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THE SHAPE OF REALITY? How Straight Lines Can Bend in Non-Euclidean Geometries If there is anything that can bind the heavenly mind of man to this dreary exile of our earthly home and can reconcile us with our fate so that one can enjoy living-then it is verily the enjoyment of the mathematical sciences and astronomy. JOHANNES KEPLER M Does space curve? FLAT athematics can help us understand the cosmic, the unapproachable, and the mysterious. Nothing is more cosmic and mysterious than the entire universe. For thousands of years, people have pondered the fundamental question: What is the shape of our universe? In any attempt to understand the world around us, it is only natural to wonder about the geometry of our physical existence. Of course, our universe is incredibly vast, and our experience is limited by time and space. Thus our question is by no means easy. Does space bend or curve? What does it even mean for space to bend or curve? Since we do not see space bending or curving around us, our initial sense is that space is fiat. However, because we exist on such a microscopic scale compared to that of the entire universe, perhaps we don't sense the reality of the "big picture." Let's apply some of our techniques of analysis and try to discover the shape of space. A First Sketch of the "Big Picture" How do we start to understand the geometry of something so large that it seems beyond our capacity to comprehend it? Start by looking at something smaller. First, we look at the ground under and just around our feet. What do we see? Flat. 289 Geometric Gems : l J.ijc L trio x In dealing with life's complex issues, start with the simple and brtildfrom there. Thus, it seems reasonable to guess that our world around us is flat like a plane. This guess is not completely crazy, especially to those who live in Kansas. The world around us does tend to look pretty flat. This observation led people throughout history to study the flat plane and its rich geometry. However, it turns out that Earth is shaped like a ball. This nontrivial fact illustrates two important points. First, there is no pressing need for the Flat Earth Society; and second, what we observe locally may not accurately depict what is occurring on a larger scale. Before taking on the whole universe, perhaps we should consider the geometry of the next simplest realm: the sphere. A Next Sketch: The Geometry of a Sphere What is the shortest distance between two points? In a flat, unobstructed world, that shortest distance is always a straight line. But in New York City the shortest distance from 5th Ave. and 42nd St. to 8th Ave. and 38th St. is not a straight line. What path does the crow take? If the crow drove a taxi, he would have to follow the grid of streets to find the shortest path. So the shortest paths between two points-the "straight lines"--depend on the shape of the space where we live. We live on Earth. So what are the "straight lines" on Earth? Let's travel around and see. ~· "' New York City Since Earth is round, we do not live on a plane; yet many travelers live in a plane a good deal of the time. Pilots have a great attachment to fuel and hate to run out of it at 35,000 feet in the air. Thus, airplanes go from place to place along the shortest routes possible. Pilots, like crows, know that Earth is round and choose their routes accordingly. Let's see what those routes are. The best method for bringing this point home would be for you to now take a nonstop plane trip from Chicago, Illinois, to Rome, Italy. We '11 wait patiently, but you had better send us a postcard. 4.6 • The Shape of Reality? • Traversing the Globe Path shown = 5300 miles Rubber band's path = 4800 miles The arc path is part of a "great circle" Chicago and Rome are both at the latitude of nearly 40° north. You might think that the shortest route from Chicago to Rome would be to stick to the 40° latitude line the whole way. Let's measure how far that route would be. We will do this by measuring distances on a globe and using the scale to tell the mileage. If we take out a tape measure and place it along the 40° latitude line, we see that the distance is 5300 miles. Is there a shorter route? If we're flying the plane, we had better find out. A good, though messy, way to find the shortest route involves using a greased globe and a rubber band. We take the globe and grease it until it is so slippery that nothing, including the rubber band, will stick to it. We next put two pins in the globe, one at Chicago and one at Rome and then stretch a rubber band over the two pins. We first hold the rubber band down so it sits on the latitude line. Then we let go. Does it stay on the 40° latitude line? We don't think so. In fact, it slides up to a shorter route. Instead of staying on the latitude line, the rubber band finds a genuinely shorter route. Notice that the route heads north and goes over Labrador and Dublin, Ireland, before heading back south on its way to Rome. Is this route