Excerpt from Visual Complex Analysis (1997) by Tristan Needham (pgs. 30-34)

Geometry Through the Eyes of Felix Klein

Even with the benefit of enormous hindsight, it is hard to introduce complex numbers in a compelling manner. Historically, we have seen how cubic equations forced them upon us algebraically, and in discussing Cotes’ work we saw something of the inevitability of their geometric interpretation. In this section we will attempt to show how complex numbers arise very naturally, almost inevitably, from a careful re-examination of plane Euclidean geometry.[10]

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Although the ancient Greeks made many beautiful and remarkable discoveries in geometry, it was two thousand years later that Felix Klein first asked and answered the question, “What is geometry?”

Let us restrict ourselves from the outset to plane geometry. One might begin by saying that this is the study of geometric properties of geometric figures in the plane, but what are (i) “geometric properties”, and (ii) “geometric figures”? We will concentrate on (i), swiftly passing over (ii) by interpreting “geometric figure” as anything we might choose to draw on an infinitely large piece of flat paper with an infinitely fine pen.

As for (i), we begin by noting that if two figures (e.g., two triangles), have the same geometric properties, then (from the point of view of geometry) they must be the “same”, “equal”, or, as one usually says, congruent. Thus if we had a clear definition of congruence (“geometric equality”) then we could reverse this observation and define geometric properties as those properties that are common to all congruent figures. How, then, can we tell if two figures are geometrically equal?

Consider the triangles in [fig 23], and imagine that they are pieces of paper you could pick up in your hand. To see if T is congruent with T’, you could pick up T and check whether it could be placed on top of T’. Note that it is essential that we be allowed to move T in space: in order to place T on top of T˜ we must first flip it over; we can’t just slide T around within the plane. Tentatively generalizing, this suggests that a figure F is congruent to another figure F’ if there exists a motion of F through space that makes it coincide with F’. Note that the discussion suggests that there are two fundamentally different types of motion: those that involve flipping the figure over, and those that do no. Later, we shall return to this important point.

It is clearly somewhat unsatisfactory that in attempting to define geometry in the plane we have appealed to the idea of motion through space. We now rectify this. Returning to [fig 23], imagine that T and T’ are drawn on separate, transparent sheets of plastic. Instead of picking up just the triangle T, we now pick up the entire sheet on which it is drawn, then try to place it on the second sheet so as to make T coincide with T’. At the end of this motion, each point A on T‘s sheet lies over a point A’ of T’‘s sheet, and we can now define the motion M to be this mapping A → A’ = M(A) of the plane to itself.

However, not any old mapping qualifies as a motion, for we must also capture the (previously implicit) idea of the sheet remaining rigid while it moves, so that distances between points remain constant during the motion. Here, then, is our definition:

A motion M is a mapping of the plane to itself such that the distance between any two points A and B is equal to the distance between their images A’ = M(A) and B’ = M(B). (22)

Note that what we have called a motion is often termed a “rigid motion”, or an “isometry”.

Armed with this precise concept of motion, our final definition of geometric equality becomes:

F is congruent to F’, written F ≅ F’, if there exists a motion M such that F’ = M(F). (23)

Next, as a consequence of our earlier discussion, a geometric property of a figure is one that is unaltered by all possible motions of the figure. Finally, in answer to the opening question of “What is geometry?”, Klein would answer that it is the study of these so-called invariants of the set of motions.

One of the most remarkable discoveries of the last century was that Euclidean geometry is not the only possible geometry. Two of these so-called non-Euclidean geometries will be studied in Chapter 6, but for the moment we wish only to explain how Klein was able to generalize the above ideas so as to embrace such new geometries.

The aim in (23) was to use a family of transformations to introduce the concept of geometric equality. But will this ≅-type of equality behave in the way we would like and expect? To answer this we must first make these expectations explicit. So as not to confuse this general discussion with the particular concept of congruence in (23), let us denote geometric equality by ~.

(i) A figure should equal itself: F ~ F’, for all F.

(ii) If F equals F’, then F’ should equal F: F ~ F’ ⇒ F’ ~ F.

(iii) If F and F’ are equal, and F’ and F” are equal, then F and F” should also be equal: F ~ F’ & F’ ~ F” ⇒ F ~ F”.

Any relation satisfying these expectations is called an equivalence relation.

Now suppose that we retain the definition (23) of geometric equality, but that we generalize the definition of “motion” given in (22) by replacing the family of distance-preserving transformations with some other family G of transformations.

It should be clear that not any old G will be compatible with our aim of defining geometric equality. Indeed, (i), (ii), and (iii) imply that G must have the following very special structure, which is illustrated[11] in [fig 24].

(i) The family G must contain a transformation ε (called identity) that maps each point to itself.

(ii) If G contains a transformation M, then it must also contain a transformation M^-1 (called the inverse) that undoes M. [Check for yourself that for M^-1 to exist (let alone be a member of G) M must have the special properties of being (a) onto and (b) one-to-one, i.e., (a) every point must be the image of some point, and (b) distinct points must have distinct images.]

