Communicating mathematical ideas requires more than just the recitation of axioms and proofs. It is necessary to give an intuitive feel for the mathematical entities and their relationships. This article will describe a variety of visual techniques for building intuition based on such design elements as color, motion and rhythm. Working from the specific to the general, I will begin with concrete examples from the Project Mathematics video series. I will have a section on each program telling the sorts of things I was thinking about when I designed each scene. This will be followed by a few general observations and ideas.
These are reviews and reprises of scenes from previous programs. The intention is that, even if the viewer hasn't seen these concepts before, they can at least get a hint at the ideas. And of course it is a reminder for those who have had it before.
There are two types of ratios, internal and external. Lines glow on the geometric diagram at the same time that the ratio expression involving them appears on the screen.
Because lengths are the focus of attention here the lines and labels are drawn in a brighter color and the background and triangle insides are in a dimmer color.
Note that labels appear on lines, not beside them. This is so that, when shapes are rotated the labels can stay put. This raises difficulties in keeping them from blending together so the labels are brighter than the lines. Drop shadows on the labels also help distinguish them from the lines. Note that drop shadows are only put on labels, not on elements of the 2D geometry.
This color scheme is repeated later when these equations are used.
This is a concept that is not often stressed in high school geometry courses. When the triangles and square shear they fade into a set of parallel lines. The line widths and spacings were carefully picked to be small enough to give the impression that they cover the entire area of the triangle but still be big enough to be perceptible as lines. Sound effects of sliding door is used whenever shearing happens. The parallelogram shears twice in two directions because we will be using this in a proof later. This is put here to soften the blow for understanding that you can mangle a quadrilateral by several successive shearings without changing its area. Note the original rectangle gets lighter when its copy is being manipulated. Transparency is used a lot to de-emphasize parts of the picture that are not being talked about currently.
A trivial example of the area of a square begins the scene. An edge is made bright. It and the area (signified by a figure in the middle of the shape) are 1. This is meant to be a reprise of scenes we are currently working on for the Similarity program. I am trying to use the same color scheme across programs. The colors for the square and pentagon are repeated later too.
The background is a mathematical function made with a frequency modulation due to Geoff Gardner. It's a visual use of the same technique used for sound synthesis in some music synthesizers. So many people complained about it looking tacky that I am now doing something different. Note the number in the corner keyed to chapter in workbook.
This is just gratuitous animation, made to look interesting and let me play with run cycles. The animation was specified as rotation angles of limb joints at several key frames. The program interpolates angles for inbetweens. Note also the rotation of the line labels as the lines rotate. Also not that positioning lines end to end is not a proof, just a means of building intuition. Note that the triangle is slightly transparent. This is done for two reasons: One, it shows the grid through so you can get a better feel for its size. Two, it makes the color closer to the background color to visually tie the shapes together.
Note that backgrounds don't have patterns. This is Euclidean space. All regions have the same properties. Using a patterned background would tend to blunt this perception.
More gratuitous animation. Character animation is done very simply with rigid bodes and hinge joints at the arms and legs. Note shark in moat to emphasize that we can't go in there. Tried to make ladder the same relative size as that which appeared in live action shots. Tried to pick numbers that were about right for the scales of the objects shown.
Similar comments to above.
When side c is broken into x and y I had to move the c's off the line so it didn't look like it labeled one of the pieces. The c moves back when they separate. Labeling moving diagrams is trickier than static ones partly because you don't want to call attention to parts of the diagram which are of lesser importance; overly actively dancing labels tends to do have this bad effect.
I worried a lot about flipping the triangles over. I didn't want it to look like the areas and sides were being scaled in some arbitrary manner, causing the viewer to lose track of exactly how long the relevant edges were. It's done as a rotation in 3D with correct perspective, so it looks like a familiar motion and hopefully won't be confusing. I brought in the ratios of sides in a manner similar to how they appeared in the prerequisite scene.
Algebra for Mathematics! is in general more deliberate than that in The Mechanical Universe. Instead of having a jump over the equal sign, I explicitly multiply both sides by a, one side at a time. When it hits one side the equation tips over and when the other one comes in the equation rights itself. I want to give a general feeling for algebra as an attempt to do things to and equation while being careful that you are always keeping both sides equal to each other.
