Space Tiling with Captain Planet

Purpose

In this unit tessellations are used as an application of angle properties of polygons. Interior angle properties of polygons are used to justify the existence of the five platonic solids.

Achievement Objectives
GM4-5: Identify classes of two- and three-dimensional shapes by their geometric properties.
Specific Learning Outcomes
  • explain why a shape tessellates
  • find the size of the interior and exterior angles in regular polygons
  • use the properties of interior angles in regular polygons to justify the existence of only five platonic solids
Description of Mathematics

Interior angles in a regular polygon are measured in the context of tessellating polygons and their size confirmed by applying the property of exterior angles of a polygon summing to 360 degrees. Interior and exterior angles of a polygon summing to 180 degrees is used.

This angle knowledge provides the platform to explore the nature of and the total number of regular polyhedra (platonic solids). The language of faces, vertices and edges of a 3D shape is reinforced and the number of each recorded. This is used to highlight Euler's formula relating the vertices (V), faces (F) and edges (E) of the platonic solids F+V =E+2. By considering the number of faces and the size of the interior angles meeting at each vertex of the 3D shapes the existence of only the five regular polyhedra (platonic solids) is discovered and justified.

The final activity is a fun extension that consolidates the work on the platonic solids and extends the investigation to an exploration of the planes that pass through the interior centre of these solids. The five platonic solids can be formed out of a set of cones each with a vertex at the center of the figure. This is in fact the set of all planes that pass through one edge of the polyhedron and its center point.

Required Resource Materials

Regular polygon model set (optional) Copymaster 1

Yogurt pottles to put the wine gums in

Coloured Cardboard

Play dough

Toothpicks or straws

Staplers or PVA glue

String

Old greeting cards

Activity

Getting started (sessions 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.
Using a protractor and a ruler how you can construct an equilateral triangle (a 3-sided regular polygon), inside the circle? (Any size radius will do.)
Rub out the circle and any construction lines, measure the interior (inside angles at the corners, called vertices) of the equilateral triangle.
How do you know if you have drawn an accurate diagram?
(Surprise – 60 degrees! if you have been accurate)
Can you construct the other 7 regular polygons?
Carefully measure the interior angles of the polygons and record in a table.
Get the students to verify the accuracy of their diagrams and angle measurements by revisiting the exterior angles in a polygon summing to 360 degrees and using the sum of exterior and interior angles making 180 degrees. (Share the calculations around the group)

 

Polygon Name No. of Sides Interior Angl Tessellate -Yes or No
Equilateral trian 3 60 degrees  
Square 4    
Pentagon 5    
Hexagon 6    
Heptagon 7    
Octagon 8    
Decagon 10    
Dodecagon 12    
  1. 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.
  2. 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.

  • Two squares with three equilateral triangles
  • Two hexagons with two equilateral triangles
  • Two squares with three equilateral triangles
  • A hexagon, two squares and two equilateral triangles
  • A hexagon and four equilateral triangles
  • Two octagons and one square
  • Two dodecagons and one equilateral triangle
  • A square, a hexagon and a dodecagon
  1. 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.
Exploring platonic solids (Session 2)
  1. In this session the students make 3-D shapes using toothpicks (or straws) and play dough.  
    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 play dough sharing the process with the class.
    What will I use the play dough, the toothpicks for? What is the mathematical name for what the play dough and the toothpicks form in this 3-D shape?
    How many triangular faces met at each vertex?
Summarise: The tetrahedron is made with 3 triangular faces meeting at each vertex (m in the table below). There are 4 play dough vertices (v in the table). It has 4 faces (f in the table) and 6 toothpick edges (e in the table).

 

 
m
v
f
e
tetrahedron
 
3
 
4
 
4
 
6
 

 

  1. Using the play dough 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.
Can you make another 3-D shape that uses equilateral triangles for each face? How many can you make using equilateral triangles? What is the name of the polyhedron you have made? Have you made all the different polyhedra you can with the equilateral triangle? Why? Can you name all the polyhedra you have made?
What is the difference between a polyhedron and a polygon?
  1. Make more toothpick and play dough 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?
Students can complete the table once they have explored the possibilities and shared possible names for the polyhedra they have made.

 

Polygon Name No. of Sides Sum of Interior Angles Interior Angle Tessellate -Yes or No
Equilateral triangle 3 180 degrees 60 degrees Yes
Square 4 360 degrees 90 degrees Yes
Pentagon 5 540 degrees 108 degrees No
Hexagon 6 720 degrees 120 degrees Yes
Heptagon 7 900 degrees 128.6 degrees (1dp) No
Octagon 8 1080 degrees 135 degrees No
Decagon 10 1440 degrees 144 degrees No
Dodecagon 12 1800 degrees 150 degrees No
  1. Can you discover the relationship between the number of sides and the interior angle? 180(n - 2)
     
  2. 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. (The platonic solids are named after Plato, a philosopher who lived in 400BC.  He wrote about their role in universe and associated each with one of the four elements of the universe.)
  3.  Questions to explore the existence of only five regular polyhedra (platonic solids)
    (You may wish to present Copymaster 3 to the class as a worksheet)
Captain Planet and the Planeteers

shapes.

  1. How many equilateral triangles meet at each vertex (corner) of
  1. the tetrahedron   (3)                      
  2. the octahedron   (4)                  
  3. the icosahedron? (5)
  1. Can we make a regular polyhedron, which has six or more equilateral triangles meeting at each vertex? Why or why not?
    6x60 =360 degrees so the sides would make a flat surface – ie 2D not 3D
  1. 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.
  2. How many regular pentagons meet at each vertex of the regular dodecahedron? (3)
  3. 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.
  4. 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)
  5. Use your answers to the above to prove that there are only five regular polyhedra (Platonic solids).
Using the regular polygons as faces only the ones in the table below can be used because the total of all the interior angles that met at a vertex must be less than 360 degrees. Also there must be more than 2 faces meeting at any one vertex to form a regular polyhedron. 
Equilateral triangle Square Pentagon Hexagon
3 faces (YES) 3 faces (YES) 3 faces (YES) 3 faces (NO)
4 faces (YES) 4 faces (NO) 4 faces (NO)  
5 faces (YES)      
6 faces (NO)      

 Hence there are only 5 platonic solids.

Consolidation and extension (Session 4-5)
  1. In this concluding session the students work in a group of four to make an octahedron out of old greetings cards (or coloured card). 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?
The diagonals of a square bisect one another at right angles.
The shape is an open-faced tetrahedron or triangular cone shape. The sides are isosceles triangles.

diagram.

  1. 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)                                                        A pyramid (square-based). 
    The base of the shape has the same area and dimensions as the original square.
    The height is equal to half of the diagonal of the original shape. 
  1. 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?                                                                      Yes, an open-faced octahedron shape. A re-entrant octahedron
  1. 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?                                                                    Two planes at right angles to one another
Extension

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

Tetrahedron           -3 planes pass through the edges and met at the center point
Cube                     -4 planes pass through the edges and met at the center point
Dodecahedron      -planes pass through the edges and met at the center point
Icosahedron          -planes pass through the edges and met at the center point

 

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