This page is a sub-page of our page on Geometric Algebra.

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**Related KMR-pages**:

• Geometric Numbers

• The Evolution of Geometric Arithmetic

• The Vector Algebra War

• Metric Geometry

• Affine Geometry

• Projective Geometry

• Differential Geometry

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**Books**:

• John Stillwell (2016), Elements of Mathematics – From Euclid to Gödel,

Princeton University Press, ISBN 978-0-691-17854-7.

• John Stillwell (1999, (1989)), Mathematics and Its History, Springer, ISBN 0-387-96981-0.

• John Stillwell (1980), Classical Topology and Combinatorial Group Theory, Springer Verlag, ISBN 0-387-90516-2.

• Jeremy Gray (2007), Worlds Out of Nothing – A Course in the History of Geometry in the 19th Century, Springer, ISBN 1-84628-632-8.

• Geometric Algebra for Computer Science – An Object-Oriented Approach To Geometry

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**Other relevant sources of information**:

• Geometric Algebra (at Wikipedia)

• The Cauchy-Binét formula (which generalizes the product formula for determinants):

Det(AB) = Det(A) Det(B))

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The rest of this page is based on Stillwell (2016, sections 5.10, 5.11 and 10.9)

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/////// **Quoting** Stillwell (2016, section 5.10, p. 177):

**Geometry / Historical Remarks**

Euclid’s *Elements* is the most influential mathematical book of all time, and it gave mathematics a geometric slant that persisted until the twentieth century. Until quite late in the twentieth century, students were introduced to mathematical proof in the style of the *Elements*, and indeed it is hard to argue with a method that survived over 2000 years.

However, we know that Euclidean geometry has an algebraic description (vector space with an inner product), so we have another “eye” with which to view geometry, which we surely should use. To see how this new viewpoint came about, we review the history of elementary geometry.

Euclid’s geometry, illustrated in sections 5.2 to 5.4, has delighted thinkers through the ages with its combination of visual intuition, logic, and surprise. We can visualize what Euclid is talking about (points, lines, areas), we are impressed that so many theorems follow from so few axioms, and we are *convinced* by the proofs – even when the conclusion is unexpected. Perhaps the biggest surprise is the Pythagorean theorem. Who expected that the side lengths of a triangle would be related to their squares?

The Pythagorean theorem reverberates through Euclid’s geometry and all its decendants, changing from a theorem to a definition on the way (in a Euclidean space it holds virtually by definition of the inner product). For the ancient Greeks, the Pythagorean theorem led to \, \sqrt{ 2 } \, and hence to the unwelcome discovery of irrational lengths. As mentioned in section 5.3, the Greeks drew the conclusion that lengths in general are not numbers and that they cannot be multiplied like numbers. Instead, the “product” of two lengths is a rectangle, and equality of products must be established by cutting and reassembling areas. This was complicated, though interesting and successful for the theory of area. However, as mentioned in section 5.3, one needs to cut volumes into *infinitely* many pieces for a satisfactory theory of volume.

It should be emphasized that the subject of the *Elements* is not just geometry, but also number theory (Books VII to IX on primes and divisibility) and an embryonic theory of real numbers (Book V), in which arbitrary lengths are compared by means of their rational approximations. Later advances in geometry, particularly Descartes (1637) and Hilbert (1899), work towards fusing all these subjects of the *Elements* into a unified whole.

The first major conceptual advance in geometry after Euclid came with the introduction of coordinates and algebra, by Fermat and Descartes in the 1620s. These two mathematicians seem to have arrived at the same idea independently, with very similar results. For example, they both discovered that the curves with equations of degree 2 are precisely the conic sections (ellipses, parabolas, and hyperbolas). Thus they achieved not only a unification of geometry and algebra, but also a unification of Euclid’s geometry with the *Conics* of Appolonius (written some decades after the *Elements*). Indeed, they set the stage for *algebraic geometry*, where curves of arbitrarily high degree could be considered. When calculus emerged, shortly thereafter, algebraic curves of degree 3 and higher provided test problems for the new techniques for finding tangents and areas.

However, algebra and calculus did not merely create new geometric objects; they also threw new light on Euclid’s geometry. One of Descartes’ first steps was to break the taboo on multiplying lengths, which he did with the similar triangle constructions for product, quotient, and square root of lengths described in sections 5.5 and 5.6. Thus, for all algebraic purposes, lengths could now be viewed as numbers. And by giving an algebraic description of numbers constructible by straightedge and compass, Descartes paved the way for the nineteenth-century proofs of *non*-constructibility, such as the one given in section 5.9.

