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Other Types of Geometry 19

Lesson

1-3 Other Types of Geometry

Lesson 1-3

BIG IDEA Points and lines in discrete geometry and in graph theory have different properties than points and lines in Euclidean geometry.

In Lesson 1-1, a point was described as an exact location. In Lesson 1-2, a point was described as a location in the plane identifi ed by an ordered pair of real numbers. In this lesson, you will examine two other common descriptions of points. This is important because different descriptions of points and lines serve as the foundation for different types of geometry.

Dots and PointsLook at the Seurat painting. In both the original of this painting and the picture reproduced here, points are represented by dots. These dots have both length and width.

Vocabularypixels

discrete point

discrete line

discrete geometry

graph theory

arc

network

node

vertices, vertex

transversable network

even node, odd node

A B C D

If you take any of the 3 paths from A to B, any of the 4 paths from B to C, and any of the 5 paths from C to D, how many different routes are possible from A to D?

Mental Math

George Seurat’s Sunday Afternoon on the Island of Grande Jatte measures 6 3 __ 4 feet by 10 feet.

SMP_SEGEO_C01L03_019-026.indd 19SMP_SEGEO_C01L03_019-026.indd 19 11/13/07 12:12:52 PM11/13/07 12:12:52 PM

20 Points and Lines

Chapter 1

Dots come in many different shapes, colors, and sizes. Consider, for example, the image you see when you look at a computer screen or digital camera. The image you see consists of thousands of tiny dots called pixels that are combined to create the image you see on the screen. These dots are so small that it is very hard to distinguish one from another.

On the other hand, consider a scoreboard. The writing on the scoreboard is made up of larger dots created by light bulbs.

Discrete GeometryWhen a point is described as a dot of some size, a line is made up of points that could have space between them. In other words, when using these descriptions, between two points on a line there is not necessarily another point. Points are called discrete points and the lines are called discrete lines. The study of discrete points and discrete lines is one type of geometry called discrete geometry.

Discrete Geometry

Description of a pointA point is a dot.

Description of a lineA line is a set of dots in a row.

One difference between Euclidean synthetic geometry and discrete geometry occurs when you consider crossing lines. In Euclidean geometry, when two lines cross, they are guaranteed to intersect at a point that is on each line, because there are no spaces or gaps in the line. For instance, in the upper fi gure at the right, lines m and n cross at point A, which is on both lines. This is not necessarily true in discrete geometry because the points on a line have space in between them. The lower diagram at the right shows two discrete lines that cross and have a point in common.

See Quiz Yourself at the right.

In both Euclidean and discrete geometry, points represent locations. However this is not true in graph theory, where points simply represent items to be connected and lines are the connectors. This will become clearer after you work through the following very famous problem.

m n

A

crossing lines havinga common point

crossing lines havinga common point

QUIZ YOURSELF

Draw an example of two discrete lines crossing in which the two lines do not share a common point.



Lesson 1-3

The Königsberg Bridge ProblemThrough the city of Kaliningrad, Russia, fl ows the Pregol’a River, and in this part of the river are two islands. In the 1700s this city was part of East Prussia and was known as Königsberg. Seven bridges connected the islands to each other and to the shores, and it was common on Sundays for people to take walks over the bridges. These walks and bridges led to a problem. Can you walk over each bridge exactly once?

The drawing at the right is based on one that fi rst appeared in an article by the great mathematician Leonhard Euler (OY ler). The islands are A and D; the bridges are a, b, c, d, e, f, and g; the shores of the river are B and C.

The Königsberg Bridge NetworkTo solve the Königsberg Bridge Problem, Euler redrew the map with the islands A and D as very small, and lengthened the bridges. This doesn’t change the problem.

Then he realized that the shores B and C could be small. This again distorts the picture but it doesn’t change the problem.

Finally—and this was the big step—he thought of the land areas A, B, C, and D as points and the bridges a through g as arcs connecting them. The result, shown at the right, is a network of points and arcs. In this network there is a path (though not necessarily direct) from any point to any other point. Networks are sometimes called graphs. This is why the geometry of networks is called graph theory. In a network, the only points are the endpoints of arcs. These endpoints have no size and are called nodes or vertices. (The singular of “vertices” is vertex.)

