Essays on Geometry: Celebrating the 65th Birthday of Athanase Papadopoulos

Preface
Contents
Contributors
1 Introduction
2 Interview with Athanase Papadopoulos
3 A Glance at S. Novikov's Theory of Multivalued Morse Functions
	3.1 From the Morse Complex to the Morse-Novikov Complex
		3.1.1 Preamble
		3.1.2 Generic Conditions
		3.1.3 The Fundamental Groupoid and the Novikov Ring
		3.1.4 The Twisted Morse Complex
		3.1.5 The Morse-Novikov Complex
	3.2 Morse Model, α-Adapted Gradients and Homoclinic Bifurcation
		3.2.1 The Morse Model in the Setting of an Adapted Gradient
		3.2.2 Final Choice of ∂v and Holonomic Factor
	3.3 Effect of Crossing Sg on the Morse-Novikov Complex
		3.3.1 A Natural Question
	3.4 Isolated Homoclinic Bifurcations
		3.4.1 Some Basics on Closed One-Forms
		3.4.2 Birth of Two Zeroes
		3.4.3 Descent Map
		3.4.4 A Homoclinic Bifurcation Attached to p
		3.4.5 The Construction Continued
	3.5 The Doubling Phenomenon
		3.5.1 An Apparent Contradiction
	References
4 A Twisted Invariant of a Compact Riemann Surface
	4.1 Introduction
	4.2 A Twisted Invariant of a Compact Riemann Surface
	4.3 Tensor Fields on the Moduli Space Mg
		4.3.1 The First Mumford–Morita–Miller Class e1 = κ1
		4.3.2 The Jacobian Map Jac: Mg →Ag
	4.4 A Proof of Theorem 4.1
	4.5 Proof of Proposition 4.1
	References
5 Directional Moduli and Pseudoconvexity
	5.1 Introduction
	5.2 Preparations from One Complex Variable
		5.2.1 The L1- and L0-principal Functions
		5.2.2 The Schiffer Span for (R,ζ)
		5.2.3 The L1- and L0-differentials
		5.2.4 The a-Span of (R,χ)
	5.3 Variation of the Schiffer Spans and the a-spans
		5.3.1 Smooth Variation of Open Riemann Surfaces
		5.3.2 Variation of the Schiffer Spans
		5.3.3 Variation of the a-spans
	5.4 Geometric Meaning of the a-span's Variation
		5.4.1 Closings of a Marked Open Riemann Surface
		5.4.2 Geometric Meaning of the a-span
		5.4.3 Restatement
	References
6 Angle Defect for Super Triangles
	6.1 Introduction
	6.2 Super Minkowski Space
	6.3 Super Geodesics
	6.4 Orthosymplectic Group OSp(1|2)
	6.5 OSp(1|2)-invariant Area Form on IH
	6.6 Line Integrals
	6.7 Area of a Super Triangle
	6.8 Angle Defect
	Appendix: Super Ideal Triangles
	References
7 Lipschitz and Quasiconformal Mappings in Cartography
	7.1 Introduction
	7.2 Formulation of the Problem
	7.3 The Formulae for the Five Maps
		7.3.1 The Projection from the Center of S
		7.3.2 The Horizontal Projection
		7.3.3 The Conical Equidistant Map
		7.3.4 The Orthogonal Projection
		7.3.5 The Best Quasiconformal Map
	7.4 Numerical Computations for the Various Maps
		7.4.1 Comparison of Maps on A
		7.4.2 Euler's Problem
	7.5 Tissot Ellipses
	References
8 Spherical Representations of the Group of Isometries of Semi-homogeneous Trees
	8.1 Introduction
	8.2 Semi-homogeneous Trees and Their Laplace Operator
	8.3 The Convolution Algebra of Radial Functions and Its Multiplicative Functionals
	8.4 Positive Definite Spherical Functions on V+
	8.5 Spherical Representations of G
	References
9 Trees of Fractions
	9.1 The Monoids  SL2 (N)  and  GL2 (N)
	9.2 The Calkin–Wilf Tree
	9.3 The Stern–Brocot Tree
	Exercises
		9.3.1 Notes
	9.4 The Drib Tree
	Exercises
	9.5 The Bird Tree
	Exercises
		9.5.1 Extension of Jimm to the Real Line
	Exercises
	9.6 More Trees
		9.6.1 Yu–Ting–Andreev Tree
		9.6.2 Hanna–Czyz–Self Tree
		9.6.3 Kepler Tree
		9.6.4 Lebesgue Tree
		9.6.5 Notes
	Exercises
	References
10 Binary Quadratic Forms: Modern Developments
	10.1 Introduction
	10.2 A Very Brief History of Binary Quadratic Forms
	10.3 Modern Era
		10.3.1 Notation and Terminology
		10.3.2 Minus Continued Fractions: Zagier Reduction
		10.3.3 Geometry of Gauss Product: Penner's Work
		10.3.4 Bhargava's Cubes: Composition Laws
		10.3.5 Çarks: A Combinatorial Viewpoint
			Actions on Çarks
	References
11 A Note on Reversibility of Unipotent Matrices
	11.1 Introduction
	11.2 Proof of Theorem 11.1
	References
12 ``Le complément supérieur'': On the Poetics of Mathematics
	12.1 Introduction
	12.2 Towers of Babel: Geometry Before Philosophy
	12.3 ``In Mathematical Language'': Geometry Before Physics
	12.4 A ``complément supérieur'' of a ``complément supérieur'': Geometry Before Geometry
	12.5 The Chaosmos of Compositions: Geometry Before Poetry
