J.M. Rees1618 Summit St. Kansas City, Missouri 64108 USA
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 The following paper is part report, part methodological speculation
on a class offered during the fall semester of 2003. The title
of the course, 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 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.
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 It is perhaps telling that Klein regarded as his most notable
achievement the unification of geometry in what is widely known
as his 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 [1968]
I take the logical progression of geometric groups to be: congruent,
similar, affine, projective, inversive, differential, and topological.[5] 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. 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
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 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.
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
[11])
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 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)
- strange, often counter-intuitive ideas;
- the ideological, historical and disciplinary context of the information;
- their unexpected, myriad relations.
The approach is spatially organized, graphically demonstrated, as technically accurate as the audience allows, conceptually sophisticated and flexible.
- 1)
*Graphical*information leads the presentation and treats geometry as a species of visual art. There is a great deal of evidence concerning a basic human competence that might be described as using images to think*with*(rather than merely*about*). That this capability is disrespected in the academy is scandalous. As Barbara Maria Stafford has written: "In the widespread postmodern denigration of the aesthetic, what is forgotten is that from Leibniz to Schiller, the term connoted the integration of mental activity with feeling.*Aisthesis*, as perception or sensation, has in post-Cartesian and especially post-Kantian thought become separated from cognition. Rediscovering its pragmatic capacity to bridge experience and rationality, emotion and logic, seems all the more important in the era of virtual reality and seemingly nonmediated media. The awareness that images can sustain the continuity of thinking, not merely serve as fictionalizing counterfeits or pseudo-intellectual goods, brings both an ethical and aesthetic dimension to the computer age" [Stafford 1996:52].[12] - The
*technical*detail may be as elaborate or as elemental as outside factors allow and I think it is important to present as much geometrical detail as possible. During the course of the semester I was able to rigorously define continuity, equivalence, closure, group, and homeomorphism based on less thorough definitions of transformation, symmetry, infinity, cross ratio, curvature, function and I suspect what held the students' interest was the unfolding story. It is a narrative [13] in which many of the details were only glimpsed yet; are we as teachers measured by the questions we inspire as well as the facts we impart? *Synthetic*in the technical geometrical definition means deduction: building a system proposition by proposition from general principles. Klein's system is synthetic in this way, (even though others filled in the details). In this sense Klein's work exists squarely in the grand tradition of Euclid. I mean synthetic in a slightly different sense, as "combining ideas so as to form a whole that is greater than the sum of its parts."[14] Klein's schema is also synthetic in this sense. Big ideas in science and mathematics are synthetic in that one gets to do more with less.[15]
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.
- Some minority, yet significant percentage of art students
want to know about matters geometrical. Of the thirteen classes
I have taught only one was
*not*over-subscribed. Partially this is because I teach trendy subjects like chaos theory and partly, I think, it is because there is a pent-up demand for science-related courses. Mathematicians sneer at the idea that mathematics could be a spectator sport but I think that too much is at stake to allow the professionals to have the last word. - The flexibility of the schema is most promising. I adapt
it to a variety of geometry related courses. In
*C is for Chaos*I draw on the schema to develop ideas of scaling, fields, symmetry and continuity in order to explain dynamical systems, sensitive dependence, irreversibility, confinement, and periodicity. The Klein schema is tailor made for the course*Space from Aristotle to Einstein*. In that class I concentrate on the geometries from projective on in order to explain, at the end, how objects follow straight lines in curved space. A topics in "Western Thought" class,*Art, Science and Rhetoric*, dwells on the geometries from Euclid to Alberti, concluding with a detailed exposition of perspective: locating painting in a scientific tradition and statics in an artistic tradition. I have also taught*Paraline Drawing*, a studio class filtered through Euclidean and Galilean geometry disguised as the tools and techniques of drafting. - Finally, I think it important to acknowledge that not everything
fits the schema. In another course,
*Color from Aristotle to Newton*, the Klein schema is irrelevant. I harbor hopes that differential geometry might provide some techniques germane to the perception of color and that a class on color in psychology and physics could be founded on a geometrical exposition, but; enough speculation. As evidence of the actual class content for*Advanced Geometry*the next section presents the final from that class, with and without answers. I think there is no better way to convey a sense of the course content than to see for what information the students were held accountable. Just for fun, take the test yourself!
To download
the final exam with and without answers, click here
[1] The tile Elementary Mathematics from an Advanced Standpoint: Geometry,
was originally published as volume 2 of Elementarmathematik:
vom hoeheren Standpunkte aus (Lepizig, 1909; 3rd ed. Berlin,
1924); translated into English in 1931 (from the 3rd edition),
it has been reprinted by Dover Publications (2004). return
to text[2]
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
( [3] 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. [4] 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. [5] 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
done. [6] 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.
[7] 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. [8] 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]).
[9] 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.
[10] 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.
[11] 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. [12] 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.
[13] 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. [14] Herbert Simon refers to what I am calling synthetic
as a "pragmatic" response to complexity: [15] 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.
Brannan, D. A., Matthew F. Esplen, and Jeremy
Gray. Doblin, Jay. 1958. Forder, Henry G. 1962. Gauss, Carl Friedrich. 1965. "Gauss's
abstract of the Hartshorne, Robin. 1997. Huggett, Stephen A. and David Jordan. 2001.
Ivins, William Mills. 1964. Klein, Felix. 1893. "A comparative review
of recent researches in geometry." ______. 2004. Klein, Felix and W. Rosemann. 1968. Kramer, E. E. 1982. Lanczos, Cornelius. 1965. Lee, T.D. 1988. Lord, Eric A. and C. B. Wilson. 1986. Maxwell, James Clerk. 1954. Ogilvy, C. Stanley 1969. Reid, Constance. 1970. Hilbert. New York: Springer-Verlag. Scheiner, Christoph. 1631. Simon, Herbert A. 1962. "The architecture
of complexity." Stafford, Barbara Maria. 1996. Todd, James T., Augustinus H. J. Oomes, Jan
J. Koenderink and Astrid M. L. Kappers. 2001. "On the affine
structure of perceptual space." Yaglom, I.M. 1979. Yates, Frances. 1966.
Architecture
of Paper (2004), the virtual nomad (1999), and Manhattan
Miniture Golf (1978). He is guest editor of the 2006 REVIEW
Architecture and Urban Planning Annual. He has projects under
construction in Colorado and Missouri. I am a generalist in what
feels like a world of specialists.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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