What is geometry? Is there geometry beyond high school gometry?
Didn't Euclid do all that there is to be done?

These are actually questions that were asked of me by a school kid
whom I met in a train.

To give a satisfactory answer is difficult since:

 - we must re-write modern geometry to make it easier to understand
 - we must provide better training in mathematics for "the rest"

First of all, high school geometry is not really Euclid's geometry!
Euclid was a purist (as a geometer) and would have been horrified
by the "Ruler placement postulate" or the "Protractor placement
postulate".

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

The fundamental objects in Euclidean geometry are points and line segments
and _all_ of mathematics (not just geometry) must emerge from this;
at least according to Euclid.

So how do we measure things without a ruler? There are three things
that help us.

The first is that a line segment can be "moved" from place to place
without changing its length. When you think about it this is a very
deep idea which is is _not_ completely reflected in physical reality!

Suppose I take a metal rod and place it a few hundred kilometers from
the sun. Will it retain its length?

So the geometric notion of "congruence" of line segments is something
imposed _by_ our imagination onto an "imperfect" physical reality.

This may lead us to understand why geometers don't draw pictures.

Aside: When I wrote my thesis and showed it to my advisor Professor
S. Ramanan, he said that some of the arguments would be easier to
understand if I inserted pictures. Unfortunately, I forgot to add
them. Ever since then I have been trying to justify myself by saying

 Actually we don't need pictures to understand geometry

For example, at http://www.imsc.res.in/~kapil/geometry/boromagic.png
you will find the following image. Now this may not look like
anything but in fact your mind's eye can create a three dimensional
image out of it! (I will leave it there for a minute so that you can
try to see the image.)

To come back to the point, geometry is a like a set of mental spectacles
through which we view physical reality. So to study geometry we do
not need to refer to physical reality but let our mind roam free
within its own vision.

% One such vision is that geometrical shapes move from place to place
% and retain their geometric properties. If you will pardon my French
% (my mother taught French so perhaps you can call it my mother-tongue!).
% 
%  plus ç change, plus c'est la même chose
% 
% This is the fundamental notion of "congruence".

The second important notion (which was actually not noted in Euclid's
books but was pointed out by Archimedes) is the big-step little-step
idea.

 one "giant leap for mankind" is made up of many small leaps by many men
 and women.

or

 more of less is more than more

Mathematically speaking

 a number of small segments combine to give a segment bigger than a big segment

% To understand this, we need to realise the importance of "order"
% in Euclidean geometry. In fact, Steiner created a version of
% Euclidean geometry in which the fundamental notions were that of
% points and a relation of "between-ness":
% 
%  a point C lies between A and B if it lies on the line segment
%  joining A and B
% 
% In Steiner's geometry this became a "definition" of a line segment
% using "between-ness" as a fundamental notion:
% 
%  the line segment joining A and B is the locus of points that lie
%  between them

The third important notion is that of "divisibility".

 given any segment we can chop it up into two (or more) equal parts

(Note that this notion of indefinite divisibility is also not
reflected in physical reality!)

Putting these three notions ("congruence", "big-step little-step"
and "divisibility") together allows us to measure the length of one
segment using another as a "unit". This act of producing a binary
or decimal expression for the result is called _measurement_.

Now that we know the results of modern physics, it looks like none of
these notions is actually applicable to the physical world.

So is geometry a flawed way of looking at physical reality? Can we
still make measurement meaningful? Do numbers emerge from geometry
in a different way?

The answer comes by looking at the way we can perform basic
arithmetic operations using geometry.

Given points O, A, B on a line l, the following construction gives
a point C so that the segment OC is the sum of the segments OA and
OB.

Take a point O' outside l and let m be the line through O' which is
parallel to l.

Let p denote the line through O and O'.

Let q be the line through A which is parallel to p and r be the line
through B which is parallel to p.

Let A' be the point of intersection of q and m. Let B' be the point
of intersection of r and m.

Let t be the line joining A' and B and u be the line parallel to t
through B'.

Then C is the point of intersection of u and l.

Perhaps a figure _will_ help!

There is a similar construction that gives us multiplication;
we need a additionl point I on l that represents the unit.

Take a line l' through O which is different from l and m be a line
through I which is different from l.

Let I' be the point of intersection of l and m.

Let p be the line through A which is parallel to m and let A' be the
point where it intersects l.

Let q be the line joining I' and B.

Let r be the line parallel to q which passes through A'.

Then C is the point where r meets l.