Activity 1MATERIALS Tracing paperKönigsberg Bridge Problem: Trace the Königsberg Bridge network and use your drawing to fi nd a way to walk across all seven bridges of Königsberg so that each of the seven bridges (a, b, c, d, e, f, and g) is crossed exactly once. (For instance, if you start at D and walk on f, b, and a, then you are at point B, and you are stuck because you have already walked on all the bridges from B.) Try to explain any diffi culties you may encounter.

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C

B

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B

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C

B

D


22 Points and Lines

Chapter 1

Arcs and NodesEuler was able to rephrase the Königsberg Bridge Problem to become: “Without lifting a pencil off the paper, can one trace over all the arcs of this network exactly once?” If the answer is yes, this kind of network is called a traversable network. Euler noticed that the number of arcs at each node provided a clue as to whether or not a network was traversable. As shown below, nodes in a network can have different numbers of arcs.

There are no points in the middle of an arc, but between two nodes there can be many different arcs connecting them.

Graph Theory

Description of a pointA point is a node of a network.

Description of a lineA line is an arc connecting either two nodes or one node to itself.

In the Königsberg Bridge Problem, there are 5 arcs at vertex A (c, d, e, b, and a) and 3 arcs at each of vertices B, C, and D. Before we show you Euler’s solution to the Königsberg Bridge Problem, here are three more networks to consider. In these networks, the arcs are drawn as segments.

Example 1How many arcs are at each node of networks I, II, and III?

IA

B

D

C

E

IIF

G

I

H

J

IIIK

L

N

M

P

NNode N with arc to itself

PO

Nodes O and P with 3 arcs each

T

RQ

SFour nodes with 3 arcs each

(Arcs QT and SR do not intersect.)

Leonhard Euler



Lesson 1-3

SolutionNetwork I: There are 4 arcs at node E, 3 at B, 2 at A and C,

and 1 at D.

Network II: There are 4 arcs at node J and 2 arcs at F, G,

H, and I.

Network III: There are 3 arcs at P, K, L, and M, and 2 arcs at N.

If the number of arcs at a node is even, the node is called an evennode. Otherwise it is an odd node. In Example 1, A, C, E, F, G, H, I, J, and N are even nodes while B, D, K, L, M, and P are odd nodes.

Activity 2Look again at the networks in Example 1.

1. Where must you start and end in order to traverse network I?

2. Where must you start and end in order to traverse network II?

3. Is it possible to traverse network III?

Euler’s Solution to the Königsberg Bridge Problem Euler realized that when a path goes through a node, it uses two arcs: one to the node and one away from it. This led him to realize that when a network has an odd node, it must be the starting or fi nishing point for a traversable path. In Activity 2, you should have found that Network I is traversable only if you start at B or D. If you started at B, then you ended at D and vice versa.

Euler realized that all four nodes in the Königsberg network are odd. Since a traversable network can only have one starting point and one fi nishing point, the Königsberg network is not traversable. Whenever a network has more than two odd nodes, it is not traversable.

Example 2Look back at the networks from Example 1. Use Euler’s reasoning to explain why networks I and II are traversable while network III is not.

Solution Network I is traversable because there are exactly two odd nodes (B and D). Network II is traversable because

there are no odd nodes. Network III is not traversable because

there are 4 odd nodes.

Graph theory and problems like the Königsberg Bridge Problem have many applications.

a

c

b

dg

e

f

A

C

B

D


24 Points and Lines

Chapter 1

Graph theory is used in computer networking, city planning, security, airline fl ight scheduling, and many other areas. For instance, a curator of a museum would be interested in fi nding the minimum number of security cameras needed to provide security for all of the exhibits in the building. A traveling salesman might want to minimize the distance on his route in order to maximize sales. A restaurant manager might want to know the most effi cient way of assigning tables to wait staff to maximize effi ciency. All of these questions could be addressed using graph theory.

