Polyhedra and Geodesic Structures

Vincent J. Matsko

One component of Polyhedra and Geodesics is a collaboration with students of the Langton Institute for Young Mathematicians (LIYM). The institute is directed by Dr. Snezana Lawrence of the Simon Langton Grammar School for Boys. A chatroom has been created through the LIYM where students can collaborate jointly on projects.

Dr. Matsko will be visiting the Langton School in June, and Dr. Lawrence will be visiting IMSA this fall. We look forward to a fruitful collaboration!

Week 1

Week 2
• Tuesday, 20 Jan: Build the Platonic solids and convex deltahedra using JovoToys [website]. Build other Johnson solids as time permits, perhaps creating some near-misses [website]. Do Exercises 7 and 8 in Chapter 2 [download] for next Tuesday.

• Wednesday, 21 Jan: Build the Platonic solids with cardstock, using the method of double-tabbing. (Nets may be found at the end of Chapter 2.)

• Thursday, 22 Jan: Discuss the geometric and algebraic proofs for the enumeration of the Platonic solids.

• Friday, 23 Jan: Take the Purdue Visualization of Rotations Test. (You will take this test at the end of the semester to see how you have progressed.)

Start Chapter 3 [download]. Draw and cut out the net for a spherical triangle on p. 45. Using the Lenart sphere, introduce spherical triangles, emphasizing the non-Euclidean nature of spherical trigonometry.

Week 3
• Monday, 26 Jan: Continue the discussion of spherical geometry. Discuss why the sum Σ of the vertex angles in a spherical triangle must satisfy 180° < Σ < 900°. Derive the cosine law for spherical trigonometry [download]. Begin the worksheet on edge angles [download].

• Tuesday, 27 Jan: Continue with the worksheets from Monday.

• Thursday, 29 Jan: Discuss edge angles of polyhedra. Work out the icosahedral case. Finish for Monday!

Also, read Chapter 3 (except Sections 4 and 5). Be prepared to answer questions Monday. Pay special attention to Section 6 on non-Euclidean geometry.

• Friday, 30 Jan: Class cancelled today.

Week 4
• Monday, 2 Feb: Work on issues from previous homework problems.

• Tuesday, 3 Feb: Finish edge angle discussion. Discuss non-Euclidean geometry as in Section 3.6. Assign building problems 9 and 10 in Chapter 3 for next Thursday.

• Thursday, 5 Feb: Discuss course selection for next year. Begin Chapter 4 [download]. Derive a construction for a geodesic dodecahedron in preparation for Friday!

• Friday, 6 Feb: Build a geodesic dodecahedron!

Week 5
• Monday, 9 Feb: We finished the geodesic dodecahedra!

• Tuesday, 10 Feb: Build the Archimedean solids with JovoToys.

• Wednesday, 11 Feb: Discuss edge angles of truncated polyhedra (see Section 6.1 [download]). Using the same technique we used for the truncated tetrahedron, compute the edge angles of the truncated octahedron and truncated icosahedron for Tuesday.

• Thursday, 12 Feb: Holiday (No Class).

Week 6
• Tuesday, 18 Feb: Having recently found out that IMSA has a CNC (Computer Numeric Control) ShopBot, I thought it would be interesting to build Dragonflies, a three-dimensional wooden sculpture by George Hart. So the sequence of topics for the next few weeks prepares us for this project.

Dragonflies is a sculpture based on a stellation of the rhombic dodecahedron. As the rhombic dodecahedron is dual to the cuboctahedron, we'll begin a discussion of duality by building a truncated tetrahedron and its dual, the triakis tetrahedron. [download net for truncated tetrahedron] [download net for triakis tetrahedron] See Chapter 9 [download] for a more complete discussion of duality.

• Thursday, 19 Feb: Finish building models! Discuss face angles of dual polyhedra.

• Friday, 20 Feb: The ShopBot requires that the shapes for the pieces be computer generated in an appropriate format. So we need to understand two- and three-dimensional coordinate systems. Read Sections 14.2 and 14.3 [download] for Monday. Pay special attention to the edge angle derivation. We'll continue the discussion on Monday!

Week 7
• Monday, 23 Feb: Begin a discussion of three-dimensional coordinates. How do we graph in three dimensions? What are the coordinates of a cube? A rhombic dodecahedron? We can find coordinates for a rhombic dodecahedron by looking at this interesting dissection.

• Tuesday, 24 Feb: Continuing the discussion of yesterday....

• Thursday, 26 Feb: Now we move on to a study of the stellation process, moving from the octahedron to the regular dodecahedron to the rhombic dodecahedron. [Click here] to see Vladimir Bulatov's excellent applet for viewing stellations of polyhedra.

