Notes — Chapter 16: Non-Euclidean Geometries

To the Student. Geometry is the science of space and shapes. We know that our surroundings are not uniform. Up and down are different. To our right we see one thing; to our left we see another. But these differences are due to the particulars of our immediate situation. The earth below us pulls us down in a consistent way, even though as the earth rotates our “down” is a direction that whirls around against the sphere of the fixed stars.

We believe that aside from such differences due to our immediate surroundings, the space that we live in is uniform — in all directions and throughout the universe. The geometry that we believe underlies what we see should be completely symmetric, giving no special role to any place or orientation. If we are concerned solely with geometry, a triangle behaves the same way wherever it is constructed and however its sides are oriented.

The method of superposition by which a figure is directly compared with another and the congruence postulates both express our confidence in the uniformity of space, in the idea that geometric properties of a configuration are independent of its position.

It seems equally clear that the geometric properties do not depend in any essential way on the size of a configuration. As a practical observation, we know we can make scale models and that the geometric properties of things seem unaffected by changes in size. This fact can be proven once we have the Parallel Postulate. But because it seems so obviously a necessary consequence of the symmetry built into our ideas, and our world,

there was a long and ultimately unsuccessful effort to prove the Parallel Postulate from the congruence postulates and other simple and irrefutable assumptions of Euclidean geometry.

Even at the beginnings of geometry, people realized that the geometry of triangles on the celestial sphere and on the surface of the spherical earth was different from the geometry of triangles in a plane. But although we now see spherical geometry as one form of non-Euclidean geometry, it was not thought of that way. Spherical geometry fitted into the Euclidean picture because the two-dimensional surfaces we are thinking of lie in a Euclidean space of three dimensions.

Even before Hipparchus (about 150 B.C.), the earth was believed spherical by astronomers based on the observation of the circular shadow the earth cast on the moon during eclipses. It was also believed to be spherical because this explained why the sun at noon on the summer solstice would be lower in the sky the farther north you go. This evidence was convincing for those who examined it. But for others, it was “flat earth all the way.”

Geometry on a Sphere (Lesson 1) is the oldest non- Euclidean geometry, but it was not recognized as such. As a geometry, it has flaws that can only be eliminated by identifying pairs of antipodal points. This dizzying operation gives us a space in which each pair of points determines precisely one line containing both points and each pair of lines determines exactly one point where both lines concur. We have all the usual congruence axioms plus AAA congruence, but there can be no pair of similar but noncongruent triangles. Lines have finite length, so we need to make an appropriate modification of our Ruler Postulate and, because lines curve back on themselves, we need to make adjustments to the order axioms. These adjustments are somewhat fussy, but their general nature is clear.

This perfected version of spherical geometry is not easy to grasp intuitively. (It is called elliptic geometry, or single elliptic geometry to distinguish it from sphere geometry.) Here we only deal with sphere geometry or, later, Riemannian geometry. This is because we wish to stick to things that are easily understood by thinking of triangles on a sphere (a regular sphere, without identifying antipodal points).

The best resource for those who want to understand the possibilities for the structure of space is The Shape of Space by Jeffrey R. Weeks, Marcell Dekker, New York 1985 and second edition 2001. Also see books and other materials Weeks has developed. (Go to the Key Curriculum Press website, for instance.) Weeks has written for general readers, but ones who want to understand and develop intuition about the geometry of curved spaces.

Weeks is applying his ideas not just to geometry, but to the problem of what geometry the universe might have,

given the data we have from astronomy. Go to the Web for this breaking story.

The big break for non-Euclidean geometry came with the exploration of hyperbolic or Lobachevskian models of.the plane. These models have infinitely long lines and our usual ruler and order postulates hold. They correspond to surfaces of constant negative curvature, as contrasted to spherical geometry, which is the geometry of surface with constant positive curvature.

Lesson 1. Geometry on the Sphere. A good source on the Web is John C. Polking’s Geometry on the Sphere, available at mathforum.org in 2003. I outlined a proof of Girard’s Theorem, which is one of Polking’s main results in Areas of Spherical Triangles which is in the supplemental notes for Chapter 11, above. The geometry of the sphere, as a surface in Euclidean threespace is easily understood. People have thought about it since they began to study the sky and stars. But the geometry of the sphere with its antipodal points identified is not intuitive and is a relatively modern creation. You can explore it using the disk model for elliptic geometry, which has been realized using Geometer’s Sketchpad™. I see references and samples on the Web. The tricky thing here is that points opposite each other on the boundary of the disk of the model are identified, so things are not as they seem. Go to the book The Shape of Space by Weeks for examples to help develop your intuition.