Descartes did not intend to construct a new foundation for geometry, with “points” being ordered pairs \, (x, y) \, of numbers, “lines” being point sets satisfying linear equations, and so on. He was content to take points and lines as Euclid described them, and mainly wished to solve geometric problems more simply with the help of algebra. (Sometimes, in fact, he tried to solve algebraic equations with the help of geometry.)

The question of new foundations for geometry came up only when Euclid’s geometry was challenged by *non-Euclidean* geometry in the 1820s. We say more about non-Euclidean geometry in the subsection below. The challenge of non-Euclidean geometry was not noticed at first, because the geometry was conjectural and not taken seriously by most mathematicians. This all changed when Beltrami (1868), building on some ideas of Gauss and Riemann, constructed *models* of non-Euclidean geometry, thereby showing that its axioms were just as consistent as those of Euclid.

Beltrami’s discovery was an earthquake that displaced Euclidean geometry from its long-held position at the foundation of mathematics. It strengthened the case for *arithmetization*: the problem of founding mathematics on the basis of arithmetic, including the real numbers, instead of geometry. Arithmetization was already under way in calculus and, thanks to Descartes, arithmetic was a ready-made foundation for Euclidean geometry. Beltrami’s models completed the triumph of arithmetization in geometry, because they too were founded on the real number and calculus.

By and large, geometry remains arithmetized today, with both Euclidean and non-Euclidean “spaces” situated among a great variety of *manifolds* that locally resemble \, \mathbb{R}^n , but with “curvature.” Among them, Euclidean geometry retains a somewhat privileged position as the one with zero Gaussian curvature

When a surface has a zero Gaussian curvature at all of its points, then it is a developable surface and the geometry of the surface is Euclidean geometry.

In this sense, Euclidean geometry is the simplest geometry, and the non-Euclidean spaces of Beltrami are among the next simplest, with constant (but negative) curvature at all points. In modern geometry of curved spaces, Euclidean spaces have a special place as *tangent spaces*. A curved manifold has a tangent space at each point, and one often works in the tangent space to take advantage of its simpler structures (particularly, its nature as a vector space).

It is all very well to base geometry on the theory of real numbers, but how well do we understand the real numbers? What is *their* foundation? Hilbert (1899) raised this question, and he had an interesting answer: the real numbers can be based on geometry! More precisely, they can be based on a “completed” version of Euclid’s geometry. Hilbert embarked on the project of completing Euclid’s axioms in the early 1890s, first with the aim of filling in some missing steps in Euclid’s proofs. As the project developed, he noticed that addition and multiplication arise from his axioms in a quite unexpected way, so that the field concept can be given a completely geometric foundation. Se section 5.11. Then, by adding an axiom guaranteeing that the line has no gaps, he was able to recover a complete “number line” with the usual properties of \, \mathbb{R} .

/////// **End of Quote** from Stillwell (2016)

/// Connect with the KMR page on Differential Geometry

• Geodesic curves

• Isometric Deformations

/// Connect with the KMR pages on Plane Curves and Surfaces:

• Conics

• Optical Properties of Conics

• Quadrics

• Confocal Quadrics

• Optical Properties of Quadrics

• Focal Surfaces

/////// **Quoting** Stillwell (2016, section 5.10, p. 180):

**Non-Euclidean Geometry**

In the 1820s, Janos Bolyai and Nikolai Lobachevsky independently developed a rival geometry to Euclid’s: a *non*-Euclidean geometry that satisfied all of Euclid’s axioms except the parallel axiom. The parallel axiom has a different character from the other axioms, which describe the outcome of finite constructions or “experiments” one can imagine carrying out:

1. Given two points, draw the line segment between them.

2. Extend a line segment for any given distance.

3. Draw a circle with given center and radius.

4. Any two right angles are equal (that is, one can be moved to coincide with the other).

The parallel axiom, on the other hand, requires an experiment that involves an indefinite wait:

5. Given two lines \, l \, and \, m , crossed by another line \, n \, making interior angles with \, l \, and \, m \, together less than two right angles, the lines \, l \, and \, m \, will meet, *if produced indefinitely* (figure 5.31).

Since the time of Euclid, mathematicians have been unhappy with the parallel axiom, and have tried to prove it from the other, more constructive, axioms. The most determined attempt was made by Saccheri (1733), who got as far as showing that, if non-diverging lines \, l \, and \, m \, did *not* meet, they would have a common perpendicular at infinity. This, Saccheri thought, was “repugnant to the nature of straight lines.” But it was not a contradiction, and in fact there is a geometry in which lines behave precisely in this fashion.

