MacTutor
Geometry as an Invariant Theory
In 1930 Oene Bottema was employed as a docent at the University of Groningen to teach the course 'Special chapters in geometry." On 20 October 1931 he delivered a public lecture De meetkunde als invariantentheorie held at his acceptance of the position of docent at the University of Groningen. We give an English version of this lecture below:
Geometry as an Invariant Theory.
Public lecture held at the acceptance of the position of Private Teacher at the University of Groningen on 20 October 1931.
by Dr O Bottema.
Anyone who, after reading and practicing geometry in the extent to which it is taught in our high schools, will concentrate on studying the extensive and multifaceted complex of geometrical investigations, which is considered to belong to the so-called higher mathematics, and will soon come to the insight that, in short, the word geometry has a plural.
It soon becomes clear to the future geometer that the subject of his choice is by no means limited to the expansion and deepening of elementary Euclidian geometry, but that it is rather concerned with the study of geometrical propositions, which differ from those of old known to him, in the sense that a different system of axioms is their foundation.
He not only learns to draw ever-more-far-reaching conclusions from the Euclidean axioms and to introduce new concepts based on them, but he also studies the consequences of other assumptions. He not only learns new methods for tackling the problems of the past, relying on other areas of mathematics, but he also becomes acquainted with geometric systems; in which at least some theorems of Euclidean geometry lose their sense and validity.
From the otherwise subjective point of view of elementary geometry, one could divide the "other" geometry into two groups.
I would like to include in the one group those geometries whose axioms are all or partially different from those that apply to Euclidean geometry. This includes the two geometries, the hyperbolic and the elliptical, which are summarized as non-Euclidean geometry in a narrower sense. They are based on a different parallel axiom than the Euclidean. In the hyperbolic planimetry there are infinitely many, one line in the elliptical, which has no point in common with a given line. This other premise naturally leads to different conclusions. While in Euclidean geometry the sum of the angles of a triangle is 180°, it is not fixed in non-Euclidean geometry. In the elliptical the sum is always larger, in the hyperbolic always smaller than 180°. Another proposition that can give an impression of the deviant nature of these geometries is that which expresses that in elliptic geometry a straight line has finite length. Non-Euclidean geometry had to wage a long-standing struggle for existence, and its interesting, often described, historical development gives a striking picture of the effort it took to overcome the supremacy of Euclidean geometry that was so very important for daily experience and seems the "true" geometry.
The second group of geometries, that of the Euclidean differences, consists of systems that occupy an entirely different place compared to it. They essentially do not contain any proposition that does not apply in the Euclidean and where at first glance a deviation from a Euclidean truth can be found, there it can be explained by a difference in terminology. We can most easily imagine these geometries as based on some, but not all, Euclidean axioms and on no others. So there is still room for the entire Euclidean system in these geometries. These larger geometries are not next to or opposite the Euclidean, but they are built around it, as it were. The complex of concepts and theorems of such geometry is less extensive than that of the Euclidean, it is formed by a certain selection from the Euclidean phenomena. Well-known examples of such geometries are projective geometry, affine geometry, and conformal geometry. We also note that projective geometry, in turn, is built around affine geometry and that non-Euclidean geometry just mentioned can also be accommodated within projective geometry.
One could compare projective geometry with a region with a little differentiated legal status, in which certain districts have laid down further rules for their area, which are not uniform for the different districts, but which never contravene general national laws. Every projective proposition is also one that applies in Euclidean geometry, but the reverse is not the case. The theorem that in each triangle the sum of the angles is 180°, does not apply in the projective, but not in the sense that this sum there would have a different constant value or would not be constant. The reason is rather that the concept of angle measure in projective geometry is not definable. The theorem also does not apply in conformal geometry, where the concept of angular measure is known, but where the expression "straight line" is not understood and the figure of two intersecting circles is thought of at an "angle". The proposition that applies in the projective plane: "two straight lines always have one point in common", appears at first glance to be contrary to what we know in Euclidean geometry. This point of difference is one based on a difference in terminology. In order to come from Euclidean or also from non-Euclidean geometry, to the projective, or from the Euclidean to conformal, one must become aware of its position with regard to the so-called infinite, but it is also possible to reach a settlement with the infinite. There are accommodations with the infinite.
To this end, so-called "ideal elements" are introduced in a purely formal way in Euclidean geometry, which for the case of projective geometry amounts to obscuring the distinction between two intersecting lines and two parallel lines and intersecting both types of figures under the collective name lines. You can consider this example as an indication of the correctness of a statement from Poincaré: "mathematics is the art of giving different things the same name."
