Platonic Solids

Emily Peters
U.C. Berkeley
Hampshire summer program for high school students

March 10, 2007

What are the 3-dimensional platonic solids?

Emily solicits answers from students and draws answers on the board:

Emily: Let's try to come up with a definition of a platonic solid. What are some properties that you think a platonic solid should have?

[Students volunteer ideas; Emily writes them on the board, using student language whenever possible and prompting for clarification when needed:]

Definition of a platonic solid
-------------------------------

- all faces are identical polygons
              regular polygons
              (equal sides, equal angles)

Emily: Is this it? Is it enough that the faces are all the same regular polygon?

Student: It's got to look the same from any face.

Emily adds:

- they ``look the same'' from any face.

Emily: What do you think? Is this enough? Let's take a vote.

[Class votes. Some people undecided. ]

Emily: I want you to break into small groups and try to come up with a shape that has the first property but not the other. So all faces are identical regular polygons, but the solid doesn't ``look the same'' from all faces.

[Class breaks into groups. No one comes up with an example solid that Emily asked for, but several students discovered:]

Emily: Someone's come up with a very interesting example, so let's all take a look.

[Emily leads class in discussion of why this example ``looks the same'' from every face]

Emily: Is this a platonic solid?

Students: No!

Emily: Why not?

[Emily and students have discussion, and a student says:] It doesn't look the same from every vertex.

Emily: Sounds like we should add something to the definition. What do you want to add?

- each vertex ``looks same''
     i.e., has equal # edges

Student: What if we use hexagons?

Emily: Let's see.

[Emily works with students to find a way to glue hexagons together and arrive at:]

Emily: Let's take a vote. Is this a platonic solid?

Students: No.

Emily: Why not?

Students: It's not 3-D.

- 3-dim.  i.e., a solid

[Emily and students discuss sum of angles around a vertex in the gluing. They decide that if the angle sum around a vertex is 360, then it determines a tiling of the plane, is therefore not a solid, and therefore not a platonic solid.]

Student: What about octagons?

Other student: That won't work.

Emily: Why not?

Student: Because it's greater than 360.

Emily: That's right. What do you think it would look like if you tried to glue octagons?

[Students and Emily discuss; conclude that octagon gluings could create crinkles--if we persisted in gluing octagons to each other, we would eventually get a ``crazy tutu'' called the hyperbolic plane.]

[Emily helps students summarize their findings:]

[Emily discusses optional project: take a bunch of equilateral triangles and glue them so that each vertex is contained in exactly 7 faces and each edge is contained in exactly 2 faces. Alternatively, take a bunch of squares and glue them so that each vertex is contained in exactly 5 faces and each edge is contained in exactly 2 faces.]

Emily: Okay, I think we are ready to prove a theorem.

Thm. There are 5 platonic solids.

Emily: How can we prove this?

[After some discussion, some student suggests looking at triangles.]

Emily: Is there a platonic solid using triangles if we glue 2 to a point?

[Students decide no-- does not form a solid.]

[Emily helps students create and discuss a case analysis. Some cases result in platonic solids, some don't form a solid. Some are ``wacky hyperbolic''.]

Emily: Now that we've done this proof, let's see if we can take this in another direction. What about 4-D?

Anyone know of a 4-dimensional platonic solid?

Student: hypercube?

[Emily draws a diagram of a hypercube on the board and then shows off a zometool model of a hypercube.]

Emily: There are vertices, edges, faces, and these 3-dimensional faces that we'll call facets so we're not confused.

[Students discuss lower-dimensional cubes. Decide that a 2-dimensional cube is a square and a 1-dimensional cube is a line segment.]

Emily: Let's tabulate something. How many vertices does a line segment have?

                2
line segment ______ vertices
 
             
                4
      square ______ edges             
 
      
                6
        cube ______ faces             
 
  
                ?
   hypercube ______ facets                   
            

[Students guess that a hypercube must have 8 facets. They verify it using the zometool model of the hypercube.]

Emily: Let's unfold a cube. What does it look like? ... What do you think unfolding a hypercube would look like?

Student: But which one stays the same when you fold it?

[Through discussion, Emily discovers that student wants 4-dimensional analog to placing a cube blueprint on a table and folding up the cube while fixing the center square in the blueprint.

[Class explores how to fold up a tesseract with ``4-dimensional paper.'']

Emily: Anyone heard of another 4-dimensional solid other than a hypercube?

Student: Is there a hypertetrahedron?

Emily: Let's see what that would be.

             #vertices
line segment   2
 
    triangle   3 

 tetrahedron   4 

 hypertetra?   

[Students guess 5]

Emily: It's kind of like a cone [and describes].

At this point, we're running out of time, so let's see if we can come up with another 4-dimensional solid.

[Emily coaxes students into guessing the name hyperoctahedron and discusses how to visualize it by analyzing cube. The corner of a cube looks like a triangle. The corner of a hypercube looks like a tetrahedron. Corners of cubical facets stick into faces of tetrahedra. Hyperoctahedron gotten by sticking tetrahedra into tetrahedra.]

Student: What's so special about tetrahedra?

Emily: That's a good question. What other shapes can we use?

Student: Cubes.

[Emily discusses use of cubes and adding up radial angles]

Student: What about octahedra?

[Discussion continues; class comes to a close.]

Emily: If you want to look at pretty pictures and read more about platonic solids, you should look at John Baez's page. [Writes url on board]

[Graduate student in back loads up page with graphics and high school students turn to look. They begin to pack up and a few stay to take a closer look at the graphics. One stays to talk to Emily about the Hampshire summer program for high school students.]