Space Tiling with Captain Planet

Getting started (Lesson 1-2)

1. Tell the students they are going to construct eight regular polygons, with sides 3,4,5,6,7,8,10,12 using a compass, protractor and ruler. They are to make their polygon set on cardboard in order to explore tessellating shapes. This can be completed as a group task producing a set per four students.
2. Get the students to draw a circle, with a compass, as a starting point for each regular polygon.

3. Cut the polygons out to make templates.
    Which of these regular polygons can you repeatedly fit together by themselves to cover a floor without leaving any gaps (except perhaps at the outer edge of the floor)?
   Play with the cardboard templates then get the groups to justify their answers from their angle calculations.
   Complete the last column in the table above.
4. Now consider combinations of the regular polygons.
   Can you tile a floor using combinations of the regular polygons?
   Find all the possible ways you can combine different regular polygons to tessellate. These are called semi-regular tessellations.
   Expect the students to justify why the shapes tessellate. Encourage the students to confirm their answers using angle calculations as well as the physical models.

1. Two squares with three equilateral triangles
2. Two hexagons with two equilateral triangles
3. Two squares with three equilateral triangles
4. A hexagon, two squares and two equilateral triangles
5. A hexagon and four equilateral triangles
6. Two octagons and one square
7. Two dodecagons and one equilateral triangle
8. A square, a hexagon and a dodecagon

5. Non –regular tessellations
   Get the students to make a non-regular triangle and quadrilateral. Trace these on paper to see if it can tessellate by themselves.
   Do non-regular triangles, quadrilaterals tessellate? Why or why not?
   Play with other shapes and see what you think.
   
   All triangles will tessellate because the sum of the interior angles in 180 degrees and
   all quadrilaterals will also tessellate because the sum of the interior angles is 360 degrees. This allows combinations of vertex angles to cover the complete 360 degrees around a point and tessellate a plane.

1. In this second session the students make 3-D shapes using toothpicks and wine gums.
   Tell the students that you are going to make together a 3-D shape using only equilateral triangles for the faces. What would be the simplest shape you could make? Make a tetrahedron using the toothpicks and wine gums sharing the process with the class.
   What will I use the wine gums, the toothpicks for? What is the mathematical name for what the wine gum, the toothpicks form in this 3-D shape?
   How many triangular faces met at each vertex?

2. Using the wine gums as the vertices (corners) and the toothpicks as the edges try to make a 3-D solid, that uses only regular triangles (equilateral) for faces, that is different to the tetrahedron.

3. Make more toothpick and wine gum solids using other regular polygon as faces. Always use the same kind of polyhedron for the faces each time.  Can you make any using a square? pentagon? hexagon?, octagon? How many different ones can you make altogether that use the same regular polygon on each face?

4. Can you discover the relationship between the number of sides and the interior angle? 180(n - 2)
    
5. Captain Planet and the Platonic Solids.
   Tell the students about the Planeteers (Captain Planet’s helpers who fight to  “save the environment”) and that the rings they wear have stones on them that represent each of the platonic solids, earth, water, wind and fire. . (I have taken some poetic license with this story)  When the planeteers combine their powers, through the joining of their rings, Captain Planet (the universe or planet, represented by the dodecahedron) miraculously arrives on the scene. 
   (They may even enjoy singing the song for some light relief!)

1. How many equilateral triangles meet at each vertex (corner) of

1. the tetrahedron   (3)                      
2. the octahedron   (4)                  
3. the icosahedron? (5)

2. Can we make a regular polyhedron, which has six or more equilateral triangles meeting at each vertex? Why or why not?

3. Can we make a regular polyhedron with four or more squares meeting at each vertex? Explain your answer.

No, 4x90 = 360 so again it would be a flat surface.

1. How many regular pentagons meet at each vertex of the regular dodecahedron? (3)
2. Can we make a regular polyhedron with four or more regular pentagons meeting at each vertex? Explain your answer.
   The interior angle of a regular pentagon is 108 degrees,and 4x 108= 432 degrees which is bigger than 360 degrees, so this would not make a vertex where the angles would concave away to join with the rest of the figure. 3 pentagons is OK at a vertex because 3x108=324 degrees which is less than 360 degrees.

5. Can we make regular polyhedra from regular octagons? What about other regular polygons as the faces? Explain your answer.

No. A 7 sided regular polygon has interior angles of 128.5 (1dp) and 3x128.5 is more than 360 degrees – so 3, 7-sided polygons will not meet at a vertex successfully, and you cannot make a polyhedron with only 2 faces meeting at a vertex. Hence you cannot have octagonal faces either as 135 x2 =270(too little), and 135 x3 = 405 (too big)

6. Use your answers to the above to prove that there are only five regular polyhedra (Platonic solids)

1. In this concluding session the students work in a group of four to make an octahedron out of old greetings cards. They create eight open-faced triangular cone shapes that fit together to form an open-faced octahedron, showing the planes that pass through the interior center point of the octahedron. Each student will need to bring one good-sized greeting card, so there are four cards per group.
2. Cut the card into a square – everyone’s square in the group needs to be the same size. Get the students to decide how big the side of their group’s square can be – make it as big as possible. Then cut the square and divide the card into two equal squares (one the patterned side and one the blank back side) by cutting down the fold line of the card. Each student has 2 squares. Two in the group use the patterned square and two use the plain square
3. Fold one square carefully along the diagonals. (The students may wish to score the fold with a pair of scissors or a ballpoint pen). What can you say about the diagonals of a square? Cut along one half of one diagonal –from the corner into the centre, lap one of the four triangles formed over the top of another and glue or staple these two triangles together.
   What shape have you made? What shape are the sides?

4. In your group fit all 4 of your shapes together to make another shape. Can you name this shape?
   How big is the base of this shape?
   How tall is the shape? (compare with you group’s original square size)

5. Repeat the process with your other square and put all 4 shapes together again with the others in your group. Can you fit all 8 together to make another solid shape?
   Can you name the shape? What is the height of this shape?

6. Look back at the octahedron made out of wine gums and toothpicks. Join the opposite vertices with string inside the shape and find the interior center point. Imagine the planes that pass through one edge of the polyhedron and this interior center point. How many different planes pass through this center point?

Make the other re- entrant polyhedra.
How many planes meet at each interior centre point for each of these re-entrant polyhedra?