• Friday, 27 Feb: Discuss stellations of the rhombic dodecahedron. [Download] the stellation diagram for the rhombic dodecahedron.

Week 8
• Monday, 2 Mar: OK, let's go! Begin by creating a small-scale, paper model of the Dragonflies sculpture. This is much harder than it looks!

• Tuesday, 3 Mar: Continue the construction.

• Thursday, 5 Mar: On student had an idea of creating another model to use as a guide.

• Friday, 6 Mar: Continued....

Week 9
• Monday, 9 Mar: Efficiently creating computer graphics requires the use of matrices to represent symmetries of polyhedra. Let's begin by looking at how we could implement geometric transformations in Mathematica. [Download] the Mathematica notebook for using matrices.

• Tuesday, 10 Mar: Continue discussing matrices in Mathematica. Then write a thread on the LIYM Forum about your project ideas.

• Wednesday, 11 Mar: [Download] the handout on rotation matrices. [Download] Chapter 15, which covers matrices and symmetry groups. Your homework for next week is to create a class mascot! Or perhaps an abstract sculpture, as long as you use matrices in some way.

• Thursday, 12 Mar: Continue working on your homework.

Week 10
• Monday, 16 Mar: This week was all about the fourth dimension: objects called polytopes or polychora rather than the three-dimensional analogues, polyhedra.

• Tuesday, 17 Mar: What is a hypercube? How many cells? Faces? Edges? Vertices? Then what happens when a four-dimensional polytope is truncated?

• Thursday, 19: So there aren't any photos of these four-dimensional objects. For obvious reasons....

• Friday, 20: Today we began working on our project proposals. A significant component of the course is a final project, poster, and presentation.

Week 11
• Monday, 23 Mar: As you can see below, pieces for the Dragonfly sculpture are being cut out and ready to be assembled. Let's go!

• Tuesday, 24 Mar: We're making a lot of progress. Will all the pieces go together?

• Thursday, 26 Mar: The sculpture was finished later in the day on Tuesday! We were all very excited. A discussion ensued about making a larger sculpture for permanent display at IMSA.

• Friday, 27 Mar: Project Day!

Week 12
• Monday, 30 Mar: Our last day of discussion of the fourth dimension. We looked at cross-polytopes in any number of dimensions. Abstract geometry meets combinatorics!

• Tuesday, 31 Mar: The Dragonflies sculpture went together wonderfully. Today, we are creating a video of this process. Stay tuned!

• Thursday, 2 Apr: More work on projects today.

• Friday, 3 Apr: Class cancelled! (Talk at ISMAA conference in Peoria.)

Week 13
• Tuesday, 14 Apr: Class cancelled! (Talk at UIUC in Urbana-Champaign.)

• Wednesday, 15 Apr: Over Spring Break, pieces for the large-scale Dragonflies sculpture were cut out. It went together beautifully! See the pictures below.

• Thursday, 16 Apr: It was a beautiful day, so we went outside to have a discussion of Flatland by Edwin A. Abbott. We had already studied the fourth dimension in class, so many of the ideas in the book were understandable.

• Friday, 17 Apr: Project Day!

Week 14
• Monday, 20 Apr: We began a discussion of Sphereland, a sequel to Flatland written by Dionys Burger.

• Tuesday, 21 Apr: We watched the animated film, Flatland. The movie was clearly different from the classic text. We also viewed the commentary by Tom Banchoff expert on Edwin Abbott, author of Flatland.

• Thursday, 23 Apr: Today we built slide-togethers based on the rhombic triacontahedron. These models are based on designs by George Hart. Click here to jump to Geroge's web page!

• Friday, 24 Apr: Project Day!

Week 15
• Monday, 27 Apr: More work on slide-togethers.

• Tuesday, 28 Apr: We completed the discussion of Sphereland.

• Thursday, 30 Apr: Getting ready for the field trip to the CUBE!

• Friday, 1 May: At the CUBE!!!

Week 16
• Monday, 4 May: Discussion of the CUBE experience.

• Tuesday, 5 May: Project Day!

• Thursday, 7 May: Building flexagons.

• Friday, 8 May: Project Day!

Week 17
• Monday, 11 May: More Flexagons.

• Tuesday, 12 May: Project Day!

• Wednesday, 13 May: Project Day!

• Thursday, 14 May: Project Day!

Week 18
• Monday, 18 May: Project Day!

• Tuesday, 19 May: Project Day!

• Thursday, 21 May: Purdue Rotations Test and course feedback.

• Friday, 22 May: Party!

Feel free to download a draft of Polyhedra and Geodesic Structures for educational purposes. You may distribute it, but you may not sell it in any form.

Several updates have yet to be made. Expect revisions in Fall 2009.

If you use this text either for your own research or in the classroom, please email me and let me know. This information would be useful in garnering support from my school.