The Greek philosopher Plato described a set of three-dimensional shapes in the Timeaus, a dialogue written in 360BC. These shapes, known as Platonic solids, are made by putting together polygonal faces that are identical in shape and size in a way where all sides and angles are equal, and an equal number of faces meet at each vertex.
Only five solids meet these criteria: the tetrahedron, cube, octahedron, icosahedron, and dodecahedron. In the Timeaus, Plato assigns the first four to fire, earth, air, and water, respectively, then writes about the dodecahedron: “And seeing that there still remained one other compound figure, the fifth, God used it up for the Universe in his decoration thereof.”
Over 2,000 years later, mathematicians have made a basic discovery about the dodecahedron, cementing its status as the oddball of the Platonic solids.
In a recent paper, mathematicians Jayadev Athreya, David Aulicino, and Patrick Hooper showed that there are 31 kinds of straight paths that begin at a corner of a dodecahedron and walk around the entire shape without touching another corner. Such paths are not possible for any other Platonic solid. An article in Quanta first reported the group’s discovery, which was published in the journal Experimental Mathematics in May.
Aulicino, a mathematics professor with appointments at Brooklyn College and CUNY Graduate Center, said that the group was inspired by two recent papers demonstrating that a straight path starting at a corner and bypassing every other corner could not be made for tetrahedra, cubes, octahedra, and icosahedra. They then adapted an algorithm to answer the straight path question for the dodecahedron.
"We were like, 'Let's prove this thing doesn't exist,' and we wrote this computer code. Regardless of how many times we debugged it, the computer code wouldn't say that there was nothing there,” Aulicino said.
“We finally took out a sheet of paper and were like, 'Let's try to draw one of these things that the computer is telling us.' We were completely blown away that the thing actually closed up, even though we drew a straight line on it.”
In their paper, the trio present a theoretical proof for why such a trip is possible on dodecahedra but not other Platonic solids. Aulicino explained that there are certain symmetries that the dodecahedron lacks and that the other solids possess, and these symmetries are what prevent you from getting back to where you started without passing another corner on these solids.
Now that this question has been answered for each standard Platonic solids, one new avenue of research would explore the generalizations of these shapes. There are no holes in the classical Platonic solids, but you can form an infinite number of shapes with holes that follow every other guideline for what a Platonic solid should be.
Aulicino said that he often works with high school students at Brooklyn College, and it’s heartening to point to this research as an accessible discovery that is much more recent than theorems by famous historical mathematicians like Euclid and Gauss.
“I've had so many students over the years that say, 'What do you mean you do math research? It's all in the textbook.' It's really nice to be able to say, 'Here's something that we didn't know, and now we know the answer to it.’”