Bolyai and Lobachevsky worked out a large and coherent body of theorems that follow from Euclid’s axioms 1 to 4 together with the axiom:

5′. There exist two lines \, l \, and \, m , which do not meet, although they are crossed by another line \, n \, making interior angles with \, l \, and \, m together less than two right angles.

Their results were eventually published in Lobachevsky (1829) and Bolyai (1832) (the latter an appendix to a book by Bolyai’s father). They found no contradiction arising from this set of axioms, and Beltrami (1868) showed that no contradiction exists, because axioms 1, 2, 3, 4, and 5′ are satisfied under a suitable interpretation of the words “point,” “line,” and “angle”. (We sketch one such interpretation below.) Thus Euclid’s geometry had a rival, and deciding how to interpret the terms “point,” “line,” “distance,” and “angle” became an issue.

As we have seen, Euclid’s axioms had a ready-made interpretation in the coordinate geometry of Descartes: a “point” is an ordered pair \, (x, y) \, of real numbers, a “line” consists of the points satisfying an equation \, ax + by + c = 0 , and the “distance” between \, (x_1, y_1) \, and \, (x_2, y_2) \, equals \, \sqrt{ {(x_2 - x_1)}^2 + {(y_2 - y_1)}^2 } .

For Bolyai’s and Lobachevsky’s axioms, Beltrami found several elegant interpretations, admittedly with a somewhat complicated concept of “distance.” The simplest is probably the *half-plane* model, in which:

• “points” are points of the upper half-plane; that is, pairs \, (x, y) \, with \, y > 0 ,

• “lines” are the open semicircles in the upper half-plane with their centers on the \, x -axis, and the open half-lines \, \{ \, (x, y) \, : x = a \, , \, y > 0 \, \} ,

• the “distance” between “points” \, P \, and \, Q \, is the integral of \, {\sqrt{ dx^2 + dy^2 }} / y \, over the “line” connecting \, P \, and \, Q .

It turns out that “angle” in this model is just the ordinary angle between curves; that is, the angle between their tangents. This leads to some beautiful pictures of non-Euclidean geometric configurations, such as the one in figure 5.32.

/// Figure 5.32 GOES HERE

It shows a tiling of the half-plane by triangles which are “congruent” in the sense of non-Euclidean distance. In particular, they each have angles \, \pi/2, \pi/3, \pi/7 . Knowing that they are congruent one can get a sense of non-Euclidean distance. One can see that the \, x -axis is infinitely far away – which explains why it is not included in the model – and perhaps also see that a “line” is the shortest path joining its endpoints, if one estimates distance by counting the number of triangles along a path.

It is also clear that the parallel axiom fails in the model. Take for example the “line” that goes straight up the center of the picture and the point on the far left with seven “lines” passing through it. Several of the latter “lines” do not meet the center line.

/////// **End of Quote** from Stillwell (2016)

/// Connect with the KMR pages on:

• Conformal Mapping

• Inversion

• Möbius transformations

• Steiner Circles

• Stereographic Projection

• The Riemann-Poincaré model

/////// **Quoting** Stillwell (2016, section 5.10, p. 182):

**Vector Space Geometry**:

Grassmann (1861), the *Lehrbuch der Arithmetik* mentioned in section 1.9, was not Grassmann’s only great contribution to the fundations of mathematics. The first was his *Ausdehnungslehre* (“Extention theory”), Grassmann (1844), in which he based Euclidean geometry on vector spaces. The *Ausdehnungslehre*, like the unfortunate *Lehrbuch*, was greeted at first with total incomprehension. The only person to review it was Grassmann himself, and its virtually unsold first edition was destroyed by the publisher. The full story of the *Ausdehnungslehre*, its genesis and aftermath, is in the Grassmann biography by Petsche (2009).

Grassmann was let down by an extremely obscure style, and terminology of his own invention, in attempting to explain an utterly new and complex idea: that of a real *n*-dimensional vector space with an *outer product*.

[**Footnote 6**: We will not define the concept of outer product, but it underlies the concept of determinant, then the center of what was called “determinant theory” and now at a less central position in today’s linear algebra]

The simpler concept of *inner* product was in Grassmann’s view an offshoot of the outer product – one he planned to expound in *Ausdehnungslehre*, volume two. Not surprisingly, the second volume was abandoned after the failure of the first.

