5 Calculating with points on lines

Newsgroups: sci.math Subject: Re: I'm looking for axioms and proof in math texts Date: 6 Aug 2003 02:53:12 -0700 From: [email protected] (prometheus666)

[...] and being passing familiar with the number line -- I'm sure if I don't say passing familiar somebody here will say, "You have to know vector tensor shmelaculus in 15 triad synergies to really understand the number line. It's not even called that, it's called the real torticular space." or something to that effect -- [...]

Found on the WWW

In the last section we have seen that cross ratios form a universal link from projective geometry to the underlying coordinate field. In this chapter we want to elaborate more on this topic. Thereby we will see that one can recover the structure of the real numbers from the purely geometric setup of the real projective plane. A line has (seemingly) by far less geometric structure than the plane. In the plane there can be collinear and non-collinear points, and we have seen that being collinear is a crucial property that stays invariant under projective transformations. On a line all triples of points are automatically collinear, so we cannot hope for this being a crucial characterizing property of projective transformations. The simplest non-trivial property that stays invariant under a projective transformation on a line is the specific value of the cross ratio of a quadruple of points. It will be the ultimate goal of this chapter to show that a map that fixes a certain value for the cross ratio of four points automati- cally fixes all cross ratios and will automatically be a projective transforma- tion. From there we will prove Theorem 3.3 that claimed that any bijective 2 transformation in RP that preserves collinearity is automatically a projective transformation. 86 5 Calculating with points on lines 5.1 Harmonic points

We will now focus on the geometric properties of quadruples of points on a line that have cross ratio of (a, b; c, d) = 1. It is clear that if we fix points a, b, c on a line then this condition uniquely−determines the position of the last point d. So, from a projective viewpoint there is essentially only one point set (up to projective transformations) that satisfies this condition. However, under the aspect of coordinates there are many interesting cases that can be considered.

Definition 5.1. Two pairs of points (a, b) and (c, d) are in harmonic position with respect to each other if the cross ratio (a, b; c, d) is 1. − First we observe that this definition is well defined, this means that it really is invariant under transposition of the points in the pairs. Lemma 5.1. If (a, b; c, d) = 1 we also have (b, a; c, d) = (a, b; d, c) = (c, d; a, b) = 1. − − Proof. This lemma is an immediate consequence of Theorem 4.2 and the fact that 1/( 1) = 1. − − #" In fact, if a = b and c = d and the cross ratio (a, b; c, d) stays invariant under interchange $ of the last$ two letters the points must be in harmonic position. This is the case since we then have 1 1 (a, b; c, d) = = (a, b; d, c) (a, b; c, d)

The only values for the cross ratio that satisfy this equality are 1 and 1. Since a = b and c = d this excludes the first case. − If a, $b, c on a lin$ e ! are given the fourth harmonic point can be constructed in the following way: Start with an auxiliary point o, • connect this point to a, b, and c, • on join(o, c) choose another auxiliary point p, • construct d by • d = meet(!, join(meet(join(o, a), join(p, b)), meet(join(o, b), join(p, s))).

Figure 5.1 shows the construction. The construction is not feasible when a and b coincide.

Lemma 5.2. Independent of the choice of the auxiliary points o and p the construction presented above will end up with the same point d. This point satisfies (a, b; c, d) = 1. − 5.1 Harmonic points 87

o

PSfrag replacements a! c! b!

p

a c b d

Fig. 5.1. Construction of harmonic points.

