## About the Platonic Solids

The five Platonic solids are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Their faces are regular polygons. These solids are perfectly symmetrical in that each face of a solid is identical to every other face of the solid, each vertex is identical to every other vertex, and each edge is identical to every other edge.

Each Platonic solid has a dual: a solid whose vertices correspond to the faces, and faces to the vertics. The dual of the cube is the octahedron. The dual of the dodecahedron is the icosahedron. The tetrahedron is its own dual.

There are 13 slightly less symmetrical solids called the Archimedean solids. The faces of these solids are regular polygons, their edges have the same length, and their vertices are identical. Their faces, however, are not identical. An Archimedean solid may have two or three different polygons as faces.

## Supplies

I have found the easiest way to pass out the supplies for this lesson is to fill baggies with the correct number of gumdrops and toothpicks. I can then place a baggie on each student's desk. I hand out extra toothpicks, but not extra gumdrops.

You will need:

• For the tetrahedron, 4 gumdrops and 6 toothpicks.
• For the octahedron, 6 gumdrops and 12 toothpicks.
• For the cube, 8 gumdrops and 12 toothpicks.
• For the icosahedron, 12 gumdrops and 30 toothpicks.
• For the cubeoctahedron, 12 gumdrops and 24 toothpicks.
• For the diamond, 14 gumdrops and 22 toothpicks. (Add one gumdrop and four toothpicks for the model of a carbon atom.)

### Other Resouces

#### Paper Patterns

Paper patterns of the five platonic solids are available as a postscript file or as two GIF files: plato1 and plato2.

#### Interactive 3D display

Buckyspin is a stand-alone DOS program that displays platonic solids, archimedean solids, and Buckminsterfullerenes. It is available in ZIP format, so you will need some way to unzip the files.

## Lesson Plans

#### Appropriate grade levels:

• Tetrahedron: All grade levels
• Octahedron: Second grade and up
• Cube: All grade levels
• Icosahedron: Third grade and up
• Dodecahedron (paper): Third grade and up
• Cubeoctahedron: Fourth grade and up
• Diamond: Fifth grade and up

### General notes on constructing these solids in the classroom

• I demonstrate each step of the construction while the students watch, and then have them do it.
• Make sure to stick the ends of the toothpicks deep in the gumdrops, but not all the way through
• When sticking a gumdrop onto more than one toothpick, I find it easier to stick the ends of the toothpicks in one at a time.
• Some students can see the symmetry and get ahead of me. As long as they are putting the solid together correctly, I encourage them to do so.
• When the students have finished constructing a solid, have them count the vertices (gumdrops), edges (toothpicks), and faces (triangles, squares, or pentagons as the case may be.)
• Point out the symmetry of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron. Explain that except for the colors of the gumdrops, each vertex is identical to every other vertex. Each edge is identical to every other edge, and each face is identical to every other face.

### The tetrahedron

The tetrahedron has four triangular faces, four vertices, and six edges.

#### Construction

1. Construct a triangle out of 3 toothpicks and 3 gumdrops.
2. Set the triangle on the desk. Stick an additional toothpick into each gumdrop so that the opposite ends of the toothpicks meet above the center of the triangle, forming what looks like a pyramid or a tent.
3. Stick the fourth gumdrop onto the ends.

Cannonballs are often stacked in a tetrahedral array. You can create such a stack by glueing marbles together. Put a triangle of 10 marbles in the bottom layer, 6 marbles in the second layer, 3 marbles in the third layer, and one marble on top.

You can create a tetrahedral lattice out of 20 gumdrops and 40 toothpicks.

### The octahedron

The octahedron has eight triangular faces, six vertices, and 12 edges.

#### Construction

1. Begin by sticking four toothpicks and four gumdrops together to form a square.
2. Take four more toothpicks and one more gumdrop. Build a pyramid on the square by sticking a toothpick into each gumdrop and joining the ends with the new gumdrop.
3. Turn the pyramid upsidedown. Take four more toothpicks and another gumdrop. Build another pyramid on the other side of the square.

### The cube

A cube has six square faces, eight vertices, and 12 edges.

#### Construction

1. Take four toothpicks and four gumdrops. Stick them together to make a square.
2. With four more toothpicks and four more gumdrops make another square.
3. Take four toothpicks. Place one square on the desk. Stick one toothpick into each of the gumdrops in the square so that the toothpicks are vertical.
4. Stick the other square on top

For students who construct both the cube and the octahedron I point out the following:

• The octahedron is rigid, but the cube is not. I ask the students to explain the difference. Usually they can figure out that the octahedron is rigid because it is made of triangles.
• The vertices of the cube correspond to the faces of the octahedron, and the vertices of the octahedron correspond to the faces of the cube. To demonstrate this, I open my cube by pulling one end of one toothpick out of a gumdrop, popping the octahedron inside, and putting the cube back together.