You have now seen examples from four different kinds of geometry: Euclidean synthetic geometry, Euclidean plane coordinate geometry, discrete geometry, and graph theory. These are presented for you as a way of showing that the word “geometry” can have different meanings depending upon how you describe the simplest components of that geometry—the point and the line. Euclidean geometry is by far the most useful and most common of these geometries, so the two types of Euclidean geometry are the ones we will examine most in the remainder of this book.

QuestionsCOVERING THE IDEAS

1. What type of geometry best describes the lines you see in each of the following diagrams?

a. b. c.

2. How are points described in Euclidean plane coordinate geometry?

3. How many lines contain any two points in Euclidean plane coordinate geometry?

4. How are points described in discrete geometry?

5. Is it possible for two lines in Euclidean synthetic geometry to cross without having a point in common?

6. How is a point described in graph theory?

7. In graph theory, what does it mean for a network to be traversable?

8. What is an odd node?

9. When you graph a line on a graphing calculator, what kind of geometry best describes the line that the calculator creates?



Lesson 1-3

10. How do you know if a network is traversable?

11. Below you see a “square” in each of the three types of geometry described in this lesson: Euclidean synthetic geometry, graph theory, and discrete geometry. Name the type of geometry represented by each “square”.

a. b. c.

In 12−14, tell whether the network is traversable.

12.

E

F

C

D 13.

E

H

B

P

I

G 14.

APPLYING THE MATHEMATICS

15. Draw a network with 2 nodes and 4 arcs. Is this network traversable?

16. A map of Frytown, Iowa is shown at the right. A snowplow driver wants to know if it is possible to plow all of the roads in Frytown without passing over the same road twice.

a. If you think of this map as a network in graph theory, what do the arcs represent and what do the nodes represent?

b. Is this network traversable? If so, where can the driver start and stop?

17. Refer to the map of Königsberg from the lesson. Euler proved that it was impossible to travel over all of the bridges without crossing over at least one bridge twice. Let us suppose, however, that you travel back in time and build one additional bridge anywhere that you choose. Where could you build the bridge so that it becomes possible to walk over every bridge without crossing over any bridge twice? Are there multiple correct answers to this problem?

18. Create a network that has 4 nodes and 7 arcs and is possible to traverse. Use arrows to show the path that you used to traverse it.

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26 Points and Lines

Chapter 1

19. Create a network that has 4 nodes and 7 arcs but is impossible to traverse. Explain why your network is impossible to traverse.

20. Consider the capital letters of the English alphabet. Find 3 letters that cannot be drawn without picking up your pencil or retracing a line and fi nd 3 letters that can. Explain your choices using the ideas from graph theory.

REVIEW

In 21-23, graph the line with the given equation, and classify it as horizontal, vertical, or oblique. (Lesson 1-2)

21. 20y = 5 22. x + 7y = 4 23. x + y = 5 + y

24. Selena’s hair is 10 cm long and growing at a rate of 2 cm per month. (Lesson 1-2, Previous Course)

a. Graph a line representing Selena’s hair length over time. b. Write an equation of the line in slope-intercept form.

25. What does it mean for a line to be coordinatized? (Lesson 1-1)

26. Suppose Megan, Namiko, and Omar are standing in a straight line. Megan and Omar are 20 feet apart and Namiko is 7 ft from Omar. What are the possible distances between Namiko and Megan? (Lesson 1-1)

EXPLORATION

27. a. Explain why it is impossible to trace the drawing of a box shown at the right without ever lifting up your pencil and without tracing the same path twice.

b. What is the fewest number of times you need to pick up your pencil in order to trace the box?

28. There are many puzzles using networks. Here is one:Start with a network that has n arcs. (The networks at the right have 4 nodes and 6 arcs.) Name each node with a different number from 0 to n. Then number each arc by the difference of the nodes it connects. For example, in the graceful network at the right, the arc connecting nodes 5 and 2 is named 3 because 5 - 2 = 3. The goal is to name the nodes in such a way that the n arcs are numbered with all the integers from 1 to n. Such a network is called a graceful network.

