BenGregsonUBC342Blog: Math Art Project

We chose Robert Fathauer's "Fungus Form" from the 2020 Bridges Art Exhibition. Here are the links and a photo of his piece:

http://gallery.bridgesmathart.org/exhibitions/2020-bridges-conference/fathauer

https://2020.bridgesmathart.org/art/

Fathauer's piece is made from ceramics. Here is how he describes it, "The mathematical basis for this abstract sculpture is a Goldberg polyhedron with 60 hexagonal faces and 12 pentagonal faces. The spiked features located on each face are inspired by fungi, particularly calvatia sculpta ("sculpted puffball")."

Here is a photo of the fungus, calvatia sculpta, that he is representing in his art.

The common name for this fungus, sculpted puffball, revealingly includes sculpted, hinting at its natural beauty.

Before we can understand the beauty of a Goldberg Polyhedron in a secondary school context, we must understand a few definitions.

First, a pentagon is a shape like a square, but instead of 4 sides and 4 vertices, it has 5 sides and 5 vertices. Penatgons look like this:

A hexagon is a similar shape but with 6 sides. Like this:

Polygons are shapes like pentagons and hexagons but more generally, with a finite number of edges and vertices. Some basic polygons look like this:

A Convex polygon means that the polygon is entirely enclosed by its own edges. It is perfectly put together and entirely closed so that there are absolutely no openings in the shape.

A polyhedron is a 3D version of a polygon, a shape with finitely many sides, such as a pentagon or hexagon. There are 5 regular polyhedra (plural for polyhedron). They are known as Platonic Solids or Platonic Polyhedra and are composed of only one type of polygon. They look like this:

Notice how polyhedrons become more spherical as the number of faces and edges is increased. You can imagine trying to roll each one of these down the street.

The following image allows you to see what you would need to start with in order to construct some of these 3D shapes from a 2D shape. Notice how they become increasingly complicated as the number of faces is increased. These are called nets.

Euler's Formula for polyhedra states that the number of faces, F, plus the number of vertices, V, equals the number of edges minus 2. It looks like this: F + V = E - 2

There are also semiregular polyhedra, irregular polyhedra and stellated polyhedra. Semiregular polyhedra include:

1. Convex Prisms, which are shapes where the top and bottom are the same polygon in the same position. The non-base faces are quadrilaterals such as squares or rectangles. There are infinitely many convex prisms and they look like this:

2. Antiprisms are shapes with a top and bottom that are the same polygon but they are twisted in relation to each other and the sides are triangles, not quadrilaterals. There are infinitely many antiprisms and they look like this:

3. Archimedean Solids, also known as Archimedean Polyhedra are polyhedra that are convex, uniform, composed of regular polygons, and have every vertex the same in the sense that every vertex has the same set of polygons touching each other in the exact same way. This makes them highly symmetrical. There are exactly 13 Archimedean Solids and they look like this:

Irregular polyhedra are polyhedra with the characteristic that not all sides are equal. Here are some examples of what they look like:

While there is beauty in the symmetry of a Goldberg Polyhedron, beauty can also be found in a lack of symmetry. Here is an artistic example of an irregular polyhedron:

Stellated Polyhedra are polyhedra that have been stellated. Stellated means that edges or faces have been extended until they meet each other so that a new polyhedron is formed. Here is an example of a polyhedron undergoing stellation:

Notice how the centre of each face is "pulled" out from the centre of the polygon to make a new shape.

Goldberg Polyhedra

The mathematics involved in Goldberg Polyhedra include the combinatorics of polyhedra. In other words, polyhedra with such large numbers of faces, edges and vertices that the mathematics of combinatorics is required to count them all.

There are 3 important aspects of Goldberg Polyhedrons.

1. Each face is either a pentagon or a hexagon

2. Each vertex touches 3 faces

3. Rotational icosahedral symmetry

Rotational icosahedral symmetry describes a type of symmetry that applies to 3D objects. It is not symmetrical in the sense of left right identicality, which would be 2 dimensional. Instead, it is symmetrical in a 3 dimensional sense, where the top 3 dimensional half is the same as the bottom 3 dimensional half.

Here are visuals showing two kinds of rotational symmetry called Tetrahedral Symmetry and, with more symmetrical dividers, Octahedral Symmetry:

Icosahedral symmetry has even more symmetrical divisions. It looks like this:

A familiar form of Goldberg Polyhedra, called a truncated icosahedron, is better known as a soccer ball. They have 12 pentagonal faces, usually in black, and 20 hexagonal faces, usually in white.

Truncated means that it has been shortened or lessened, usually by removing part of the end or the top.

An icosahedron is a polyhedron with 20 faces. An icosahedron looks like this:

A truncated icosahdron has 32 faces, not 20, because the truncating process increases the number of faces.To go from an icosahedron (above) to a truncated icosahedron (soccer ball), truncate each of the 12 vertices so that one third of each edge is removed at both ends of each edge. This creates the 12 black pentagons and turns each of the 20 triangles of the icosahedron into the white hexagons.

Here is a net for constructing a soccer ball:

To introduce the notation used to describe Goldberg Polyhedra, think of the numbers as the number of jumps needed to get from one pentagon to another. The first number is the number of jumps in one direction and the second number is the number of jumps in a second direction that is 60 degrees different than the first direction. Here are some examples of Goldberg Polyhedra and their notation:

(1,0) and (2,0)