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# Dots, Lines, Regions

For Dr. Martin's Theory of Math Class, S 10, Westminster College

by

Tweet## Andrew McHugh

on 8 November 2010#### Transcript of Dots, Lines, Regions

Dots, Lines, and Regions Andrew R McHugh

Theory of Math This is just one point in space Dots:1

Lines:0

Regions:1 We will define "regions" as the space enclosed by lines plus an initial region. If we add a dot, we must connect it with a line. Dots:2

Lines:1

Regions:1 But our lines cannot cross through other lines. THIS IS BAD! Dots:2

Lines:?

Regions:? Definitions Rules Dots:2

Lines:2

Regions:2 Dots:2

Lines:3

Regions:3 1 2 3 2 1 Purpose Is there a relation between how many dots, lines, and regions are in a graph? YES! The answer is this way Ex: Two Dots Ex: Three Dots Simple Complex Dots:3

Lines:2

Regions:1 Dots:3

Lines:3

Regions:2 Dots:3

Lines:9

Regions:8 If you don't mind, I'll stop writing the letters now... D:4

L:3

R:1 D:4

L:4

R:2 D:5

L:4

R:1 D:5

L:4

R:2 Ex: Four...Five...n D

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R 0=D-L+R-2 D=R =( + ) D

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R ≠( + ) OH NO! +1 +1 Base Theory First I thought that there was a base that was added upon. Base Base Base The base would be the irreducible configuration, one in which there is only one region. PROBLEM: The configurations become recursive. This means that configuration "k" is built off of the previous configurations, all they way down to the base =( + ) =( + ) +1 +1 +1 +1 +1 +1 I thought there should be a better way... I made a list... +1 +1 +1 +1 +1 +1 D

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R 0= - + -2 I looked at the numbers and tried to add and subtract them to find a solution. That is when I found: D=L-R+2 Which can be rewritten as: 0=-D+L-R+2

0=D-L+R-2 Let's Try this Again: 0=D-L+R-2 D:4 (found by counting)

L:13 (found by counting)

R:11 (R=-D+L+2) D:7 (D=L-R+2)

L:8 (found by counting)

R:3 (found by counting) D:12 (found by counting)

L:17 (L=D+R-2)

R:7 (found by counting) D:4

L:3

R:1 D:4

L:4

R:2 D:5

L:4

R:1 Checking the rule: Why It Works In order to add a line, you must add either a dot or a region. (from the rules of configuration) +1 Region +1 Dot ...to try and find a relationship. 0=D-L+R-2

Full transcriptTheory of Math This is just one point in space Dots:1

Lines:0

Regions:1 We will define "regions" as the space enclosed by lines plus an initial region. If we add a dot, we must connect it with a line. Dots:2

Lines:1

Regions:1 But our lines cannot cross through other lines. THIS IS BAD! Dots:2

Lines:?

Regions:? Definitions Rules Dots:2

Lines:2

Regions:2 Dots:2

Lines:3

Regions:3 1 2 3 2 1 Purpose Is there a relation between how many dots, lines, and regions are in a graph? YES! The answer is this way Ex: Two Dots Ex: Three Dots Simple Complex Dots:3

Lines:2

Regions:1 Dots:3

Lines:3

Regions:2 Dots:3

Lines:9

Regions:8 If you don't mind, I'll stop writing the letters now... D:4

L:3

R:1 D:4

L:4

R:2 D:5

L:4

R:1 D:5

L:4

R:2 Ex: Four...Five...n D

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D L

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L R

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R 0=D-L+R-2 D=R =( + ) D

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R ≠( + ) OH NO! +1 +1 Base Theory First I thought that there was a base that was added upon. Base Base Base The base would be the irreducible configuration, one in which there is only one region. PROBLEM: The configurations become recursive. This means that configuration "k" is built off of the previous configurations, all they way down to the base =( + ) =( + ) +1 +1 +1 +1 +1 +1 I thought there should be a better way... I made a list... +1 +1 +1 +1 +1 +1 D

1

2

2

2

2

3

3

3

3

4

4

4

4

4

:

D L

0

1

2

3

4

2

3

4

5

3

4

5

6

7

:

L R

1

1

2

3

4

1

2

3

4

1

2

3

4

5

:

R 0= - + -2 I looked at the numbers and tried to add and subtract them to find a solution. That is when I found: D=L-R+2 Which can be rewritten as: 0=-D+L-R+2

0=D-L+R-2 Let's Try this Again: 0=D-L+R-2 D:4 (found by counting)

L:13 (found by counting)

R:11 (R=-D+L+2) D:7 (D=L-R+2)

L:8 (found by counting)

R:3 (found by counting) D:12 (found by counting)

L:17 (L=D+R-2)

R:7 (found by counting) D:4

L:3

R:1 D:4

L:4

R:2 D:5

L:4

R:1 Checking the rule: Why It Works In order to add a line, you must add either a dot or a region. (from the rules of configuration) +1 Region +1 Dot ...to try and find a relationship. 0=D-L+R-2