The uniform positive curvature of a sphere causes a spherical triangle’s spherical excess (defined as the angle sum minus a half rotation) to be proportionate to the triangle’s area. Additionally, it alters the way the circumference and area of a circle vary with the circle’s radius r. Consider a sphere of radius R. Measure the radius r of a circle on this sphere along the surface of the sphere so that the arc this radius gives has radian measure r/R, then the circumference is 27 sin(r/R) and its area on the surface of the sphere is 2πr²(1 —cos(r/R)). Johann Heinrich Lambert (1728-1727) in his researches on the parallel postulate noticed that formulas related to the sphere (positive curvature = R-?) go over to analogous formulas for hyperbolic space when the radius R is replaced by iR, where i is a square root of −1.

The circular functions with the variable in radians are expressed in terms of exponentials thus

pi - 2-10 gi? 4 ot

sin i ng = —————_ F and__ cos? = ——~—-. 5

Replacing @ by i@ we get the corresponding hyperbolic functions sinh (pronounced “sinch” ) and cosh,

e° 6 —e -0 err’ 6 -9

sinh Oma and cosh 6 = ——. 5

There is a wealth of material on hyperbolic trigonometry on the Web, including applets that show you the areas of hyperbolic triangles in the Poincaré disk model of the hyperbolic plane. See pages 116 and 117 of Geometry for this model.

Set I. Our Spherical Earth (1-4) and Basic Differences (5-11) familiarize students with sphere geometry and how it differs from plane geometry. We have not identified antipodal points here. Shortest Routes (12-15). In going from A to B on the surface of the earth, the shortest path will be the one that minimizes its curvature at all points along the way. Since there is no way you can avoid curving with the earth’s surface, you must accept that curvature, but avoid other curvature by following a great circle. This is what airplanes do, although they will deviate from the shortest path to take advantage of favorable winds. Beach Ball (16-24) is an exploration of angle sums and areas in sphere geometry.

Set II. Spherical Triangles (25-28) relates to AAA congruence. Seemingly Parallel (29 and 30) involves visual and critical thinking. Euclidean and Sphere Geometries (31-40) involves discovery, conjecture, and critical thinking.

Set III. Antipodal Points is a marvelous map.

Lesson 2. The Saccheri Quadrilateral. Andrew M. Gleason points out that Euclid’s parallel postulate, given here, has advantages over the Playfair postulate used in Geometry. Euclid tells you that you will observe something, the intersection of two lines. He also tells you where to look for this intersection. In contrast, the Playfair postulate is neither constructive nor explicit. Current practice favors Playfair’s version and it works, so that is what Jacobs used.

Saccheri developed the conclusions of assuming that the summit angles of an isosceles Saccheri quadrilateral are acute — in modern terms, the conclusions of of working in a Lobachevskian or hyperbolic plane. But the significance of his 1733 work, featured in the chapter opening image, was not understood for perhaps a hundred years, as Jacobs notes. See Morris Kline’s Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, 1972, page 879 et seq.

Set I. These exercises show how Saccheri developed the properties of this geometry using the careful reasoning we have used — proofs. This allowed Saccheri to reach consistent but counterintuitive conclusions. Set II brings in Nasir Eddin, mentioned in A Colorful Contributor to Trigonometry in the supplemental material for Chapter 11 in these notes. Set III, Offsets, is a stunning and inevitable consequence of trying to live by the square on a spherical planet. Quadrangle maps that show Universal Transverse Mercator girds will also illustrate this phenomenon, and these are available for most areas. GPS units give your location with respect

to the local UTM grid. Mapping and working out coordinates are geometric problems involving technical details once you get beyond the basics. There are good projects to be done in this area.

Lesson 3. The Geometries of Lobachevsky and Riemann. Bernard Riemann was assigned the topic “the foundations of geometry” by Gauss for his Habilitationsvortrag as Privatdozent at Gottingen. Riemann delivered this address in 1854 and in it he outlined what has become our modern approach to geometry. Building on Gauss’s use of calculus to understand the geometry of surfaces (differential geometry), Riemann outlined a theory allowing n-dimensional geometric spaces whose curvature was defined by a local idea of distance (a Riemannian metric) whose nature could vary from one part of the space to another. Riemann’s ideas allowed his spaces (now called Riemannian manifolds) to have positive curvature in some regions and negative curvature in others. Using the same side-lengths we might notice spherical excess for our triangle in one region, but if we assembled the same length sides in another region, we might find a spherical deficit. See the note at the end of this lesson on Curved Space and Spacetime.