Thus, Grassmann’s contributions to geometry might well have been lost – if not for a marvelous stroke of luck. In 1846, the Jablonowskian Society of Science in Leipzig offered an essay prize on a question only Grassmann was ready to answer: developing a sketchy idea of Leibniz about “symbolic geometry.” (The aim of the prize was to commemorate the 200th anniversary of Leibniz’s birth.) Grassmann (1847) duly won the prize with a revised version of his 1844 theory of vector spaces – one that put the inner product and its geometric interpretation at the center of the theory. He pointed out that his definition of inner product was motivated by the Pythagorean theorem, but that, once the definition was given, all geometric theorems follow from it by pure algebra.

Despite its greater clarity, Grassmann’s essay was not an overnight success. However, his ideas gathered enough momentum to justify a new version of the *Ausdehnungslehre*, Grassmann (1862), and they were gradually adopted by other mathematicians. Peano was among the first to appreciate Grassmann’s ideas, and was inspired by him to create the first axiom system for real vector spaces in Peano (1888), section 72. Klein (1909) brought Grassmann’s geometry to a wider audience by restricting it to three dimensions. Klein mentioned the inner product, but his version of Grassmann relied mainly on the determinant concept, which gives convenient formulas for areas and volumes.

/////// **End of Quote** from Stillwell (2016)

/// Connect with the Cauchy-Binét formula which generalizes the product formula for determinants.

/////// **Quoting** Stillwell (2016, section 5.11, p. 184):

**Non-Euclidean Geometry**

In this book I have made the judgement that non-Euclidean geometry is more advanced than Euclidean. There is ample historical reason to support this call, since non-Euclidean geometry was discovered more than 2000 years after Euclid. The “points” and “lines” of non-Euclidean geometry can be modeled by Euclidean concepts, so they are not advanced in themselves, but the concept of non-Euclidean distance surely is.

One way to see this is to map a portion of the non-Euclidean plane onto a piece of surface \, S \, in \, \mathbb{R}^3 \, in such a way that distance is preserved. Then ask: how simple is \, S \, ? Well, the simplest possible \, S \, is the trumpet-shaped surface shown in figure 5.33 and known as the *pseudosphere*.

**Henry Segerman on the pseudosphere** (YouTube):

The pseudosphere is obtained by rotating the *tractrix* curve with equation

\, x \, = \, \ln {\dfrac{1 + \sqrt{1 - y^2}}{y}} - \sqrt{1 - y^2} \,

about the \, x -axis.

The formula is complicated enough, but the conceptual complication is much greater. It is possible to compare only small pieces of the non-Euclidean plane with small pieces of a surface in \, \mathbb{R}^3 , because a complete non-Euclidean plane does not “fit” smoothly in \, \mathbb{R}^3 . This was proved by Hilbert (1901). The pseudosphere, for example, represents just a thin wedge of the non-Euclidean plane, the edges of which are two non-Euclidean lines that approach each other at infinity. These two edges are joined together to form the tapering tube shown in figure 5.33.

In contrast, Euclidean geometry is modeled by the simplest possible surface in – the plane!

[**Footnote 7**: It may be thought unfair to the hyperbolic plane to force it into the Euclidean straightjacket of \, \mathbb{R}^3 . Might not the Euclidean plane look equally bad if forced to live in non-Euclidean space? Actually, this is not the case. Beltrami showed that the Euclidean plane fits beautifully into non-Euclidean space, where it is a “sphere with center at infinity.”]

/////// **End of Quote** from Stillwell (2016)

**Tractrix and Catenary – Involute and Evolute of each other**

The catenary is the evolute of the tractrix, and the tractrix is an involute of the catenary:

/// Connect with the Beltrami-Klein model

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/// Connect with Horocycles in Hyperbolic Geometry

/// Connect with Pseudospherical Surfaces

**Concentric ‘ordinary’ circles in the Beltrami-Klein model **

with a common center (inside the hyperbolic space) and varying radius:

**Concentric horocycles in the Beltrami-Klein model**

with a common center (at infinity)and varying radius:

**Concentric hypercycles in the Beltrami-Klein model **

with a common center (outside the hyperbolic space) and varying radius:

**Illuminating hyperbolic geometry**

(Henry Segerman and Saul Schleimer on YouTube):

**Stereographic projection of the Riemann sphere** (by fsgm1):

Stereographic projection takes a circle on the sphere to a circle in the plane, but it does NOT map the center point of the original circle to the center point of the image circle. As shown in the movie below, the point that is mapped to the center point of the image circle is the vertex point of the cone that is tangent to the sphere along the original circle.

