Modular Origami and the Trefoil Knot

Amy Winslow with Eve TorrenceRandolph-Macon College

Department of Mathematics

Abstract

Thomas Hull invented the PHiZZ origami module in order to make mathematical

structures involving pentagons and hexagons. I will explain how I used these units to

make a trefoil knot tessellated by polygons and then I will show how to use graph theory

to explore the possibility of creating a proper three coloring for this structure.

Background Information

Before we get into the background of this project, we should first look at what

inspired it to understand what was being made. The

inspiration for this project was the July and August issue of

American Scientist in 2006, in tribute of the World Cup of

soccer that year. It displayed a graphic, made by Michael

Trott of Wolfram Research, of a knotted torus, tessellated

with pentagons and hexagons, like that of a soccer ball.

Basically, it showed a branched covering of a soccer ball

onto a trefoil knot. A branched covering is where the pattern from one shape can be

molded to another; this is used in rubber sheet geometry.

A branched covering can be done with just one soccer ball by this method: first,

choose two antipodal points, and a seam

that runs between them on the edges of

the polygons. Next, take that seam,

straighten it out, and pull it apart so it only covers a hemisphere. Finally, take a copy of it,

and stitch it to the first copy, so the antipodal points are together. This is a branched

covering and the antipodal point where six edges meet instead of three are branch points

(Kotschick, 350).

In order to attempt to make the tessellated trefoil knot, it was necessary to find the

units to do so. The PHiZZ units of Thomas Hull were the ones we turned to. The term

PHiZZ is an acronym created by Thomas Hull for Pentagon Hexagon

Zig Zag unit. It is a unit made of paper that can be connected strongly

with two other units to make a vertex. Each unit is in essence an edge of

a polygon. These units can be used to create many

mathematical structures including a dodecahedron, a

buckyball, better known as a soccer ball, and a torus, which

looks like a ring, or donut.

Since it takes three of these units to make a vertex, then we know that the graph of

each of these structures would be cubic. A cubic graph, or sometimes called a trivalent

graph, is a graph whose vertices have degree three, in

other words, there are three edges coming out of each

vertex (Holton and Sheehan, 235). The cubic graph of the

dodecahedron is also a planar graph. A planar graph is a

graph that can be drawn in the plane without any edges crossing. This can only be done

for some of the graphs in which the PHiZZ units make, and we will see later, that there is

a way to draw a graph without edges crossing for structures which are not planar.

Something interesting that can be found in

certain cubic and planar graphs are Hamiltonian

circuits. A Hamiltonian circuit is a path through a

graph that hits every vertex exactly once and

returns to where it started, shown in red on the

dodecahedron (Rubin, 576). A Hamiltonian circuit can be used to properly three color a

cubic graph. An edge coloring is the fewest number of colors needed to color a graph so

that no two edges incident on the same vertex have the same color (Jensen and Toft, 33).

A proper three coloring would then be a graph in

which there are only three colors used to color the

graph and no two edges on any one vertex have the

same color. The Hamiltonian circuit for a graph

makes it easy to do this by alternating between two

colors on the circuit, and coloring all edges not on the circuit the third color. Vizing’s

theorem tells us that a graph can be edge colored in Δ or Δ+1 colors, where Δ is the

maximum degree of the graph (Weisstein). Since we know all structures made of PHiZZ

units are cubic graphs, with degree three, we know they can all be colored with either 3

or 4 colors. If a Hamiltonian circuit can be found, it can be colored with three. If not, it

must be colored with four, and would be a snark.

The term snark was first created by Lewis Carroll in his book, The Hunting of the

Snark from 1876. It was a nonsensical poem that never actually said what a snark was;

only that it was very elusive and rare. Martin Gardner, a well known mathematician, then

coined the term for mathematics, since this was true for what they were being named for.

A snark in mathematics is a bridgeless cubic graph of edge chromatic number four (Pegg

and Weisstein). A bridge is just an edge of a connected graph

whose removal disconnects the graph. Therefore, a snark is a cubic

graph for which one can

not find a Hamiltonian

circuit, and must be colored with four colors. The

smallest snark, and the only known one up until

1946, is the Petersen Graph. The Petersen graph is

made up of all pentagons but is not planar (Holton and Sheehan, 15). However, the graph

can be drawn on a torus without crossings, thus tessellating the torus with pentagons. We

will see later that this is interesting and relevant to the question as to whether we had

found a snark.

Methods and Results

In order to make the trefoil knot, tessellated by polygons, we needed to take into

account the curvature of the structure. The curvature of the trefoil knot does not allow for

only pentagons and hexagons to be used, as with a normal soccer ball structure. PHiZZ

unit pentagons have only positive curvature, and PHiZZ unit hexagons have only zero

curvature. We needed the trefoil knot to have areas of negative curvature, on the inside of

the loops. This can be accomplished with other polygons. Heptagons, octagons, and

polygons with more sides do, in fact, create negative curvature. Also, the branch points

seen on the trefoil knot would not be possible for a PHiZZ unit, and therefore, we had to

use other polygons in place of them.