There was an inconsistency in some situations however. Sometimes the expression a^2 splits into two copies of a to show they are products and sometime a solitary a splits into two copies just to make another copy of it.
Substitution is always tricky. You must draw viewer's attention to what is going to change by shaking it (with cowbell sound effects). It might have been better to actually fly the expression in; I did this on some later substitutions.
The number 2 shakes to draw attention to the squares in the equation Note that the orientation and labeling of the scene is set up at beginning so that when the big square nudges equation out of the way and the squares slide off they are in the proper a+b=c order. Lots of animations are designed back to front. You find the desired configuration for the punchline at the end of the scene and you arrange the shapes at the beginning of the scene so that there is minimal motion requires to get there.
I originally toyed with the idea of making the three square shapes have different colors so that the larger square represented a mixture of the two different colors of the smaller squares. This bothered me because color (in fact brightness) affects your perception of size. Brighter colored shapes look larger than dimmer colored shapes even if they are the same size. So I used the same color for all shapes whose areas were being compared or added.
At the end of the scene we are subtracting a^2 from both sides. Again, instead of hopping over equal sign the -a^2 and a^2 jump up and down to call attention to themselves before they fade out. If they just faded out they would be gone before viewer realized what was changing. Where b merges with b it violates the b^2 principle.
Lengths are measured with digital counters to emphasize the event when integers are present. This is something students can do themselves with three rulers. The color was picked to look sort of wood like.
This starts with the same orientation and coloration as the geometric interpretation scene. A larger square is built with copies of the yellow triangle. The trick is that, as we slide the yellow triangles around, the blue region maintains a constant area. I had to reorient the yellow triangles in this proof from how they are typically shown to avoid needing to rotate them. The c^2 labeling the large blue square splits into two c's labeling each of its edges. Likewise the a and b edges are later labeled and the labels come together to form a^2 and b^2. At the end of the scene, a quick adjustment of parameters is made to show that a right triangle of any shape will give the same result. This is an important part of proving anything, the variation of parameters that still satisfy the conditions of the theorem.
This proof as usually given in textbooks is riddled with construction lines and labels for lots of points. We give the same proof here with fewer distractions since the construction lines can come an go as they are needed. The background color was selected to look sort of like a parchment page in an old book. It's the same color as the shearing demo at the beginning of the program. We show the dark blue auxiliary triangle in the proof as rotating to two different positions; Euclid showed them in both positions on top of each other. With animation this was made less confusing. Note the sound effects for rotation (different from those for translation and shearing). When the triangle rotates, its edges are labeled with b and c to emphasize that these edges do not change. At one end of the rotation each edge lines up with the sides of a square. At the other end each lines up with the other sides of the same squares. The whole thing is repeated for the green side to reinforce what is going on.
The shearing version is where the prerequisites scene pays off. Again, parallel shear directions (construction lines) can be put on or taken off the scene as needed.
This is the most popular proof for the audiences we've shown this to. There are actually two ways to rearrange the pieces. The way we used simulates hinge joints to unfold the pentagonal pieces. An alternate way is to just slide the pieces around with rotating anything. This has the advantage of not confusing the viewer with keeping track of the size of a shape as it rotates, but it adds to confusion since the shapes have to either overlap as they move (due to space constraints) or having to have them move off the screen temporarily. Note that as the five pieces rearrange to form the smaller squares, a ghost image of their original positions remains on the larger square for reference.
Each shape shown on the edges of our triangle has its own color. The sequence roughly follows the spectrum as we go from rectangles to the pentagons. Capitol letters in the centers of the shapes stand for areas, lower case letters on the edges stand for edge lengths. Coordinate grids are used only when measurements are being compared. The flipping over of the if-then slate was maybe not a great idea since you tend to forget which clause was which as it rotates.
The big potential for confusion in this scene is the fact that the three similar triangles used for the proof are all on the inside of the triangle, and that they are all the same shape as the original triangle. Also, when they are on the inside they all overlap so it's difficult to see that there are three triangles there. For this reason a prolog is shown with the pentagonal shapes flipping back and forth from outside to overlapping inside the triangle. This gets viewers used to the idea of the shapes being inside the triangle. The shapes flip back and forth on a hinge line made of the edges of the triangle to avoid 3D shape alteration confusions. Note the transparency used to show overlapping shapes while keeping them distinct.