really shorter? Let's measure. We place our measuring tape along our new route and measure about 4800 miles. This new route saves about 500miles! Let's take a closer look at this rubber-band route. If we extend the route, we get a great circle that is as big as possible going around the whole globe-that is, a circle whose center is at the center of the globe and whose circumference is as long as the equator. Indeed, the shortest path between any two points on the globe is always on a great circle that contains them. The segments of great circles are the shortest distance between two points on the globe. Why? Why Great Circles Are the Way to Go Let's think about why the great circle segments are the shortest paths. If Earth were hollow, the shortest path from Chicago to Rome really would be a straight line inside Earth. So for our purposes, let's imagine a straight-line tunnel connecting Chicago and Rome burrowing right through the planet. The shortest route on the surface would deviate as little as possible from that straight, underground Chicago-Rome tunnel. Let's notice something about circles and lines. If we take two points and make a circle that contains them, then bigger circles are flatter and therefore stay closer to the straight line between the two points. So, among the paths that stay on circles, taking the biggest circle on the globe containing Chicago and Rome-that is, the great circle-will stay closest to the straight line. The latitude circle, being smaller, bends out more from the straight line and is therefore longer. "Straight lines," that is, the shortThe larger the circle, the "!latter" the segment. est paths on Earth, are great circle segments. Geometric Gems Latitude is longer Distances in a Different World We live on Earth, which is essentially a ball, but how about a bug on the wall? If our bug doesn't fly, its world is in the shape of the walls. So, when it sees its dining destination on some other wall, it has some serious calculations to perform. What is the shortest distance from here to dinner? Take a guess. A good guess would be the path shown in the figure below. Food Food I • Bug Bug J_Jfe .1tJJOJi .fhink about some ~imple cases. :o ~•--- 1Bug e· Food L-------Shortest path = Straight line ~_..L-------- An initial guess for the shortest path Shortest path? ~~ I I Is this guess the shortest path? Suppose dinner is on the same wall. This situation is easy. The bug simply walks in a straight line. The bug is off to a great start. How about if dinner is on an adjacent wall? It is pretty clear that the bug needs to go straight to the boundary edge and then straight on the next wall to its dinner. The question is, Where on that edge should it cross? Describe a method for locating the best crossing place. Notice that, if the walls were at some angle other than 90°, the distances from the bug to the crossing point and the crossing point to dinner do not change. So let's consider a different question. Suppose the bug is on an open door, and its dinner is on the wall. "Hinged door" • (.<.· Food I:IFood Shortest path I Where to cross the edge? Imagine that, as the bug is considering its shortest route, someone comes along and closes the door. Suddenly the question becomes much easier. Now the bug and its dinner are on the same wall, and the bug simply proceeds in a straight line. Now the bug is on a roll. Let's now return to the scenario where the bug's dinner is on the opposite wall. How will it figure out the shortest route? Having experienced the closing of the 4.6 • The Shape of Reality? • door, surely our bug cannot resist the idea of unfolding the walls. The problem is that there are many ways to unfold them. Which one should the bug choose? The straight lines from bug to dinner vary in length, depending on the route. Can you think of other ways of unfolding the walls, keeping the food and the bug in the same relative locations? -30ftI • 12ft • ~- tr· -3on- -30ft- - 12 It - 12 It I - 42 It 'J ,, Some straight paths are shorter than others. Shortest path -3ott-- 40 It What to do? One method of deciding would be to unfold the room in all possible ways and measure the straight-line distances. We've seen several different roomunfolding possibilities. Notice that different unfolding scenarios result in different placements of the food and the bug. Visualize the reassembly of the flattened rooms and verify that the relative positions of the bug and the food are always the same. Once we find the shortest flattened path, we can draw the straight line on the unfolded model and then refold it. In this case, the shortest path takes the bug over five walls-a dramatic departure from our first guess. Now we have a better sense of shortest paths and straight lines in various worlds, including our own earthly sphere. Putting these straight lines together allows us to explore some basic geometry that captures the essence of the graceful curvature of the sphere. Let's put three straight lines together to make a triangle. Geometric Gems r-.1ife -1crJ1J}{ _____, Curvy Geometry · Triangles on a Sphere Sometimes, when we are faced with a problem and we don't know what to do, we should just consider everything. -~----. ..