Proof. For the labelling we refer to Figure 5.1 . The point o can be considered as the center of projection from ! to the line spanned by b! and a!. Thus we get (a, b; c, d) = (a!, b!; c!, d). Similarly, p could be considered as a center of projection which implies (a!, b!; c!, d) = (b, a; c, d). Taking these two relations together we obtain (a, b; c, d) = (b, a; c, d) which implies (a, b; c, d) = 1. This also implies the independence of d from the specific choice of o and p−. #" There are a few remarkable collections of relative positions of points that generate a harmonic position. The following collection of equations collects some of them. For sake of simplicity we identify points on a line with the corresponding real numbers on R (including the point for the point at infinity). ∞

Lemma 5.3. Let x R and y R be arbitrary numbers. We have ∈ ∈ (i) ( x, x; 0, ) = 1, (ii) (0−, 2x; x, ∞) = −1, x+y∞ − (iii) (x, y; 2 , ) = 1, (iv) ( 1, 1; x, 1∞/x) = −1, (v) (−x, x; 1, x2) = −1. − − Proof. We start with a proof of property (v). We can simply check it via a calculation. representing determinants as differences of numbers (as described in Section 4.4.1.) We can calculate:

( x 1)(x x2) x( x 1)(1 x) − − − = − − − = 1 ( x x2)(x 1) x( 1 x)(x 1) − − − − − − − Statement (iv) is a consequence of statement (v) and arises by dividing all entries by x (this is a projective transformation and does not change the cross ratio). Statement (i) can be considered as a limit case of (iv) if x 0. Statement (ii) arises from (i) by adding x to all entries (this is again→a projective transformation). Statement (iii) arises from (i) also by scaling and shifting. #" 88 5 Calculating with points on lines 5.2 Projective scales

We now fix three distinct points on a line, and call them 0, 1 and . Every point x on the line can be associated with a unique cross ratio (0,∞ ; 1, x). This construction allows us to equip the line with a scale that behave∞s “as if” point 0 is the origin, 1 is a unit point, and point is infinitely far away. In particular we have: ∞ Lemma 5.4. The following equations hold (i) (0, ; 1, 0) = 0 (ii) (0, ∞; 1, 1) = 1 (iii) (0, ∞; 1, ) = ∞ ∞ ∞ Proof. The lemma is a direct application of the definition of the cross ratio. #" The next lemma shows that the cross ratios with respect to these three points can be used to reconstruct the coordinates of a point from its geometric location. For this we set 0, 1 and to the position 0, 1 and on a real number line. ∞ ∞

Lemma 5.5. Assume that we have the specific homogeneous coordinates 0 = (0, 1), 0 = (1, 1), = (1, 0), x = (x, 1) for x R, then we have (0, ; 1, x) = x. ∞ ∈ ∞

Proof. We can proof this fact by direct calculation.

0 1 1 x det det 1 1 0 1 (0, ; 1, x) = ! " ! " = x ∞ 0 x 1 1 det det 1 1 0 1 ! " ! " #" If we single out three distinct points on a line we can always consider them as a projective basis and refer all measurements to these three points. By this we can assign to every point on the line a real number. The resulting number is then invariant under projective transformations. In a way this is like reconstructing the original scenery from a perspectively distorted photograph if the original positions of three points are given.

5.3 From geometry to real numbers

From 1850 to 1856 the mathematician Karl Georg Christian von Staudt (1798- 1867) published a series of books under the title “Beitra¨ge zur Geometrie der Lage”. In these books von Staudt develops a completely synthetic (this 5.3 From geometry to real numbers 89 means there are no calculations) setup for projective geometry. In his setup for projective geometry he works on a similar axiomatic level as Euclid did it for Euclidean Geometry. One of the major achievements of von Staudt’s work was to provide a method that starts with a purely projective setup and reconstruct an underlying algebraic structure. In particular, he was able to reconstruct the field structure of the underlying coordinate field from properties of geometric constructions. We will not follow exactly these lines here. However, we will demonstrate how intimately the concepts of real numbers, cross ratios and projective trans- formations are interwoven. In particular, we will provide the promised proof 2 of Theorem 3.3 that claimed, that whenever we have a bijective map in RP that maps collinear points to collinear points, then it must be a projective transformation. We will even work a little harder and provide the proof for a similar fact on the projective line from which Theorem 3.3 will easily follow. We will show:

1 1 Theorem 5.1. Let τ: RP RP be a bijective map with the property that harmonic quadruples of poin→ts are mapped to harmonic quadruples of points. Then τ is a projective transformation.