It is undoubtedly much easier to accept a system such as projective geometry than to unite with non-Euclidean geometry. Somewhat simply proposed, one could say that the projective geometer is only interested in certain propositions of Euclidean geometry, propositions that also apply if one omits a portion of the Euclidean axioms, and which he can generally simplify by using the ideal elements more than is possible in Euclidean geometry. Nevertheless, projective geometry has long needed to be recognized as a system independent of Euclidean geometry, and it must be admitted that the beginner always has some difficulty in realizing that a purely projective concept such as the cross ratio of four collinear points is indeed independent of the Euclidean concept of distance.
Since Euclidean geometry is infused with propositions that have a more general character, the elementary geometer is therefore in fact also concerned with the wider geometry. As one can speak in prose without realizing it, one is, for example, in the chapters on proportionality of line segments or in the treatment of different properties of surfaces and centres of gravity, unknowingly taught in some fundamental propositions of affine geometry.
In order to get an overview of the various measuring techniques, it would suffice to identify each of them with the system of axioms that determines it. But apart from the fact that such a classification would lack a certain clarity, for example because various systems of axioms can yield the same geometry, there is the great disadvantage that a frequently used method of treating geometry does not fall directly to its axioms. Professor Schaake in his inaugural address about the construction of geometry, gave an overview of the two methods that are at the service of the geometer, the axiomatic-synthetic and the analytical method. The latter is based on axiomatics and the successes of arithmetic and algebra, further "geometric" axioms are indispensable for it, its definitions consist of giving geometric names to arithmetic variables. We come to the question of how the distinction between the different analytical treatments should be considered.
For this, let us again start from Euclidean geometry, and let us ask ourselves what we think about a geometric figure, for example a triangle, as being interesting. This includes all sorts of properties concerning the size of sides and angles, remarkable points, etc. But this does not include all those properties that distinguish the triangle from one that is congruent to it. We do not care whether the triangle on the left or right is drawn on a board or printed in a book. This is so obvious that very often the concepts congruently and identically merge. When the students of a class have each drawn a triangle and these triangles are congruent, it is very common to say that they all have "the same" triangle. And when it is noticed that there is only one triangle with three given sides, then it is also meant that all triangles with three given sides are congruent, and it is also clear how this confusion of congruence and identity could arise: if one moves a material triangle, one can rightly maintain that one has preserved the same triangle.
But if one has adapted to the saying of the geometer that one triangle changes into another triangle that is congruent under displacement, then one understands what it means to say, if one declares that elementary mathematics is interested in those characteristics of the figure, which can withstand a displacement without changing, which are said to be invariant with respect to a movement transformation. All this may seem somewhat artificial in the case of an elementary, that is to say, synthetic treatment of geometry; with the analytical method this conception appears naturally. In analytical geometry one can most easily identify a triangle by the coordinates of its vertices. A second triangle will only be congruent with the first if there are certain relationships between their two corner point coordinates. It is clear of what nature these relationships are. They are apparently nothing but the translation, in the language of algebra, of the displacement, which is our synthetic criterion of congruence. In the analytical, formal treatment, congruence therefore amounts to the existence of certain expressions, by means of which the coordinates of one point are added to those of another point. These expressions are of the first degree and are therefore among the simplest the mathematician knows, but they are not general linear expressions and in so far as explicitly drawing up these equations, at least for spaces of more than two dimensions, it still provides some difficulty.
The theorems of geometry, practiced purely analytically, are proven by demonstrating the existence of certain equations and arithmetic results. What we now emphasize is that only those comparisons have a geometric meaning, which appear to have retained the same shape after the application of a random movement transformation. To give an example: At first glance it may seem important that one of the coordinates of the point {3, 0} located in the flat plane is equal to zero. Apparently this is only appearance. As soon as the plane is subjected to a motion transformation, the point will, for example, move to the point {5, 7} and our comparison appears to have the character of an accidental coincidence. If one has two points whose coordinates and satisfy the equation
,
then it will appear after a transformation that this, for example, has passed into
and our relation is loosing the right to be called a geometric property. However, if one considers the equation
,
then it appears that, after any transformation, it will change to
and is therefore elevated to a geometric theorem.
Every comparison that presents itself as a geometric theorem must be tested for its constancy; if it passes the test, if it emerges unharmed from the transformation machinery, then it is worthy of being included in the list of propositions.
