Category Archives: Geometry and Topology

Make A Flexagon for your Valentine

What better way to show your Valentine you care, than to make them a flexagon! You can print out this cool Valentine tri-tetraflexagon card, featuring teddy bears, at Aunt Annie’s Crafts. Or, you can choose the plain frame template and add your own pictures. But, if you want to make them for an entire class, make sure you start now! It takes some time to print, cut out, and fold. (My son is making these, and he has made enough for about a third of his class so far.)

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If this seems like too much work, there are tons of cute math-themed pictures on the internet you can turn into Valentines. Last year, we found cute a Sierpinski Valentine.

Rubik’s Cube Math

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Over the Christmas break, while on vacation with extended family, my son learned how to solve the Rubik’s Cube (a.k.a. magic cube) from his older cousin. He became somewhat obsessed and worked on it until he could solve it himself. One of the things he told me when he got his new cube was, “Mom, the colors can be arranged in over 43 quintillion ways!” This is why it is so hard to solve; there are too many different permutations. You’d never land on the solution by randomly turning the sides. The number of permutations is calculated with the equation:

 {8! \times 3^7 \times (12!/2) \times 2^{11}} = 43,252,003,274,489,856,000

The first term, 8!, comes from the fact that there are 8! (read 8 factorial) ways to arrange the 8 corner “cubies” (8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1). The next term is 37, because there are 3 colors on a corner cubie and 7 of the corners can be oriented independently. (“Oriented” differs from “arranged” in that arranged refers to moving a cubies to different locations on the cube, while oriented refers to the cubie being at the same corner but rotated to show a different color on top.) The orientation of the eighth cubie depends on the preceding seven. The next term, 12!, is the number of ways to arrange the edges. This term must be divided by 2 because it is an even permutation. Eleven edge cubies (with 2 colors each) can be flipped independently, with 12th depending on the preceding ones, giving 211.

If you are interested in a more in-depth explanation, there is a powerpoint presentation here, and paper here.

Most people solve the Rubik’s cube by memorizing sequences of moves called “algorithms”. In mathematics, the term algorithm means a step-by-step procedure for calculations. Many algorithms, especially those used later in the solution, are designed to move a cubie to a specific spot without moving parts of the cube that have already been moved into place. If you are interested in learning basic algorithms for solving the cube, there is a solving guide available at Rubiks.com. People into “speedcubing” learn algorithms that make it possible to solve the cube in fewer moves, but requires learning more algorithms used with specific patterns on the cube.

Puzzles like the Rubik’s Cube can also be investigated through mathematical group theory. Group theory is the study of the algebraic  structures called groups. A group is a set of elements together with an operation that combines any two of its elements to form a third element also in the set while satisfying the group axioms (these are closure, associativity, identity and invertibility). For example, the set of integers together with the addition operation; the addition of any two integers forms another integer. The Rubik’s group is the group corresponding to possible moves.

October is the Month of the Flexagon

A few years ago, recreational mathematician, Vi Hart, declared October “Fexagon Month” in honor of Martin Gardner. Gardner, born October 21, 1914, wrote a monthly column on “recreational mathematics” for Scientific American from 1965 to 1981, and it all began with his article on hexaflexagons in the December 1956 issue.

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A flexagon is a flat model that can be flexed or folded to reveal faces besides the two that were originally on the front and back. Watch these two videos from Vi Hart to learn more about the history of flexagons and how they work:

Flexagons are very cool to make and fun to play with. You can try making your own hexaflexagon from a strip of paper like Vi Hart shows in her video, or you can go to one of the tons of websites with colorful, printable templates that you can print out and fold according to the directions. Here are just a few:

You can even make a flexagon template using your own photos using the tool here.

Happy Flexagoning!

How to Make an Envelope Tetrahedron

My previous post about platonic solids included a link to a site with printable templates for making your own solids. However, there are other easy ways to make paper platonic solids. Here are directions for making a tetrahedron using an envelope.

