Modular Origami and the Trefoil Knot
Amy Winslow with Eve Torrence
Randolph-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