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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
Introduction
Difficult Questions
Is there geometry beyond high school geometry?
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
2 / 14
Introduction
Difficult Questions
Is there geometry beyond high school geometry?
Didn’t Euclid do all that there is to be done?
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
2 / 14
Introduction
Difficult Questions
Is there geometry beyond high school geometry?
Didn’t Euclid do all that there is to be done?
What does a geometer do to earn his salary?
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
2 / 14
Introduction
Difficult Questions
Is there geometry beyond high school geometry?
Didn’t Euclid do all that there is to be done?
What does a geometer do to earn his salary?
To give a satisfactory answer is difficult since:
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
2 / 14
Introduction
Difficult Questions
Is there geometry beyond high school geometry?
Didn’t Euclid do all that there is to be done?
What does a geometer do to earn his salary?
To give a satisfactory answer is difficult since:
I
we must re-write modern geometry to make it easier to understand
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
2 / 14
Introduction
Difficult Questions
Is there geometry beyond high school geometry?
Didn’t Euclid do all that there is to be done?
What does a geometer do to earn his salary?
To give a satisfactory answer is difficult since:
I
we must re-write modern geometry to make it easier to understand
I
we must provide better training in mathematics for the rest.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
2 / 14
Introduction
Difficult Questions
Is there geometry beyond high school geometry?
Didn’t Euclid do all that there is to be done?
What does a geometer do to earn his salary?
To give a satisfactory answer is difficult since:
I
we must re-write modern geometry to make it easier to understand
I
we must provide better training in mathematics for the rest.∗
“The rest” is how mathematicians usually refer to non-mathematicians!
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
2 / 14
Introduction
Ruler and Protractor Placement
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
3 / 14
Introduction
Ruler and Protractor Placement
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
3 / 14
Introduction
Numbers and Geometry
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
4 / 14
Introduction
Numbers and Geometry
Numbers are not imposed on geometry
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
4 / 14
Introduction
Numbers and Geometry
Numbers are not imposed on geometry, rather they emerge from it.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
4 / 14
Introduction
Pictures and Geometry
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
5 / 14
Introduction
Pictures and Geometry
We do not need pictures to understand geometry.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
5 / 14
Introduction
Pictures and Geometry
We do not need pictures to understand geometry.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
5 / 14
Introduction
Do you see it?
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
6 / 14
Introduction
Do you see it?
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
6 / 14
Introduction
Key Notions
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
7 / 14
Introduction
Key Notions
Congruence: Figures can be moved around without
changing their shapes.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
7 / 14
Introduction
Key Notions
Congruence: Figures can be moved around without
changing their shapes.
Small Steps: A number of small segments combine to give
a segment bigger than a big segment.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
7 / 14
Introduction
Key Notions
Congruence: Figures can be moved around without
changing their shapes.
Small Steps: A number of small segments combine to give
a segment bigger than a big segment.
Divisibility: Given any segment we can chop it up into two
(or more) equal parts.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
7 / 14
Introduction
Arithmetic
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
8 / 14
Introduction
Arithmetic
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
8 / 14
Introduction
Euclid’s Fifth Postulate
Given a line and a point outside it, there is a unique line through that
point which does not meet the given line.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
9 / 14
Introduction
Euclid’s Fifth Postulate
Given a line and a point outside it, there is a unique line through that
point which is parallel to the given line.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
9 / 14
Introduction
Euclid’s Fifth Postulate
All coplanar lines meet!
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
9 / 14
Introduction
Projectivise
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
10 / 14
Introduction
Projectivise
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
10 / 14
Introduction
Projectivise
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
10 / 14
Introduction
Projectivise
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
10 / 14
Introduction
Geometry on a Piece of paper
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
11 / 14
Introduction
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.
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
12 / 14
Introduction
Non-Euclidean Distance
log(|AP|/|AQ|) − log(|OP|/|OQ|)
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
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Introduction
Thank you for your attention
Kapil Paranjape (IISERM)
What is Geometry?
KVPY Camp 2010
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