**Stereographic projection of a circle on the sphere onto the plane** (moving camera):

The interactive simulation that created this movie.

=======

/////// **Quoting** Stillwell (2016, section 5.11, p. 186):

**Numbers and Geometry**

Now let us return to the axioms of Hilbert (1899), and what they tell us about the relationship between numbers and geometry. Hilbert’s axioms probably capture Euclid’s concept of the line (with the finer structure explored in Book V of the *Elements*). So, given the Descartes “model” \, \mathbb{R}^2 \, of Euclid’s axioms, Hilbert has shown Euclidean geometry to be essentially *equivalent* to the algebra of \, \mathbb{R} .

However, algebraists and logicians now prefer not to use the full set \, \mathbb{R} \, in geometry. They point out that the set of constructible numbers suffices, because Euclid’s geometry “sees” only the points arising from straightedge and compass constructions. Thus

one can get by with an algebraically defined set of points, which is only a “potential” infinity, in contrast to the “actual” infinity \, \mathbb{R} .

Logicians also prefer the theory of constructible numbers because its “consistency strength” is less than that of the theory of \, \mathbb{R} . That is, it is easier to prove the consistency of constructible numbers (and hence the consistency of Euclid’s axioms) than it is to prove the consistence of the theory of \, \mathbb{R} \, (and hence the consistency of Hilbert’s axioms).

/////// **End of Quote** from Stillwell (2016)

/////// **Quoting** Stillwell (2016, section 5.11, p. 186):

**Geometry and “Reverse Mathematics”**

In recent decades, mathematical logicians have developed a field called *reverse mathematics*, whose motivation was stated by Friedman (1975) as follows:

*When the theorem is proved from the right axioms,
the axioms can be proved from the theorem*.

As logicians understand it, reverse mathematics is a technical field, concerned mainly with theorems about the real numbers (see section 9.9). However, if we understand reverse mathematics more broadly as the search for the “right axioms,” then reverse mathematics began with Euclid.

He saw that the parallel axiom is the right axiom to prove the Pythagorean theorem, and perhaps the reverse – that the Pythagorean theorem proves the parallel axiom (given his other axioms). The same is true of many other theorems of Euclidean geometry, such as the theorem of Thales and the theorem that the angle sum of a triangle is \, \pi . All of these theorems are equivalent to the parallel axiom, so it is the “right axiom” to prove them.

To formalize this and other investigations in reverse mathematics we choose a *base theory* containing the most basic and obvious assumptions about some area of mathematics. It is to be expected that the base theory will fail to prove certain interesting but less obvious theorems. We then seek the “right” axiom or axioms to prove these theorems, judging an axiom to be “right” if it implies the theorem, and conversely, using only assumptions from the base theory.

Euclid began with a base theory now known as *neutral geometry*. It contains basic assumptions about points, lines, and congruence of triangles but *not* the parallel axiom. He proved as many theorems as he could before introducing the parallel axiom – only when it was needed to prove theorems about the area of parallelograms and ultimately the Pythagorean theorem. He also needed the parallel axiom to prove the theorem of Thales and that the angle sum of a triangle is \, \pi . We now know, conversely, that all of these theorems imply the parallel axiom in neutral geometry, so the latter is the “right” axiom to prove them.

Neutral geometry is also a base theory for *non*-Euclidean geometry, because the latter is obtained by adding to neutral geometry the “non-Euclidean parallel axiom” stating that there is more than one parallel to a given line through a given point.

Grassmann’s theory of real vector spaces, as we have seen, can also be taken as a base theory for Euclidean geometry. It is quite different from the base theory of neutral geometry because the Euclidean parallel axiom *holds* in real vector spaces, and so does the theorem of Thales. Nevertheless, this new base theory is not strong enough to prove the Pythagorean theorem, or indeed to say anything about angles. Relative to the theory of real vector spaces, the “right” axiom to prove the Pythagorean theorem is existence of the inner product, because we can reverse the implication by using the Pythagorean theorem to define distance, hence angle and cosine, and then define the inner product by

\, u \cdot v \, = \, |u| \cdot |v| \cos \theta .

This raises the possibility of adding a different axiom to the theory of real vector spaces and obtaining a different kind of geometry, just as we obtain non-Euclidean geometry from neutral geometry by adding a different parallel axiom. Indeed we can, and simply by asserting the existence of a different kind of inner product. The inner product introduced by Grassmann is what we now call a *positive-definite* inner product, characterized by the property that \, u \cdot u = 0 \, only if \, u \, is the zero vector.