The house and its skeleton cast a shadow on the ground plane to enhance the 3D effect. Note that the diagonal line is brighter than the edge lines. Also note that the line quality helps see what is in front of what. Each line is drawn with darker edges so it, in effect, looks like a little cylinder. Therefore, when a line crosses in front of another you can see which is in front. All labels in 3D cast appropriate shadows (especially the h and d). This was difficult to choreograph so that things didn't get on top of each other. Again note translucency of the yellow triangles. And substitution here is done more literally than elsewhere.
We used the letter t for the angle because some feedback indicated that many high schoolers might be intimidated by Greek letters. Also note that the square of the sine function is written as sin(t)^2 instead of the more conventional but weirder sin^2 t.
This is also a bit gratuitous; we happen to be able to easily make pictures of the earth, so we worked it in. The beginning of this scene is meant to look like an abstract bunch of colored blotches. It's only after the earth pulls back that you realize that it is the earth. The top vertex of the triangle is situated at the latitude and longitude of the island of Samos.
For this program the section slides all showed geometric constructions that used or related to circles. They were all cycles that could repeat indefinitely; only about 1.5 cycles were actually shown.
The big problem here was distinguishing between a special case and the general case. The problem actually becomes significant in a later scene where this prerequisite is used. (This scene was designed after the scene that used it to make sure it foreshadowed the right thing.) Here it is only important to see that the special cases are shown with white figures while the general case is shown as yellow column headings. The ides is to convey the difference between the specific value of a particular edge length and the general concept of edge length.
Note the use of approximate equal signs with physical measurements. When pi comes on it changes to a real equal sign. Pi does a dance that introduces a sound effect that we use repeatedly to call attention to it in later scenes.
The ruler zoom is meant to show the obvious, that pi has infinitely many digits. My original idea was to have the ruler zoom around the arrow that indicates the value of pi and have the digits appear at the top of the screen for each power of ten of zoom. This had the problem that the viewer had to move their eye back and forth between the numbers and their geometric representation. In the final version the decimal expansion slowly shifts to the left so that the newly forming figures are always situated just over the arrow. Viewers can focus their attention to just that one area of the screen and still see both of the things going on (digits and zoom).
This section was originally going to be just a series of equations containing pi. It expanded itself into several sections.
This became a bit of a 3D geometry course in areas and volumes. We have received some feedback that high schools are weak in 3D geometry so we should put a bit in wherever possible. In these scenes, however, the pi in the equations gets lost so we accentuated it with a sound effect. Every time pi appears in the program it performs a squeaky dance. Some scenes, like this one, are complex enough visually that the viewer might not catch the pi on the screen visually. The audible squeak notifies them that it is there somewhere.
Each sub-scene starts with a 2D shape. The background is a flat color and the area and length are shown on rectangular slates, color keyed to the parts they measure. When the shape falls back into 3D the background becomes shaded and the slates get beveled edges, all mechanisms to enhance the three dimensionality. Now the shape either extrudes or sweeps around an axis to make a surface of revolution. The area slate changes to volume and the length slate changes to area. When this happens the expressions are multiplied by the appropriate factor from Pappus' theorem (distance to center of mass). This whole thing is to be multiplied by 2pi for the surface of revolution; actually the 2pi grows slowly as the sweep occurs becoming complete at the completion of the sweep.
The colors for the 3D shapes were picked to look like an apple with skin. These were necessary since the area equations we showed didn't always apply to the entire surface, for example it excluded the end caps on the cylinder and cone. So the red equation pertains only to the part of the surface area colored red.
Flight from NY to Tokyo scene timed for approximate duration of flight so movement of sun (light source) is correct. Night side is artificially brighter than real night side. Helps to make trajectory show up better too.
The labels on the Gaussian curve were originally to be ABCDE but we didn't know if schools give E or F these days. I decided to change it to the faces to lighten the emotional impact of grading. I originally had the bar chart expand sideways as the area under the curve filled up sideways (like an integral). This didn't seem to give the right message to some test viewers. Now the Gaussian empties into a vertical bar with unit width; its height then is a measure of the area.