If one now asks how analytical geometry came to the invariant expression, which I just mentioned, it must be, to its slightly compromising answer, that it took a loan to this end from synthetic geometry, which is more based on contemplation.
It is namely the formula based on the Pythagorean theorem, which expresses that the square of the distance of the two points is 9. If the analytical geometer wanted to give up everything that the spectator and the synthetic deduction had taught him from the figures, then he would have to continually determine from each relationship he found between coordinates whether it contained a geometric truth, or is no more than a coincidental circumstance dependent on the particular choice of the coordinate system.
The investigation into the question whether his relationship could suffer from a movement transformation would be similar to a game that might initially be attractive, but soon exhausting, with constantly changing results. It is therefore understandable that a systematic investigation has been made into the whole of relationships that can withstand certain transformations and, if possible, to find a finite number of these invariants, from which all others can be distinguished. This chapter of algebra developed primarily in the 19th century and is called invariant theory. Because of its results, the analytical geometry has formally become completely independent of the synthetic. It enables her to choose, from the confusing abundance, the phenomena that have lasting value. Seen from this point of view, Euclidean geometry is the invariant theory of certain transformations, which we will continue to call motion transformations, but where all view can be missed.
However, if we see our elementary geometry from this point of view, its very relative significance becomes clear to us. The transformation formulas, which represent the Euclidean movement, are no more interesting and certainly no simpler than a large number of other formulas to which the coordinates of a figure could be subjected. It is even certain that if our synthetic past were not there, these formulas would not have attracted our special attention in any way. There is therefore nothing more obvious than the geometer also considering other coordinate additions than those we have come to know as the algebraic analogue of Euclidean displacement and are interested in the invariants of other transformation sets. The statement by Poincaré would thereby grant him the freedom to give the same word of 'geometry' to this other case. However, the geometer imposes certain restrictions on himself. In his new system he will call two figures "congruent", as soon as his collection of transformations contains a copy that adds one to the other, but he now continues to argue that one figure is congruent with itself, furthermore that the congruence of A with B follows the congruence of B with A, and finally that the congruence of A with B and that of B with C, follows the congruence of A with C. If our transformations meet such analytical conditions, that the continuation of these propositions is guaranteed, then the transformation collection is called a transformation group, and we have arrived at a general principle that was first clearly expressed by Klein in 1872 in a publication that is often quoted as the "Erlanger Programme": each transformation group gives rise to a geometry that is essentially nothing but the invariant theory of the group.
The possibility outlined here to arrive at new measurement techniques via a group and the concept of invariants is not to be regarded as a result of historical development. Nor should it be imagined that the analytical treatment of these geometries dates from the moment that the Erlanger Programme was born. On the contrary, the most famous transformation groups give rise to geometries, which had long been known and studied, and Klein was therefore able to demonstrate the correctness of his principle through various examples. The great significance of his principle can be seen in the possibility that it allows us to classify the various systems treated in an analytical way and to determine their mutual relationships. As such, it has been extremely illuminating.
If we want to take a closer look at a single transformation group, then one of the most important ones is the one where the new coordinates of a point are found by substituting the original in broken linear expressions, all of which have the same denominator.
The geometry associated with this group is no different from that which we have referred to above as projective geometry. It gets its major development in the first half of the 19th century, and can be regarded in various respects as one of the most important geometries for the geometer to study. So important that one of its practitioners, the English mathematician Cayley, was able to explain in his time without contradiction for reasons that will be further explained: "projective geometry is all geometry." As another argument, I mention the fact that when the mathematician speaks of invariant theory, he always means the invariant theory of the projective group.
The projective group comprises a wider collection of transformations than the Euclidean. It makes more extensive changes in the figures that are subjected to it. To give an impression of the nature of these changes, we return to the more illustrative geometry and thereby limit ourselves for the sake of simplicity to flat figures. As in Euclidean geometry, the figures are permitted to be displaced and if we are only interested in those properties that can withstand a displacement, the projective geometry also permits the figure to move from a point outside its plane to project on another plane. And one pays attention only to those properties that have been preserved in these more comprehensive transformations. It is clear that various concepts and propositions from Euclidean geometry are now meaningless. After all, after such a projection, the distance between two points will generally not have remained the same, nor the angle between two lines. Parallel lines can merge into intersecting lines through projection, and vice versa; the peculiar points of a triangle lose all meaning. It can even be shown that a triangle can always pass through a suitably chosen projectivity into an arbitrary other one, that is to say that all triangles are projectively congruent, or in other words, that a triangle has no invariants. In order not to draw the impression, after having mentioned these negative characteristics of projective geometry, that I would not have any proposition, I would like to point out to you that a straight line through projection again becomes a straight line; so that the collinear position of some points is a concept that has not lost its meaning. A circle can pass through central projection into one of the figures, which in Euclidean geometry is referred to as ellipse, parabola or hyperbola, but it does not lose the property of having with a straight line at most two points in common. Projective geometry therefore does not know these figures separately, but studies their common properties, with no word whatsoever of centre, focal point or asymptote.
