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Identities Generalizing the Theorems of Pappus and Desargues

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Identities Generalizing the Theorems of Pappus and Desargues S S symmetry Article Identities Generalizing the Theorems of Pappus and Desargues Roger D. Maddux Department of Mathmatics, Iowa State University, Ames, IA 50011, USA; [email protected] Abstract: The Theorems of Pappus and Desargues (for the projective plane over a field) are general- ized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory. Keywords: Pappus theorem; Desargues theorem; projective plane over a field 1. Introduction Identities that generalize the theorems of Pappus and Desargues come from two primary sources: lattice theory and invariant theory. Schützenberger [1] showed in 1945 that the theorem of Desargues can be expressed as an identity (the Arguesian identity) that holds in the lattice of subspaces of a projective geometry just in case the geometry is Desarguesian. Jónsson [2] showed in 1953 that the Arguesian identity holds in every lattice of commuting equivalence relations. The further literature on Arguesian lattices includes [3–22]. Two identities similar to those in this paper were published in 1974 [23] (with different meanings for the operations), as examples in the context of invariant theory. For proofs see [24] (p. 150, p. 156). Further literature on such identities includes [25–39]. Citation: Maddux, R.D. Identities The identities in lattice theory use join and meet and apply to lattices. The identities Generalizing the Theorems of Pappus in invariant theory use the wedge (exterior) product (corresponding to join), bracket and Desargues. Symmetry 2021, 13, (a multilinear altering operation like determinant), and an operation corresponding to 1382. https://doi.org/10.3390/ meet. The identities apply to Cayley-Grassmann algebras (exterior algebras with a bracket sym13081382 operation). The connections between the two kinds of identities are not straightforward and are studied in several of the papers listed above. Academic Editor: Ivan Chajda The identities presented in this paper have a more mundane origin. In the spring semester of 1997, the author taught the second semester of college geometry out of the text Received: 1 July 2021 by Ryan [40]. The text’s analytical approach included the fact that three points (or lines) in Accepted: 22 July 2021 the projective plane over a field are collinear (or concurrent) if and only if the determinant Published: 29 July 2021 of their homogeneous coordinates is zero; see (6) below. Pappus and Desargues both assert that if certain collinearities or concurrences hold, then others do, that is, if some numbers Publisher’s Note: MDPI stays neutral are zero, then so are others. Clearly, then, there must be algebraic formulas that assert such with regard to jurisdictional claims in dependences between determinants and from which the two theorems can be deduced. published maps and institutional affil- iations. Such formulas were found and included in the course notes [41]; parts of those notes appear in this paper. Section2 presents the two identities along with definitions of their operations. They use cross products and determinants rather than the joins, meets, and brackets of invariant theory [24]. Section3 is a review of the projective plane over a field. Section4 derives Copyright: © 2021 by the authors. the theorems of Pappus and Desargues from the identities. Section5 shows how cross Licensee MDPI, Basel, Switzerland. product relates to join and meet, followed by a Discussion, an Acknowledgement, and an This article is an open access article AppendixA containing proofs of the identities. Good survey papers on the theorems of distributed under the terms and conditions of the Creative Commons Pappus and Desargues include [42,43]. Attribution (CC BY) license (https:// 2. Cross Products and Determinants creativecommons.org/licenses/by/ 3 4.0/). Let F be a field. The elements of F are written as column vectors. Let Symmetry 2021, 13, 1382. https://doi.org/10.3390/sym13081382 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 1382 2 of 10 2 3 2 3 2 3 x0 y0 z0 3 3 3 r 2 F, a = 4x15 2 F , b = 4y15 2 F , c = 4z15 2 F . x2 y2 z2 1. The scalar product of r and a is 2 3 rx0 3 ra = 4rx15 2 F . rx2 2. The inner product of a and b is ha, bi = x0y0 + x1y1 + x2y2 2 F. 3. The cross product of a and b is 2 3 x1y2 − x2y1 3 a × b = 4x2y0 − x0y25 2 F . x0y1 − x1y0 4. The determinant of a, b, c is x0 y0 z0 det[a, b, c] = x y z 1 1 1 x2 y2 z2 = x0y1z2 + x2y0z1 + x1y2z0 − x0y2z1 − x2y1z0 − x1y0z2 2 F. Identity (1) in the next theorem generalizes the theorem of Pappus and identity (2) generalizes the theorem of Desargues. They can be proved by expanding both sides in 18 variables (3 for each of 6 vectors) and obtaining the same result. A proof of Theorem1 using properties of cross products and determinants is in the AppendixA. In Theorem2, the theorems of Pappus and Desargues are proved using Theorem1. 