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.
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.)
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.
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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 has four triangular faces, four vertices, and six edges.
Construction
Construct a triangle out of 3 toothpicks and
3 gumdrops.
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.
Stick the fourth gumdrop onto the ends.
Comments
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.
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The octahedron has eight triangular faces, six vertices, and 12 edges.
Construction
Begin by sticking four toothpicks and four gumdrops together
to form a square.
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.
Turn the pyramid upsidedown. Take four more toothpicks and another
gumdrop. Build another pyramid on the other side of the square.
A cube has six square faces, eight vertices, and 12 edges.
Construction
Take four toothpicks and four gumdrops. Stick them together
to make a square.
With four more toothpicks and four more gumdrops make another square.
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.
Stick the other square on top
Comments
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 has 20 triangular faces, 12 vertices, and 30 edges.
Construction
Begin by taking five toothpicks and five gumdrops and sticking them
together to make a pentagon.
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.
Repeat the first two steps to make another pyramid like the first.
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.
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.
Comments
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.
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The cubeoctahedron is an Archimedean solid. It has fourteen faces, six
square and eight triangular, 12 vertices, and 24 edges.
Construction
Take six toothpicks and six gumdrops. Stick them together to make a
hexagon.
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.
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.
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.
Comments
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.
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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.
Now begin the construction of the diamond crystal by taking six toothpicks
and six gumdrops and sticking them together to make a hexagon.
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.
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.
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.
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.
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.