Another important transformation group is the affine; its elements subject the point coordinates to whole linear substitutions. Geometrically, this means that, in addition to Euclidean movements, parallel projections are permitted. The changes brought about by this in the figures are less far-reaching than those induced by the projective group. We are getting closer to home, there are now fewer Euclidean concepts which are meaningless. Affine geometry knows the concept of parallel, in the sense of non-intersecting, it contains the propositions about proportionality of line segments, at least the centre of gravity of the peculiar points of the triangle, it also knows the distinction between ellipse, parabola and hyperbola. It does not yet have the basic metric concepts of "distance" and "angle measure".
In addition to the projective and affine groups, I also mention those that, in addition to the Euclidean movements, contain the transformations of conformity and inversions, and which form the basis of conformal geometry. It is also the place here to point out that topology is that geometry which is linked to the very large group of continuous transformations.
The definition given above of a geometry as the invariant theory of an arbitrary transformation group, would at first sight lead to the expectation that the geometer deals with very abstract systems that are alien to reality and which have only the name in common with ordinary geometry. The examples given have shown that some simple and widely studied transformation groups form the analytical background of well-known, synthetic methods to be treated and that the various elements of these groups can be interpreted as illustrative geometric operations.
We have already observed that the great significance of the group theoretical conception of geometry did not seem to us to be primarily based on the possibility that it offers to add new geometries to the previously studied, or in the help that it provides us in determining the mutual relationship of the systems treated. These are two important concepts from group theory, which come to the fore, namely the concept of subgroup and the concept of isomorphism.
The group G' is called a subgroup of the group G, if all its elements belong to G. All invariants of G are a fortiori invariants of G', all concepts and propositions of G-geometry, also occur in G'-geometry, but the reverse does not seem to be the case. We initially considered two types of geometry that are distinct from the Euclidean. We can now briefly define the one species as consisting of geometries whose transformation group contains the Euclidean movement group as a subgroup. Surrounding Euclidean geometry, the equiform, equiaffine, affine and projective geometry are situated in concentric, increasingly large circles. The projective group also has the non-Euclidean movement group as a subgroup, the Euclidean is again a subgroup of the conformal group. One could illustrate the mutual location of the various measurement techniques by means of a kind of map.
With regard to the subgroup concept, one has to think of the relationship between a broader, less differentiated geometry and a more specialized one. The word isomorphic indicates a relationship of equivalence. The groups and are called isomorphic when a one-to-one correspondence can be found between the elements of and so that the product of two elements and from , maps under the correspondence to the product of the image elements and from . Abstractly speaking, the groups and are identical, the geometries belonging to and are essentially the same, but they can differ in nomenclature, because the two groups have a different embodiment. The one geometry can be seen as a mirror image of the other; image and object differ in appearance, but have the same inner structure. Such comparable measurements are of exceptional importance for the geometer. They enable him, when dealing with geometry, whether systematically or according to the demands of the moment, to ask how the analogous problem occurs in the geometry and how he can determine it, which of the two realizations makes him suspect the best way to solve his problem. Between the two geometries there is a connection as between two perfectly equivalent languages, he can choose in which his full sentences sound the clearest.
The projective geometry in the space of three dimensions is isomorphic with the non-Euclidean in ; the conformal in the plane with the non-Euclidean in ; the group of projectivities in , which leaves a straight invariant, is isomorphic to the uniform group in , see some examples there to illustrate the concept of isomorphism. It may be that in a geometry that is equivalent to a geometry, the problems become so much simpler that they lose some of their charm. That method of treating non-Euclidean geometry, which uses its isomorphic copy, if not identity, with Cayley's metrics is certainly the easiest, but it lacks the attractiveness of a historic treatment. Various problems of non-Euclidean geometry are solved so easily with the help of the absolute quadratic variety that Coolidge rightly compares the role that this figure plays with that of a 'Deus ex machina'.