Number the nodes in these three networks to make them graceful. a. b. c.

2

4

55 3

16

0

6

2

graceful (arcs are numbered from 1 to 6)

3

3

55 2

16

0

6

3

not graceful (two arcs are numbered 3)

QUIZ YOURSELF ANSWER

Answers may vary. Sample:


UCSMP Geometry, vol. 1UCSMP ProjectsProgram OrganizationTable of ContentsTo the StudentChapter 1Lesson 1-1Lesson 1-2Lesson 1-3Lesson 1-4Lesson 1-5Lesson 1-6Lesson 1-7ProjectsChapter Wrap-UpSummary and VocabularySelf-TestChapter Review

Chapter 2Lesson 2-1Lesson 2-2Lesson 2-3Lesson 2-4Lesson 2-5Lesson 2-6Lesson 2-7ProjectsChapter Wrap-UpSummary and VocabularySelf-TestChapter Review

Chapter 3Lesson 3-1Lesson 3-2Lesson 3-3Lesson 3-4Lesson 3-5Lesson 3-6Lesson 3-7Lesson 3-8Lesson 3-9ProjectsChapter Wrap-UpSummary and VocabularySelf-TestChapter Review

Chapter 4Lesson 4-1Lesson 4-2Lesson 4-3Lesson 4-4Lesson 4-5Lesson 4-6Lesson 4-7Lesson 4-8ProjectsChapter Wrap-UpSummary and VocabularySelf-TestChapter Review

Chapter 5Lesson 5-1Lesson 5-2Lesson 5-3Lesson 5-4Lesson 5-5Lesson 5-6Lesson 5-7ProjectsChapter Wrap-UpSummary and VocabularySelf-TestChapter Review

Chapter 6Lesson 6-1Lesson 6-2Lesson 6-3Lesson 6-4Lesson 6-5Lesson 6-6Lesson 6-7Lesson 6-8Lesson 6-9ProjectsChapter Wrap-UpSummary and VocabularySelf-TestChapter Review

Chapter 7Lesson 7-1Lesson 7-2Lesson 7-3Lesson 7-4Lesson 7-5Lesson 7-6Lesson 7-7Lesson 7-8Lesson 7-9Lesson 7-10ProjectsChapter Wrap-UpSummary and VocabularySelf-TestChapter Review

PostulatesTheoremsFormulasSelected AnswersAdditional AnswersGlossaryIndexPhoto CreditsSymbolsHelpStudent EditionChapter 1Lesson 1-1Lesson 1-2Lesson 1-3Lesson 1-4Lesson 1-5Lesson 1-6Lesson 1-7ProjectsChapter Wrap-Up

Chapter 2Lesson 2-1Lesson 2-2Lesson 2-3Lesson 2-4Lesson 2-5Lesson 2-6Lesson 2-7ProjectsChapter Wrap-Up

Chapter 3Lesson 3-1Lesson 3-2Lesson 3-3Lesson 3-4Lesson 3-5Lesson 3-6Lesson 3-7Lesson 3-8Lesson 3-9ProjectsChapter Wrap-Up

Chapter 4Lesson 4-1Lesson 4-2Lesson 4-3Lesson 4-4Lesson 4-5Lesson 4-6Lesson 4-7Lesson 4-8ProjectsChapter Wrap-Up

Chapter 5Lesson 5-1Lesson 5-2Lesson 5-3Lesson 5-4Lesson 5-5Lesson 5-6Lesson 5-7ProjectsChapter Wrap-Up

Chapter 6Lesson 6-1Lesson 6-2Lesson 6-3Lesson 6-4Lesson 6-5Lesson 6-6Lesson 6-7Lesson 6-8Lesson 6-9ProjectsChapter Wrap-Up

Chapter 7Lesson 7-1Lesson 7-2Lesson 7-3Lesson 7-4Lesson 7-5Lesson 7-6Lesson 7-7Lesson 7-8Lesson 7-9Lesson 7-10ProjectsChapter Wrap-Up

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