We subdivide the proof of this rather strong theorem in several smaller parts. A bijective map with the property that harmonic quadruples of points are mapped to harmonic quadruples of points will be called a harmonic map. Our strategy will be to first show that harmonic maps are very rigid objects. We will see that harmonic maps that preserve the base points of a projective scale will induce a field automorphism. For this we will first fix a projective 1 1 scale 0, 1 and on RP . We can identify each point x of RP by its ∞ 1 1 \ {∞} real coordinate (0, ; 1, x) R. If τ: RP RP is any bijective map with ∞ ∈ → τ( ) = this induces a bijection fτ : R R according to ∞ ∞ → τ(p) = q f ((0, ; 1, p)) = (0, ; 1, q). ⇐⇒ τ ∞ ∞ The behaviour of f is mainly determined by the fact, that τ is harmonic. We obtain

Lemma 5.6. Let τ be a harmonic map that in addition satisfies. τ(0) = 0, τ(1) = 1, τ( ) = . Then the function fτ : R R satisfies the following relations: ∞ ∞ →

(i) fτ (0) = 0, (ii) fτ (1) = 1, x+y fτ (x)+fτ (y) (iii) fτ ( 2 ) = 2 , (iv) fτ (2x) = 2fτ (x), (v) fτ (x + y) = fτ (x) + fτ (y), (vi) fτ ( x) = fτ (x) −2 − 2 (vii) fτ (x ) = fτ (x) , (viii) f (x y) = f (x) f (y), τ · τ · τ 90 5 Calculating with points on lines

Proof. We can prove the statements one after another. Thereby we make heavily use of the fact the if the positions of points a, b, c on a line are fixed, then the fourth harmonic point d is determined uniquely. Within this proof 1 we will freely identify points on RP and the corresponding real numbers with respect to the basis 0, 1, . Statements (i) and (ii)∞are direct reformulations of the facts that τ(0) = 0 and τ(1) = 1. Statement (iii) holds for the following reason. For any x, y the pairs of points (x, y) and ((x + y)/2, ) are in harmonic position (com- pare Lemma 5.3 (iii)). Thus if we apply∞a harmonic map the image pairs (f(x), f(y)) and (f((x + y)/2), f( )) are harmonic again. Since we have ∞ fτ (x)+fτ (y) f( ) = the point f((x + y)/2) must be 2 . Several of the other sta∞temen∞ts follow by a similar reasoning. In these cases we only refer to the corresponding harmonic sets. Statement (iv) holds since the pair (0, 2x) is harmonic with the pair (x, ). Statement (v) is now a consequence of (iii) and (iv). Statement (vi) is a∞consequence of (v) and (i). Statement (vii) holds since ( x, x; 1, x2) is harmonic together with (ii) and (vi). Statement (viii) require−s a little calculation. By (vii), (v) and (iv) we can conclude:

2 2 2 2 fτ (x) + 2fτ (x y) + fτ (y) . = fτ (x ) + 2fτ (x y) + fτ (y ). · 2 ·2 = fτ (x + 2xy + y ) 2 = fτ ((x + y) ) 2 = (fτ (x + y)) = f (x)2 + 2f (x) f (y) + f (y)2. τ τ · τ τ Comparing the first and the last expression in this chain of equalities we obtain f (x y) = f (x) f (y). τ · τ · τ #"

