Thursday, April 22, 2010

Soma Towers

   In an April 2009 post I introduced the seven Soma puzzle pieces that fit together to form a 3 x 3 x 3cube. The seven pieces and the names I’ve attached to the pieces are shown below.
   Two of the Soma pieces look alike but are actually left-right copies of each other. To differentiate between the two, I call one a left foot and the other a right foot.
   In a later post (June 2009) I described the time I was watching my grandson Asher stack the pieces and how it dawned on me that the seven Soma pieces are useful for exploring stable and unstable structures. Perhaps, to a child, the towers of the architect are more interesting than the cube of the mathematician.
   Grandmothermath and grandadscience recently visited our oldest son and his family in Ronquiéres, Belgium. Their two sons (two of our four grandsons) are John age four and Andrew age three. They love to build with Lincoln Logs and Lego and even find a set of dominoes useful for building fences.
   I had made and sent them a set of Soma pieces. For a couple of days they ignored the Lincoln Logs and Legos and focused on building towers with the Soma pieces.
   The only Soma piece that isn't vertically stable is the 'Z' piece (see above, first picture). The 'Z' can be made stable by adding the 'C' piece as shown in the following picture.
   In fact, the  'C' piece can be placed in more than one configuration that stabilizes the 'Z' piece.
   Will any or all of the other 5 pieces stabilize the 'Z' piece?
   To motivate John, all I had to do was ask the question, “What’s the tallest tower you can build?”
   Here John has used the flat Soma pieces (C, L, T and Z) to build a tall structure but now has to deal with fitting the left-foot, right-foot, and crystal piece into the structure.
   With a little rearranging he finishes with all seven pieces forming a 12-story tower!
  As an educator I’ve witnessed the failed attempts to ‘give away’ self-esteem to kids. Gold stars, smiley faces, no red ink, etc. only fool adults, not kids. The satisfaction earned by solving a problem is all the reward John needs as is shown by the smile on his face in the previous picture.
   Andrew too built many towers but unfortunately grandadscience could not find the camera (misplacing things seems to happen a lot these days).
   In theory, what's the tallest tower that be built from the seven Soma pieces?
   Can this tower actually be built?
I leave it as a question for you, your kids, or your grandkids to answer.

Tuesday, April 13, 2010

Planar Soap Films and Spherical Bubbles

Kids love to create the bubbles made by dipping a circular frame into a soap solution and then blowing on the film. A circle is a geometric figure confined to a plane. In other words, a circle is flat, two-dimensional. When dipped, a circular frame creates a disk-shaped soap film. What happens if 3-dimensional shapes are dipped in a soap solution?

Years ago I made a set of wire-frame shapes for dipping into a soapy solution. The shapes include a wire cube, triangular-based pyramid, triangular prism, and a Möebius strip. The ends of the wire frames sit in holes drilled in a block of wood and make a nice geometrical bouquet for my office desk.

In the picture below, Joshua dips his skewed variation of a cube. Unfortunately (or fortunately) you can’t see the soap films in the photo. [To see the films, you will have to make the shapes and dip them!]

Several forces are present in a soap film. There’s surface tension, adhesion, and cohesion. Dipping the frames in soap solution and observing Water droplets are attracted to other water droplets. This is called cohesion. Water can also be attracted to other materials. This is called adhesion.Surface tension is the name given to the cohesion of water molecules at the surface of a body of water. Blowing and dipping bubbles is a good introduction to these science topics.

Multiple soap films intersect in whatever way minimizes the energy of all of the films taken together. This causes the films to pull together into the patterns seen in the dipped edge frames.

Perhaps the most unusual and unexpected shape is made by first dipping a cube edge-frame (blue cube) and then quickly dipping it a second time to capture a bubble in the center. The bubble in the center forms a smaller cube (red cube) with its 8 vertices connected to the vertices of the larger cube (green lines). The trapezoidal planes formed by a blue base line, the opposite red line, and the two green edges are beautiful to behold.

Wrinkle in Time by Madeleine L'Engle is a very popular science fantasy book read by millions of upper elementary school children. The main plot device is a tessaract, the edge-frame projection onto 3-dimensional space of a 4-dimensional The cube within a cube is that 3-space projection.

In a later post I will refer back to the soap frames as we learn about the numerical relationship between the number of edges E, vertices V, and faces F in any edge-frame model.

What's the shape of the bubbles produced by blowing on the shape-frames? Try this activity and find out!