--- -- - (/ ------- , Draw any triangle in the plane. Add up the three angles. The result is 180°. Cut off angles. ') / _:-' ____ _ ~ The sum is just a smidge greater than 180". Sum is 180. But we've seen that, in different realms, we have different ideas of straightness. Let's now explore the angles of a triangle made out of straight lines on a sphere. We begin by drawing a large triangle on a sphere. For example, let's put one vertex on the north pole, one vertex on the equator at oo longitude, and the third vertex on the equator at 90° longitude. The edges of this triangle consist of two longitudinal segments from the north pole to the equator and a segment that goes one-fourth of the way around the equator. What are the angles at each vertex? Each one is 90°. So what is the sum of the three angles of that triangle on the sphere? 90° + 90° + 90° = 270°. Yikes! This result is slightly disconcerting. The sum of the angles of this triangle on the sphere is not 180°, but 270° (90° too much). Is it possible that all triangles on the sphere have angles that sum to 270°? Let's see. Take the big triangle above and break it into two by drawing in the longitudinal segment from the north pole down to the equator at 45°. Now each of the half-sized triangles has angles of 90°, 90°, and 45°. That sum is 225°, 45° more than the 180° we would have in a triangle on the flat plane. This result is stranger still, since not only do the angles fail to add up to the comfortable 180° we know and love, but now we see that on a sphere, different triangles have different sums of angles. As always, we must look for patterns. Is there any regularity among our measurements? The big triangle had 270°,90° too much. When we divided it in half, each half had 225°, 45°. more than 180°. Did you notice that the total surplus of angle for the two smaller triangles stayed at 90°? In other words, when we took the big triangle and measured the surplus angle bigger than 180°, we got 90°. Then, when we divided the big triangle into two smaller triangles, each of the halves had a surplus of 45°, or 90° altogether. Suppose we start with any triangle on a sphere and cut it in half by bisecting one of the angles. What is the relationship between the angles of the original triangle and the angles of each of th~ two subtriangles? Well, the new angles •o and ••o add up to 180° since they are on a straight line. So, the total excess of the two triangles must be equal to the excess for the original big triangle. Notice what happens if we take a small triangle on the sphere. What is the surplus of its angles? Not very much. A small part of a sphere is basically flat, so the angles of a triangle there will have almost exactly the same angles as the angles of a triangle on a flat plane. It seems that larger triangles have greater excess in the sum of their angles than small triangles do. Furthermore, if a large triangle is divided into smaller triangles by adding edges, since all the added angles created are along straight lines or divide existing angles, the total excess of all the 4.6 • The Shape of Reality? • subtriangles making up a bigger triangle must be the same as the excess of the big triangle. What corresponds to the excess? It turns out that the excess increases as the area of the triangle increases. Thus, we see that the sum of the angles of a triangle on a sphere will always exceed 180° but that small triangles will just barely exceed 180° and large triangles will exceed 180° by a more substantial amount. Extra Degrees Through Curvature The sum of the angles of any triangle on a sphere exceeds 180° because of the curvature of the sphere. And, since every triangle on a sphere has a sum of angles exceeding 180°, we will say the sphere has positive curvature. Notice that the curvature on a sphere can be determined by measurements taken on _, __ - -- - - _ _ _ the sphere itself. It is not necessary to see the sphere from outside. For l (_ J, ) example, suppose we were bugs whose whole universe was a sphere. ~ Perhaps light stayed right along the sphere so that we could see things. We would not see a horizon, because the light would bend around the sphere, providing us with ever more distant vistas. Nevertheless, we could determine that our world has positive curvature by drawing a triangle and Curvature of the sphere measuring the sum of the angles. Even though the individual lines would causes "straight lines" to bow out a bit. appear completely straight, the sum of the angles would be more than 180°, clinching the positive-curvature claim. We now have one space, the plane, where all triangles have angles that add up to 180°. That constant sum is our benchmark, so we will say the plane has zero curvature-it is flat. We saw another space, the sphere, with positive curvature where all triangles have angles that add up to