	12.6 Conclusion: Of Mysteries, Dreams, and Divine Madness
	References
13 Pythagorean Book II of the Elements Restored and Pythagorean Incommensurabilities Reconstructed
	13.1 Introduction
	13.2 Incommensurability, the Great Mathematical Discovery of the Pythagoreans, and the Question on the Method of Proof Employed by the Pythagoreans to Prove It
	13.3 The Arithmetical Proof of Incommensurability, Hinted by Aristotle and Fully Described as ``Proposition X.117'' of the Elements, with Adherents from Heath Heath21 to Netz Netz Is Tentatively Rejected as the Original Pythagorean Proof in Favor of an Anthyphairetic Proof
		13.3.1 An Elementary Arithmetical Method of Incommensurability Hinted in Aristotle's Analytics Prior 41a and Fully Preserved in ``Proposition X.117'', a Later Anonymous Addition to the Elements
		13.3.2 Several Historians of Greek Mathematics Are Adherents to the Arithmetical Original Method of Proof
		13.3.3 Some Doubts About Whether Aristotle Was in Possession of the Simple X.117 Proof of Incommensurability
		13.3.4 Our Interpretation of the Various Versions of the Pythagorean Story Involving the Disrespectful Pythagorean Correlate Incommensurability with the Infinite, Suggesting a Pythagorean Anthyphairetic Proof of Incommensurability by the Use of Proposition X.2
		13.3.5 The Pythagorean Infinite Musical Anthyphairesis in Philolaus Fragment 6 + Boethius, De Institutione Musica III.8 Implies a Musical Incommensurability
		13.3.6 The Pythagorean Association of Incommensurability with the Infinite Suggests an Anthyphairetic Original Proof of Incommensurability
		13.3.7 The (Anthyphaireses of the) Side and Diameter Numbers Are Finite Initial Segments of the Infinite Anthyphairesis of the Diameter to the Side of a Square, an Indication that the Original Pythagorean Proof of Incommensurability Is Not Arithmetical but Anthyphairetic and the (Fundamental Pell Property of the) Side and Diameter Numbers Are Connected with Proposition II.10 of the Elements, an Indication that the Original Pythagorean Proof of Incommensurability Is Connected with Book II of the Elements
		13.3.8 Tentative Rejection of the Arithmetical Reconstructions in Favor of an Anthyphairetic One
	13.4 The Anthyphairetic Proof of the Diameter to the Side of a Square by Theaetetus Which Is the Model for Meno 80e-86c, 97a-98b Employs the Theory of Ratios of Magnitudes and Is Certainly Not the Original Pythagorean Proof
	13.5 Heuristic Discussion: The Restoration of Book II of the Elements to Its Original Pythagorean Form Is Expected to Reveal the Original Pythagorean Proof of Incommensurability
	13.6 Book II of Euclid's Elements
	13.7 Partial Restoration of Book II in Its Original Pythagorean Form: Proposition II.4/5=the Pythagorean Theorem, Proposition II.5/6=the Pythagorean Application of Areas in Defect, Proposition II.6/7=the Pythagorean Application of Areas in Excess
	13.8 The Controversy on Pythagorean Geometric Algebra
	13.9 The Proof by Mathematical Induction, Using Propositions II.10, that the Side pn and Diameter qn Numbers Satisfy the Pell Property qn2 = 2pn2 + (-1)n for Every Natural Number n
	13.10 The Chrystal, Tennenbaum, Fowler Anthyphairetic Proofs of the Incommensurability of the Diameter to the Side of a Square, Their Reliance on Propositions II.9 & 10, and Their Rejection as Reconstructions of the Original Pythagorean Proof of Incommensurability
		13.10.1 Ancient and Modern Proofs of the Subtractive Elegant Theorem II.9/10; if d2=2s2, then (2s-d)2=2(d-s)2
		13.10.2 The Anthyphairetic Proof of the Incommensurability of the Diameter to the Side of a Square from the Subtractive Elegant Theorem
		13.10.3 Ancient and Modern Proofs of the Elegant Theorem II.10/11.a: if d2 = 2s2, then (d + 2s)2 = 2(d + s)2
		13.10.4 The Anthyphairetic Proof of the Incommensurability of the Diameter to the Side of a Square from the Elegant Theorem
	13.11 The Structure of the Partially Restored Book II of the Elements Suggests the Rejection of the Anthyphairetic Proofs Based on Propositions II.9 and 10, and Leads Us to the Expectation of a Reconstruction of the Original Pythagorean Proof of the Incommensurability of the Diameter to the Side of a Square at the Position II.8/9.a of Book II
