Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June A new era of Interactive Visualization Matthias Kawski Department of Mathematics Arizona State University Tempe, Arizona U.S.A.

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Outline 0. It is not just drawing / showing pictures 1. JAVA vector field analyzer 2. Examples from many courses linear algebra, diff equns, elect circuits, writing proofs, complex anal, diff geom, applied discr.event syst. research 3. General observations, questions, tasks 4. More small JAVA samples to take home...

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Thanks for generous support by Department of Mathematics Center for Research in Education of Science, Mathematics, Engineering, and Technology Arizona State University INTEL Corporation through grant National Science Foundation through the grants DUE Vector Calculus via Linearization: Visualization and Modern Applications EEC Engineering Foundation Coalition

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Vision This is the beginning of a new era in which an interactive visual language not only complements, but often supersedes the traditional, almost exclusively algebraic-symbolic language which for generations has often been confused with mathematics itself, and which may be largely responsible for the isolation, poor public perception, and extremely difficult re-entry into mathematics due to the imposed vertical structure.

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Changing environment New opportunities! foremost: information technology New needs, expectations demands for higher efficiency/productivity –Case in point: Attitude towards “black boxes”, not just graphing calculators and CAS numerical integration of any dynamical system… e.g. “record a macro” (EXCEL, Visual Basic/C/Java) Op-amps (PSPICE, SIMULINK)

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June What is our mission? Goal? Objective? Keep math alive -- raise next generation of mathematicians (React to changing demands/needs/environ’s, but don’t betray our tradition) Applications: service to other disciplines/society… (what are willing to compromise, and what will we not compromise?) Math as a twin of philosophy, search for truth learn to argue, prove beyond any doubt... Math as a science Experiment and discover... Which of these (and others) require x and y symbols, and which may (possibly better?) be served via interactive graphical/visual languages?

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Outline 0. It is not just drawing pictures 1. JAVA vector field analyzer 2. Examples from many courses linear algebra, diff equns, elect circuits, writing proofs, complex anal, diff geom, applied discr.event syst. research 3. General observations, questions, tasks 4. More small JAVA samples to take home...

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Curl: Coherence or fragmentation?

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June hand-drawn brick-walls…. A fragmented curriculum: VC, LA, DE

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Coherence: VC, LA, DE The visual language provides the glue that connects different “aspects” of the same mathemat. objects!

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Outline 0. It is not just drawing / showing pictures 1. JAVA vector field analyzer 2. Examples from many courses linear algebra, diff equns, elect circuits, writing proofs, complex anal, diff geom, applied discr.event syst. research 3. General observations, questions, tasks 4. More small JAVA samples to take home...

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Multi-variable calculus Changing orders of integration in iterated integrals : Practically impossible using algebra alone… Extremely hard to visualize for beginners w/o tools Can the students “check” her/his work? Do teachers’ need devote much time / effort for grading? Language of this task is graphic: “slices”, and “projections”. MAPLE makes it interactive...

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Linear algebra The program as an interactive discovery tool…… The picture as a (list of) final exam questions…?! All coming from one picture: –linear mapping –under/overdetermined…. –eigen vectors and eigen values –linear combinations –singular value decomposition –orthogonal transformations –area magnification and determinant –…. A simple MATLAB program

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Differential Equations example from: Jerry C. Hamann, U Wyoming E.g. Rossler attractor: System of equations and the MATLAB-SIMULINK sample screen Are the descriptions equivalent? Which is more … useful for…? … accessible w/o many extra courses? … gets the message across? … doing analysis / math? experimentation / modeling / design …?

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Engineering: Electric Circuits Kirchhoff’s laws R, C, L, i, V, …. matrices, systems DEs? PSPICE sample screen (functionality similar to SIMULINK) Analysis? Design? Simulation? --- Interactive?! From: Hadi Sadaat Milwaukee School of Engineering

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Writing proofs, induction Summing finite sums? Towers of Hanoi? Coin-weighing?  write program!! fun part = animation Tiling with triominos

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Complex analysis: inversion z -> 1/z The visual impact of dragging moving the object is a compelling connection to physics and E-statics Construct from basic geometric principles observe and formalize, then rigorously establish general properties of conformal mappings

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Mappings of the (complex) pane Very hard to visualize w/o tools Almost ideal for study via interactive visualization The details -- e.g. color-coding & scales make all the difference

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Complex functions color-coded Color coding adapted from Frank Farris at Santa Clara U, see also Needham’s Visual Complex Analysis WWW-site

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Complex singularities Start each class w/ some pictures to start discovery and developing mathematical concepts The space and color scales have been manipulated. Want JAVA zooming tool with lots of scaling control

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Complex analysis: Laurent series Convergence of a Laurent series of rational function. Why the asymmetry? Uniform convergence on compact subsets of open annulus !!?

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Uniform convergence of ((the error term of)) a Laurent series on compact subsets of annulus Mesmerizing beauty! Observations, Questions, Conjectures, …..