more than 180°. Surely we cannot resist asking the question, "Is there a space with negative curvature-that is, where the sum of the angles of a triangle is less than 180°?" U Geometry on a Saddle Horseback riding provides us not only with a sore bottom but also with an interesting geometrical opportunity. The surface of a saddle has an appealing shape and provides a surface ripe for experiments using rubber bands and butter. Suppose we place three pins as shown in the diagram on the next page, one near the front of the saddle and two near the stirrups. (This is a poor time to actually sit on the saddle.) We now grease the saddle with butter and put a rubber band around every pair of pins. The rubber bands will slide to the shortest distances between pins. So we will have a rubber band triangle on the surface of the saddle. We now wish to measure the angles. If you don't happen to have a saddle handy, estimate using the angles in the diagram, and compare the sum of the angles to 180°. The sum of the angles of this triangle is less than 180°. Of course, the rubber-band method will not always work on a saddle because the line between some pairs of points, like the center front to the center back of the saddle, would leave the surface and float in the air. However, rubber bands are good tools for finding the shortest distances between some pairs of points on the saddle. Whether you use rubber bands or another method for finding the shortest distances between points, triangles on the saddle will have Geometric Gems 30 picture of a saddle (Use your Heart of Mathematics 30 glasses to view.) Triangle on the surface of a saddle (Angles sum to less than 180°.) A Sum of the angles is less than 180°. sums of angles less than 180°, because the sides of the triangles curve inward and thus cause those angles to shrink. So this SJ?ace has negative curvature and is an example of the exotic world known as hyperbolic geometry. We have seen three different types of geometry: plane geometry, which we say has zero curvature (all triangles have angle sums of exactly 180°); spherical geom- 4.6 • The Shope of Reality? • etry, which we say has positive curvature (angle sums vary depending on the size of the triangle, but always exceed 180°); and hyperbolic geometry, which we say has negative curvature (angle sums vary depending on the size of the triangle, but always are less than 180°). It certainly appears as though hyperbolic geometry is exotic and foreign to our real-world existence, which brings us back to our original question: What is the shape of our universe? The Shape of Our Universe We have just caught glimpses of three types of geometry: planar, spherical, and hyperbolic. Which type models our universe? Think of an experiment that we could perform to answer this. (Hint: What property distinguishes the three?) Let's measure the angles of triangles. Suppose we make a big triangle and measure its angles. If the sum of those angles equals 180°, then we would conjecture that our universe has zero curvature. If the sum of those angles exceeds 180°, then we'd guess that our universe has positive curvature. If the sum of those angles is less than 180°, then we'd guess that our universe is curved negatively. Would anyone actually attempt this experiment? Yes! The great mathematician Carl Friedrich Gauss tried this experiment in the early 1800s. He formed a triangle using three mountain peaks near Gottingen, Germany: Brocken, Hohenhagen, und Inselsberg. He had fires lit on each mountain top (Smokey the Bear would not have been amused) and used mirrors to reflect the beams of light to form a triangle having side lengths roughly 43, 53, and 123 miles. He carefully measured the angles of the triangle and added them up. His sum was within 1/180 of a degree of 180°. That small difference could easily have been caused by errors in measurement. This evidence certainly leads us to think that our universe is neither positively nor negatively curved and that the universe is flat. What is the problem with this conclusion? Think about this question in view of our spherical geometry observations. Recall that, in spherical geometry, if we have a small triangle, then the triangle is nearly flat, and thus the sum of its angles is nearly 180°. Thus, although Gauss's triangle was big, compared to the entire universe it wasn't even a speck. Thus, on such a microscopic scale, it is not surprising to see that the evidence points to a geometry having zero curvature. We would need an enormous triangle to detect the existence of any actual curvature. Is this experiment even practical? • Geometric Gems And if it were, would anyone actually attempt it? The answer to the first question is possibly and to the second is yes. Today scientists believe that the universe exhibits two important properties. The first is that it is homogeneous, which basically means that any two large sections of space will look the same-of course, here "large" needs to be LARGE. The second is that the universe is isotropic, which