One of the most common fractal patterns found in nature is branching. Fractal branching is seen in tree branches and leaf veins, lightning, river deltas, mountaintops, blood vessels and bronchi in the lungs, and many other places. Can you think of any examples?
A great activity for teaching kids about fractals is making Sierpinski triangles – one of the easiest fractals to draw.
The Sierpinski triangle is formed using this process:
- Take a point-side-up triangle
- Connect midpoints of the three sides to create a point-side-down triangle.
- Repeat process with the resulting point-side-up triangles.
- Repeat as many times as you want.
This example shows taking the process through 4 iterations.
You can print out the starting triangle from:
Fractal Foundation: Sierpinski Triangle (includes worksheet and instructions)
Spitz: Fractal Pack 1: Educators’ Guide (includes Sierpinski triangle template and other information for teaching about fractals)
If you have a classroom of kids, you can take their Sierpinski triangles and put them together to make a larger triangle – you just need a power of 3 to make a complete Sierpinski triangle (3, 9, 27, etc.).
Also, you can blow their minds by showing this Sierpinski Zoom video to show that the triangles can go on forever!
Whether or not you’ve heard of fractals before, the NOVA documentary, Hunting the Hidden Dimension, will amaze you with how cool they really are. A fractal is a geometric pattern that is repeated at smaller and smaller scales, producing shapes that can’t be represented by classical geometry. When mathematicians first started toying with the idea of fractals, they seemed so strange and foreign they were known as “monsters”. Now we see that they aren’t so foreign. In fact, they are everywhere – the branching of trees and blood vessels in our bodies, coastlines, clouds. Isn’t it amazing how even the strangest mathematical concepts seem to lead back to the natural world?
Hunting the Hidden Dimension is a fascinating look at fractals, covering the history of their study, from the 19th century, when they were known as “monsters”, to current applications, such as CGI and cell phone antennae. We also learn about the life and work of Benoit Mandelbrot, the man who developed fractal geometry as a field of mathematics and coined the term “fractal”, his advantage being that he came along at a time when computers were becoming available to tackle such problems. The “hidden dimension” in the title refers to the “fractal dimension”. You’ve heard of things being 2-dimensional or 3-dimensional, but fractal geometry can describe shapes with non-integer dimensions like 1.3 or 2.6.
I watched this documentary with my kids and I have shown clips of it to my after-school math group. The kids especially like seeing the beautiful images of the Mandelbrot set and seeing how fractals were used in the making of the latest Star Wars movies.
Hunting the Hidden Dimension is available for free on Hulu and is currently available on YouTube (embedded below).
My family and I spent this past weekend exploring all the wonderful sights and activities at Maker Faire Bay Area. Maker Faire is billed as a family-friendly festival of invention, creativity and resourcefulness, and a celebration of the Maker movement. We saw flaming sculptures, robots, giant musical tesla coils, people on stilts, Tapigami, the EepyBird Coke & Mentos show, and much more. The kids polished rocks; did lots of art projects, including a claymation video and an octopus made from gloves; helped build a beehive; made soap and silly putty; and climbed into a life-sized bejeweled flying saucer. If there is a Maker Fair event near you, GO!
Here a few math-themed photos I took at the event.
A couple of weeks ago, I did a post about origami. In origami, figures are made with only folding. But, there is a variation of origami, called kirigami, that involves folding and cutting and then opening up the folded paper. Familiar examples are the snowflake and paper dolls, that you may have made as a kid.
In the field of mathematics, Erik Demaine of MIT has been working on what is called the fold-and-cut problem. The fold-and-cut process involves folding the paper, making one straight cut, and then unfolding the paper. Any figure formed from straight lines can be produced this way. The problem is figuring out how to fold the paper – that’s where the computational geometry comes in. Demaine has several patterns on his website, and the video below shows the process for creating a swan.
WNYC’s Radiolab recently aired a segment called “Dealing with Doubt”. Jad and Robert spoke to Professional poker players, Annie Duke and brother Howard Lederer, and learned that reading other players’ “tells” is a very small part of the game. The way to win at poker is to use math – to calculate the probability that you will get a winning hand, your “hand odds”, and compare it to the “pot odds”, the ratio of the current size of the pot to the cost of a call.
They also discuss how 17th century French mathematician, Pascal, applied probabilities to a very big question.
You can listen to the segment here: Dealing with Doubt. (Sorry I can’t embed – WordPress won’t allow it. )
- Take any four-digit number, using at least two different digits. Repdigits, such as 1111, will not work, because you will just end up with 0 after step 3.
- Arrange the digits in ascending and then in descending order, adding leading zeros if necessary. Add leading zeros if necessary – for example, 4560 in ascending order is 0456 and 6540.
- Subtract the smaller number from the bigger number.
- Go back to step 2 and repeat the process.
This process, known as the Kaprekar routine, will always reach the number 6174, within 7 iterations. Once 6174 is reached, the process will continue yielding 6174 because 7641 – 1467 = 6174.
For example, choose 6532:
6532 – 2356 = 4176
7641 – 1467 = 6174
Another example, choose 4905:
9640 – 0469 = 9171
9711 – 1179 = 8532
8532 – 2358 = 6174
7641 – 1467 = 6174
6174 is known as Kaprekar’s constant, named after Indian mathematician D. R. Kaprekar.