Visualizing in 3D

I introduced my precalculus students to vectors in 3D space yesterday. They have a hard time visualizing the 3-dimensional axis system, especially since the familiar XY-plane is now on the “floor”:

3D axes

To help them understand how 3D coordinates work, I paired the students up, and gave them a slip of paper that read:

This landmark is ( ______, ______, ______ )

steps from the northwest corner of Mr. Wert’s classroom.

Using the northwest corner of my room as the origin, they went all over campus and picked a spot to put their label on. They had to keep track of how many steps they walked in the x-direction, the y-direction, and the z-direction. Here’s an example:

3D landmark

Actively keeping track of their position using 3D coordinates really helped them understand how the x, y, and z coordinate system. works. After this, they were ready to work with vectors in 3 dimensions.

Fractals and Kindles

I just can’t help myself. Whenever I get a new gadget, I have to customize it. When my wife and daughters gave me a Kindle four years ago, I was thrilled. It’s a Kindle 3, and it opened to me the amazing world of ebooks. My library now includes collections of G. K. Chesterton, Charles Dickens, Fyodor Dostoyevsky, and Shakespeare. I like the fact that the screen isn’t backlit, so there is no eye strain. It’s a wonderful device that has completely changed the way I purchase and read books.

However, I thought the screensavers that Amazon preloaded on the Kindle were really unattractive, so I tried to replace them with images more to my liking. Easier said than done! I assumed that all I had to do was locate the folder containing the screensaver files and dump my own in there. It turns out Amazon does not want you poking around in there, so that folder is hidden.

Fortunately, after a little research online, I was able to hack into my Kindle and change the screensavers. There are thousands of great images online to choose from (just Google “Kindle screensavers”), and I had a blast exploring them. Then it occurred to me that I could create my own fractal screensavers – all that is necessary is to make sure they are gray-scale images that are 600 by 800 in size.

I decided to use Chaoscope to create my screensavers. (I posted a tutorial on how to use Chaoscope here.) Make sure you render them in either Gas or Liquid mode. My first batch is posted below. They are already correctly sized – just click on a thumbnail to access the full-size image, and then save it to your computer. Enjoy!

The Golden Rectangle and Fibonacci

One of the best things about teaching math is sharing all of the connections that exist between seemingly unrelated topics. Take, for example, the Golden Rectangle. Often described as the most visually “pleasing” quadrilateral, it is a rectangle whose length, l, and whose width, w, are in the the proportion \displaystyle \frac{l}{w}=\frac{l+w}{l} .

Here’s an example:

Golden Rectangle4.69 / 2.9 = 1.62, and (4.69 + 2.9) / 4.69 = 1.62. The number they equal, 1.62, is called the Golden Ratio, and it has its own Greek letter, phi. It’s actually irrational, like pi and e, and it equals \displaystyle \frac{1+\sqrt{5}}{2} . You can derive phi algebraically by solving \displaystyle \frac{l}{w}=\frac{l+w}{l} for l in terms of w. From the quadratic formula, \displaystyle l=\left( \frac{1+\sqrt{5}}{2} \right)w, so \displaystyle \frac{l}{w}=\frac{1+\sqrt{5}}{2}.

A Golden Rectangle can be constructed with a straightedge and compass (or GeoGebra!):

1. Construct a square ABCD: Square

2. Construct the midpoint, E, of side AB.

3. Construct a circle with center at E, passing through C: Circle

4. Extend rays through AB and DC. Construct the intersection point, F, of the circle and ray AB.

5. Construct a line perpendicular to ray AB, through point F. Construct the intersection point, G, of this perpendicular line and ray DC: Constructed Rectangle

6. Quadrilateral AFGD is a Golden Rectangle.

Another way to construct a Golden Rectangle is to begin with two adjacent squares:


Add another adjacent square with dimension equal to the combined sides of the first two:


Repeat the process, adding larger and larger adjacent squares:


This not an exact Golden Rectangle, but the more squares you add, the closer it approximates a true one.

Here is where Fibonacci comes in. He was a 13th century mathematician who published one of the first math textbooks, Liber Abaci. One of the topics he covered involved a famous sequence that can represent the idealized population growth of rabbits: 1, 1, 2, 3, 5, 8, 13, 21, 34, …. Each term is obtained by adding the previous two terms together. Can you see how the rectangle above is a visual representation of the Fibonacci Sequence? What is amazing is the fact that the quotient of any two consecutive terms in the sequence approaches phi the further up the sequence you go!

Connecting opposite corners of each square with a circular arc creates a Golden Spiral, which increases a distance of phi from the origin for each square it crosses:


There are lots of exaggerated claims made about the use of the Golden Rectangle and phi in ancient architecture, but it definitely pops up in the world of nature. Flowers generally have petals in numbers of 3, 5, 8, 13, 21, etc. For an excellent explanation of the mathematical reasons behind this pattern, check out Ian Stewart’s book, The Mathematics of Life.

A rectangle, a medieval Italian mathematician, a famous sequence, and the number of petals on flowers – all brought together through the beauty of mathematics.