Non-positive-definite inner products also arise quite naturally. Probably the most famous one is the one on the vector space \, \mathbb{R}^4 \, that defines the *Minkowski space* of Minkowski (1908). If we write the typical vector in \, \mathbb{R}^4 \, as \, u = (w, x, y, z) \, then the Minkowski inner product is defined by

\, u_1 \cdot u_2 = -w_1 w_2 + x_1 x_2 + y_1 y_2 + z_1 z_2 .

In particular, the length \, |u| \, of a vector in Minkowski space is given by

\, {|u|}^2 = u \cdot u = -w^2 + x^2 + y^2 + z^2 ,

so \, |u| \, can certainly be zero when \, u \, is not the zero vector.

Minkowski space is famous as the geometric model of Einstein’s *special relativity theory*. In this model, known as *flat spacetime*, \, x, y, \, and \, z \, are the coordinates of ordinary three-dimensional space and \, w = c t , where \, t \, is the time coordinate and \, c \, is the speed of light. As Minkowski (1908) said:

*The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality*.

Undoubtedly, relativity theory put non-positive-definite inner products on the map, making them as real and important as the ancient concept of distance. But in fact such inner products had already been considered by mathematicians, and one of them is involved in a model of non-Euclidean geometry discovered by Poincaré (1881) – the so-called *hyperboloid model*.

To see where the hyperboloid comes from, consider the three-dimensional Minkowski space of vectors \, u = (w, x, y) \, with one time coordinate \, w \, and two space coordinates \, x, y . In this space, where \, {|u|}^2 = -w^2 + x^2 + y^2 , we consider the “sphere of imaginary radius”

\, \{ \, u \, : \, |u| = \sqrt{ -1 } \, \} = \{ \, (w, x, y) \, : \, -w^2 + x^2 + y^2 = -1 \, \} .

This “sphere” consists of the points \, (w, x, y) \, in \, \mathbb{R}^3 \, such that

\, w^2 - x^2 - y^2 = 1 ,

so it is actually a hyperboloid; namely, the surface obtained by rotating the hyperbola \, {|u|}^2 = w^2 - y^2 = 1 in the \, (w, y) -plane about the \, w -axis. If we take “distance” on either sheet of the hyperboloid to be the Minkowski distance, it turns out to be a model of the non-Euclidean plane. As with the other models of the non-Euclidean plane, calculating distance is a little troublesome so I omit the details.

/////// Figure 5.34 GOES HERE

Instead, I offer figure 5.34 – a black-and-white version of a picture due to Konrad Polthier of the Freie Universität, Berlin – which shows what the triangle tesselation of figure 5.32 looks like in the hyperboloid model. (It also relates the hyperboloid model to another model – the * conformal disk model* – which is a kind of intermediary between the hyperboloid model and the half-plane model shown in figure 5.32.)

This elegant relationship between Minkowski space and the non-Euclidean plane has been used for some textbook treatments of non-Euclidean geometry, such as Ryan (1986). Just as the positive-definite inner product is the “right axiom” to develop Euclidean geometry over the base theory of real vector spaces, the Minkowski inner product is the “right axiom” to develop non-Euclidean geometry.

/////// **End of Quote** from Stillwell (2016)

/////// **Quoting** Stillwell (2016, section 5.11, p. 190):

**Projective Geometry**

Another wonderfully “right” axiom is the theorem discovered by Pappus a few hundred years after Euclid. Pappus viewed his theorem as part of Euclidean geometry, but it does not really belong there. It is unlike typical Euclidean theorems in making no mention of length or angle, so its home should be a geometry that does not involve these concepts. The statement of the theorem is the following, which refers to the configuration shown in figure 5.35.

/////// Figure 5.35 GOES HERE

**Theorem of Pappus**. *If A, B, C, D, E, F are points of the plane lying alternately on two lines, then the intersections of the pairs of lines AB and DE, BC and EF, CD and FA, lie on a line*.

The Pappus theorem has a Euclidean *proof* using the concept of length, and also a coordinate proof using linear equations to define lines. But it seems to have no proof using only concepts “appropriate” to its statement: points, lines, and the membership of points in lines. The appropriate setting for the Pappus theorem is the *projective geometry* of the plane, a geometry which tries to capture the behaviour of points and lines in a plane without regard to length and angle. In projective geometry, configurations of points and lines are considered the same if one can be projected onto the other. Projection can of course change lengths and angles, but the straightness of lines remains, as does the membership of points in lines.

