Category Archives: Geometry and Topology

Fractal Branching

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?

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Fractal Education: Sierpinski Triangle

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:

  1. Take a point-side-up triangle
  2. Connect midpoints of the three sides to create a point-side-down triangle.
  3. Repeat process with the resulting point-side-up triangles.
  4. Repeat as many times as you want.

This example shows taking the process through 4 iterations.

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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.).

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Also, you can blow their minds by showing this Sierpinski Zoom video to show that the triangles can go on forever!

NOVA Exposes the Hidden Dimension of Fractals

Mandelbrot Set created by Wolfgang Beyer with the program Ultra Fractal 3.

Mandelbrot Set created by Wolfgang Beyer with the program Ultra Fractal 3.

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).

Origami + Scissors = Kirigami

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.

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This Kirigami for Kids website has directions for making several kirigami figures, including 5
and 6-pointed stars, snowflakes, and paper doll chains and rings.

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.

There can be only 5 Platonic Solids

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Of all the activities we’ve done at my afterschool math club, learning about and making platonic solids is one of the kids’ favorites.

Platonic solids are convex polyhedra with the following characteristics:

  1. all its faces are the same
  2. all faces are regular polygons (all edges and angles of the faces are the same)
  3. none of its faces intersect except at their edges, and
  4. the same number of faces meet at each of its vertices.

platonicThe 5 platonic solids are:

  1. Tetrahedron – 4 triangular faces
  2. Cube (Hexahedron) – 6 square faces
  3. Octahedron – 8 triangular faces
  4. Dodecahedron – 12 pentagonal faces
  5. Icosahedron – 20 triangular faces

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You can get foldable platonic solid printouts at the Interactives: Platonic Solids page at Learner.org. This webpage also has an interactive activity where kids can identify the number of faces, edges, and vertices for each solid and try to determine the relationship between these three properties.

Below are parts 1 and 2 of an out-of-print video about platonic solids, that is available on YouTube. The first explains the properties of platonic solids and why there only five. The second shows different views of the platonic solids, platonic solids found in nature, and the property of “duality”.

Origami: Art and Math

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Origami is the Japanese art of paper folding. But it is more than an art form; it is a teaching tool and a field of mathematical study.

Origami is a great tool for teaching math.  Origami can increase spatial skills and help students understand many geometric concepts, from shapes and geometric forms, to more complicated concepts like intersecting planes, area, volume, symmetry, and mirroring images. Origami can even be used to teach number theory concepts like fractions and powers of 2. Best of all, origami is fun and creative, and kids love it.

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There are tons of online resources for learning origami and using math as an educational resources, especially on YouTube. Here are videos for origami folding and origami as a teaching tool.

In addition to being a fun tool for teaching math, origami has become its own field of mathematical study. Paper-folding can be used to solve mathematical problems (check out Vi Hart’s video on the origami proof of the Pythagorean Theorem), and math can be used to create incredibly intricate origami designs. In his TED talk, Robert Lang explains the mathematical “laws” behind origami, shows some amazing creations, and talks about how to go from an idea to an origami design using a program called TreeMaker that he offers free on his website. He also discusses some real-world applications that have grown out of the study of origami.