AMATYC National Math Conference, Phoenix : Workshop W16 -- 2002

Seeing Math in a Whole New Dimension
by Dan Bach - Diablo Valley College - Pleasant Hill, CA

e-mail: dbach@dvc.edu website: www.dansmath.com 0. Formulas vs pictures -> Dan's 20th-Century Slide Show Book list for Mathematica and 3D Sample Graphics and Movies   1. Using Color as Information -> Springy Sine Wave Color and Saturation Color Me In(creasing) Eigenvector in a Haystack From 2D to 3D (Gradually) Contour and Density Plots

2. Three Ordinary Dimensions ->
Regular coords - xyzpt.mov Cylind. and Spher. Coords
Infini-D's Info Window Amazing Intersecting Cubes
Real-Time 3D in Mathematica "Monte Carlo" Volumes

3. Time as Fourth Dimension ->
Moving Tangent Planes Quadric Surfaces
Color Planets.Movie Morphunctions
The Digital Lava Lamp Skiing Curves on z = f(x, y)

4. Color and 3D Graphics -> Conic Sections Light 3D Primitives and Morphs Seeing Double (integrals) Surfaced Curves   5. Fifth Dimension - 3D animation -> Roller Coaster ­ Bach's Car Osculating Circles from Space! Rotating Curve Shadows Parametric 'Chia Surfaces' Dan's 3D Golf Green

6. What Else Is There? ->
Sound as a New Dimension Apple's Graphing Calculator vs iCalc
Dan's Talking Calculator Finale: Woodhead and Safir Shape Up

Detailed information and additional pictures appear lower on this page; scroll down!

Handouts from this talk available soon in PDF format on the conference website at www.amatyc.org.

 

 

Introductory Remarks

This computer lecture/demonstration will take you, the audience, on a visual tour of several

familiar, and some new, math topics. The emphasis will lie on illuminating formulas and

properties of numbers by representing them with pictures and animations.

Most students have a preferred learning style. The more senses and styles that we stimulate,

the greater chance the student has of learning and becoming excited about the topic under

discussion, indeed, mathematics in general. Because the visual aspect of learning is seen as

difficult to demonstrate, it is often given short shrift. This presentation will explore

mathematics through this visual portal.

Categories of examples will include:

1 - Using color to impart information
2 - Three ordinary dimensions: (x, y, z)
3 - Time as the fourth dimension
4 - Color and 3D graphics
5 - The fifth dimension: 3D color animations
6 - What else is out there?

Examples, such as Dan's 216-color cube, Moving Tangent Planes, or Osculating Bach's Car,

bring concepts to life by displaying, respectively: (1) all 216 web-safe colors in a three-dimen-

sional array, (2) a grid sliding along a surface created in Mathematica, or (3) a moving T-N-B

frame along a space curve (complete with headlights and four wheels). Further examples

planned include: Monte Carlo Volume Method, Finding an Eigenvector in a Haystack, and the

popular Spinning Space Curve with Shadows.

 

The feature animation is the two-minute math story, rendered in 3D, of Woodhead, a math-phobic,

and Safir, his 'spherical adviser.' This cartoon is shown to students and fans to explain the anima-

tor's use of (x,y,z) coordinates to create the characters, their movements, and their surroundings.

Software selected to create and display these images includes: Mathematica (Wolfram),

Authorware (Macromedia), Graphing Calculator (Apple), Infini ­D (MetaTools), and Photoshop

& Premiere (Adobe).

 

I will show you how to download graphics and the code of many of my Mathematica notebooks

on my website (www.dansmath.com) and then you can experiment and tweak them all you want!

A key technique in many of these notebooks is the ability to replace parts with other parts. For

example a sphere or other displayed surface is made up of hundreds of small polygons; adding

a normal vector will make hair grow from the surface. Or for a curve made up of small line

segments, the normal and binormal vectors can be used to create roller coaster tracks or

mind-boggling geometric art.

For us professionals and self-proclaimed experts, ideas we've carried for a long time can renew

their shine with an unexpected visual approach. I hope you enjoy seeing the mathematics of our

dusty textbooks come to life. Entice others into the 'club' by showing them how math can be used

to illustrate ideas and create pictures that you never expected or imagined!

 

­ Dan Bach dbach@dvc.edu www.dansmath.com

AMATYC National Math Conference ­ Phoenix ­ Nov. 15, 2002

Seeing Math in a Whole New Dimension:

More detailed notes and pictures!

by Dan Bach - Diablo Valley College - Pleasant Hill, CA

0. Formulas vs pictures (which are designed to look at?)

Dan's 20th-Century Slide Show
For a Photoshop class in 1996 I decided to contrast the often confusing world of formulas with the illustrative world of pictures. View the slides in pairs.

Books on Mathematica and 3D:
Animating Calculus, by Ed Packel and Stan Wagon
Exploring Mathematics with Mathematica, by Jerry Glynn and Theo Gray
Beginner's Guide to Mathematica 4, by Theo Gray and Jerry Glynn
Mathematica Graphics, by Tom Wickham-Jones
User's Manual and Tutorial, Infini-D v4.0, Specular Software

Sample Graphics and Movies
From the simple 2D world of graphs of y = f(x), it's an easy jump to color, thickness, 3D curves and then surfaces. Put in some motion, hide the coordinates, and you've got art!