The most important statements in the last theorem were the facts that we have f (x + y) = f (x) + f (y) and f (x y) = f (x) f (y). In other τ τ τ τ · τ · τ words the function fτ is a field automorphism of R. However, the only field automorphism of R is the identity, as the next lemma shows: Lemma 5.7. Let f: R R be a function that satisfies f(x+ y) = f(x)+ f(y) and f(x y) = f(x) f(→y) the f is the identity. · · Proof. Since 0 is the only element in R that satisfies x + x = x and we have f(0) + f(0) = f(0 + 0) = f(0) we must have f(0) = 0. Similarly, f(1) f(1) = · f(1 1) = f(1) implies f(1) = 1. Now consider an integer n N. We can write n =·1 + 1 + . . . + 1. And obtain ∈

n-times

f(n#) = f($1%+ 1 +&. . . + 1) = f(1) + f(1) + . . . + f(1) = 1 + 1 + . . . + 1 = n

n-times n-times n-times

The fact# that $f%(x y)&= f#(x) f(y) im$%plies that &for a#ny rat$i%onal n&umber · · q = n/m, n, m N we have f(q) = q since q is the only number that satisfies ∈ 5.4 The fundamental theorem 91 the equation f(n) q = f(m) and since we have f(n) = n and f(m) = m. For · similar reasons we have f( x) = f(x) for any x R. − − ∈ Now the key observation is that one can characterize positivity in R by multiplication. A number is positive if it can be written as x2 for x = 0. Thus from f(x2) = f(x)2 we obtain that f maps positive numbers to pos$ itive numbers. This implies that x < y implies f(x) < f(y) (to see this observe that x < y is equivalent to y x > 0). If f were not the identity−, then we could find a real number a such that f(a) = a. Then we have either a < f(a) or a > f(a). Consider the first case. $ Since the set of rational numbers is dense in R, the open interval (a, f(a)) contains a rational number q. This leads to a contradiction to the order pre- serevation property of f since we then have a < q, but f(a) > q = f(q). The case a > f(a) is analogous. #" Taking together Lemma 5.6 and Lemma 5.7 we can finally prove Theorem 5.1, the fact that harmonic maps can be expressed by matrix multiplications. 1 1 Proof of Therorem 5.1 Assume that ψ: RP RP is a harmonic map. The → special points 0, 1 and are mapped by ψ to some points 0!, 1! and !. ∞ 1∞ Theorem 4.1 implies that there exists a projective transformation φ: RP 1 1 →1 RP with φ(0!) = 0, φ(1!) = 1, φ( !) = . The composition τ: RP RP defined by τ(x) = φ(ψ(x)) is again∞a har∞monic map, that in addition→leaves 0, 1 and invariant. By Lemma 5.6 it induces a field automorphism on R which (by∞Lemma 5.7) can only be the identity. Thus τ itself must have been the identity map. This in turn implies that the original harmonic map ψ must be the inverse of the projective transformation φ. Hence it is itself a projective transformation. #"

5.4 The fundamental theorem

Theorem 5.1 is of fundamental importance. It links a structural geometric concept (harmonic point quadruples) to the algebraic structure of the under- lying field: On the level of coordinates every harmonic map can be expressed as a matrix multiplication. Von Staudt originally went even one step further: He started with an inicdence system that satiesfies the axoims of a projec- tive plane and required the additional presence of another incidence property (namely that Pappos’s theorem holds within the incidence structure) and re- constructed an underlying field K from this incidence structure. One can show 3 (0,0,0) that the original projective plane is isomorphic to K −{ } . We will not K 0 prove this here. For a prove of this fact see for instance [−?{]. }However, we will later on show what is the relevance of Pappos’s Theorem in this context: The presence of Pappos’s Theorem results in the commutativity of the underlying field. 92 5 Calculating with points on lines