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Diff Geometry: Gauss curvature Which color-code? After proving (!) e.g. the Theorema Egregium with MAPLE ((-- is that really doing mathematics?? ask Thurston!)), the real exciting math is just beginning!

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Diff Geometry: Viewing tensors What is a Riemannian manifold? Viewing and making sense o/ tensor fields Compare Geometry Center at U Minnesota

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Diff Geometry: Exponential map Geodesics on Riemannian manifolds (here a paraboloid) the exponential mapping and geodesic spheres, conjugate points, and the distance graph. Another almost ideal playing ground for interactive visual studies drag the mouse.. and study how the spheres depend on initial condition….

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Diff Geom: Curvature & geodesics The visual language makes a compelling “connection” -- compare Christoffel symbols?? Key interactive innovation: “drag the mouse” to study the influence of the initial conditions

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Outline 0. It is not just drawing / showing pictures 1. Sample: JAVA vector field analyzer 2. Examples from many courses linear algebra, diff equns, elect circuits, writing proofs, complex anal, diff geom, applied discr.event syst. research 3. General observations, questions, tasks 4. More small JAVA samples to take home...

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Excite and attract students Which classes are full, which cancelled? Do we want students to major in math, or let them all go to CS or …? Can we teach honest math also w/ …? (“How much math is in this picture?”)

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Research: Discrete Event Systems Simulation of Control strategies for re-entrant manufacturing systems (semiconductor manufacturing) Discrete event system on quotients of spaces of finite sequences... F(t,x)=0 ???? “Near-periodicity” obvious for eye/brain. This serves to define formal measure (metric). Teaching uses tools of real research and industrial work!

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June x and y needed? any advantage? Research/insights: Interactive visual Discussion/dissemination: Visual

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Programming - formulas??? Even the programming uses the more efficient “visual language”: macros are recorded, via point&click, color, resize, etc… Thousands of lines of “symbolic code”, here for a large discrete-event-system simulation, are visually gene- rated in a very short time (w/ little text-editing at the end). The programmer concentrates on the essence of the mathe- matical system using the more efficient visual representation almost throughout... Compare: “training a “robot arm” to perform certain movements: “by-hand” moving the arm along a path that is the initial value for an optimization routine…..

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Visual language as a productivity tool Single-letter variable names?? Rapid development relies on a visual interface and language. Precision. Efficiency. Full of mathematics, but traditional algebraic-symbolic language is moved far into the background! What can we, in math, learn from CS (until recently a part of math… )?

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Who writes the code? … rune, papyrus, books, photos, TV, overhead proj’s,.bmp, animated.gif,.vrml, … What is the purpose / the objective???? shall student WRITE the code / USE the program/tool ? Major investments (money, time to learn syntax) or freeware? General purpose tools or dedicated single-use throw-aways? hope for a reasonable set of competing languages to evolve no MS tyranny, no Tower of Babel McNealy ”the network is the computer!” Easy access to lots of tools. Rely on WWW search-engines...

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Visual language ? Syntax, grammar, semantics … ?, but What purpose? Most efficiently/effectively use: color, orientation, scale, perspective, size, window, texture and reflectivity (!) drag, rotate, resize, stop, restart, rite/left-click, copy, (un)group, …. Expect different languages, cultures to evolve

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June How to use interactive visualization? Compare playful enjoyment versus serious study to casual concert-goers versus professional musicians… The educator’s job: Provide guidance to explore the depths... Develop the effective exercises… –“graph”, “plot”, “print”: … very little gain! –“inverse questions” -- (“find parameter such that….”) !!! one-parameter often do-able by trial&error two- or three parameter problems usually good motivation for analysis! –experiment, observe, conjecture, test, formulate desired theorem, define, prove, …. generalize, experiment, ….. axiomatize…. In visual language, too, insist on just as much precision...

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Outline 0. It is not just drawing / showing pictures 1. Sample: JAVA vector field analyzer 2. Examples from many courses linear algebra, diff equns, elect circuits, writing proofs, complex anal, diff geom, applied discr.event syst. research 3. General observations, questions, tasks 4. More small JAVA samples to take home...

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June JAVA applets... Famous curves applets many others, just use WWW-search engine, e.g. start at: MK’s current project: provide template (“beans”) for student “honor’s projects”, e.g. –time-varying linear systems (Mathieu equation: but unstable) –reorientation of satellite (gymnast, falling cat) via internal shape changes –complex mappings (winding numbers, branch-cuts) –analytic continuations (click = center, image coded by color) –exponential mapping (geodesic spheres, co-states…, curvature) –connections / parallel transport along user drawn-curves… – … many more... Common: amenable to inverse questions that lead to the development (discovery) of core mathematical concepts!

Matthias Kawski Interactive Visualization AMS / 23 rd Scandinavian Congress Odense June Winding number & branch-cuts What are the exercises, questions, desired insights and discoveries? This is just a simple JAVA proof-of-concept, plan to include full “parser”… typical honors visualization project