The first thing to do for this project

was to make hundreds of PHiZZ units to

work with. The first attempt at making the

trefoil knot was a very badly planned one, which was taken apart after only two days. The

second time, we relied some on the graph of a PHiZZ unit

torus that can be found in Hull’s book Project Origami. The

graph shows a torus made of octagons, hexagons, and

pentagons. We decided to use this for some structure for the

three curves of the trefoil knot. We then used pentagons, hexagons, heptagons, and

octagons to make the knot twist and pretzel the way it needed to. Once it was made, we

then worked on forming the graph of the knot, so we could reproduce it. There are areas

of regularity where the Hull’s torus graph was employed, and the irregularity is where we

made the knot

twist in the

directions it needed

to.

After we had the graph, the next question was can it be properly three colored?

Just by looking at the graph, this does not seem to have an obvious answer. Using

Vizing’s theorem, we know that if we find a Hamiltonian circuit, it can be, but if not, we

have a snark. In general, finding a Hamiltonian circuit on any graph is an NP-complete

problem, in other words, it is very hard, and would take a computer a very long time to

try and figure out. This was proven by Garey, Johnson, and Stockmeyer in 1976.

Therefore, to find a Hamiltonian circuit, it was just trial and error. After many tries, we

did find a Hamiltonian circuit.

Then, we only needed to follow the method for three coloring using a

Hamiltonian circuit to get the three colored graph.

The last step was to recreate the trefoil knot, using a proper three coloring, by way

of the three color graph.

Using Euler’s formula, we tested this structure. We know that the trefoil knot is

just a torus that twists, so knowing Euler’s formula is V+F-E, and Euler’s characteristic

for a torus is 0, the full formula for the knot is V+F-E=0. We know that there are three

edges to each vertex, and then each edge has two vertices on it, so we know 3V=2E, or

V=2/3 E. By substituting this in for Euler’s formula for a torus, we get F-1/3 E=0. Letting

F5 be the symbol for a pentagon, F6 be the symbol for a hexagon, etc, we can then write

that F5+F6+F7+F8=F, where F is the total number of faces. We also know that

5F5+6F6+7F7+8F8=2E, where E is the total number of edges, which are counted twice.

Then, by solving for E in the second equation, and substituting these two equations for F

and E into Euler’s formula, we get F5+F6+F7+F8 – 1/3(5F5+6F6+7F7+8F8)/2=0. By

reducing this equation, we get F5=2 F8+ F7, in other words, the number of pentagons in

the knot is equal to twice the number of octagons plus the number of heptagons. We

counted the faces in the trefoil knot and found the number of pentagons to be 27, and the

number of octagons and heptagons to be 9. By plugging in these numbers, we have 27=

(2*9) +9=27. Thus the structure is a torus with three crossing, and we have successfully

created a three colored trefoil knot.

The other part of my project, which was more based on art than solely on math,

was to create a CD torus. Using CD’s, cable ties, and a pattern for a tessellated torus that

we created, we constructed a rather large CD sculpture in the form of a torus, and then

found the Hamiltonian circuit on it. However, there was insufficient time to brace the

structure, which had too much pressure on the sides, and we were forced to dismantle it.

We do however have pictures to document it.

References

Haray, Frank. “Some Historical and Intuitive Aspects of Graph Theory.” SIAM Review

V. 2 No. 2 (1996): 123-131.

Holton, D.A., and J. Sheehan. The Petersen Graph. New York: Cambridge University

Press, 1993.

Hull, Thomas. Project Origami: Activities for Exploring Mathematics. Massachusetts: A

K Peters, Ltd. 2006.

Hull, Thomas. Origami3. Massachusetts: A K Peters, Ltd. 2002.

Jensen, Tommy R., and Bjarne Toft. Graph Coloring Problems. New York: John Wiley

and Sons, Inc. 1995.

Kotschick, Dieter. “The Topology and Combinatorics of Soccer Balls.” American

Scientist V. 94 No. 4 (2006):350-357.

Orlean, Susan. “The Origami Lab.” The New Yorker February 19 (2007):112-120.

Pegg, Ed Jr. and Weisstein, Eric W. "Snark." From MathWorld--A Wolfram Web

Resource. http://mathworld.wolfram.com/Snark.html

Rubin, Frank. “A Search Procedure for Hamilton Paths and Circuits.” Journal of the

Association for Computing Machinery V. 21 No. 4 (1974): 576-580.

Weisstein, Eric W. "Vizing's Theorem." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/VizingsTheorem.html