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What is Geometry?

(an introduction to modern geometry)

Kapil Hari Paranjape

Indian Institute of Science Education and Research, Mohali

KVPY Summer Camp, 2nd June 2010

Kapil Paranjape (IISERM) What is Geometry? KVPY Camp 2010 1 / 14

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Introduction

Difficult Questions

Is there geometry beyond high school geometry?


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Introduction

Difficult Questions


Didn’t Euclid do all that there is to be done?


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Introduction

Difficult Questions



What does a geometer do to earn his salary?


I d i

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Introduction

Difficult Questions




To give a satisfactory answer is difficult since:


I t d ti

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Introduction

Difficult Questions





we must re-write modern geometry to make it easier to understand


Introduction

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Introduction

Difficult Questions






we must provide better training in mathematics for the rest .


Introduction

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Introduction

Difficult Questions






we must provide better training in mathematics for the rest .∗

“The rest” is how mathematicians usually refer to non-mathematicians!


Introduction

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Introduction

Ruler and Protractor Placement


Introduction

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Introduction

Ruler and Protractor Placement


Introduction

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Numbers and Geometry


Introduction



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Numbers are not imposed on geometry


Introduction

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Numbers are not imposed on geometry, rather they emerge from it.


Introduction

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Pictures and Geometry


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We do not need pictures to understand geometry.


Introduction

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We do not need pictures to understand geometry.


Introduction

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Do you see it?


Introduction



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Do you see it?


Introduction



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Key Notions


Introduction

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Key Notions

Congruence: Figures can be moved around withoutchanging their shapes.


Introduction

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Key Notions


Small Steps: A number of small segments combine to givea segment bigger than a big segment.


Introduction

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Key Notions


Small Steps: A number of small segments combine to givea segment bigger than a big segment.

Divisibility: Given any segment we can chop it up into two(or more) equal parts.


Introduction



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Arithmetic


Introduction



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Arithmetic


Introduction

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Euclid’s Fifth Postulate

Given a line and a point outside it, there is a unique line through thatpoint which does not meet the given line.


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Given a line and a point outside it, there is a unique line through thatpoint which is parallel to the given line.


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All coplanar lines meet!


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Projectivise


Introduction



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Projectivise


Introduction



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Projectivise


Introduction

P j i i



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Projectivise


Introduction

G Pi f



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Geometry on a Piece of paper


Introduction

N E lid Fif h P l

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Non-Euclidean Fifth Postulate

Given a line and a point outside it, there are infinitely many lines through

that point which do not meet the given line.


Introduction

N E lid Di t

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Non-Euclidean Distance

log(|AP |/|AQ |) − log(|OP |/|OQ |)


Introduction

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Thank you for your attention

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