If one seeks axioms for projective plane geometry, the following come easily to mind:

1. Any two points belong to a unique line.

2. Any two lines have a unique point in common.

3. There are four points, no three of which are in the same line.

The first axiom is also one of Euclid’s. The second disagrees with Euclid in the case of parallel lines, but projective geometry demands it, because even parallel lines can be projected so that they meet – on the “horizon.” The third axiom is there to ensure that we really have a “plane,” and not merely a line. However, these simple axioms are very weak, and it can be shown that they do not suffice to prove the Pappus theorem. They do, however, form a natural base theory to which other axioms about points and lines can be added.

What are the “right axioms” to prove the Pappus theorem? The answer is no less than the Pappus theorem itself, thanks to what the Pappus theorem implies; namely, that the abstract plane of “points” and “lines” can be given *coordinates* which form a *field* as defined in section 4.3. Thus geometry springs fully armed from the Pappus theorem! The Pappus axiom (as we should now call it) is the right axiom to prove coordinatization by a field, because such a coordinatization allows us to prove the Pappus axiom. As remarked above, this follows by using linear equations to define lines. The field properties then enable us to find intersections of lines by solving linear equations, and to verify that points of intersection lie on a line.

The idea of reversing the coordinate approach to geometry began with von Staudt (1847), who used the Pappus theorem to define addition and multiplication of points on a line. Hilbert (1899) extended this idea to prove that the coordinates form a field, but he had to assume another projective axiom, the so-called *theorem of Desargues*, which was discovered around 1640. (Roughly speaking, the Pappus axiom easily implies that addition and multiplication are commutative, while the Desargues axiom easily implies that they are associative.)

Quite remarkably, considering how long the Pappus and Desargues theorems had been around, Hessenberg (1905) discovered that the Pappus theorem implies the Desargues theorem. So the single Pappus axiom is in fact *equivalent* to coordinatization of the plane by a field.

This reversal of the Pappus theorem also tells us something remarkable about algebra: the nine field axioms follow from four geometric axioms – the three projective plane axioms plus Pappus!

/////// **End of Quote** from Stillwell (2016)

/// Connect with the KMR pages on:

• Projective Geometry

• Projective metrics

• The Euclidean Degeneration

/////// **Quoting** Stillwell (2016, ch 10.9 p. 384):

**Groups and Geometry**

Geometry is one of the most convincing cases where the group concept captures a phenomenon that mathematicians wish to study. At the same time, it is a sign of the depth of the group concept that its relationship with geometry was not uncovered until geometry had been studied for over 2000 years. One does not notice the group concept in geometry until several kinds of geometry have come to light – most importantly, projective geometry. It was projective geometry in particular that led Klein (1872) to notice the role of groups in geometry, and to *define* geometry as the study of *groups and their invariants*

The real projective line \, \mathbb{R} \cup \{ \, \infty \, \} \, studied in section 10.4, involves perhaps the simplest example of an interesting group and an interesting invariant. The group is the group of linear fractional functions,

\, f(x) = \dfrac{ax+b}{cx+d} \, where \, a, b, c, d \in \mathbb{R} \, and \, ad - bc \neq 0 ,

under the operation of function composition. That is, given functions

\, f_1(x) = \dfrac{a_1 x + b_1}{c_1 x + d_1} \, and \, f_2(x) = \dfrac{a_2 x + b_2}{c_2 x + d_2} ,

we form the function \, f_1(f_2(x)) , which corresponds to performing the projection corresponding to \, f_2 , then the projection corresponding to \, f_1 . This group is not commutative. For example if \, f_1(x) = x + 1 \, and \, f_2(x) = 2 x \,

then

\, f_1(f_2(x)) = 2 x + 1 , whereas \, f_2(f_1(x)) = 2 (x + 1) = 2x + 2 ,

so \, f_1 f_2 \neq f_2 f_1 .

Nevertheless, we can find the invariant of the linear fractional transformations without knowing much about their group structure. It suffices to know that they are generated by the simple functions \, x \, \mapsto \, x + b \, , \, x \, \mapsto \, a x \, for \, a \neq 0 , and \, x \, \mapsto \, 1/x .

As we saw in section 10.4, traditional geometric quantities such as length, or the ratio of lengths, are *not* invariant under all linear fractional transformations. However, some simple computations with the generating transformations show the invariance of the cross-ratio

\, [ \, p, q \, ; \, r, s \, ] \, = \, \dfrac{(r-p) (s-q)}{(r-q) (s-p)}

for any four points on the real projective line. The invariance of the cross-ratio under projection was already known to Pappus, and was rediscovered by Desargues around 1640. However, its *algebraic* invariance had to wait for the identification of the appropriate group by Klein.

