Knotical – Mathematical Connections

The story

Herons and penguins fish for knots in the bay–part of a tag-and-release program to study the behaviors of different species of knots.

Look closely; what do you see?

Can you find examples of knotted objects in…

• the net the herons are hauling up?
• the cattails along the shore?
• the sea serpent swimming past the boat?
• the octopus emerging from the bay?

Focus on… mathematical knots

Knot theorists define knots as tangled loops. The simplest of these is an untangled circle–or square, or oval, or even a squiggly closed loop that can be untangled into a circle without cutting it–called the “unknot.”

The simplest mathematically non-trivial knot is the trefoil knot. Tre means three, and the trefoil knot has three crossings. Non-trivial knots cannot be untied without cutting the loop.

The herons are hauling in a net full of a particular variant of knots called theta curves. Theta curves are tanglings of a loop with a bar across it–a shape like the Greek letter theta ϴ. The theta curves in the net are not only knotted in their overt shape, but also in their construction: crochet creates fabric by using a hook to cross yarn over itself, building tangles whose complexity increases with every stitch.

Can you guess why the knot above is called a theta cinquefoil?

The Knotted Dragon or Knottie for short (a lesser-known cousin of more famous Nessie), has slipped her body into what is colloquially called a knot ( a true mathematical knot has no free ends, like a loop).  Some eel-shaped animals really do tie themselves into knots like this, as one can see a hagfish do here.

In addition, as befits a Mathemalchemy creature, Knottie has decidedly mathematical quirks: look at the knobs on her spine: 1, then 1 again, then 2, then 3, next 5, …

The sea creatures being pulled up in the net are shapes called theta curves. While a mathematical knot is a tangling of a single closed loop, theta curves are tanglings of a shape resembling the Greek letter theta (θ), that is, a loop with an additional strand connecting two points on the loop.

The three varieties of theta curves which can be seen in this installation are theta-variants of three simple knots: the trefoil knot, the figure-eight knot, and the cinquefoil knot. Each of these is converted to a theta curve by the addition of a single strand between two “lobes” of the knot as it’s traditionally drawn in the plane. The coloration of each sea creature illustrates that any theta curve can be covered twice by three ordinary knots: each individual color traces out a simple knot, such that every strand in the theta curve is covered by two colors.

The double-cone baskets on the deck of the boat show that the Bay also holds swimming creatures that are just simple knots – of different types!

Apart from their knotted content, the double-cone baskets on the boat connect to a different mathematical nugget: together with the cylinder-shaped barrels and the spherical buoys, each of which is present in the same three diameter versions as the baskets, they illustrate the computation by Archimedes of the volume of the sphere. The height of each cylinder is equal to its diameter, which means that it can just contain a ball of equal diameter. As sketched in the sheet ‘Archimedes: Volume of the sphere’ in the Cavalcade, it then follows from Pythagoras’ theorem that the volume of such a cylinder is equal to the sum of the volumes of the inscribed ball and of the inscribed double cone.

The Bay is replete with starfish, a representative of the Echinoderms, the only phylum of animals that exhibit 5-fold radial symmetry. (Among plants, this symmetry is not as rare.) Regular polyhedra, most of the Johnson type, can be found a bit everywhere – more details about them can be found in the mathematical description of the Garden and Reef.

At the edge of the Bay one can see some half-broken up gear wheels of a mysterious machine. These are a reproduction of what is known as  the Antikythera Mechanism, a mechanical computing device for the path of planetary orbits as observed from Earth. The artifact was recovered in 1901 from a shipwreck dating back to Greek Antiquity, and illustrated a sophistication in astronomical computation as well as mechanical precision that was far beyond what was generally assumed for ancient Greek scientists.

The strange propulsion system of the boat is, for now, an engineering feat that is entirely confined to the Mathemalchemy world – but maybe less esoteric than one might think: there are completely new approaches to using wind power on boats, different from standard sails. In all their designs, as in the designs of modern airplane wings, mathematical modeling plays an essential role.