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But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident...
Felix Klein [1893:243]
The courses I teach are all designed to introduce scientific ideas to arts students through the visualizations that are such an important part of discourse in science. I describe the intellectual context, define selected concepts using geometry (classically, a liberal art) and introduce elementary mathematical formulae--all relying on graphic visualizations to make fundamental ideas clear. My goal is to provide a means by which visually sophisticated persons may think with geometry about culture. On good days I am a storyteller in the history of ideas.
The following paper is part report, part methodological speculation on a class offered during the fall semester of 2003. The title of the course, Advanced geometry from an elementary standpoint: Topology, is a play on words, a variation on the title of a famous work by Felix Klein. The title announces both my indebtedness to Klein and the content of the class.
Topology is the geometry of continuity, the last in a series of geometries whose definitions of equivalence become progressively more difficult to describe to students with little formal mathematical education. Topology, conventionally rendered as "rubber sheet geometry", is the geometry of stretching, squeezing, or extruding but not of cutting, folding or tearing as long as neighboring points remain neighboring points [Huggett and Jordan 2001]. This course is designed specifically for graphically sophisticated students in the arts  and is intended, in the main, to introduce geometry as a discipline of great visual and intellectual beauty. (It helps that we can visit the rare book room of The Linda Hall Library of Science and handle a dozen antique books renowned for their scientific and artistic significance; see http://www.lhl.lib.mo.us.) In class, graphic visualizations and geometrical demonstrations (mostly) take the place of a postulational, or, if you will, axiomatic, presentation. In the end it is hoped that students will unite intellectual inquiry and artistic endeavor according to their own interests.
This essay offers samples of class content highlighting the visual approach in sections 2 and 3. Section 4 details my assumptions about teaching geometry, by which I mean "things being tested in the classroom," and a course outline. Section 5 records some observations based on my experience teaching over the last seven years. Section 6 returns in detail to the content of the topology class, where I present the final examination, with answers, for your intellectual entertainment.
2 FELIX KLEIN'S GEOMETRY SCHEMA
What makes for a great mathematician may not be exclusive of what makes for a great teacher and Klein was, by all accounts, a great teacher. He wielded considerable influence over one of the great mathematical schools of the late nineteenth and early twentieth centuries, the University at Göttingen. He established a research center there that was, for a time, a focus of the mathematical universe, attracting David Hilbert from Königsberg. During his tenure the student body included Hermann Weyl, Richard Courant, and Max Born. The first woman D.Phil., Grace Chisholm Young, graduated in mathematics from a German university, graduated under his auspices.
It is perhaps telling that Klein regarded as his most notable achievement the unification of geometry in what is widely known as his Erlangen Programm of 1872 [Klein 1893]. Based on the concept of a mapping, Klein showed how the geometries of his age (metrical, projective, line) could be joined into a single geometry using the theory of groups.
A group is a set of elements filtered through an operation. To be a group the elements and their mapping must be closed, associative, contain an identity element and have an inverse. For instance, the integers are a group with respect to addition. The integers are closed since an integer plus an integer is always another integer; associative because (a+b)+c = a+(b+c); they have an identity element -- zero; and the inverse of an integer is its negative. Therefore; the integers are said to be mapped onto themselves. Pregnant with promise, the theory of groups unified geometry, unified discrete and continuous mathematics and forecast new approaches in algebra and number theory.
Following Klein's lead, filtered through Lord and Wilson  I take the logical progression of geometric groups to be: congruent, similar, affine, projective, inversive, differential, and topological. This is a logical arrangement because the operations at the core of each group are progressive. In other words congruence is a special case of similarity, is a special case of affinity. etc. Each geometry is simpler than the one that comes before. Simpler means that "within the hierarchy of possible geometries, affine structure is more primitive than Euclidean structure because it is based on a smaller set of underlying assumptions, and is therefore invariant over a larger set of possible transformations" [Todd 2001: 195]. What is specifically true for affine in the context of Euclidean geometry is generally true of topology in which there are the fewest restrictions on what constitutes a legal mapping.
Fig. 1 proposes emblems for each geometric group. The emblems, besides being simple place holders, store information about the nature of each geometry. Far less arbitrary than icons, the emblems must be associated with some conceptual content to be effective. A fragment of the content for each group is presented in the accompanying Glossary of the geometries. (Please make every effort to synthesize the graphic descriptions and the textual descriptions in what follows. The intelligence of the material and the efficacy of the method depends on it.)
Fig. 1. Klein's geometry schema in emblems and a glossary of the geometries
3 THE ARRAY: ORGANIZED LOGICALLY
Fig. 2. The Array organized logically
The array demonstrates how the emblems are used to structure the presentation of information and how class content is delivered in a memorable order. The column labelled "characterizing transformation" is intended - working hand in glove with the emblems -- to elucidate the nature of transformation in each geometry. For instance, similarity is described using the idea of scale, a concept with which students are (already) well acquainted. I use the emblem to show that similarity contains congruence as a special case (note the rotation and scaling of the dashed square relative to the dotted square). I use the "scaling" graphic from an historic source (Scheiner's Pantographice (Rome, 1631) which the students get to inspect first hand at The Linda Hall Library) to demonstrate what property remains invariant (shape or more accurately angle) and what property is relaxed (length).
