Transformations: Mathematics in Animations

Mathematics, and more specifically, geometry, transformations and coding are used in computer (and other) animations.  This blog is to help you explore the mathematics of animations, particularly the use of transformation in animations.

https://www.youtube.com/watch?v=IjsbdeRJl5E

Here  are a couple things to think about with the applications of transformations to computer animation. You'll note that some of the transformations are isometries (preserve congruence, like when Mr Incredible is working out....) and some are homotheties (preserve similarity, like when Mr. Incredible is falling from the sky) and some are neither, but are still transformations.

If you look carefully in this fun you-tube video, you'll be able to see that there's some 3-d geometry here along with programming and, of course, art. Have you seen Madagascar? You'll like this one.

Five types of animation are shown in this you-tube video.  You'll see how coding is used to create a computer animation (type 3 of animation). Type 5 animation (Stop Motion) you can do a little of at the museum downtown, Explora.

And, of course, there's a Ted Talk by a Pixar animator that discusses science with the animation that Pixar does.

Your task (the "prompt") here is to find some element of mathematics in animation then discuss it in your blog.  Do a search, see what you find.  Watch one (or more) of the fun video links I've included here.  You all have some great searching skills; find some additional information about how art and math are combined in animation.  Have fun!

Self-Similarity

The first day of school this year, we drew two fractals: Cantor Dust and a Koch Snowflake.  To refresh your memory, Cantor Dust was formed when we drew a segment, then removed the middle third of the segment.  Then, for the next iteration, we removed the middle third of the remaining two segments.  We repeated the process several times, noting a paradox: if we could continue this process infinitely, we would have an infinite number of "segments" with a total length of zero.

We also drew a Koch Snowflake.  To refresh your memory, from a regular triangle, we removed the middle third of each side of the triangle and constructed a smaller regular triangle using the removed segment as the base.  We repeated this process several times.  We noted that the perimeter would be infinite, but the area would approach a specific value, thus a paradox similar to that found with Cantor Dust.

http://www.stsci.edu/~lbradley/seminar/fractals.html

What's amazing about fractals is that they have a property called "self-similarity."  Given an infinite number of iterations, if you zoom in on a piece of the fractal, it looks identical to the original. The fractal is composed of infinitely smaller shapes each similar to the larger shape.

The word "fractal" was coined in 1975, so the study and experience of fractals is relatively new.  Computers facilitated understanding of these shapes. 

And also common culture has given us accessibility to these ideas:

Your task for this blog could be one of  a variety of activities.  (1) you could go to "First Friday Fractals" at the Museum at Natural History (This Friday is a first Friday of the month, so you could go that early!) and write very briefly about not just the spectacular art, but something about the mathematics behind the art (that's discussed in the show).  (2) You could read about the history of the development of fractals and discuss that.  (3) You could find and discuss another fractal (Sierpinski triangle or Menger Sponge for examples)  Regardless of which of the three you chose, you need to include the topic of similarity in your blog  -- after all that is the topic we've been studying.

(As always, if you come up with a different idea for this blog, ASK!  I'm always inclined to say yes, but before you chose to deviate from this topic, please ask to verify that your idea is an ok one to use for this blog post. )

Quadrature of the Lune

My sources:  A great introduction to the concept of quadrature can be found on Wikipedia HERE. 

And of course, Leonardo da Vinci was fascinated with "squarable figures."  A fun couple of puzzles that you can think about and even solve are HERE. 

But because I grew up pre-internet, I go to books.  Leonardo's Dessert: No Pi, written by Herbert Wills, III and published by the National Council of Teachers of Mathematics in 1985, was my first venture into quadrature.  Also, Will Dunham wrote a book with a first chapter that discusses Hippocrates' Quadrature of the Lune: Journey Through Genius, published in 1990.


Both Hippocrates and Leonardo da Vinci were interested in squaring "lunes" or regions defined by two intersecting arcs.  I believe each of them viewed this task as a fun pastime -- puzzles to be created and solved for great entertainment.  In class, saw and rectified the following images:

The Pendulum, rectified:

 The Axe, rectified:

If you remember, the Axe appeared in the 2001 SAT test.  Feel free to share this blog with your parents; I used this area problem on Parents' Night last fall.

Let's start by considering The Lune of Hippocrates.

To construct the lune, start with a circle with two radii drawn at right angles and the chord connecting the points on the circle met by the two radii, as shown. Find the area of sector AOB; it's one quarter of the whole circle with radius r.   (A = pi r^2/4)

Finding the midpoint of segment AB gives us the center of the circle with diameter of segment AB, shown as the red circle below. Find the area of  half of the red circle.  (A = pi r^2/4) Notice it's the same as the quarter of the larger circle above.

The Lune of Hippocrates is in the picture above; I've shaded it in the image below:

You could find the area of this shaded lune by taking half of the red circle and subtracting the section of the large circle defined by chord AB.  (Remember that a "section" is a region defined by an arc of a circle and its chord.)

Notice that if you took the quarter of the larger circle (same area as half the red circle) and subtract the same section defined by chord AB, you'd get triangle ABO.

Therefore, triangle ABO has the same area as the blue shaded lune.

Consider for a moment what's happened. You've found two regions that are quite unlikely candidates for having the same area as having the same area. The lune shaded above has the same area as triangle ABO. 

For your blog, find another puzzle to rectify, and rectify it.  You can use the links I've provided above for my sources.  (The Lune of Alhazen might be a good choice in Wikipedia.) You can also look into the history of rectification of shapes or perhaps how that applies to ideas in calculus. Google away, find something to teach me.  Have fun with this one!  If you're a puzzle person, play with a puzzle. If you're a history buff, then look into how this idea of rectification really furthered some mathematical thought.  Or if you are an artist, see if you can find how Leonardo or anyone else used these ideas in art.

Here's a sample of one puzzle with its solution, for a demonstration. It's a challenging solution, so don't feel as though you need to do one this challenging.  Consider a "Motif" that Leonardo used, and the corresponding region that I've shaded:

And here is how you might rectify that shaded region (at least, you'd cut and rotate until you get a pendulum, which you know to be rectifiable).

Paul Erdos and Beautiful Mathematics

"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is." -- Paul Erdős 

Paul Erdos was a brilliant 20th century mathematician born in Hungary.  A unique individual, he was revered by many mathematicians and traveled the world, staying with colleagues with whom he could work on math problems.  He published over 1500 pieces of original mathematics. It was a privilege to have Paul Erdos stay in your home, but he was loathe to stay long. He lived out of a scantily and poorly packed suitcase; even charity did not want this suit after he died.

In class, we discussed the Erdos number and the concept of an Erdos problem.

There are many fun stories about Paul Erdos, find one (or more) and share it in your blog. You can discover what he wanted his epitaph to be, or how he cut grapefruit with the dull edge of a knife.  You can discover what degrees he earned, where he studied, where he traveled, and how he moved around in the world.  You can find some stories in the following NYT article. You can also search on what interested him most in mathematics; his ideas were deeply complex, but see what you can glean from what you find about him on the internet.

 He died in 1996