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 non-Euclidean 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 near-perfect 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 land-surveying 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 zero-curvature 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 non-Euclidean 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:

  1. Reflexivity. Everything must be congruent to itself.
  2. Symmetry. If A is congruent to B, then B must be congruent to A.
  3. 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:

  1. The trivial transformation that sends every element of our set into itself must be included in our family of transformations.
  2. 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.
  3. 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 non-Euclidean 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:

  1. 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.
  2. 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 non-invariant 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 one-to-one correspondence between the sets S and T and a one-to-one 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.
          S --------> T
          |           |
          |           |
          |g          |h
          |           |
          |           |
          V      φ    V

(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 naturally-occuring 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.

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