Goldberg Polyhedrons

In geometry, a polyhedron is a 3D shape with polygon faces, straight edges, and sharp corners or vertices. It derives from the Greek word “many-base.” Polyhedrons are convex, meaning that their faces are curved outward. A Goldberg polyhedron is a convex polyhedron made of pentagon or hexagon faces, ie. five or six sided polygons and exactly three faces meet at each vertex.

 Goldberg Polyhedrons have notation of the form GP(x,y), where x is the number of steps in one direction, and y is the number of steps taken after turned 60^o. We can use x, y to calculate the triangulation number, T, where T = x² + xy + y² = (x+y)² – xy. Multiplying the triangulation number and the number of original faces the polyhedron has will determine the number of triangles the new polyhedron will have. This process is a type of subdivision known as principal polyhedral triangle (PPT), which breaks down the structure of a polyhedron. By subdividing the Goldberg polyhedron and slicing the spherical shape in half, we create a geodesic dome. The triangular shapes are structurally rigid and create a sturdy and durable structure. Any pressure applied to one triangle is evenly distributed to other triangles.  

Figure 1:  A GP(5,5) Goldberg Polyhedron (left).

Figure 2: A GP(5,5) subdivided by triangulation number (left). 

For this project, I tried to recreate creating several different kinds of Goldberg polyhedrons by colouring golf balls. The divots in the golf balls are all hexagon shaped. A problem I ran into while working on this project, was that divots were not entirely uniform, ie. same size. Nonetheless, it was an interesting opportunity to try and draw my own Goldberg polyhedron. An extension of this activity in the classroom could be to give students toothpicks and marshmallows and challenge them to first create a Goldberg polyhedron, then subdivide the faces to create a geodesic dome and compare strengths between their creations. This activity teaches students teamwork and cooperation, but also lets them explore and experiment on creating integral structures resistant to falling. There is another interesting resource online that is found on https://levskaya.github.io/polyhedronisme/, which generates different polyhedrons. It would be useful to integrate this resource in the digital classroom as it is a great visualization tool that demonstrates easily how polyhedrons change when different operators are applied such as truncating, orthogonality, duality, reflecting and snubbing. 

Figure 3: A GP(5,0) created from a golf ball. 

Figure 4: Virtual recreation of Fathauer's Goldberg Polyhedron using polyHedronisme. 

This was my first group project since starting the teacher education program at UBC, and I had a blast. I’ve learned so much through creating Goldberg polyhedral patterns on golf balls and researching polyhedrons. My favourite part of this project would be the interactive session using polyHedronisme. As a teacher, I hope to integrate many different digital tools in the classroom for learning. We are living in such a technologically advanced generation, I feel that incorporating these technologies into the classroom can be opportunities for students to engage with mathematics on a deeper level than equations seen in a textbook. 

Sources:

Fathauer, R. 2019. Robert Fathauer. Retrieved September 27, 2020 from http://gallery.bridgesmathart.org/exhibitions/2020-bridges-conference/fathauer. 

Hart, G.W. n.d. A Twelve-Part Puzzle Based on the (4,2) Goldberg Polyhedron “Tectonic Plates.”  Retrieved September 27, 2020 from https://georgehart.com/puzzles/GBpuzzle-4-2/goldberg-puzzle-2.html. 

Levskaya, A. 2019. polyHedronisme. Retrieved September 27, 2020 from https://levskaya.github.io/polyhedronisme/.