With hindsight, we can also see how length, and the ratio of lengths, are also algebraic invariants. The length of the line segment from \, p \, to \, q , \, | \, p - q \, | , is the invariant of the group of *translations* \, x \, \mapsto \, x + b \, of \, \mathbb{R} . We call \, \mathbb{R} \, the *Euclidean line* when it is subject to these transformations, because they make any point “the same” as any other point, as Euclid intended.

The ratio of lengths, \, \dfrac{p - r}{p - q} , for any three points \, p, q, r , is the invariant group of similarities \, x \, \mapsto \, a x + b \, where \, a \neq 0 . These transformations are called *affine*, and when \, \mathbb{R} \, is subject to these transformations it is called the *affine line*.

The deep difference between the Euclidean and affine lines and the projective line is of course the point at infinity \, \infty . The point \, \infty \, arises on the projective line because the point \, 0 \, otherwise has no place to go under the map \, x \, \mapsto 1/x . It is appropriate to call this point “infinity” because \, 0 \, is the limit of \, 1/n \, as \, n \rightarrow \infty , so the image of \, 0 \, under \, x \, \mapsto 1/x ought to be the “limit” of \, n \, as \, n \rightarrow \infty . Nevertheless, it is also appropriate to view the projective line as a finite object; namely, the circle.

This is because \, 0 \, is also the limit of \, -1/n \, as \, n \rightarrow \infty , so the image of \, 0 \, should be the limit of \, -n \, as \, n \rightarrow \infty . Thus we “approach \, \infty ” as we travel along the line in either direction. We can realize the common “limit” of \, n \, and \, -n \, by an actual point if we map \, \mathbb{R} into a circle as shown in figure 10.15.

///// Figure 10.15 GOES HERE

The topmost point of the circle neatly corresponds to the “limit” of both \, n \, and \, -n , because it is the actual limit of their images on the circle. So if we let the topmost point correspond to \, \infty \, we can view the circle as a continuous and bijective image of the real projective line.

/////// **End of Quote** from Stillwell (2016)

/// Connect with the concept of a group acting on a set.

/////// **Quoting** Stillwell (2016, ch 10.9 p. 386):

**Affine Geometry**

The affine transformations of the line \, \mathbb{R} , mentioned above, extend to the projective line \, \mathbb{R} \cup \{ \, \infty \, \} \, simply by sending the point \, \infty \, to itself. Similarly, there are affine transformations of the plane, and of the projective plane. They are the projections that send finite points to finite points, and points at infinity to points at infinity. In particular, *they send parallels to parallels*. The affine geometry of the plane studies the images of the plane obtained by such projections. The term “affine” was introduced by Euler (1748b), motivated by the idea that images related by affine transformations have an “affinity” with one another.

Like projective geometry, affine geometry has an artistic counterpart. It may be seen in classical Japanese woodblock prints of the eighteenth and nineteenth centuries, such as Harunoby’s *Evening Chime of the Clock* shown in figure 10.16. This print, which dates from around 1766, clearly shows the preservation of parallel lines, since all parallel lines in the scene actually look parallel. This makes the scene look “flat,” though perfectly consistent. In fact, the picture shows how the scene would appear when viewed from infinitely far away, with infinite magnification.

///// Figure 10.16: Affine geometry in art. GOES HERE

Image courtesy of www.metmuseum.org

The art of representing parallel lines consistently demands some skill, as it is possible to go badly wrong. Figure 10.17 shows an example, *The Birth of St Edmund*. I have previously used this example, for example in Stillwell (2010, p. 128) – as a failure of projective geometry, because the parallel lines do not converge towards any horizon. But now I think it is better seen as a failure of affine geometry. The artist really *wants* the parallel lines to look parallel, but he has utterly failed to do so consistently.

Since affine maps pair finite points with finite points, *an affine image cannot include the horizon*. Most Japanese prints comply with this condition, but sometimes even an eminent artist slips up. Figure 10.18 shows the print *Cat at Window* by the nineteenth-century master Hiroshige. The interior is fine, but it is not compatible with the exterior, which includes the horizon.

///// Figure 10.18 Cat at Window – subtle failure of affine geometry GOES HERE

/////// **End of Quote** from Stillwell (2016)

/// Connect with the KMR pages on Oscar Reutersvärd and M.C. Escher.

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