(iii) If M and N are members of G then so is the composite transformation N ∘ M = (M followed by N). This property of G is called closure.

We have thus arrived, very naturally, at a concept of fundamental importance in the whole of mathematics: a family G of transformations that satisfies these three[12] requirements is called a group.

Let us check that the motions defined in (22) do indeed form a group: (i) Since the identity transformation preserves distances, it is a motion. (ii) Provided it exists, the inverse of a motion will preserve distances and hence will be a motion itself. As for existence, (a) it is certainly plausible that when we apply a motion to the entire plane then the image is the entire plane — we will prove this later — and (b) the non-zero distance between distinct points is preserved by a motion, so their images are again distinct. (iii) If two transformations do not alter distances, then applying them in succession will not alter distances either, so the composition of two motions is another motion.

Klein’s idea was that we could first select a group G at will, then define a corresponding “geometry” as the study of the invariants of that G. [Klein first announced this idea in 1872 — when he was 23 years old! — at the University of Erlangen, and it has thus come to be known as his Erlangen Program.] For example, if we choose G to be the group of motions, we recover the familiar Euclidean geometry of the plane. But this is far from being the only geometry of the plane, as the so-called projective geometry of [fig 24] illustrates.

Klein’s vision of geometry was broader still. We have been concerned with what geometries are possible when figures are drawn anywhere in the plane, but suppose for example that we are only allowed to draw within some disc D. It should be clear that we can construct “geometries of D” in exactly the same way that we constructed geometries of the plane: given a group H of transformations of D to itself, the corresponding geometry is the study of the invariants of H. If you doubt that any such group exists, consider the set of all rotations around the center of D.

The reader may well feel that the above discussion is a chronic case of mathematical generalization running amuck — that the resulting conception of geometry is (to coin a phrase) “as subtle as it is useless”. Nothing could be further from the truth! In Chapter 3 we shall be led, very naturally, to consider a particularly interesting group of transformations of a disc to itself. The resulting non-Euclidean geometry is called hyperbolic or Lobachevskian geometry, and it is the subject of Chapter 6. Far from being useless, this geometry has proved to be an immensely powerful tool in diverse areas of mathematics, and the insights it continues to provide lie on the cutting edge of contemporary research.

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[10] The excellent book by Nikulin and Shafarevich [1987] is the only other work we know of in which a similar attempt is made.

[11] Here G is the group of projections. If we do a perspective drawing of figures in the plane, then the mapping from the plane to the “canvas” plane is called a perspectivity. A projection is then defined to be any sequence of perspectivities. Can you see why the set of projections should form a group?

[12] In more abstract settings it is necessary to add a fourth requirement of associativity, namely, A∘ (B∘ C) = (A∘ B)∘ C. Of course for transformations this is automatically true.

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See also:

1. The Hyperbolic Geometry of DMT Experiences (@Harvard Science of Psychedelics Club)
2. Materializing Hyperbolic Spaces with Gradient-Index Optics and One-Way Mirrors
3. An Intuitive Explanation of the Symmetry Theory of Valence
4. Principia Qualia: Part II – Valence
5. Problems you can solve just by looking at them: The meaning of Noether’s Theorem
6. Quantifying Bliss

Note: QRI‘s Symmetry Theory of Valence (STV) refers to the claim that valence (the pleasure-pain axis) manifests in the symmetry of the mathematical object that corresponds to each experience such that the mathematical features of that object are isomorphic to the phenomenal character of that experience. Using the lens of Klein’s conception of geometry, one could in turn give to STV a strictly geometrical interpretation. Namely, that the shape of one’s experience will be of high valence when it contains geometric invariants. In addition, Noether’s theorem (one of QRI’s lineages, which states that “every differentiable symmetry of the action of a physical system has a corresponding conservation law”) along with physicalism of consciousness suggests that for every symmetry in the mathematical object that corresponds to consciousness there is a corresponding preserved quantity. Thus, one could posit– assuming that the STV is correct– high-valence states might be extremely energy-efficient in addition to feeling good.

We are currently preparing a paper that ties all of these threads together in a way that, we believe, may turn out to have tremendous explanatory power. In particular, this formal account of valence will be able to explain succinctly a wide range of disparate and exotic empirical phenomena such as:

1. Why phenomenal symmetry during psychedelic experiences is correlated with more extreme valence values.
2. How and why Jhanas present the way they do, namely, as having:
   1. Extremely positive valence,
   2. Profound phenomenal simplicity, and
   3. Unusually high levels of energy-efficiency all at once.
3. The fact that moments of eternity are typically extremely blissful.
4. Why the phenomenal character of MDMA and 5-MeO-DMT manifest as both extremely symmetrical and high-valence.
5. Why intense regular stroboscopic stimulation has deep emotional effects.
6. And so much more…

Stay tuned!

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Thanks to Alfredo Valverde de Loyola for pointing me to this excerpt in the book Visual Complex Analysis.