Riemann turned to the study of spaces of uniform positive curvature. He found that these had the geometry of the sphere, so spherical geometries are called Riemannian, and also elliptic, following a later classification set up by Felix Klein. Morris Kline’s oft-cited History of Mathematics from Ancient to Modern Times tells the story.

We can classify these geometries in terms of their uniform curvatures as follows: Lobachevskian or Elliptic —negative curvature (no two lines are parallel), Euclidean — zero curvature (unique parallels), Riemannian or Elliptic — positive curvature (several parallels to a given line through the same point off that line).

Set I. Theorems 1 and 2 of Lobachevskian geometry (1-18). Here we can use our methods of proof, except we replace the Parallel Postulate by the Lobachevskian Postulate.

Set II. Lobachevsky and His Geometry (19-23) is a good review of the ideas and easy. Lobachevskian Quadrilateral (24-30), another proof, as above. Riemannian Quadrilateral (31-34), only requires a general and visual understanding of the spherical geometry model. Good review and practice. This also holds for Riemannian Triangle (35 and 36). Triangular Pyramid (37-40) and Open Box (41-46). Both involve informal reasoning about Lobachevskian geometry.

Set III. Pseudosphere. This introduces one model of the Lobachevskian plane. Note that this model is not smooth and it runs off to infinity along an axis. It is impossible to find a model for the Lobachevskian plane that is a smooth surface in space and with the usual notion of distance along curves in the surface. But see

Lobachevsky Meets Escher on page 716 of geometry for the Poincaré disk model.

▶ Curved Space and Spacetime. This ties into the material of the next lesson. Our best understanding in 2003 is that the paths of light through space are as close as we can come to straight lines. Light paths through space do not behave like straight lines in Euclid’s geometry. The strongest evidence is that we can see the same distant galaxy by looking in two (or even four) slightly different directions. For examples of this you can search the Web for gravitational lensing or Einstein cross. In 2003, Goddard Space Flight Center had some good material (gsfc.nasa.gov), including the picture of the day for 2000 April 23. There was also good material at astron.berkeley.edu, and an excellent photograph of the Einstein cross of gravitational lens Q2237 + 030 at www.astr.ua.edu/keel/agn/qso2237. Being able to see the same thing by looking in different directions violates the postulate that two points determine a line — if we take light paths to represent lines. This departure from Euclid is explained by Einstein’s Theory of General Relativity, which states that mass and energy determine the curvature of spacetime. Even the less mind-bending idea of spacetime is a departure from Euclidean ideas.

According to Einstein’s special relativity, we should think of time as an essential coordinate in describing events. The Interval, between events is defined by the quadratic expression (x) Interval? = (Az)* + (Ay)? + (Az)? — (cAt)?,

where Az is the observed difference of the x-coordinates for the two events, and so on for y and z, while At is the difference in the observed times of the two events and c is the speed of light. Observers whose relative velocity is unchanged between the occurrence of the two events will compute the same Interval relating these two events.

This is true even though they may observe differing time intervals or space intervals between these events. If the first event is the flash of a laser on earth aimed at the moon and the second event is the arrival at the moon of that flash, then Interval? = 0, because the space component is exactly matched by the negative time term, —(cAt)”. In Einstein’s special relativity, light travels along null-paths between events whose interval of separation in space and time will always give us Interval = 0.

The spacetime geometry of special relativity is different from Euclidean geometry. For example, we have seen that two different events can be separated by an Interval of length 0. We have a difference in (x), so the Interval? can be positive, negative, or zero. When it is positive, the Interval is called spacelike and its length is VInterval” > 0. When (*) gives us a negative value, we call the interval, timelike, and take the length to be —Interval? > 0. If two events are separated by a timelike interval, then one can influence the other by

signals traveling at or below the speed of light. If the interval between two events is spacelike, then neither can influence the other. If Interval? = 0, then a nullinterval separates the two events. One can influence the other only by a signal that travels at the speed of - light. All of this is very different from the Pythagorean distances of Euclidean geometry.

The counterintuitive surprises of relativity arise from the very different natures the geometry underlying special relativity and Euclidean geometry. To feel at ease with this geometry you must become good at and trust the calculations or retrain your geometric intuition. Both take considerable doing and would distract from mastery of Euclidean geometry, especially at the introductory level.

Good and relatively elementary treatments of special relativity are available. A proven resource is Spacetime Physics by Edwin F. Taylor and John Archibald Wheeler, W. H. Freeman and Company, First Edition, 1963, Second Edition, New York, 2000. Both editions are highly recommended, and Taylor has software packages that let you see how things behave in special relativity; (Spacetime and Rellab are two of these packages.

