Topos the mornin' to ya

Saint Hamilton March 17th is celebrated by many as St. Patrick's Day, in honour of none other than the patron saint of Ireland, William Rowan Hamilton, who drove the discretization requirement out of classical mechanics... or something like that.

So listen to Finnegan's Wake Topology or some Danny Boy Surface, wear your Green's Function with pride else face a pinch point, have a pint of Beer–Lambert law, and enjoy a scholarly St. Patrick's Day.

Lucky Trefoil

Right-Handed Trefoil In knot theory, the trefoil (or clover-leaf) knot is the simplest nontrivial knot. Being the unique prime knot with three crossings, the clover-leaf knot has many exciting properties that make a little recreational knot theory perfect for St. Patrick's Day. The clover-leaf knot cannot be deformed continuously into its mirror image without tearing it, but it is invertible. It is the intersection of the unit 3-sphere S3 in C2 with the complex plane curve of zeroes for the complex polynomial z2 + w3. It can be described as a (2,3)-torus knot and its complement in S3 is a fiber bundle over the circle S1.

If you're looking for some knot(y) suggestions, and the trefoil is too boring on it's own, consider playing with its homotopic form or projection.

Fig 3. from 'Turning a penrose triangle inside out' Fig 7. from 'The knotted hexagon '

Turning a penrose triangle inside out, Thaddeus M. Cowan, Journal of Mathematical Psychology. Volume 26, Issue 3, December 1982, Pages 252-262. doi:10.1016/0022-2496(82)90004-9

The knotted hexagon, Adrian Baddeley, Lecture Notes in Mathematics; Combinatorial Mathematics V. 1977, Pages 55-60. doi:10.1007/BFb0069176

If knot theory isn't your style, physicists have been making use of the trefoil for years.

It turns out, the clover-leaf/trefoil leads to some beautiful equipotential designs. For more pages of pretty pictures: Cutoff space of cloverleaf resonators with electric and magnetic walls, J. Helszajn and D.J. Lynch, IEEE Transactions on Microwave Theory and Techniques, Volume 40, Issue 8, 1992, Pages 1620-1629. doi:10.1109/22.149540

Fig 13. from 'Cutoff space of cloverleaf resonators with electric and magnetic walls ' Fig 16. from 'Cutoff space of cloverleaf resonators with electric and magnetic walls '

If you want a little combination knot theory and condensed matter physics:

Fig. 3. from 'Observation of laser-induced microscale knotted and unknotted vortex filaments on vaporizing tantalum surface' Fig. 3. from 'Observation of laser-induced microscale knotted and unknotted vortex filaments on vaporizing tantalum surface'

LEFT: Observation of laser-induced microscale knotted and unknotted vortex filaments on vaporizing tantalum surface, S. Lugomer, Physical Review B, Volume 54, Issue 7, 1996, Pages 4488-4491. doi:10.1103/PhysRevB.54.4488

Fig 2. from 'SURF_ER—surface electron spin resonance (ESR) of the surface domain of large objects'

For a more natural feel, electron spin resonance spectroscopy even allows for some actual clover visuals.

RIGHT: SURF_ER—surface electron spin resonance (ESR) of the surface domain of large objects, Th. Herrling, J. Rehberg, K. Jung and N. Grotha, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, Volume 58, Issue 6, April 2002, Pages 1337-1344. doi:10.1016/S1386-1425(01)00723-5

Fig 138. from 'Geometry, Relativity, and the Fourth Dimension'

If a little relativity and analogies are more your speed, consider a circular “Lineland” (a closed, 1-D space); with singularities, it happens to look an awful lot like a clover.

LEFT: Geometry, relativity, and the fourth dimension, Rudy von Bitter Rucker, Dover Publications, 1977. ISBN-10: 0486234002

And that's not all the St. Patrick's themed fun from physics...

Kiss me, I'm Green's Function

So maybe a good integral kernel doesn’t scream “Happy St. Patrick’s Day” to you, but well all right, you’re probably not alone there. There are some mathematical functions, however, that do instill the spirit of the holiday: spherical harmonics, for example.

While an electron can technically be found at any distance from its associated atom, there are regions where the probability of finding it is greater. These regions are the familiar boundary surfaces (shown here in festive green). It is the Azimuthal quantum number (or the orbital angular momentum quantum number), l, that determines an orbital's shape (ie. it’s boundary surface) with the 3-dimensional spherical harmonics, and the magnetic quantum number, ml, that determines its orientation. The first four “shapes” are known as the s (l = 0), p, (l = 1), d, (l = 2), and f (l = 3), orbitals.


The relevant St. Patrick’s Day orbitals are obviously the five “d” orbitals. They are sometimes described as “dumbbells” or, often more accurately, as the “cloverleaf” orbitals. In honour of the holiday, instead of using the tired mnemonic “some poor dumb fool” to remember the atomic orbital order, I think “St. Patrick’s Day Fun” might be more appropriate.

Pot o' Goldbach's Conjecture

March 17th, 1764, Leonhard Euler wrote his final letter addressed to Christian Goldbach, finishing the famous 196 letter Euler-Goldbach correspondance series [ref]. Euler may or may not have been aware that it was also St. Patrick's Day. In case the death of the patron saint of Ireland wasn't enough for you to throw a party, March 17th holds many important anniversaries in the mathematical/physical sciences too.

Affine day to you, too!

But hell, I still went with Guinness Remember, like most holidays, St. Patrick’s Day is best celebrated with beer, a substance that is historically well appreciated by physicists. While the story of Donald Glaser being inspired by seeing the bubbles in a pint of beer to build the first Bubble Chamber is now known to be apocryphal, some of the early Bubble Chamber prototypes did actually use beer as the superheated liquid. Unfortunately for beer and Bubble Chamber enthusiasts, Glaser’s “experiments with beer left nothing but a stench in the room and raised a few eyebrows.” [ref]

Personally, I still think beer’s greatest achievement in physics was the 1920s opening of the Niels Bohr's Institute for Theoretical Physics, which was sponsored by the Carlsberg Brewery. The Carlsberg Foundation also paid for Bohr’s travel to England in 1911 to study with J.J. Thomson.

And St Patrick's Day...