means that, as we look around, things look about the same in every direction. It turns out that we can find geometrical objects that are homogeneous and isotropic that are either planar, spherical, or hyperbolic. This fact leads to a question of great interest to scientists today: Does the universe have zero, positive, or negative curvature? A large group of scientists now believes that the universe is negatively curvedthat is, that the geometry of the universe is actually the exotic hyperbolic geometry suggested by the saddle. In fact, a conference was held in October 1997 at Case Western Reserve University that brought together 20 cosmologists and 5 mathematicians to discuss the possible shape of the universe and ways to measure its curvature. NASA very recently used MAP-the Microwave Anisotropy Probeto measure microwave background radiation, which is a residue of the "big bang." They have determined that within the limits of instrument error the universe is, in fact, fiat. European scientists have sent up the Planck Probe. This probe should be able to make even more careful measurements of the variations in microwave radiation. These modern experiments capture the spirit of Gauss's attempts to measure the curvature of the universe. · So, what is the shape of our universe? In the first edition of this text, we wrote, "Although many experts believe it may be hyperbolic and negatively curved, no one knows for certain." However, the new evidence appears to favor the theory of a fiat universe. Twenty-first century science and technology together with mathematics will continue to enable us to measure the curvature of our vast space and understand its subtle and beautiful geometry. A Look various shapes. We can distinguish how space bends by examining the shortest paths-straight lines, although they may not necessarily be straight. Three different kinds of geometry are planar, spherical, and hyperbolic. The flat plane, round sphere, and saddle are good models for planar, spherical, and hyperbolic geometries, respectively. On a very small scale all look nearly the same, and thus we have not yet been able to determine the shape of our universe by taking measurements of our local environment. The distinguishing feature of the three different geometries is their curvature. If a space has zero curvature (the sum of the angles of any triangle is exactly 180°), then the space is flat. If a space has positive curvature (the sum of the angles of any triangle exceeds 180°), then the space is spherical. Finally, if the space has negative curvature (the sum of the angles of any triangle is less than 180°), then the space is hyperbolic. When we wish to consider big issues it is often valuable to start with simple and familiar models or examples and build from there. By identifying both similarities and differences in our various examples, we can often discover the underlying structure that determines the general case. SPACE CAN HAVE 4.6 • The Shape of Reality? ~--·-------Life • LeJJ1JJtJ __________..., Start with the simple and build from there . • When you don't know what to do, consider everything . • Look for patterns. ' L · ----------- --------~ ~'--~~~-~~~---~~-~~----~--- MINDSCAPES ~~----~---------- --~~·----- Invitations to frl.rlrther~hou;ht In this section, Mindscapes marked (H) have hints for solutions at the back of the book. Mindscapes marked (ExH) have expanded hints at the back of the book. Mindscapes marked (S) have solutions. Developing Ideas 1. Walking the walk. Here are three walks from corner X to corner Yin a city. The first walk is seven blocks long. How long are the other two? If you only travel east or north, how long is any other walk from corner X to corner Y? y North East- y y North North X East- East- Geometric Gems 2. Missing angle in action. The triangles below are drawn in the plane and the numbers represent the degrees of the angles. In each figure, compute the unknown angle(s). X 70 50 50 3. Slippery X. A triangle is drawn on a sphere. Can you determine the size of the angle x? Why or why not? 4. A triangular trio. The sphere below has three triangles on it. For which triangle is the sum of the angles largest? For which triangle is the sum smallest? 5. Saddle sores. The triangle at right is drawn on a saddle surface. Can the angle x be as large as 90°? Why or why not? 4.6 • The Shape of Reality? • Solidifying Ideas Travel agent. In each of the following three Mindscapes, get a globe and trace the shortest paths between the pairs of cities. For each pair on the left, find the location on the right that is on the shortest path between them. 6. Austin, Texas-Tehran, Iran Reykjavik, Iceland 7. Williamstown, Massachusetts-Beijing, China Denali, Alaska 8. Austin, Texas-Beijing, China Near the north pole Latitude losers (H). In each of the following three Mindscapes, you are given a pair of cities that are on the same latitude. Fill in the table by measuring the distance from city to city, first staying along the latitude and then measuring the distance taking the great-circle route. City Pair Latitude Distance Great-Circle Distance Beijing, China-Chicago, Illinois 9. 