In general relativity, the four dimensional spacetime picture of special relativity holds locally, but the Riemannian geometry of spacetime is determined by the effects mass and energy exert on space itself. As John Archibald Wheeler puts it on the first page of A Journey into Gravity and Spacetime, SCIENTIFIC AMERICAN Library, HPHLP, New York 1990,

“Spacetime grips mass, telling it how to move; and mass grips spacetime, telling it how to curve.”

In general relativity, particles or planets move along the straightest paths available in a spacetime manifold whose geometry is determined by Einstein’s field equations. Such paths on curved surfaces or in curved space are geodesics. The geodesics on a sphere are great circles. In the Euclidean plane, they are ordinary straight lines. On the surface of a right circular cylinder, the geodesics are paths that make a constant angle with the vertical generating lines.

The density of massenergy (the words are mashed together as in spacetime) that determines the curvature of spacetime. The ideas are subtle and the formalism is daunting. But Wheeler’s Journey book will give the essential ideas to a persistent general reader. Julian Schwinger wrote Einstein’s Legacy, for the same ScIENTIFIC AMERICAN Library series, published in 1986. Schwinger kept the outline of how Einstein equations were derived, despite resistance by his editor. See his Preface, where he hopes that this material may tempt some readers to work through the mathematics itself, and see his pages 184-186 for the material.

The benefit of the relativistic point of view is that it reveals many things that were unimagined: for instance, the equivalence of mass and energy via the equation E = mc’. It allows us to understand the orbital motion of the planets as resulting from these bodies moving along geodesics in our spacetime manifold at a constant four-dimensional velocity — a velocity that has a time component as well as its three space components.

Free float describes the trajectory of particles moving under the influence of gravity and inertia alone. These trajectories are geodesics in our curved spacetime. Each moving particle is considered to carry with it an ideal clock that records what is called proper time. Even though these clocks keep perfect time, two actors, each carrying a clock from event A to event B, may show different elapsed times for the journey. The elapsed time will be longer for the path that is closest to a geodesic from A to B. This gives us the principle of maximal ageing — the longest possible time interval measured in going from A to B is the time measured along a geodesic path. In their book, Exploring Black Holes: An Introduction to General Relativity, Benjamin Cummings, 2000, Edwin F. Taylor and John Archibald Wheeler use the principle of maximal ageing and calculus to work out relativistic effects near black holes. This book begins where their Spacetime Physics leaves off.

To work with relativistic ideas one must master a certain amount of mathematical machinery. Because there are practical applications of finding geodesics that include tent design and manufacturing of clothing, commercial software is available that automatically solves some of these geometrical problems. In 2003 the Elements Research website had material on geodesic computation (elesoft.com/geodcomp). The site had nice examples, although it also mentions the daunting machinery of Christoffle symbols.

The challenge in working with curved spaces is not in grasping the idea of curvature of space. That idea is clear from the analogy with curved surfaces. Rather, the challenge lies in having machinery that allows you to define specific curved spaces and work out their properties in detail. Software now allows us to explore and display properties of curved and higher-dimensional spaces.

Lesson 4. The Triangle Angle Sum Theorem Revisited. In supplemental notes at the end of my comments on Chapter 11’s lessons, I cover Similarity and the Angle Sum Theorem, Similarity and Non-Euclidean Geometry, and related topics. From the note above on Curved Space and Spacetime, you can see that the geometry of our universe seems to be curved, with a curvature that varies. The way we understand these more intricate models is by smoothly patching together pieces of spaces of the sort we have already studied. We have seen the basic structures, and we understand the idea

of curvature of space and how it makes lines (geodesics) converge or diverge depending on whether the local curvature is positive or negative.

Our space is non-Euclidean, but it is by our study Euclidean geometry that we came to understand this fact. The models of non-Euclidean geometries are constructed within Euclidean geometry, so our understanding of the non-Euclidean is dependent on our understanding of the Euclidean. This chapter on non- Euclidean geometry is both a fitting climax of this book and a hopeful beginning for further geometric adventures.

Set I. These exercises are proofs related to angle sums of triangles.

Set II exercises are more informal that Set I and offer students a wider range of opportunities to look, see, reason, and draw conclusions. Even if you usually assign exercises from Set I, you should consider these for assignment.

Set III. Astronomical Triangles. You can never prove that the curvature of space is exactly 0 by measurements because of their finite precision.

Summary and Review. This gives a clear overview of the material of the chapter and introduces the beautiful Poincaré disk model for the Lobachevskian plane. If your students are using the Web to explore and research things, they will be able to find good models of non-Euclidean geometry there. Software packages such as Geometer’s Sketchpad™ and Cabri Geometry can help, too.