Step 1. Seal the envelope and then fold length-wise to mark the center line. I have marked the center line with a dotted line. (Marking the line is optional, I was just trying to make it more visible.)
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Step 2. Fold the top corner down to the center line. The fold should extend down to the bottom corner.
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Step 3. Fold opposite side over until it touches the corner that was folded down. This is to find the line perpendicular to the center line that touches point where the corner touches. I’ve marked this with another dotted line.

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Step 4. Cut along the vertical line and unfold the corner.

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Step 5. Fold the two corners on the open end of the envelope from the center line to the opposite top corner. These folds are marked with the highlighted lines. Fold forward and back to make a good crease.

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Step 6. Open up envelope at the cut end and pop out the 4 folds you made (the highlighted folds plus the corresponding ones on the opposite side). Tape the open end together and you have your completed tetrahedron.  IMG_1467

Make Your Own Conic Sections

Conic sections are the curves created when a cone is intersected by a plane. There are 4 types of conic sections: circle, ellipse, parabola, and hyperbola. Sometimes, you may see it stated that there are 3 types of conic sections, because the circle can be considered a special kind of ellipse (we’ll discuss that more another time). The picture below shows how the curves are formed from the intersecting planes. As you can see, cutting the cone so that you miss the upper cone, but slice through the bottom gives a parabola. Cutting the cone so that you do not intersect the top/bottom of either cone gives a ellipse (a circle if the cut is parallel to the top/bottom). And, cutting the cone so that you slice through both the top and bottom cones gives a hyperbola.

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As with many concepts, doing a hands-on activity can help demonstrate conic sections, so why not make your own? You can easily make your own cone with modeling clay and slice some conic sections. To make your cone:

  1. Cut a half-circle from sturdy paper or card stock and a second half-circle from parchment paper or wax paper.
  2. Curl cardstock half-circle into a cone and secure with tape and then place parchment paper inside (do not tape parchment paper).
  3. Pack cone to the top with air-dry modeling clay. Use small pieces and press down to the tip of the cone. You can pull the parchment paper out a little to peek as you go, to make sure there aren’t any big gaps or air bubbles.
  4. Flip over onto a paper plate or sheet of parchment paper. Remove cardstock cone, then peal away the parchment paper.

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Once you have your cone you can use dental floss to make your sections. Cut straight across near the top to make a circle.

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Below your circle, make a slanting cut to make your ellipse.

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Below your ellipse, make a cut that slants enough to pass through the bottom of the cone to make your parabola.

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You could do another cut that goes straight down, but without the top cone, it is hard to see it as a hyperbola. So with my students, I chose to just show them a picture of the hyperbola. Once the pieces dry, you can color your curves. Below, is the cone that my 6-year old daughter made. She chose to color her circle blue (she also colored the bottom blue, since that is also a circle). She colored her ellipse yellow and her parabola orange.

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Fractal Education: The Dragon Curve

I recently wrote about using the Sierpinski triangle to teach students about fractals. Another relatively easy fractal for young students to make is the dragon curve fractal. The dragon curve is created by folding a strip of paper in half over and over (in the same direction) and then placing all folds at 90o. The pictures below show what the curve looks like after 1, 2, 3, and 4 folds. (This is a great time to introduce the term ‘iteration’ to students. Iteration means the process of repeating. The curve with 4 folds is the 4th iteration.)

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If you start with a really long piece of paper, you may be able to fold it 5 or 6 times, but that is about the limit. If you could fold it 14 times, it would look like the picture below. If you kept folding, the curve would have the same shape, it would just be folded in on itself even more. Notice how each “section” of the curve looks like a smaller version of the whole curve.

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It can be difficult to make all the folds to lie at 90o angles, even with a small number folds, so I like to have students tape their curve down onto a piece of paper. If you size things correctly, you can put your folded paper dragon curve onto a picture of a high-iteration dragon curve, as shown below – a great example of how a fractal looks similar at different scales.  You can print my directions and handout here.

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You can also use colored paper or cardstock to make some dragon fractal art. Take a look at these lovely dragon curves at CutOutFoldUp.com. If you are interested in the math behind the dragon curve, this page at The Fractal Umbrella explains how to mathematically describing the fold sequences.

Numberphile has a great video about the dragon curve (below) that covers many fascinating facts about the curve, including why it is also known as the Jurassic Park fractal.