ly ly

y y PSfrag replacements ∞ PSfrag replacements ∞

y l l ∞ p ∞ 1y 1y 1 1

0 x 0 x ∞ lx ∞ lx 1 1 yx x x x p

2 Fig. 5.2. Fixing a projective Framework in RP .

Theorem 5.1 is a 1-dimensional counterpart of Theorem 3.3 for which we 2 still have not presented the proof. This theorem stated that in RP any bi- jective map, that maps collinear points to collinear points can be expressed by a projective transformation. We will obtain this in a similar way as Theo- rem 5.1, however we will have to find a way to encode 2-dimensional positions 2 in term of cross ratios. A harmonic map in RP is any bijection that maps harmonic points on a line to harmonic points (perhaps on another line). A 2 collinearity in RP will be any bijection that maps collinear points to collinear points. It is obvious that a harmonic map is a collineation (collinear points are mapped to collinear points). However, the opposite is also true. We have: 2 Lemma 5.8. Any collineation in RP is a harmonic map. Proof. Consider four harmonic points a, b, c, d on some line l. By Lemma 5.2 there exist four auxiliary points not on l that form (together with a, b, c, d) the incidence pattern of Figure 5.1. Under a collinearlity this incidence pattern is mapped to an incidence pattern with the same combinatorial structiure. This implies that (again by Lemma 5.2) the image points are harmonic, as well. #"

2 We now use two values of cross rations in RP as “coordinates” for 2- 2 dimensional points. For this we single out a projective basis in RP consisting of four distinct points 0, x, y and 1. These four points will play the role of the origin, an infinite ∞point∞on the x-axis, an infinite point on the y-axis and a point with coordinates (1, 1) respectively. Furthermore we define

1 = meet(join(0, ), join(1, )) x ∞x ∞y and 1 = meet(join(0, ), join(1, )) y ∞y ∞x

The triple (0, 1x, x) forms a projective basis on the line lx spanned by 0 and , and a poin∞t x on l is uniquely determined by the cross ratio ∞x x 5.5 A note on other fields 93

x := (0, ; 1 , x). ∞x x

Similarly, the triple (0, 1y, y) forms a projective basis on the line ly spanned by 0 and , and a point ∞y on l is uniquely determined by the cross ratio ∞y y y := (0, ; 1 , y). ∞y y 2 Any point p of RP that does not lie on the line l := join( x, y). Defines uniquely two points ∞ ∞ ∞

x = meet(l , join(1, )) and y = meet(l , join(1, )) x ∞y y ∞x from which it can be reconstructed by means of:

p = meet(join(x, ), join(y, )). ∞y ∞x In other words, the pair of numbers (x, y) with x := (0, ; 1 , x) and y := ∞x x (0, y; 1y, y) uniquely determines the postition of p. ∞Now we are ready for the proof of Theorem 3.3: Every collineation is a projective transformation.

2 2 Proof of Therorem 3.3 Assume that ψ: RP RP is a collinearity. The special → points 0, 1 and and are mapped by ψ to some points 0!, 1!, ! ∞x ∞y ∞x and y!. Theorem 3.4 implies that there exists a projective transformation ∞2 2 φ: RP RP with φ(0!) = 0, φ(1!) = 1, φ( x!) = x!, φ( x!) = x!. The compos→ition τ(x) = φ(ψ(x)) is again a collin∞eation, ∞that in a∞ddition∞leaves 0, 1, and invariant. ∞x ∞y By Lemma 5.6 it induces a field automorphism on R which (by Lemma 5.7) can only be the identity. Thus τ itself must have been the identity map on 2 the lines lx and ly. Our above considerations imply that all points of RP that are not on l must stay invariant. As a consequence all points on l must stay invarian∞t as well (any such point can be encoded as the intersec∞tion of l and a line line through two points not on l ). Thus τ is the identity. This in∞ turn implies that the original harmonic ma∞p ψ must be the inverse of the projective transformation φ. Hence it is itself a projective transformation. #"

5.5 A note on other fields

The last three sections established a close connection of the field that under- lies a projective geometry and projective transformations. In our approach we entirely focused on the real projective plane. Almost all our constructions (like projective transformations, cross ratios, harmonic points, collieation, etc.) could as well be carried out over arbitrary fields. However, with respect to the fundamental theorem a little care is necessary. 94 5 Calculating with points on lines