0 0 0 Theorem 1. For any field F and any a, b, c, a , b , c 2 F3, det[(a × b0) × (a0 × b), (b × c0) × (b0 × c), (c × a0) × (c0 × a)] (1) = det[a0, b0, b] det[b0, c0, c] det[c0, a0, a] det[a, b, c] + det[b, a, a0] det[a, c, c0] det[c, b, b0] det[a0, b0, c0], det[(a × b) × (a0 × b0), (b × c) × (b0 × c0), (c × a) × (c0 × a0)] (2) = det[a × a0, b × b0, c × c0] det[a, b, c] det[a0, b0, c0]. 3. The Projective Plane over F The points A, B, . of the projective plane over F are the one-dimensional sub- spaces of the three-dimensional vector space F3 over F, and its lines l, m, . are the two-dimensional subspaces of F3. A point A is on a line l if A is a subset of l: A is on l iff A ⊆ l. (3) Every non-zero vector in F3 determines both a point and a line. Suppose 0 6= a 2 F3. Then A = fra : r 2 Fg is the 1-dimensional subspace generated by a, so A is a point in the projective plane over F, and l = fx : hx, bi = 0g is the 2-dimensional subspace of vectors perpendicular to a, so l is a line in the projective plane over F. The vector a is referred to as homogeneous coordinates for point A and line l. The incidence relation between a point and a line holds when their homogeneous coordinates are perpendicular, for if 0 6= a, b 2 F3 then Symmetry 2021, 13, 1382 3 of 10 point fra : r 2 Fg is on line fx : hx, bi = 0g iff ha, bi = 0 (4) An axiom of projective geometry (every two points are on a unique line) takes the form in the projective plane over a field that every two 1-dimensional subspaces are contained in a unique 2-dimensional subspace. The 2-dimensional subspace containing two 1-dimensional subspaces is the set of sums of two vectors, one from each subspace. To notate this, we extend the notion of addition from individual vectors to sets of vectors. For arbitrary sets of vectors A, B ⊆ F3, let A + B be the set of their sums: A + B = fa + b : a 2 A, b 2 Bg. (5) If A and B are distinct points, then there are linearly independent non-zero vectors a, b 2 F3 such that A = fra : r 2 Fg and B = fsb : s 2 Fg. The unique line containing both A and B is the 2-dimensional subspace generated by a and b, consisting of all their linear combinations, A + B = fra + sb : r, s 2 Fg. The cross product a × b is perpendicular to both a and b, so any vector perpendicular to a × b will be in the unique line A + B containing A and B (and conversely), that is, A + B = fx : hx, a × bi = 0g. Thus, points with homogeneous coordinates a and b determine a line with homo- geneous coordinates a × b. Suppose two lines, l and m, have homogeneous coordinates a, b 2 F, respectively: l = fx : hx, ai = 0g, m = fx : hx, bi = 0g. Note that a and b are linearly independent, since otherwise l = m (and there is only one line, not two, as postulated). An axiom for projective planes is that distinct lines meet at a unique point. In the projective plane over a field, this axiom asserts the fact that distinct 2-dimensional subspaces contain a unique 1-dimensional subspace, namely, their intersection. Thus, the intersection l \ m of the lines l and m is the unique point that occurs on both lines. When l and m have coordinates a and b, their intersection is the 1-dimensional subspace of F3 generated by any non-zero vector perpendicular to both a and b, such as a × b: l \ m = fr(a × b) : r 2 Fg. Thus, two lines with homogeneous coordinates a and b meet at a point with homoge- neous coordinates a × b. Suppose 0 6= a, b, c 2 F3. Let A, B, and C be the points, and l, m, and n the lines, with homogeneous coordinates a, b and c, respectively: A = fra : r 2 Fg, B = frb : r 2 Fg, C = frc : r 2 Fg, l = fx : hx, ai = 0g, m = fx : hx, bi = 0g, n = fx : hx, ci = 0g. By definition, A, B, C are collinear iff A + B = A + C = B + C, and l, m, n are concur- rent iff l \ m \ n is a point.
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