Download THE SHAPE OF REALITY?

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
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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.
~·
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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?
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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
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Food
L-------Shortest path = Straight line
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An initial guess for the shortest path
Shortest path?
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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?
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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?
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Some straight paths are shorter than others.
Shortest path
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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.
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, Draw any triangle in the plane. Add up the three angles. The result is 180°.
Cut off angles.
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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?
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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 ·
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--------~
~'--~~~-~~~---~~-~~----~---
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?)