NAMING POLYHEDRA

Objective:

Students will be able to explain how polyhedra are named by understanding the relationships between polyhedra.

KaleidoTile software on Macintosh computers
Naming Polyhedra Activity Sheet
Physical models of the five Platonic solids

• Load the KaleidoTile program.
  
• Press the (2,3,5) Symmetry Group button.
  
• Move the Basepoint to the center dot and listen to the computer say the name rhombitruncated icosadodecahedron.
  
• Have the class say rhombitruncated icosadodecahedron together a few times until it sounds correct.
  
• Tell the students that they will be learning what that name means and how other polyhedra are named.

• Pass out the Naming Polyhedra Activity Sheet.
  
• With the Platonic solid models, have the students name each based on the number of sides.
  
• When each student has given a name to each solid (not necessarily correct), have the student load the KaleidoTile program and continue with the Naming Polyhedra Activity Sheet.

Have the students share their sketches with others, either in small groups or as a class.
Pose these questions to the class:
• What are the names of the five Platonic solids?

  Answer:
    tetrahedron		( 4 faces)
  
    cube or hexahedron	( 6 faces)
  
    octahedron		( 8 faces)
  
    dodecahedron		(12 faces)
  
    icosahedron		(20 faces)
  
    
  What does truncated mean in the name of a polyhedron?  How do you show truncation with KaleidoTile ?
  
  

  Answer:	Truncation means "cutting off the vertices".  You show the
  	truncation of a Platonic solid on KaleidoTile by dragging the
  	Basepoint along the edge in the Basepoint window.

• How did cuboctahedron and the icosadodecahedron get their names?

  Answer:	The cuboctahedron is obtained by complete truncation of either
  	the cube or the octahedron.  Likewise for the icosadodecahedron.

• What does rhombi mean in the name of a polyhedron?

  Answer:	The true answer to this is a bit complex.  Students should 
  	make a connection between the red (medium shaded) squares that arise 
  	in the polyhedra with rhombi in the naming.  You could make 
  	the connection that the etymology of rhombi meant a square.

• In the (2,3,3)-family, why did you name some of the polyhedra differently than KaleidoTile says to? Which is correct?

  Answer:	Both namings are correct.  The octahedron does not really 
  	belong to the (2,3,3)-family, however it is formed by the complete 
  	truncation of the tetrahedron.  By naming this octahedron a 
  	tetratetrahedron, you are understanding that the color (or shading) 
  	matters.  KaleidoTile's naming system doesn't take that into 
  	consideration.

• Who can describe what the rhombitruncated icosadodecahedron looks like without actually looking?

  Answer:	You should get answers that tie the different parts of the 
  	name together.  Dodeca (12 sides) and icosa (20 sides) 
  	combine to produce an icosadodecahedron.  This is then 
  	truncated (vertices cut off) producing a square (rhombi) face where 
  	each vertex was.

It is recommended that you show the video The Platonic Solids by Key Curriculum Press. The video describes the Platonic solids, explains why there are only five of them, and discusses duals of polyhedra.