10. Mexico City, Mexico-Bombay, India 11. Sydney, Australia-Santiago, Chile Triangles on spheres. For each of the following five Mindscapes, on a globe, draw triangles whose vertices are the following sets of cities. For each such triangle, measure the sum of the three angles of the triangle. 12. Minneapolis, Minnesota; Austin, Texas; Williamstown, Massachusetts. 13. (S). Panama City, Panama; Nome, Alaska; Dublin, Ireland. 14. Quito, Ecuador; Monrovia, Liberia; Thule, Greenland. 15. Quito, Ecuador; Bangkok, Thailand; the south pole. 16. Wellington, New Zealand; Moscow, Russia; Rio de Janeiro, Brazil. 17-21. Spider and bug. For each pair of points on the boxes, describe the short- est path from one point to the other. 17. Left Front • Geometric Gems 19. 20. On back Fro nt .._,___., Bac k wall 1 Front wall 2 21. (S). 1 Side 4 ~{ ~{ Left 1 Right 4 22. Becoming hyper. Professor William Thurston found a neat way to build a model of hyperbolic geometry. Photocopy many equilateral triangles. (Enlarge the sheet given here on a copier.) Cut them out and tape them together so that seven triangles meet at each vertex. You will have to bend the triangles to fit 4.6 • The Shape of Reality? • them together. Continue attaching the triangles so that seven come together at each vertex. You will notice that your model will become floppy. The larger you make it, the floppier and more accurate your model will be. 23. Deficit angles (H). Draw a big triangle on your floppy sheet constructed in Mindscape22. Span several of the pieces by flattening a section on the ground and drawing a straight line, then flatten another section and draw another straight line, and then complete the triangle in the same way. There is a lot of squashing involved. Now measure the three angles and add them up. What do you get? 24. Same old. Go to a vertex on your floppy plane. Look at the pattern of all the triangles that you can reach from there passing through at most two triangles. Now go to another vertex and do the same. Are the patterns the same or different? 25. Gauss II. Try Gauss's experiment. Select three tall objects that are reasonably far away from one another (for example, three buildings or trees) and measure the angles between them. What are your measurements? Sum the angles. Creating New Ideas 26. Big angles (H). What is the largest value we can get for the sum of the angles of a triangle drawn on a sphere? Experiment with larger and larger triangles and compute the largest sum value. 27. Many angles (S). Draw three different great circles on a sphere. How many triangles have you made on the sphere? Compute the sum of the angles of all the triangles. Draw another group of three different great circles and answer the questions again. What do you notice? Make some conjectures. 28. Quads in a plane. Measure the sum of the angles of the quadrilaterals below. Why is the sum of the angles of any quadrilateral in a plane the same? 29. Quads on the sphere. Below are quadrilaterals on spheres. Measure the sum of the angles of each quadrilateral. Make a conjecture about the relationship between the sum of the angles of a quadrilateral and its area. • Geometric Gems 30. Parallel lines (ExH). On a plane, if you draw a line and then choose a point off the line, there is one and only one line that goes through that point and misses the line. Is this true for a sphere? Take a line on a sphere (which, remember, is a great circle) and take another point. How many lines-that is, great circles--can go through the point and miss the first great circle altogether? Explain your findings. Given line Exact\y one line parallel to given line through given point 31. Floppy parallels. On the floppy plane you constructed in Mindscape 22, draw a line and then choose a point some distance off the line. How many lines can go through the point and miss the first line altogether even if they are extended indefinitely? Given "line" 32. Cubical spheres (ExH). Take a cube. Put a point in the middle of each face. Now draw the straight lines to the middles of each of the sides of that face, thus producing a plus ( +) sign on each face. The kinked line that goes from the center of one face to the center of an adjacent face forms a bent edge on this cubical world. Thus we have created eight "bent" triangles whose vertices are the centers of the faces of the cube. Now what is the sum of the angles for each of those triangles? What is the sum of all the angles of all the triangles? Compare your answer to the answer to Mindscape 27. 33. Tetrahedral spheres. Let's do a similar calculation for the tetrahedron. Put a vertex at the center of each face of a tetrahedron and connect adjacent faces over the center of each edge. Answer all the questions in Mindscape 32. 34. Dodecahedral spheres. This Mindscape is the same as the previous two, except start with a dodecahedron. 