From: Christoph Scheiner, Pantographice, Rome, 1631.
The column "transformation of the square" is an attempt to engage the students' considerable patterning skills. A square chessboard is presented as a gauge figure and "deformed" in a way that is consistent with the rules of transformation for each geometry. These patterns are intended to be evocative, rather than rigorously mathematical, a shameless appeal to students' design sensibilities.
The important idea regarding the array is that Klein's logical progression of geometries can be elaborated in any of a number of ways, depending on what needs to be presented in order to clarify and extend geometrical and scientific concepts. Other columns of information that are included in extended versions of the array are: fundamental theorems, analytical expressions, associated geometers, representative transformations and/or optical analogs. This is to name only a few of the possible themes that may be included in a class.
4 COURSE OUTLINE: ADVANCED GEOMETRY FROM
AN ELEMENTARY STANDPOINT; TOPOLOGY
Sometimes related ideas can be about the rules behind the rules. Under congruent geometry, I develop symmetry as a related idea. One way to define symmetry is through a demonstration of proper (identity, translation and rotation) and improper (reflection ) rigid motions. This approach is always effective because students can visualize the processes that lead to a superimposition of figures and thereby strengthen their geometric intuition. This Euclidean notion of symmetry (I), however, is not very robust. Therefore, it is important to present symmetry (II) more abstractly, as one of the three conditions that has to be met in order for there to be an equivalence relation between sets. Equivalence relations require sets to be reflexive (a set D must be equal to itself, D=D), symmetric (if transformation t maps E' to E" then t' maps E" to E') and transitive (if A is congruent to B and B is congruent to C then C must be congruent to A). Linking the superimposition of figures (symmetry I) to the idea that the mapping must be reversible (symmetry II) is one instance of the way concepts are "grounded" in geometry. It is an example of what makes geometry so beautiful -- the evolution of ideas towards their simplest, oftentimes most abstract expression.
Finally, related ideas allow themes to be developed over the course of a semester. The theme developed in the topology class was infinity. The infinity of points on a line; the twin infinities of the very small and the very large; the role of infinity in the development of projective geometry; infinity as a point in the complex number plane; the infinity of a figure that is bounded but not closed, etc. Related ideas add density to the course. More importantly the model of 1) a taxonomy of geometries providing the structure and 2) a stew of related ideas providing the variety, is adaptable to a cluster of science courses. For instance a class with a kinematic emphasis would present physical concepts as related ideas. As a bonus students who elect two different classes may begin to appreciate geometry as the "language of physics" and (possibly) begin to compare that to the role geometry plays in art.
I. Preliminaries (note: RI = Related Idea)
II. Klein's Schema (see figs. 1 and 2)
b. Mnemonic. I assume that a mnemonic association of images and concepts in a structured hierarchy fosters assimilation of:
The approach is spatially organized, graphically demonstrated, as technically accurate as the audience allows, conceptually sophisticated and flexible.
c. Methodological. I assume the presentation of content may be adjusted to fit the audience without "dumbing down" the material. The method triangulates among graphical, technical and synthetic information.
To be sure, the approach to teaching geometry herein described is synthetic for students in so far as they can appreciate the connective tissue unifying geometry, but; it may be more than that. On good days I see flashes of insight that joins geometry and physics and, every now and again, a glimmer that promises an implementation of geometric techniques and scientific ideas in their own work. By this measure there remains more to be done in adapting advanced mathematical ideas for artists, and I derive inspiration from the students, who often turn out to be excellent teachers.
d. Voice. I think the voice in which geometry is presented is important. I favor the voice of revelation trading on geometry as an hermetic tradition, with one important caveat. The beauty in economy that proofs display, the elegant foundations and of the chain of logic, is mystery enough. There is no need to trade in some variety of Rosicrucian mysticism because the details of modern scientific geometry are as demanding, as hidden from untutored consideration and as full of wonder as any esoteric teaching.