One crucial ingredient for Theorem 5.1 and Theorem 3.3 was to derive a field automorphism from a harmonic map. The fact that R does only have the trivial field automorphism translated into the statement that every collineation can be expressed as a matrix multiplication. Over other fields (like C) we have non-trivial field automorphisms. These field automorphisms induce collineations that cannot be expressed as a projective transformation. Let us 2 focus for a moment on the complex projective plane CP . In complete analogy 2 2 (0,0,0) 2 (0,0,0) to RP we can define this space by = C −{ } and = C −{ } . PC C 0 LC C 0 A point (x, y, z) is on a line (a, b, c) if a x + b−{ y}+ c z = 0. Matrix−{m}ulti- plications are again the projective transfo·rmation· s and ·induce a collineations. However, over C we have additional collineations. We obtain the simplest such collineation by taking the map that assigns to each coordinate entry its complex conjugate:

(x, y, z) (x, y, z) and (a, b, c) (a, b, c). *→ *→ We have a x + b y + c z = 0 a x + b y + c z = 0. The gen·eraliza·tion of· the fu⇐nd⇒ame·ntal th·eorem· that applies to all fields can be stated as follows. 1 1 Theorem 5.2. Let τ: KP KP be a collineation. Then τ can be factored as φ ψ where φ is a proje→ctive transformation and ψ is induced by a field autom◦orphism.

5.6 Von Staudt’s original constructions

In Section 5.3 we introduced a way to mimic multiplication and addition by 1 geometric constructions. Since we wanted to stay within RP our basic prim- itive were harmonic point quadruples rather than collinearity. The construc- tions performed in Lemma 5.6 can also be turned into geometric constructions for addition and multiplication by using just join and meet operations. For this every harmonic quadruple of points is forced to be harmonic using the construction of Figure 5.1. From an “economic” point of view this way of en- coding geometric addition and multiplication is by far too complicated. Each harmonic set used in the construction requires four additional points. There is a much more direct way to encode geometric multiplication and geomet- ric addition by a few construction steps only. These were the constructions originally given by von Staudt. They require only four additional points for addition or multiplication. Since these constructions are nice little gadgets that are useful for many geometric constructions we will present them here. As a “warmup” we start with a construction that from 0, 1 and (which are assumed to be given on a line !), constructs the points corresp∞onding to the integers. We start with one more arbitrary point p not on ! and a point q on the line joining 0 and q. We construct a point r by 5.6 Von Staudt’s original constructions 95

r = meet(join(p, x ), join(q, x1)) ∞

Then we construct a point q1 according to

q1 = meet(join(q, x ), join(q, 1)) ∞ and a point 2 according to

2 = meet(!, join(r, q1)).

For the special choice of coordinates 0 = (0, 0, 1), 1 = (1, 0, 1) and = (1, 0, 0) Figure 5.3 shows the corresponding situation (in the usual view o∞f the Euclidean plane). Since is a point at infinity the three lines !, join(q, ) and join(p, ) are parall∞el in the picture. Thus the (oriented) segment leng∞ths 0, 1 , 1, 2 ∞and q, q are related by: | | | | | 1| 0, 1 1, 2 | | = | | . q, q q, q | 1| | 1| Multiplying by both sides by q, q we obtain | 1| 0, 1 = 1, 2 . | | | | Hence, the location of 2 is independent of the choice of p and q. It is as far from 1 as 1 is from 0. Hence it must have homogeneous coordinates (2, 0, 1). Iterative application of this construction can be used to construct a sequence of points with homogeneous coordinates (n, 0, 1); n N. Figure 5.4 shows the result of the construction if is chosen to be a∈ finite point. Still our considerations imply that the posi∞tions of 0, 1 and determine the positions of 2, 3, 4, . . . uniquely, independent of the choice of∞p and q. As a next aim we will model the operations of addition and multiplication by geometric constructions. For this we again fix points 0, 1 and and chose points p and q as before. The constructions for addition and m∞ultiplication are given in Figure 5.5.