35. Total excess. Using the observations from the previous Mindscapes and Mindscape 27, make a conjecture about the total excess of sums of angles of triangles that cover up polyhedra as described. 4.6 • The Shape of Reality? • Further Challenges Geometry on a cone (H). In Mindscapes 36 through 39 we will consider the following construction: Take a piece of paper and cut out a pie-shaped piece of angle z. Now put the two ends of the cut-out piece together to construct a cone. Let's see what happens if we look at triangles that surround that cone point. To draw a triangle, we have to determine what a straight line is on the cone. That is fairly easy, since the cone is made from a piece of paper that was originally flat. Take two points on the cone, flatten the cone in such a way that both points are on the flat part, and connect them with a straight line. A triangle is just made of three straight lines. First measure the angles in a triangle that does not go around the cone point. &~LL . --- - . Flattened Angle z - Cone Triangle that goes around the cone 36. What is the sum of the three angles? Why? Consider the more interesting case of a triangle that goes around the cone point. Draw the three sides sepa. rately by flattening in different ways. 37. What is the snm of the angles of your triangle? Is the sum the same for all triangles that go around the cone point? Let's try some more experiments, this time by removing thinner and fatter pie slices before making our cone. 38. Removing a slice of the pie. Complete the following table by making the cones, drawing triangles around the cone points, measuring the angles of the triangles, and adding them up. Angle of Pie Removed Sum of Angles of Triangles Difference from 180° ao· 60° 90° 180° 39. Conjuring up a conjecture. Make a conjecture about the relationship between the angle of the slice removed to make the cone and the excess above 180° of the sum of the angles of a triangle that goes around the cone point. 40. Tetrahedral angles. What is the sum of the angles around each vertex of the tetrahedron? For each vertex, compute 360° minus the sum of the angles at that vertex. Multiply that number by four since there are four vertices of the tetrahedron. What do you get? Are you surprised? Geometric Gems In Your Own Words 41. Personal perspectives. Write a short essay describing the most interesting or surprising discovery you made in exploring the material in this section. If any material seemed puzzling or even unbelievable, address that as well. Explain why you chose the topics you did. Finally, comment on the aesthetics of the mathematics and ideas in this section. 42. With a group of folks. In a small group, discuss and actively work through the relationship between the sum of the angles of a triangle on a sphere and a plane. After your discussion, write a brief narrative explaining the relationship in your own words. 43. Creative writing. Write an imaginative story (it can be humorous, dramatic, whatever you like) that involves or evokes the ideas of this section. 44. Power beyond the mathematics. Provide several real-life issues-ideally, from your own experience-that some of the strategies of thought presented in this section would effectively approach and resolve. For the Algebra Lover Here we celebrate the power of algebra as a powerful way of finding unknown quantities by naming them, of expressing infinitely many relationships and connections clearly and succinctly, and of uncovering pattern and structure. 45. Taxi total. You're in Manhattan for a job interview. A taxi picks you up from your hotel and takes you x blocks north, then x/2 blocks east, then x + 3 blocks north, then x/6 blocks east to your destination. Write an expression in terms of x that gives the total number of blocks traveled. Simplify the expression as much as possible. If the total number of bJocks you traveled is 19, determine how many blocks you traveled north between the time you entered the cab and the time the cab first turned east. 46. Angle x. A triangle in the Euclidean plane has angles measuring x, 2x, and 3x degrees. Find x. 47. Measuring moment (H). While lounging around your room one day, avoiding your history reading, you notice a tissue box and decide to do some measuring. The box has dimensions l, w, and h, measured in inches. Your measurements reveal that the perimeters of the faces have the following lengths: 21 + 2w = 26, 21 + 2h = 24, and 2w + 2h = 14. While it might not help you with your history assignment, solve these equations simultaneously to find l, w, and h. 48. Is x big enough? A triangle drawn on the sphere has angles measuring 80, 90 and x degrees. If x satisfies the equation x 2 - 30x + 200 = 0, find x. (There's only one value for x that works. What is it and why?) 49. Negative x? A triangle is drawn on a surface having negative curvature. Two of the angles measure 80 degrees. If x is the measure of the third angle, and x satisfies the equation x 2 - 25x + 100 = 0, find x. (There's only one value for x that works. What is it and why?)