# 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.