If after this discussion of two important concepts of group theory we return to the great significance of projective geometry, we can now understand it by the fact that the transformational groups are the best known geometries, either identical or isomorphic with a projective group or one of its internal subgroups.
That Klein's principle not only has a historical significance, but has a stimulating effect up to our time, may be seen from the fact that during the last decades differential geometry has increasingly detached itself from the Euclidean point of view. In addition to classical differential geometry, systems have also developed in the other geometries, as I have told you, which deal with concepts such as "arc length" and "radius of curvature". They are more or less complete, for example, projective, equiaffine, and conformal differential geometry.
In elementary geometry, a relatively strict distinction is made between planimetry and stereometry. Although in general the complications of geometrical propositions and the multiplicity of figures increase with the number of dimensions, a division of the geometries according to their dimensions is nowadays without essential significance and the time has passed that multi-dimensional geometry formed a separate chapter of geometry. In some projective investigations, however, the different behaviour of otherwise analogue figures in spaces of an even and of an odd number of dimensions remains interesting.
We wish to comment on a different distinction in geometry with a single word. The Klein principle refers to groups defined by transformation comparisons between coordinates. In addition, we have left out the types of numbers that people wish to allow as coordinates. In the geometry of the projective group, one obtains the simplest and most generally editable theorems if one derives its coordinates from the set of complex numbers. Restriction to the body of real numbers leads to many interesting questions about the existence of certain figures.
Among the geometries that have yet another number field for their coordinates, I tell you about the case where this field is finite. These peculiar geometries apparently also possess only a finite number of points and they are therefore by no means distinguished by great wealth of forms. Yet in the projective cases of these finite geometries all relevant axiom's of projective geometry apply. There are also isomorphisms between finite geometries and some known inflection point configurations. - It is also possible for the geometer to derive its coordinates from a set of numbers whose mutual operations do not comply with the usual rules. Thus, the so-called non-Pascal geometry is based on a number system that lacks the commutativity of multiplication.
In the foregoing it has been attempted to give you an impression of the way in which the mathematician has come to immerse himself in other geometries than the ancient Euclidean. It has been found, however, that it does not occupy a special place among the many and varied geometric systems. Apart from the fact that it has not been able to maintain its unique significance for the description of the physical world concept, it must be admitted that its importance is relative in the whole of geometric science. Because its transformation group is a subgroup of some other important groups, it is one of its different positions, which have a broader meaning. The resulting wealth of concepts and propositions is obtained at the expense of inner unity. It would be no good to speak evil of such a venerable science as Euclidean geometry, but it is a fact that it lacks its own cachet to some extent. The majority of propositions regarding proportionality of line segments, inclusion and centres of gravity belong to affine geometry, as well as, for example, the propositions of Menelaus and the Ceva; a theorem such as that of Ptolemy finds its foundation in conformal geometry, while others have a generally projective meaning. Autochthon theorems of Euclidean geometry, on the other hand, are, for example, those about the sum of the angles of a triangle and the theorem of Pythagoras.
When one of several theorems from Euclidean geometry, for example, has acknowledged its affine character, this does not mean that one could simply incorporate it into a textbook of affine geometry. The affine propositions are often not proven with affine axioms. If one consistently wanted to implement the principle of not using any means to prove a certain proposition, which are alien to the essence of the case, that would not rely on superfluous conditions, then you must reject the usual proof for the proposition: "in a parallelogram the diagonals bisect each other in the middle" This statement is an affine statement, so it must be possible to prove it without any theory about congruence.
It would be ungrateful not to speak here of the temptation, which can lie in the treatment of problems of Euclidean geometry, which cannot really be subsumed directly under Klein's principle. The volume of a physical body is an affine concept; if one asks the question to what extent two physical bodies with the same volume can be divided into pairwise congruent pieces, then that problem has a somewhat ambivalent character. If it were set aside, we would miss the interesting investigations that, for example, Dehn made regarding these questions.
I thought I should choose a general geometric principle as the subject for this public lecture. Klein's principle gave me reason to tell you about some geometric systems that deviate from the Euclidean, have their own charm, and can serve to better understand the geometry with which we all grew up. I highly appreciate that the benevolence of the Trustees and of the Faculty of Mathematics and Physics has allowed me to give lectures at this University, in which "special chapters" of geometry can be further discussed.
Last Updated November 2019