The successive approximations are shown on our number line. Ideally it would zoom in as history progresses, showing better and better approximations. Unfortunately this wasn't the case historically; some later approximations were worse than earlier ones. We ultimately decided to skip over these.
The extra labels on the number line (in little boxes) were necessary to give context of where we are in the accuracy scale, since we didn't show a whole zoom during each part of the scene.
Here is where the care in special case delineation paid off. Two examples of circles are shown. The ratio of c over d is the same for both, C. We then show the general equation, built up out of the column headings. While this is being done the grid disappears to remove the impression of specific measurement. The text names shrink down do the---now general case symbols c and d. We have therefore used d as a specific diameter at the beginning of the scene and as a generic diameter at the end of the scene, hopefully keeping them straight.
The same general vs. special case situation applies here as above. Note the unit square grows on the circles at the same time the right hand column radius changes to square of radius. This also gives a visual comparison of their sizes.
The scene begins with the entire circumference shrinking to just half the circumference, pi r. When this unfolds you can see it is 3 plus units long. As the pie sections subdivide within the rectangle they have to open up to maintain the uniform interspersing. This looks potentially confusing.
The colors here are about the same as the above but a somewhat different shade of orange. I originally tried to make it a different color but couldn't find one that didn't look too dingy.
When the rings unwind, the radius follows them to emphasize what is going on. At the end, I rearranged the triangle into a rectangle to show the total area. I could have done it by simply invoking the equation for the area of the triangle, but I decided in general not to use algebra when there was a visual way of doing the same thing.
This broke down into two parts.
Here the sound track doesn't match up so well with the visuals since we had to record the sound before we had the visuals done. In particular the discussion of perimeters of circle and hexagon are reversed between sound and picture. I had to zoom in to the polygonal approximation and to the number line to show the increasing accuracy, but each one needed a different scale factor. Removal of the grid helped remove the exact quantitative nature of the sizes of the objects. The tick marks on the ruler show the only numerically accurate measurement. Note that the accuracy bounds are arrows with only half an arrow head.
The main visual thing here is the more complex pattern for the background. Since this is not meant to represent a Euclidean plane, texture is appropriate.
Here, the final scene would be a distant view of the plane marked with two different types of points. I had to make the visible dots a color brighter than the background and the non-visible dots darker so that, when we pull back, there is still a lot of contrast between them. Also, even though the grid is important here, I set its contrast as low as I could get because bright grids are very overpowering. The view of the points from the origin is also a powerful visual image as well as an excuse for going to three dimensions. Note the squeak when the pi comes on.
I tried to make the two categories of needles glow with different colors, but the effect was too subtle. I wound up using the same color here as I did for the ring proof of the circle area, but with light and dark brightnesses. This balances with the lattice point color being the same as the pie slice area proof. Also, I had to simulate quite a lot of needles to get good enough statistics to get a reasonable approximation to pi.
I feel I am still learning how to do this. With each project I discover more general principles about the animation process as applied to education. Many of the ideas here are either provisional or are reiterations of ideas presented above with respect to the specific productions I've worked on to date.
The process of designing a scene seems to go through three stages.
The ideal situation is to have enough time to progress through the first two stages and get to the third. It has not always been possible.
I spend a lot of time picking colors. Use of color has become a somewhat dangerous tool since it is easy to go overboard and make the images too garish. You should have a reason for the colors you use rather than assigning them randomly. Also, since some people will only see these tapes in black and white and some people are colorblind, colors should not be the only distinguishing feature between parts of an image (see also the readability section below). The color schemes used in Mathematics! haven't evolved quite as much as they have for the Mechanical Universe, partly since we have not produces so many programs and partly since the dimensional analysis in MU isn't so appropriate to mathematics. The main conventions I have arrived at are, as mentioned previously:
Shapes whose areas are being compared should be the same color.
Review-of-prerequisite scenes should use the same color scheme as the scene where that prerequisite is used.
In the field of graphic design the term "readability" refers to whether an image is understandable and whether the important elements stand out from the unimportant ones. Here are some ideas.