### The icosahedron

The icosahedron has 20 triangular faces, 12 vertices, and 30 edges.

#### Construction

1. Begin by taking five toothpicks and five gumdrops and sticking them together to make a pentagon.
2. Take five more toothpicks and one more gumdrop. Build a pyramid on the pentagon by sticking a toothpick into each of its gumdrops so that the ends of the toothpicks meet above the center of the pentagon. The resulting shape resembles a funny hat.
3. Repeat the first two steps to make another pyramid like the first.
4. Take 10 toothpicks. Pick up one of the pyramids and hold it upsidedown. Stick two toothpicks into each of the gumdrops in the pentagon so that pairs of toothpicks form a V pointing straight up. Tips of the toothpicks should meet neighboring toothpicks to form triangles.
5. Take the other pyramid and stick the gumdrops of the pentagon onto the tops of the triangles. The finished icosahedron should be made entirely of triangles. Each gumdrop should have five toothpicks sticking out of it.

The icosahedron is a "geodesic" dome. Students may have seen such domes, made entirely of triangles, in playgrounds or on covered ampitheaters.

Ask students to count the pentagons in the icosahedron. (There are 12.)

I recommend having the students make a paper model of the dodecahedron to compare with the icosahedron.

The faces of the icosahedron correspond to the vertices of the dodecahedron, and the faces of the dodecahedron correspond to the vertices of the icosahedron. Students often notice that the dodecahedron looks like a soccer ball. A soccer ball is, in fact, a truncated icosahedron.

### The cubeoctahedron

The cubeoctahedron is an Archimedean solid. It has fourteen faces, six square and eight triangular, 12 vertices, and 24 edges.

#### Construction

1. Take six toothpicks and six gumdrops. Stick them together to make a hexagon.
2. Place the hexagon on the desk. Take 6 toothpicks and 3 gumdrops. Stick a toothpick into each of the gumdrops in the hexagon. Join pairs of toothpicks at the top with the three gumdrops to form three triangles sticking up. It looks like a broken crown, or like teeth.
3. Take three toothpicks. Join the three gumdrops at the top of the triangles. You should now have a dome made of four triangles and three squares.
4. Turn the dome over and build an identical dome on the other side, making sure to build triangles next to squares, and squares next to triangles. In the finished solid, each triangle shares its edges with three squares, and each square shares its edges with four triangles. Each gumdrop has four toothpicks sticking out of it.

If you don't mind making the investment, have the students construct a cube and an octahedron to compare with the cubeoctahedron. Have them count square faces and triangular faces. Show how all three solids have the same basic symmetry.

The cubeoctahedron can be constructed mathematically by connecting the midpoints of the edges of a cube, or by connecting the midpoints of the edges of an octahedron. A cubeoctahedron is formed naturally by the close packing of spheres. You can make one by glueing marbles together.

### The diamond

Diamond crystal consists of interlocking hexagons with an underlying tetrahedral symmetry.

#### Construction

1. Begin by having students construct a tetrahedron.
2. Have the students take out one gumdrop and four toothpicks. Ask them to stick the toothpicks in the gumdrop so that the ends of the toothpick are as far apart as possible, as if they repel each other. This is a model of the bonds in a carbon atom. Students tend to keep the toothpicks in one plane. Show them that they can get the toothpicks further apart if they think in three dimensions. If they hold the gumdrop in the center of the tetrahedron and have the toothpicks stick out through the centers of the four triangles, they will create the correct model.
3. Now begin the construction of the diamond crystal by taking six toothpicks and six gumdrops and sticking them together to make a hexagon.
4. Take three toothpicks and three gumdrops. Stick the toothpicks straight up in three of the gumdrops of the hexagons, choosing alternate gumdrops so that each gumdrop with a toothpick has gumdrops without toothpicks on either side. Stick a gumdrop on the end of each vertical toothpick. They will look a bit like lollipops.
5. Take four toothpicks and one gumdrop. Stick the toothpicks in in the carbon atom arrangement. Stick three of these toothpicks into the "lollipop" gumdrops on the ends of the vertical tootpicks.
6. Take three toothpicks. Stick a toothpick into each of the toothpicks of the hexagon that has a toothpick stick straight up out of it. As you do so, raise these gumdrops, so that each is now the center of a "carbon atom" arrangement. The other three gumdrops, with only two toothpicks sticking out of them, remain on the surface of the desk.
7. Take four gumdrops and stick a gumdrop on the free end of each of the four toothpicks that is sticking out. The resulting shape reminds me of a space probe, but it is in fact the basic structure of diamond.
8. Have the students look for tetrahedrons in the diamond. In addition to the tetrahedral arrangement of the toothpicks in the gumdrops, they ought to be able to see that the four outer gumdrops form a tetrahedron, as do the gumdrops with four toothpicks in them. The six gumdrops with two toothpicks in them form an octahedron.