The kilogram is the only metric unit still defined by an artifact. The International Prototype Kilogram, IPK or “Le Grand K”, is a golf-ball-sized platinum-iridium cylinder in a vault outside Paris. This year I expect the General Conference on Weights and Measures to replace the IKP by an electronic realization that balances gravitational and electrical power.

The Kelvin or Ampere balance suspends a horizontal wire loop of mass m, length \ell, and current I, by a radial magnetic field B. Integrate the magnetic force \overrightharpoon{F}=q\overrightharpoon{v}\times\overrightharpoon{B} around the loop to find the force balance

m g = F = \left| \oint d \overrightharpoon{F} \right| = \left| \oint dq \,\overrightharpoon{v} \times \overrightharpoon{B}\right| = \left| \oint I d\overrightharpoon{\ell} \times \overrightharpoon{B}\right| =I \ell B

and solve for m. Unfortunately, \ell and B are difficult to measure accurately.

In 1975, Bryan Kibble proposed the calibration step of moving the current-less wire loop vertically at speed v. Integrate the force per charge \overrightharpoon{F}/q=\overrightharpoon{v}\times\overrightharpoon{B}  around the loop to find the induced voltage

V = \oint \overrightharpoon{E} \cdot d\overrightharpoon{\ell} = \oint \frac{\overrightharpoon{F}}{q}\cdot d\overrightharpoon{\ell} = \oint \overrightharpoon{v} \times \overrightharpoon{B} \cdot d\overrightharpoon{\ell} = v B \ell.

Eliminate \ell and B from the force and voltage expressions to find the virtual power

P = V I = v B L I = m g v

in Watts, and again solve for m. Accurately measure voltage V by comparing to the superconducting Josephson-effect voltage

V =\frac{n_J f}{K_J},

where K_J = 2 e / h = 0.48~\text{THz} / \text{mV} is the Josephson constant, n_J is the number of Josephson junctions, and f is their microwave frequency. Convert current I = V_R / R to voltage and resistance by Ohm’s law. Accurately measure resistance R by comparing to the quantum Hall-effect resistance

R =\frac{R_K}{n_L},

where R_K = h /e^2 = 26~\text{k}\Omega is the von Klitzing constant, and n_L is the number of filled Landau levels. Accurately measure velocity v and acceleration g using interferometers.

Hence the mass

m=\frac{VI}{gv}=VV_R\frac{1}{R}\frac{1}{gv} =n_J f\left(\frac{h}{2e}\right) n_J f_R\left(\frac{h}{2e}\right) n_L\left(\frac{e^2}{h}\right)\frac{1}{gv} =\frac{n_L n_J^2 f f_R h}{4gv}\propto h,

where h=0.66~\text{zJ} / \text{THz} is the Planck constant. The Kibble or Watt balance thus defines mass in terms of the rate of change of a photon’s energy with its frequency.

The NIST-4 Kibble balance has measured the Planck constant to 13 parts per billion and is thus accurate enough to help redefine the kilogram. Credit: Jennifer Lauren Lee.

One of my favorite illustrations is the cannon thought experiment from volume three of Isaac Newton‘s Principia Mathematica. Johannes Kepler argued that planets orbit elliptically with Sol at one focus. Galileo Galilei argued that terrestrial bodies fall parabolically in space and time. Living in the next generation and standing on their shoulders, Newton realized that Kepler’s ellipses and Galileo’s parabolas were extremes of the same continuum, the Newtonian synthesis, which he dramatized by imagining a cannon on a tall mountain shooting cannon balls at increasing horizontal speeds: a falling apple orbits Earth (but collides with its surface); the orbiting Luna falls toward Earth (but its tangential velocity prevents a collision).

In Newton’s famous thought experiment, subsuming both Galileo and Kepler, cannonballs shot at ever increasing horizontal speed eventually fall around Earth.

Low-resolution photograph of page 6 volume 3 of Newton’s Principia, as it appears in the Voyager interstellar record now en route to the stars.

I’ve been fascinated by aperiodic tilings of the plane since Martin Gardner first wrote about them in Scientific American. In the 1960s, Robert Berger discovered a set of 20 426 prototiles or tile-types that can tile the plane but only with no translational periodicity — a wonderful mix of the expected and the surprising, a kind of visual music.

Over the years, the number of required prototiles has been greatly reduced. In the 1970s, Roger Penrose discovered a set of just two concave aperiodic prototiles. Robert Ammann then dissected these to discover a set of three convex aperiodic prototiles. Can a single prototile, one tile or stone, literally ein stein in German, force a nonperiodic tiling? Despite several near misses and potential applications to quasicrystals, the existence of an ein Stein remains a fascinating unsolved problem.

Three convex Ammann tiles force a nonperiodic tiling of the plane.

Two concave Penrose tiles force a nonperiodic tiling of the plane.