Gather round the campfire, children. It is time to hear a story, a tale
about geometry and how it came to be the way it is today. Find a good natural log to
sit on and get comfortable, for we are about to journey back to the nineteenth century. Your Euclidean straightedge and compass are optional.
Our story begins in 1872, in the German university of Erlangen. The time was
ripe for the marvellous discoveries that were to follow. Mathematicians such as K. F. Gauss, János Bolyai, and Nikolai Lobachevski had succeeded in the past couple of decades in bringing nonEuclidean geometries of the likes of hyperbolic geometry into respectable ground. Elsewhere, Bernhard Riemann, made considerable work in the geometry that bears his name, where distance properties can be affected by the curvature of the space in question. The work of Renaissance painters in projective geometry had been carefully studied and catalogued. Mathematicians like the "invariant twins" Arthur Cayley and James Joseph Sylvester were discovering the powerful principle of invariants in algebra, while Hermann Grassmann explored the worlds of multidimensional affine geometry. It was an exciting time for mathematics, waiting for someone to come by and synthesise it all. Someone who could package all of geometry into a neat little box and tie it with a simple and elegant blue ribbon. The person who did this was Felix Klein.
The Erlanger Programm is Klein's brainchild. The name was chosen because of the speech given at Erlangen during Klein's inauguration. It is a nearperfect unification
of all geometries. The way in which it achieves this is by reversing the way we look at geometry.
What is geometry? A simple etymological interpretation of the Greek roots of
the word would answer that geometry is the study of "Earth measurement." This
is a woefully anachronic definition of the subject, which may have served well
the landsurveying purposes of the Egyptians when they were partitioning lands
around the Nile river for cultivation. Today, this definition will simply not
do, because at least since Euclid's time we have an abstract view of geometry.
Although Euclid cast the definitions, axioms, and postulates of geometry with the
hope of modelling a physical reality, the game we play when we deduce theorems
from these foundations has little use of empirical evidence for its
justification. Nevertheless, let us not yet discard this naïve definition of
geometry, for there is something we can salvage here.
Let's take a closer inspection. The key word is measurement. This is
indeed an important notion in Euclidean geometry. How do we determine if two
triangles are congruent? Determine if corresponding sides have equal
measurement. What about similarity of two pentagons? Check the equality of
the corresponding five angles. Sometimes we are also interested in finding the
measurement of a specific angle or distance given other measurements of
distance, angles, areas, or volumes; sometimes measurement is an interesting
question in its own right. And why would we care about the measurements of
things; why assign numbers and magnitudes to these abstract notions of points,
lines, planes? For the same reason that we ever assign numbers to anything
else: for the purpose of comparison. We want to know if this side is longer,
shorter, or equal to that other side over there. In fact, if we could simply
determine if it was equal or not, we would have already made considerable
progress. This may be enough to satisfy us. In short, in the classical spirit
of geometry, we begin with a notion of measurement and from that derive a notion
of equality; more precisely, a notion of congruence.
This paradigm works well for the zerocurvature world that Euclidean geometry
is. Unfortunately, it falls short of providing a good method for dealing with
all other geometries. We may perhaps be able to continue our pursuit in
hyperbolic and elliptic nonEuclidean geometries using more sophisticated
methods of measurement, but once we try to apply this idea to projective
geometry, the geometry of Renaissance perspective painters, we will run into an
impasse. It is impossible to define a distance function (that is, given any two
points in projective space, assign a distance between them) except for the
trivial distance function, the one that states that everything is zero distance
away from everything else. This leads nowhere useful. To geometrise
everything with distance will become increasingly complicated and ultimately
impossible as we try to subsume more geometries into our theory.
So, if Plan A doesn't work, let's try Plan B. Let's reverse the roles of
congruence and distance. Let us now declare that congruence is the fundamental
concept in all geometries. At this moment, the Erlanger Programm comes into
play. Suppose we have some set, some collection of things on which we want to
do geometry. How shall we define what is congruent to what? Felix Klein
suggests that we use the principle of superposition: a subset of our
collection is congruent to another subset in our collection if there is a
transformation that takes one subset onto the other. Now, in order to fully
specify a geometry on our set, we must now describe which are the permissible
transformations on our set. Because we want congruence to be an
equivalence relation, there are three requirements it must satisfy:
 Reflexivity. Everything must be congruent to itself.
 Symmetry. If A is congruent to B, then B must be congruent to A.
 Transitivity. If A is congruent to B, and B to C, then A must be congruent
to C as well.
In terms of transformations, this requires that:
 The trivial transformation that sends every element of our set into itself
must be included in our family of transformations.
 If a transformation T is in our set, then the inverse T^{1} must also be in
our set. In particular, all the transformations we consider must be invertible.
Our family of transformations must be closed under inverses.