1. Using Color as Information (it's like another coordinate!)

Springy Sine Wave
When a spring is compressed, the stresses build up in the metal. Use color to impart this important information. See a few other choices as well.

Color and Saturation
Each hue in the rainbow is assigned a number from 0 to 1, the same with saturation and brightness. You can use this HSB or the RGB color space. I have 216 cubes showing both.

Color Me In(creasing) Ever wish the road maps showed how steep a road was? Just add a color coordinate!   Eigenvector in a Haystack A linear transformation from R2 to itself leaves certain directions invariant; pick them out of a color wheel and spot those eigenvectors! From 2D to 3D (Gradually) Pull yourself out of those flat maps and use numbers to illustrate heights, temperatures, or population densities at points on a region.

Contour and Density Plots
Before there was real 3D, there were level curves and contour maps. Density plots also made up for the dimensional deficiency.

2. Three Ordinary Dimensions (length, width, and angles?)

Regular coords - xyzpt.mov
This movie, small enough to fit on a website, shows the simple move out of the xy-plane that really launched the 3D revolution.

Cylindrical and Spherical Coordinates
Enough of this xyz tyranny! See each coordinate, r, q, z, r, and f, do its own thing. Partial derivatives suggest their true meaning when one at a time changes.

Infini-D's Info Window
What do 3D animators work with? See this basic interface with xyz coordinates, scaling, RGB color, lighting, texture maps, transparency, and motion, all controlled by math!

Amazing Intersecting Cubes
What if you had two solid cubes, each of which could allow the other to pass through it without coming apart. How is this possible, you ask? Infini-D will show you!

Real-Time 3D in Mathematica
Ok, this is the coolest thing. You can put in raster images and convert them into 3D, then rotate them (or any other 3D object) in real time with your mouse!

"Monte Carlo" Areas and Volumes
To calculate the area of a plane shape, throw darts at it blindfolded, then compute the percentage that hit inside the region, compared to your wall. This can be done in 2D and 3D with virtual darts and Mathematica.

3. Time as the Fourth Dimension (but it doesn't have to be!)

Moving Tangent Planes
Using a gradual progression from surface to trace curves, to tangent lines, to spanning plane, to motion of the base point, we can create moving planes along any surface!

Quadric Surfaces
The only thing holding tangent planes back in this 3D world is their flatness. By using the quadric surface approx. at each point, these movies become much more animated.

Color Planets Movie What if our solar system was the five Platonic solids, orbiting a polyhedral sun? Morphunctions What is the real difference between y = x2 and y = x4 ? How can I see the stages of Taylor or Fourier approximations? Use a space-time homotopy, known as the com- mon morph. The Digital Lava Lamp I got this out of the inaugural issue of the Mathematica Journal; enough said. Skiing Curves on z = f(x, y) When looking at a surface, we usually see a grid from the xy-plane. But why not other paths on the surface, defined by general parametric curves in the base plane?

4. Color and 3D Graphics (making four dimensions if you're counting)

Conic Sections Light
Ok, the algebra books draw planes intersecting a cone, and some of us bring flashlights to class, but can a software wall and a virtual cone of light pass the ellipse test?

3D Primitives and Morphs
What are those basic shapes, the elements of geometric model building, and how can we make them change shape from one to the other at will?

Seeing Double (integrals) Anybody can do an iterated integral (well, most of us and some of our students), but can you see the volume as the sum of a bundle of french fries for any z = f(x, y)? Surfaced Curves Along with the tangent vector, the normal vector and the binormal vector are attached. Using these three vectors we can create all kinds of things on and around the curve.

5. Fifth Dimension - 3D animation (color and time as well as position)

Roller Coaster ­ Bach's Car If you go for the extreme in everything, you'll want to draw a track perpendicular to the binormal vector, four wheels, and choice of car bodies. Add a driver, and off you go! Rotating Curve Shadows Create any rotating 3D curve inside a box. Do the curve's shadows rotate on the walls? Maybe not literally, but they sure indicate what's going on in the center of the room! Osculating Circles from Space! Take a trip along any 3D curve: The tangent vector, normal vector, and circle of curvature at each point, leave a trail of centers that any differentiating eye can follow! Parametric 'Chia Surfaces' Donuts, spheres, Möbius bands, and other 3D surfaces are made up of polygons that can be replaced at your whim by lines, spots, and yes, even hair!

Dan's 3D Golf Green
I don't play real golf or even much video game golf, but I like making white spheres roll along parametrized greens toward a dark hole!

6. What Else Is There? (let me tell you, there's plenty!)

Sound as a New Dimension
Ever hear a trig function? Sine waves, square waves, and sawtooth waves all sound different, but how do the different frequencies interact? What about the Fourier Series?

How Do You Calculate? Apple's Graphing Calculator, iCalc, and Dan's Talking Calculator are compared side by side. Finale: Woodhead and Safir Shape Up I've worked on this one for a while. It's the story of a math-phobic, Woodhead, and the round wise one, Safir, who explains and shows how they are literally made of math! Modeling, rendering, and animation done in Infini-D, edited in Adobe Premiere.

Visit www.dansmath.com often, and tell your students about the free math

lessons, weekly contest problems (with leader boards), sample copy-paste-

able Mathematica notebooks, and many feature pages!

 

Handouts from this talk will continue to be available at this location,
and soon in PDF format on the conference website, www.amatyc.org.