You can turn off the color of a monitor to see the picture in black and white. If you do so, do key parts of the image become the same shade of grey? If so you should change it.
Decide what is the most important part of picture. Now look away from screen and then quickly look back again. What do you see first? Is that the most important thing in the picture? If not, make it so, usually with color or size changes. The eye is also attracted to sharp contrasty edges or fine detail.
Look at screen from across the room. Is it still readable? Don't try to put too much fine detail in the picture. Many people will be seeing it on small screens or from far away. Also some people won't be able to rewind the video so must get what they can the first time they see it.
Drop shadow are used to distinguish between foreground and background. It's remarkably easy for things to disappear into the background. Note that the shapes themselves never cast shadows (especially if there is a grid). This would be bad since it removes them from the Euclidean plane.
The algebraic ballets were generally more deliberate than for The Mechanical Universe to emphasize basic algebraic operations instead of shortcuts.
I never know what speed to make things. I have decided to err on the side of too fast rather than too slow. Making it slow enough to see everything explicitly the first time through would make it too draggy on repeated viewings. Most people think its going slower the second time they see it anyway. Since students will have this on videotape they can play it at any speed they like. Some possible future interactive versions of these projects will provide even more flexibility in this regard.
It is common in the animation business to economize by making only 15 frames per second and recording each frame twice. I avoid this as it tends to make the animation look too frenetic. The smoother motion somewhat alleviates the "too fast" viewer response to the timing.
Any parameter (position, size, rotation angle) can be made a function of time. Its values are typically specified at particular key frames and the computer interpolates between them for the inbetween frames. Linear interpolation as a function of time is the most straightforward method. Conventional wisdom in the animation business is that this looks too mechanical and jerky. I typically use cubic spline interpolation where the parameter is a cubic polynomial function of frame number, and where the slopes are usually set to zero at the keyframes. I am beginning the think, however, that this looks too mushy. The eye is attracted to contrast, both in time as well as color. If the motion is too smooth it's hard to track. It's the time equivalent of a blurry picture. (The eye is good at tracking uniform motion; I don't know how good it is at accellerated motion.) More experimentation needs to be done here.
Most scenes show some activity constructing some result, then pause on the result, then go on to the next activity. The placement and duration of pauses takes some thought. Pausing on an incomplete result---an equation with a term missing or label that is not centered---creates tension. Pausing on the completed equation or the centered title creates release. This is a principle used in conventional filmmaking (suspense and resolution) and in music (suspended and dominant 7th chords resolving to tonic chords).
Must pick the one set of parameters (e.g. a triangle shape) that shows the most general case of the object. That is, if you are discussing an arbitrary triangle you shouldn't illustrate the discussion with an equilateral triangle.
To understand a proof you must show
Three dimensions is (usually) only used if it's necessary to show an inherently three dimensional concept, not to decorate essentially 2D concepts. The worst example of this I have seen published is the extruded pie chart drawn in perspective. Three dimensional animation is much harder to design, not so much because of the increased computational load, as because there are more parameters to specify. The animator must pick lighting directions, surface qualities and viewing positions that properly show the situation. It is often difficult to show three dimensional shapes without some essential feature being hidden by another one.
This is very simple since it's not my forte. Amazingly simple things can still work though. Characters can turn their heads or bodies in one frame. Arms and legs can be simple hinge joints. This is similar to paper cutout animation I did in high school.
The backgrounds section titles for later programs are interesting moving geometric constructions; often varying a parameter in some locus problem. These are meant to just look interesting but they can also stimulate discussion with the teacher. They are drawn with low contrast fuzzy lines to keep from distracting too much from the text on the screen.
Almost all scenes use a simple rendering program that handles colored lines and colored transparent polygons. (Several other 3D rendering programs are available.) All rendering is anti-aliased. Not using anti-aliasing can give the scene a video-game like quality. This has become a design style in its own right but the eye gets tired looking at it for too long.
In scenes where lines are the main character, the lines are brighter, but not as bright as the labeling letters. Letter labels are on the line, not next to it, so that it looks OK when the shape rotates or moves.
In scenes where areas are the main character the lines are dark and the areas are bright.