If we have allowed T and S to be permissible transformations, then the
composition T • S must also be permissible. In other words, our family of
transformations must be closed under composition.
The cognoscenti will realise that we have just enumerated the
axioms for an abstract group (it is not necessary to explicity require
associativity, because the group operation is composition of functions, which is
always associative). Let us make a brief pause to bundle all these notions into
a concise
Definition: A geometry is an ordered pair (S,G), where S is
some set and G is any group acting on S.
The set S is the underlying space of the model; the group G is the
transformation group. I invite you to marvel at the economy of
this definition. Geometry is nothing more than the study of groups actions!
When doing group theory, one is never too far from pure geometry. This is
important, a deep and fundamental fact of modern mathematics that working
geometers keep in the back of their minds while they go about conducting their
research.
It is a sad situation to have to admit that the full reach of the Erlanger
Programm cannot be appreciated until several examples are studied in
detail. We shall leave those examples for another time. For now, keep in mind the
following nontrivial and familiar example: A model of Euclidean plane geometry
consists of the Cartesian plane and the group of rigid motions. A rigid motion
is any translation, reflection, or rotation, and their compositions.
Colloquially, this group is sometimes known to consist of slides, flips, and
turns. In this model, two figures such as triangles are congruent if there is a
way to slide, flip, or turn one triangle and superimpose it onto the other. We
could also use a smaller group of transformations and define a nonEuclidean
geometry. For instance, if we took the group of all translations, then we get
translational geometry. In this simple but strange geometry, two triangles are
not congruent if one is a rotation of the other. No notion of measurement is
necessary to define congruence here.
With this example in mind, I shall
explain a bit more the content of Felix Klein's maxim:
Geometry is the study of
invariants of a particular transformation group.
Measurement can now be defined in terms of congruence. This is a
familiar notion to anyone who has ever used a ruler. To determine the width
of this couch that you would like to fit into your living room, you take your
measuring tape; slide it; flip it; and turn it until aligns with the armrest of
the couch, and see which marking on the measuring tape the couch matches.
From the point of view of the Erlanger Programm, the reason why this is a valid
way to define the width of the couch is that your measuring tape will not get
longer or shorter under the flipping, sliding, or rotating actions of the
transformation group. The mathematical way to phrase this is by saying
"distance is invariant under the Euclidean group of rigid motions."
Invariants such as distance are at the heart of geometry. An invariant is
exactly what it sounds like: it is a magnitude that does not change
under the action of the transformation group, or a set that gets mapped into
itself by the same group. An example of a magnitude that is not invariant in
Euclidean geometry is that of slope: the ratio of "rise" over "run" of a line.
This magnitude is define relative to a choice of coordinates of the Cartesian
plane; the artificiality of this choice of coordinates is revealed by the fact
that slope of a line changes if we rotate said line. Slope is not an
intrinsic property of Euclidean geometry; distance is.
Here are a couple of examples of invariant sets and functions of Euclidean
geometry:

The set of all triangles is an invariant, as is the set of all
quadrilaterals, the set of all lines, the set of all circles, the set of all
dodecahedrons... It is not possible to use rigid motions and transform a
triangle into a polyhedron; we cannot leave the set of straight lines through
reflections, rotations, or translations. Therefore, each of these sets is an
invariant and deserves our careful attention.