	13.12 The Reconstructed Original Pythagorean Proof of Incommensurability at the Position II.8/9.a of Book II of the Elements, with the Use of the Application of Areas in Excess
		13.12.1 Proposition II.8/9.a and Its Pythagorean Proof
		13.12.2 Proposition: The Preservation of Application of Areas (In Excess)/The Preservation of Gnomons
		13.12.3 Proposition: If Aa2 = Bab + Cb2 and Ac2 = Bcd + Cd2, then Anth(a,b) = Anth(c,d)
		13.12.4 The Two Pythagorean Uses of the Pythagorean Application of Areas in Excess II.6/7 in the Original Pythagorean Incommensurability Proof: (a) Geometric Algebra for Steps 2 and 3, and (b) Preservation of Gnomons for Step 4
	13.13 The Anthyphairetic Nature of the Two Pythagorean Philosophic Principles Infinite and Finite at the Position II.8/9.b Provides a Confirmation of Our Reconstruction of the Original Pythagorean Proof of Incommensurability in Sect. 13.12
		13.13.1 According to Aristotle, the Two Philosophic Pythagorean Principles, Infinite and Finite, Have Mathematical Origin
		13.13.2 The Pythagoreans Gave the Symbolic Names Even/Artion for the Principle of the Infinite/Apeiron, and Odd/Peritton for the Principle of the Finite/Peras
		13.13.3 Aristotle and Simplicius Clarify the Symbolic Description of the Pythagorean Principle of the Finite as Odd: An Odd Number of Units in the Figure of an Arithmetical Gnomon Surrounding a Square and thus Forming a Larger Square, thus Preserving the Square Figure; the Preservation of the Square Arithmetical Figure During This Process Ad Infinitum Is Precisely the Symbolic Finitization
		13.13.4 The Symbolic Description of the Infinite as Even Implies that the Real Pythagorean Principle of the Infinite Is a Principle of Binary Division Ad Infinitum, and in Fact, as Simplicius and John Philoponus Make Clear, a Principle of Binary Division Ad Infinitum of Magnitudes into Equal or Unequal Parts
		13.13.5 The Symbolic Arithmetic Gnomons Refer to the Real Geometric Gnomon of Book II of the Elements, and thus the Two Pythagorean Principles Arose from the Mathematics in Book II
		13.13.6 The Geometric Gnomons in the Real Principle of the Infinite Must Be Not Increasing, as in the Symbolic Description, but Decreasing, as Hinted by Aristotle and Specified Explicitly by Proclus
		13.13.7 Anthyphairetic Interpretation of the Two Pythagorean Principles Infinite and Finite/Finitizing
		13.13.8 According to Philolaus, Simplicius, Hesychius, Eustratius, Anonymon Scholion to Elements, the Gnomon Possesses the Power to Provide Knowledge
	13.14 The Two Philosophical Principles Infinite and Finite Reappear in Zeno's Fragment B3, with Substantially the Same Anthyphairetic Meaning, Establishing an Early Dating of the Pythagorean Proof of Incommensurability and a Further Confirmation of Their Method of Proof
	13.15 The Second Confirmation of Our Reconstruction: Our Interpretation of the Pythagorean Definition of the Three Kinds of Angles at the Position II.13/14.a and of the Postulate 4 of the Elements at the Point II.13/14.b, in Terms of the Two Pythagorean Principles Infinite and Finite
		13.15.1  The Cause of the Obtuse and Acute Angles on the One Hand Is Said by Proclus to Be the Pythagorean Principle of the Infinite, and on the Other Hand by Our Interpretation Is the Infinite Anthyphairesis of the Diameter to the Side of a Square, Suggesting that the Pythagorean Principle of the Infinite Coincides with the Infinite Anthyphairesis of the Diameter to the Side of a Square
		13.15.2 The Cause of the Right Angle on the One Hand Is Said by Proclus to Be the Pythagorean Principle Finite, and on the Other Hand by Our Interpretation Is the Preservation of the Gnomons in the Anthyphairesis of the Diameter to the Side of a Square, Suggesting that the Pythagorean Finite Coincides with the Preservation of the Gnomons in the Anthyphairesis of the Diameter to the Side of a Square
		13.15.3 The Meaning that the Pythagoreans Assigned to the Fourth Postulate in Euclid's Elements: All Right Angles Are Equal According to Proclus, In Euclidem 191,5–15
		13.15.4 Proclus, In Euclidem 191,5–15
		13.15.5 Our Interpretation of Proclus' Passage on Postulate 4
	13.16 Our Answers to the Questions About Pythagorean Mathematics and Philosophy
	References
Index