length turns out to be 0.962 . . . in contrast with the hyperbolic length 1 of the horizontal segment AB. More drastically, the arc of the semicircle centered at (50, 0) that joins the points A(0, 1) and X(100, 1) has hyperbolic length 9.21, a mere 9% of the hyperbolic length of the segment AX.

Given any two points, their hyperbolic distance is the minimum of the hyperbolic lengths of all the curves joining them. As was the case for spherical geometry, the geodesic segments of hyperbolic geometry are those curves that realize the hyperbolic distance between their endpoints. Loosely speaking, hyperbolic geometry, or the hyperbolic plane, is the geometry that underlies the upper half-plane when distances are measured in accordance with Equation (1). More precise definitions will be given in Sections 2.2 and 2.3. The upper half-plane is only one manifestation, or model, of hyperbolic geometry, and several others exist [see Greenberg, Stahl]. Nevertheless, for the purposes of this introductory exposition, readers may identify hyperbolic geometry with its representation as the distorted upper half-plane.

Figure 1.9. A hyperbolic geodesic

Figure 1.10. Six hyperbolic geodesics

Proposition 1.2.1 (Hyperbolic geodesics) The geodesic segments of the hyperbolic plane are arcs of circles centered on the x-axis and Euclidean line segments that are perpendicular to the x-axis.

The geodesics of the first variety are called bowed geodesics, whereas the vertical ones are the straight geodesics (see Fig. 1.10). This distinction is only meaningful to the outside observer. The inhabitants of this geometry perceive no difference between these two kinds of geodesics.

It so happens that as the inhabitants of the hyperbolic plane approach the x-axis they shrink at such a rate as to make the x-axis unattainable. Technically speaking, the hyperbolic lengths of all of the geodesic segments in Figure 1.10 diverge to infinity as the endpoints Q approach the x-axis. This claim will be given a quantitative justification at the end of this section.

A hyperbolic angle is the portion of the hyperbolic plane between two geodesic rays (Fig. 1.11). The measure of the angle between two geodesics is, by definition, the measure of the angle between the tangents to the geodesics at the vertex of the angle. Accordingly, two geodesics are said to form a hyperbolic right angle if and only if their tangents are perpendicular to each other as Euclidean straight lines (Fig. 1.12). Given any three points that do not lie on one hyperbolic geodesic, they constitute the vertices of a hyperbolic triangle formed by joining the vertices, two at a time, with hyperbolic geodesics (Fig. 1.13).

Figure 1.11. Three hyperbolic angles

The geometry of the hyperbolic plane has been studied extensively. Its trigonometric laws are surprisingly, not to say mysteriously, similar to those of spherical geometry.

Proposition 1.2.2 (Hyperbolic trigonometry) Let ABC be a hyperbolic triangle with sides a, b, c and interior angles α, β, γ. Then

Example 1.2.3 Solve the hyperbolic triangle with sides a = 1, b = 2, and c = π/2.

Figure 1.12. Hyperbolic right angles

Figure 1.13. Three hyperbolic triangles

It follows from Formula i) of hyperbolic trigonometry that

and so α = cos−1(.9461) ≈ 18.89°. Similarly β ≈ 87.67° and γ ≈ 39.34°.

Example 1.2.4 Solve the hyperbolic triangle with two sides of lengths 2, 3 respectively, if they are to include an angle of 30°.

Set α = 30°, b = 2, c = 3. It follows from