Length, area, volume. To each figure for which each of these measurements
apply, the measurements will not change under rigid motions, a fact that infants
learn around age five or six, according to studies conducted by Piaget.
One example of invariants in translational geometry is the set of all lines
parallel to a specified line; the only way to leave this set is through
rotations or reflections, which are not allowed in translational geometry.
Observe that this set is not invariant in Euclidean geometry, for the same
reasons that slope is not an invariant of Euclidean geometry. Also observe
that slope is an invariant of translational geometry.
The Erlanger Programm does not forbid the usage of noninvariant properties of a
geometry. Sometimes, an argument, proof, or exercise becomes easier to express
if we allow ourselves to talk about slopes. Instead, what the Erlanger Programm
recommends is to only consider those statements about invariants as belonging to
the particular geometry under study.
A last technicality that should be addressed before we go out into the world, definitions blazing, ready to classify and subsume all geometries under the Erlanger
Programm (this can and has been done). We need to recognise when we are working
essentially the same geometry but with different models. When are the names different, yet the essence the same?
We must look at the
intrinsic, the reality that underlies the arbitrary human notions we use to
understand the world. The concept of an isomorphism does this. Apologies to
anyone who is offended by the following pinch of abstract nonsense. A more
tame explanation and example will follow shortly after:
Definition. Suppose that (S,G) and (T,H) are two geometries. Then (S,G) and
(T,H) are models of the same abstract geometry if there is a single function φ
that sets up a onetoone correspondence between the sets S and T and a onetoone
correspondence between the groups G and H, such that for any
element s of S and any transformation g in G corresponding to h in H we have that h(φ(s)) = φ(g(s)).
The function φ is an isomorphism.
φ
S > T
 
 
g h
 
 
V φ V
S>T
(The statement that follows after the "such that" in the above definition can be
made more palatable by saying that in the above diagram we obtain the same
function if we first follow the top and right arrows or if we follow the left
and bottom arrows. In category theory, this is phrased by saying that "this
diagram commutes". Fans of abstract nonsense will recognise φ to be a
naturallyoccuring functor between the concrete categories of geometry.)
What the above definition and scholium try to formulate is this: if we have two
models of a geometry, viz, two ordered pairs of sets with groups acting on them,
then we are looking essentially at the same geometry if all the elements of one
set in one model correspond uniquely to all the elements of the set in the other
model, and likewise for the transformation groups, provided that the nature of
the transformation groups does not change under this correspondence. For
example, we could fit all of Euclidean geometry into a disc of radius 1, and
define appropiate rigid motions of this circle. (For those interested, such a
distorted Euclidean geometry can be obtained by the function f(z) = z/(1 +
z^{2})^{1/2}, if we identify the Cartesian plane with the complex plane, as is
customary.) If you have ever looked at the world through a crystal ball and
seen the deformations that result by looking at all the world through this
spherical prism, you can understand what is happening here. The geometry of the
world hasn't changed because you have decided to look at it through a crystal
ball; only the way you model the world has suffered any alterations.
The disc model for Euclidean geometry can be adapted to a disc model for
hyperbolic geometry, with a very different transformation group, to be sure, and
even for a disc model for elliptic geometry. This leads us to suspect that
hyperbolic, Euclidean, and elliptic geometries are in fact expressions of a
single geometry, and indeed they are; such a geometry is known as an absolute
geometry.
The Erlanger Programm allowed us to see that all geometry could be treated in
the same way, and that geometries that at first glance looked disparately
different were in fact expressions of the same underlying principles. These are
the sort of discoveries that fill me with wonder of the structure of the universe.
Story time is over, kids. I hope you enjoyed yourselves. Get into your tents, you noisy and adorable rabble. I will put out the fire, and we may finish the marshmallows another night. I want you all in your sleeping bags and torches put out within fifteen minutes. Tomorrow we have a longer day. We shall hike through the disc models of absolute geometry. Distances can be misleading in those lands.