# Transcript for ASL interpretation Note: slides 62, 63, 94, 95 are dense in math expressions. ## 1 ## 2 Today I’m presenting excerpts from my ongoing text project titled A First Course in Figurative Painting. ## 3 A painting manual written in the language of modern math. ## 4 Parts. One. Sample text with chapters one and two. Physical copies are on the table in person. PDF at link in Zoom chat. ## 5 Two. Front and back cover. Printed and displayed on the wall in person. Included in the PDF. ## 6 Three. Slideshow presentation of highlights from chapters 1 through 4 of the text. On the Zoom call. Slideshow and transcript at link in Zoom chat. ## 7 Highlights. Content warning: violent and sexual content. ## 8 Front and back cover. ## 9 A First Course in Figure Painting was the first text the collect and organize the fragmented movement of figuration in the mid-twentieth century. It presents, instead of a historical account of the movement, a self-contained outline of the mathematical constructions that played a key role in the foundational theory and practice of what became known as the field of figurative painting. ## 10 It boasts the inclusion of over 600 figures to the originally unillustrated text, and solutions to selected exercises, offering a more suitable introductory text for freshmen in figurative painting than its previous printings. ## 11 Preface. One. Prerequisites to this text are reading in English and readiness to learn new vocabulary. ## 12 Two. Examples of what symbols, nouns, and adjectives look like in the text. ## 13 Subset of or equal to. ## 14 Union. ## 15 Intersection. ## 16 Empty set. ## 17 The set of natural numbers. ## 18 The set of rational numbers. ## 19 The set of real numbers. ## 20 The three-dimensional Euclidean space. ## 21 The n-dimensional Euclidean space. ## 22 The open ball of radius r around point p. ## 23 Function f from domain X to codomain Y. ## 24 Equivalence relation. ## 25 Metric. ## 26 Subspace. ## 27 Star-convex. ## 28 Injective. ## 29 Surjective. ## 30 Bijective. ## 31 Finite. ## 32 Countably infinite. ## 33 Uncountable. ## 34 Three. Some describe the reading experience as reading a detective novel in French while learning French. ## 35 Four. Before Renaissance, math is divided into two branches: Arithmetic and Geometry. ## 36 Five. Today's classification system name ninety-eight branches of math. ## 37 Six. This text uses undergraduate level material in branches dealing with spatial properties, namely ## 38 Differential geometry, general topology, and manifolds and cell complexes. ## 39 Seven. Modern math is characterized by generality, ## 40 Favoring abstract definitions and statements that extend to arbitrary dimensions and sizes. ## 41 Example of generality in a definition. The standard metric on R n encapsulates the concept of distance between two points in n-dimensional space. ## 42 The concept is defined for any natural number n, not just for dimensions one, two, and three which we are familiar with. ## 43 Eight. Generality gives deeper insight into structures, simpler presentation of material, and minimal duplication of effort. ## 44 Nine. Modern math texts make assumptions clear by following the structure of definition, axiom, proposition, proof. ## 45 Ten. What objects to define, which axioms to accept, and which logical rules to go by are all assumptions. ## 46 Eleven. Which assumptions to make is an aesthetic choice of the practitioner. ## 47 Twelve. This text provides figures illustrating low-dimensional and finite cases of every concept. ## 48 Example. The standard metric on n-dimensional Euclidean space can't be drawn for n greater than three. ## 49 The case of n equals two can be drawn easily since the space is flat. ## 50 In the topologist’s earring figure, an infinite number of circles can’t be drawn and are replaced by ellipses. ## 51 Chapter one. Sets, relations, and functions. Fundamental definitions used in all of math, such as ## 52 Set intersection. ## 53 Set difference. ## 54 Chapter two. Metrics and topologies. ## 55 Open ball. ## 56 S one. ## 57 S two. ## 58 Hedgehog space. ## 59 Exercises. Two point one. This problem asks you to make an infinite number of drawings, each with a different number of lines through a point, on the most general space possible for the task. ## 60 A partial solution is in the back of the book. ## 61 Chapter three. Anatomy. Definitions used to describe figures, with examples. ## 62 Stick-figure spaces. Let n be a natural number. Define the n-mother axial skeleton Omega n as the wedge sum of S n and the closed interval from zero to infinity in the reals, where the S n component is called the head of the n-mother. The figure below shows the case when the head is S one. ## 63 Given a subset big A in Q, define the mother A-appendicular skeleton psi A as a collection of trees psi little a theta for all little a in big A, and theta in the closed interval from zero to two pi intersect Q. The quotient space Omega n disjoint union psi big A mod tilde is called the n-mother big-A-stick-figure space, where little a is equivalent to psi b theta if and only if little a equals b for all little a, b in big A. ## 64 A figure illustrating the n-mother-A-stick-figure space. ## 65 A stick figure space is any subspace of the mother stick-figure space. Examples. ## 66 ## 67 ## 68 This definition is sometimes dissatisfactory, because: ## 69 One. It suggests that each stick-figure is a part of a certain definite whole. ## 70 Two. It doesn't account for stick-figures with multiple heads. For example. ## 71 ## 72 ## 73 ## 74 In rare cases, stick-figures can have symmetries. Reflectional, or bilateral. ## 75 Radial. ## 76 The definition of stick-figure spaces can be extended to 2D figure spaces with its boundary, or skin, enclosing an inside, ## 77 like making fonts bold. ## 78 Similarly, we can extend the 2D figure space definition to define 3D figure spaces. ## 79 The boundary of a 2D figure is made of lines. The boundary of a 3D figure is a 2D surface. ## 80 Chapter bonus. An uncommon variant of the stick-figure space called the snow-figure space. ## 81 Chapter 4. Surfaces and vector fields. One. 3D figure spaces are among the most common subjects of figurative painting. ## 82 Two. By the classification theorem, any surface of a closed figure is a connected sum of these surfaces: Sphere, torus, projective plane, and klein bottle. ## 83 Illustration of the operation of connected sum. ## 84 Chapter four. Surfaces and vector fields. Three. Classification, sums, and decompositions of surfaces determine the mode of abstraction in figurative painting. ## 85 Triangulation is an example of surface decomposition. ## 86 Three. Vector fields on surfaces determine the modes of markmaking in figurative painting. ## 87 Examples of vector fields on surfaces. ## 88 ## 89 Chapter five. Operations on figure spaces. ## 90 Aside from triangulation, there are many classes of figure decompositions. For example, ## 91 Kinbaku theory studies figure decompositions given by graph embeddings on surfaces. ## 92 Other common decompositions: ## 93 Flaying. Fracture. ## 94 If a stick-figure X is not headless. I.e. O equals p where p is a single point, then a decomposition of X is called a decapitation if the interval from C naught to C one in R is not connected. The figure shows a decapitation of a bilateral H one symmetric double-story one-stick-figure. ## 95 A decomposition of a connected figure space X is a quartering if it consists of exactly four connected components, U one through U four, where X is the union of U one through U four. Furthermore, the decomposition is a regular quartering if U i equals U j for all i and j from one to four. ## 96 Remark. Many constructions are named after real-life practices, though not all practices correspond to math definitions. Example. Mummification. ## 97 This is the end of my presentation. Thank you.