 The design for this course is a product of the process I went through (in fits and starts) to grasp simple mechanical concepts. Graced with a facility for geometry, I was often frustrated in my attempts to understand analytical physical expositions. Unwilling to give up the appreciation of statics for obvious reasons (I am an architect) I found the giants of physics often presented their insights in geometrical forms relatively easy to understand (F=ma is due to Euler not Newton). My experience suggests that if there is a reason (and a rational) to teach physics geometrically it may also be possible to teach geometry graphically. I am not proposing a reform of technical education for scientists or mathematicians, just a different emphasis, one that may play to the strengths of non-specialists more effectively. return to text
 The clear-eyed, no holds barred, appraisal of the work of mathematicians by other mathematicians has always delighted me. It stands very much in contradistinction to the relativistic discussion of art common in American schools and is often harsh even by architectural school standards of critique. return to text
 See the MacTutor History of Mathematics archive of the University of St. Andrews, Scotland, for an excellent biography of Klein, his intellectual accomplishments and mathematical context. return to text
 This stratification of Klein's schema is by no means
unproblematic. Many mathematicians omit congruence and similarity
as discrete geometries, subsuming the whole of Euclidean geometry
in affine. I do not favor this approach because it is important
to ease students into the details of mappings using transformations
with which they already have experience. Since I take pains to
show how similarity is a special case of affinity, no harm is
 This is well trod territory in science museum displays and I have never seen the whole schema played out with anything near the graphical sophistication accorded the rigid motions. I expected to find this kind of a detailed presentation, since most historians call attention to the central role of Klein's schema in teaching mathematics, yet I found such a schema only in Lord and Wilson whose simple diagrams became touchstones. As my grasp of the geometries developed I took to revising the diagrams, often formalizing a sketch from my notes or re-drawing graphics from particularly helpful sources. The discipline required to construct an hyperbola or draw, with construction lines, a pair of inverse points, is an important part of the method here espoused. I often give drawing problems as homework and revise lectures according to students' progress measured by their drawings. return to text
 Vanishing point in this context refers to the point where converging lines intersect. Converging lines in perspective constructions are parallel. When the vanishing point is moved infinitely far away, lines that appear to converge in a finite field are said to intersect at infinity but; they no longer converge and are therefore said to be parallel. As a consequence affine geometry is established as a special case of projective geometry. The necessity for this unexpected reformation of Euclidean geometry -- any two lines, in a plane, always intersect (at an imaginary point if necessary) -- has to do with the reformulation of geometric foundations by David Hilbert and with the introduction of homogeneous coordinates. However; I often refer to it as a strategy to preserve the duality of lines and points. return to text
 I think it important to distinguish between strong and weak forms of non-Euclidean geometry. Inversive is weakly non-Euclidean because it shares every fundamental geometric characteristic but that of the fifth postulate. Strong non-euclidean geometries violate the principle of rigid motion. (cf. [Hartshorne 1997]). return to text
 Gauss called non-Euclidean geometry anti-Euclidean [Gauss 1965], a usage I favor because curved spaces violate the spirit of Kant's a priori regarding Descartes's coordination of Euclid; by which I mean the automatic assumption of embeddedness. return to text
 I think it impossible to overestimate the importance of the tendency to ever greater generalization often evident in geometry. To express this idea as a gross generalization: in the humanities intellectual progress is often evident as the differentiation of ever narrower domains. Art history is divided into Ancient, Renaissance and Modern. Renaissance Art history is divided into Proto-, High-, Baroque and Mannerist. High-Renaissance is distinguished according to its Venetian and Florentine varients -- and so it goes, ever narrower, ever more specialized. It seems that in mathematics there are (at least) more instances of major intellectual breakthroughs that unite discrete practices, than in any other discipline. Klein's Erlangen Programm is such a breakthrough. Another example is the way projective geometry provides a unified treatment of circles, ellipses, parabolae and hyperoblae as conic sections. Another example, drawn from geometrical physics, is the way Newtonian relativity (itself a generalization of Galilean relativity) is a special case of a more general rule -- Special relativity. "Synthetic" as I use the word later in the essay, is akin to this process. return to text
 Reflection is an improper rigid motion because it requires the figure to move outside of its plane. I make much of this distinction early on so that when discussing attitude transformations (in differential geometry) as translations and rotations only, it is clear that reflections are excluded because they change the handedness of the coordinate system. return to text
 By the way, I think the "Institute" (art education) has made the inverse error, ignoring "mental activity" and fixating the "feeling" component of aesthetics. It is my conviction that architectural education presents a "third way," combining intellection and emotion, aesthetic and scientific education in an effective synthesis. return to text
 I am careful in class to draw a distinction between "a" story and "the" story. There is no question that I am only telling one of many possible stories. return to text
 Herbert Simon refers to what I am calling synthetic
as a "pragmatic" response to complexity:
 For example, before Klein developed the concept
of a group, Euclidean and non-Euclidean geometry were treated
as fundamentally different geometries. After he developed the
group concept they can be treated as parts of a greater whole,
the geometry of invariants. A large part of his stated motivation
for the Erlangen Programm was, in fact, this unification of geometry.
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Hartshorne, Robin. 1997. Companion to Euclid: a Course of Geometry Based on Euclid's Elements and its Modern Descendants. Providence; Berkeley: American Mathematical Society.
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______. 2004. Elementary Mathematics from an Advanced Standpoint: Geometry. New York: Dover (1st ed. 1909).
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Sources for the illustrations:
ABOUT THE AUTHOR
He writes about himself: "As I often find myself a purveyor of unpopular ideas, allow me a short apology. My avocation is history of geometry which means that I occasionally read geometric proofs for entertainment. Architecture is entertaining precisely opposite the way mathematical proofs are entertaining. Geometry (like painting) requires a highly focused contemplation towards an occasionally ecstatic reward. Architecture, on the other hand, received in a state of distraction, is thicker than the eureka moment geometry and painting share. Architecture is thick the way play is deep, a somatic thrill as opposed to an intellectual reward."
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