Geometric Models Investigation

Have you ever built model airplanes, cars or boats? Models are often scaled-down versions of real objects. Physical models have many of the same features as the original but are often more convenient to study or to play with.

Theoretical or mathematical models are very common. Geometric ideas such as points, lines, planes, faces, edges, vertices, polygons, and diagonals can be used to represent physical objects. In the following investigation you will discover the rules that describe geometric situations.

Task

Part 1

Regular polyhedra have intrigued mathematicians for thousands of years. They were important to ancient Greek scholars who placed great emphasis on the study of science. Admired and exalted from earliest times, the five Platonic solids—tetrahedron, octahedron, icosahedron, cube, dodecahedron—are the most perfectly symmetric of all solids. How perfect? In nature, the cube, tetrahedron, and octahedron appear in crystals. The dodecahedron and icosahedron appear in certain viruses and radiolaria.

For the first part of this investigation you will be asked to make three-dimensional geometrical models by building ‘wire frame’ platonic solids using sticks and connectors.

A simple version of this is as follows: use clay or plasticine for the connectors and use toothpicks for the sticks. Then, build the five platonic solids.

I would like to make a suggestion here:

  1. I would recommend that you use materials other than marshmallows.
    • Marshmallows are not robust. Perhaps use clay that dries, plasticine or use cut pieces of hoses with holes bored in them.
    • Use kebab sticks and cut them to the desired length.
  1. As you build the Platonic solids keep track of your observations. While you are building these beautifully regular shapes you might want to consider:
    • what do all these five shapes share in common
    • why would they be called regular
    • what observations can you make about the angles of the adjoining faces on each solid
    • what observations have you made about the vertices, edges and faces of each of these solids

Part 2

Gold-plated lion from the front of the Gate of Heavenly Purity, Closeup of Ball. Forbidden City, Beijing. From Qing Dynasty (1736-1796)

If just one of the requirements for a solid to be regular is removed, a large collection of other highly symmetrical forms can be discovered. Using the rule below, make other shapes.

Explicitly construct the dual of each of the Platonic solids by adding ‘pyramids’ onto the faces

Part 3

  1. If we continue to require that all vertices be identical and that the solid be convex, but we remove the requirement that only one kind of regular polygon be used, the family of solids that results is called the Archimedean Solids (also called the semiregular solids), of which there are 13. Seven of the Archimedean solids are derived from the Platonic solids by the process of “truncation”, literally cutting off the corners. For example, we can start with a dodecahedron, and trim off its corners to change each face from a pentagon to a decagon, leaving an additional small triangular face at each corner.
  2. Construct as many Archimedean solids as you can.
    • As you build the Archimedian solids keep track of your observations. While you are building these solids you might want to consider:
      • what do all these solids share in common
      • how do they differ from the Platonic solids
      • what do they share in common with the Platonic solids
      • what observations can you make about the angles of the adjoining faces on each solid
      • what observations have you made about the vertices, edges and faces of each of these solids

What have you learned about the Archimedian solids?

Note: when I was a child (age 11), I did something like this purely for fun. I build out of sticks and cut up hose with holes in it a complete third frequency geodesic sphere. It was fun. My recollection of it was that the materials that I used were not great. Thus, I would recommend testing the materials prior to giving this project out. -Charles (creator of this investigation)

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