matthias kawski interactive visualization ams / 23rd …kawski/conferences/visual.pdf ·...

Matthias Kawski Interactive Visualization AMS / 23rd Scandinavian Congress Odense June 2000

http://math.la.asu.edu/~kawski [email protected]

A new era of Interactive Visualization

Matthias Kawski

Department of Mathematics

Arizona State University

Tempe, Arizona U.S.A.



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...



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 98-34

National Science Foundation through the grants

DUE 97-52453 Vector Calculus via Linearization: Visualization and Modern Applications

EEC 98-02942 Engineering Foundation Coalition



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.



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)



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?



Outline

0. It is not just drawing pictures








Curl: Coherence or fragmentation?



hand-drawn brick-walls…. hand-drawn brick-walls…. hand-drawn brick-walls…. hand-drawn brick-walls….

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.

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s….



ha

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.

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A fragmented curriculum: VC, LA, DE



Coherence: VC, LA, DE

• The visual language

provides the glue that

connects different

“aspects”

of the same

mathemat.

objects!



Outline









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...



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



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 …?



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



Writing proofs, induction

•Summing finite sums?

•Towers of Hanoi?

•Coin-weighing?

write program!!

fun part = animation

Tiling with triominos



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



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



Complex functions color-coded

xyIyxIyx 2)( 22

Color coding adapted from Frank Farris at Santa Clara U, see also Needham’s Visual Complex Analysis WWW-site



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



Complex analysis: Laurent series

• Convergence of

a Laurent series

of rational function.

Why the asymmetry?

Uniform convergence

on compact subsets

of open annulus !!?



Uniform convergence of ((the error term of)) a

Laurent series on compact subsets of annulus

Mesmerizing beauty!

Observations, Questions, Conjectures, …..



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!



Diff Geometry: Viewing tensors What is a

Riemannian

manifold?

Viewing and

making sense o/

tensor fields

Compare Geometry Center at U Minnesota



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….



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



Outline


1. Sample: JAVA vector field analyzer







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?”)



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 http:/ /math.la.asu.edu/~kawski/kawski.html

INFORMS New Orleans, October 95: Set-up times lead to conflict

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1SA1IBL1,2,4.XLS STARTS 1 / 10, A only ONES, B PULL

Linear “convergence” to ‘almost empty’ buffer

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x and y needed? any advantage?

Research/insights: Interactive visual

Discussion/dissemination: Visual



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…..



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… )?



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 ...



Visual language ?

Syntax, grammar, semantics … ?

xyyx

ABBA

BuAsI

y

1

BuAyy

, 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



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...



Outline


1. Sample: JAVA vector field analyzer




4. More small JAVA samples

to take home...



JAVA applets...

• Famous curves applets http://www-history.mcs.st-and.ac.uk/~history/Java/

many others, just use WWW-search engine, e.g. start at: http://sunsite.ubc.ca/

• 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!

Ct)(



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

matthias kawski interactive visualization ams / 23rd …kawski/conferences/visual.pdf ·...

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