r p r p PSfrag replacements PSfrag replacements

q q1 q q1

0 3 1 2 0 1 2 3 4 5 4 5

Fig. 5.3. Constructing the integers 96 5 Calculating with points on lines

r r

PSfrag replacements p PSfrag replacements p

q1 q1 q q 3 4 0 1 2 4 0 1 2 3 5 ∞ 5 ∞

Fig. 5.4. Constructing the integers projectively

We assume that on the line ! two additional points x and y are given with homogeneous coordinates (x, 0, 1) and (y, 0, 1), respectively. We demonstrate how to construct the points x + y and x y with homogeneous coordinates (x+y, 0, 1) and (x y, 0, 1), respectively. Th·e constructions can be read off in a straight forward wa· y from the pictures. Using the parallel lines in each picture and considering length ratios we obtain for the first picture the relation

0, x y, x + y | | = | |. q, q q, q | 1| | 1| Thus the distance from 0 to x is the same as the distance from y to x + y. This implies the desired relation for addition. For the picture on the left we obtain the relations: 0, 1 r, s 0, y | | = | | = | | . 0, x r, p 0, x y | | | | | · | The first relation comes from the relations induces by the lines passing through q the second comes from the lines passing through q1. Comparing the first and the third term gives the desired relation. If we alter the role of the points, the same configurations can be used to perform geometric geometric subtraction or geometric division. By combin-

r s p PSfrag replacements r pPSfrag replacements 1 q1

x y q q1 q · x + y

0 x y x + y 0 1 x y x y · s

Fig. 5.5. Geometric addition and multiplication 5.7 Pappos’s Theorem 97 ing several von Staudt constructions the evaluation of arbitrary quotients of polynomials can be modeled geometrically.

5.7 Pappos’s Theorem

As a final topic of this chapter, we want to demonstrate the important role that is played by Pappos’s Theorem in relation to underlying fields of a projective plane. We will see that Pappos’s theorem is equivalent to the commutativity of the underlying field. To see this we will have a closer look at the von Staudt constructions for addition and multiplication. If we interchange the role of x and y in the von Staudt addition (see Figure 5.5 left) we end up with a drawing that has exaclty the same combi- natorics. This implies the commutativity of addition. This is not the case for von Staudt multiplication. Figure 5.7 overlays a different constructions for the point x y. The blue lines are exatly the same as presented as “von Staudt multiplic·ation” in the last section. The green construction corresponds to the “von Staudt multiplication” for y x (the roles of x and y are interchanged). The black lines belong to both con·structions. Since over the real numbers the multiplication is commutative and we have x y = y x both constructions must result in the same point. This fact can be in· terpre·ted as a purely geometric incidence theorem (without ever referring to algebra). In fact the decisive incidence theorem that hides behind commutativity of multiplication is nothing else as Pappos’s Theorem (recall the first chapter of this book). In Figure 5.7 The nine lines that correspond to Pappos’s Theorem are drawn in bold and the nine points are emphasized in red. If we look back at the collection of proofs for Pappos’s Theorem that we presented in the first chapter of this book, then we will recognize that each of these proofs at one point made use of commutativity of the underlying coordinate field. The reason for this is that the presence of Pappos’s Theorem in a projective plane implies that the plane can be considered as ( , , ) PK LK IK for some (commutative) field K. Projective planes in which Pappos’s Theorem holds are called Pappian Planes. With this terminology we can reformulate the above fact in the following manner:

Theorem 5.3. A projective plane is Pappian if and only if it is of the form ( , , ) for some field K. PK LK IK We do not prove this theorem here since the main focus of this book is not in the distinction of projective planes that come from fields from those that do not come from fields. All projective planes used further on will come from a field and hence we can (and will) freely make use of Pappos’s Theorem. PSfrag replacements PSfrag replacements

98 5 Calculating with points on lines

x + y x + y

t r s p t r s p

q1 q1

q2 q2 q q

0 1 x y x y = y x 0 1 x y x y = y x · · · ·

Fig. 5.6. Commutativity of multiplication and Pappos’s theorem.