Again a figure may help!

Both of these figures make extensive use of (a consequence of) Euclid's infamous "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. (I.e. is parallel to
 the given line).

Since this postulate is controversial, let us take a more
controversial postulate. It is evident (for example by looking at
railway lines) that parallel lines meet at the horizon! So we include
the horizon or the "line at infinity" inside our geometry and declare
that 

 All pairs of coplanar lines meet!

This leads to "projective" geometry which is certainly more visual
than Euclidean geometry. Let us transform the above pictures into
projective pictures.

Suddenly things look more 3-dimensional! In fact some developmental
biologists have argued that the brain handles inputs from the retina
through projective transformations and that is what leads to our 3-d
imagination.

It turns out that above operations make the algebraic operations of
addition, subtraction, multiplication and division possible for
points on a line in projective space (of three or more dimensions).
(We avoid applying these operations to the point at infinity!)

This becomes our new "number system". In fact, there is nothing new
about it and one can show that it is just our familiar
real/decimal/binary number system!

Every point on the projective line can then be represented as a
ratio (p:q) for numbers p and q which are not both zero.

Note that:
 - (p:q) is the same ratio as (ap:aq) where a is non-zero
 - the projective line _includes the point (1:0); which is the point at infinity

One can go further and show that points on the projective plane can be
represented by a triple (p:q:r) where at least one of p,q,r is non-zero.
As before the triples (p:q:r) and (ap:aq:ar) represent the same point
when a is non-zero.

This shows that numbers, in the sense of co-ordinates _are_
fundamental to the way we view the world. Note that these are _only_
co-ordinates and do not have an obvious connection to distance or
measurement! It is thus possible to ask:

 - is the notion of distance between points still meaningful?
 - is there a formula for this distance in terms of the co-ordinates?

One of the ways that the eye gauges distance is by the extent that
the muscles of the eye have to stretch in order to view an object! So
even if "absolute" distance is cannot be defined (or so Einstein told
us!) there is a well-defined notion of "angular displacement". Indeed
this notion can be used to give a positive answer to the above
questions about the projective plane.

All this talk of points at infinity and horizons will sometimes make
one's head hurt! (Even a geometer's head!) So let us try to be
practical and look at the geometry of lines and points in a
smallish region, like this room or on a sheet of paper.
sheet of paper. 

Since we have often done high-school geometry drawings on a piece of
paper, we are reasonably sure that the basic notions of points and
lines make sense. The notion of parallellism falls down however,
since now we cannot decide whether a pair of lines meets outside the
piece of paper or is _actually_ parallel! Being practical people, we
can declare that such lines _never_ meet! We thus get a replacement
for Euclid's fifth postulate:

 Given a line and a point outside it, there is are many lines through
 that point which do not meet the given line.

In fact, there are infinitely many such lines. We do not call them
parallel lines since we shall see that parallel lines have a slightly
different definition.

Unfortunately, other axioms will break down since our little steps
will quickly take us out of the paper. So we need some trick to make
our steps smaller and smaller as we approach the edge of the paper.

Let us cheat and put "usual" co-ordinates on a line l through a point
O on the seet of paper. Let P and Q be left and right "edge" points of
this line on the sheet of paper. If A is a point on the line l then
we can define distance as

 log (|AP|/|AQ|) - log (|OP|/|OQ|)

As usual, the positive side is to the right and the negative side is
to the left --- which is against left-handers like me!

With this definition of distance one can do many things similar to
Euclidean geometry.

A word of warning is that circles that we draw with a compass are no
longer the collection of points equidistant from a given point!

Moreover, angle measures are no longer what they appear to be.

One can show that this gives nothing other than the non-Euclidean
geometry of Bolyai and Lobachevsky. Warning: this model
is different from the one usually found in text books on
non-Euclidean geometry.

How does one justify the formula for distance _without_ using the usual
co-ordinates?

One can introduce "ideal" points "outside" the paper where lines do
meet even when they do not meet on the sheet of paper. Similarly, we
can introduce "ideal" lines where planes meet when they do not meet
within this room. Desargues' theorem can be used to show that this
gives a _consisten_ answer. However, we have no way to distinguish
"really" parallel lines from those that meet outside this room! So
the points, and lines at infinity are _automatically_ part of this
ideal geometry. We then show that this ideal geometry is actually our
now familiar projective space and so we can put co-ordinates on it.

I hope I have convinced you that

 Projective geometry is indeed in the eye of the beholder

Thank you for your attention.
