The periodic table of the elements is almost as old as The College of Wooster, and I am a big fan. As we approach next year’s sesquicentennial of Dmitri Mendeleev‘s 1869 periodic table, I present a modest addition to the over 600 known periodic tables, which include 2D, 3D, and even 4D designs!

I wanted the table to reflect an element’s electron configuration, its energy and angular momentum quantum numbers n, l, m_l , m_s, and its mass M and nuclear charge Z. I also did not want to alienate viewers familiar with the classic twentieth-century short-form periodic table, but problems with the short form include gaps and jumps in the presentation and the way the lanthanide and actinide elements appear as footnotes to the main table.

I first created a large electron-configuration periodic table for the Taylor 111 physics lecture hall about 15 years ago. Recently, I learned that French amateur scientist Charles Janet (pronounced “sharl shuh nay”?) first created such designs circa 1930. I have a paper Janet table rolled into a spiral and a wooden one arranged like a stepped layer cake.

In my 2D version, rainbow colors code the principle quantum number n. The levels and blocks tilt slightly to emphasize that lower means larger mass M and charge Z. Lutetium and lawrencium are directly below scandium and yttrium, as they should be, not appended to the end of the lanthanide-actinide footnote. The placement of helium is nontraditional, and if it makes you uncomfortable, I feel your pain, but chemistry literature exists to support it.

Black corner triangles tag elements with exceptional electron configurations, the lightest being chromium and copper. Grayed symbols indicate elements without stable isotopes: technetium, promethium, and all elements heavier than lead. (Bismuth and uranium are nearly stable having isotopes with half-lives much longer than the age of the universe and about the age of Earth.)

A focused light beam can trap a small particle, such as a micron-sized latex sphere (or biological cell). If the sphere is much larger than the light’s wavelength, ray optics suffices to explain the trapping.

Light bends as it passes through the sphere, as in Fig. A. The piconewton forces (red and blue arrows) on the sphere are equal and opposite to the rate of momenta transferred to the light rays: the rays deflect one way, and the sphere deflects the opposite way. The net force (black arrow) is longitudinal and pushes the bead along the beam. A beam that is less intense at its edge nudges a laterally offset sphere back towards its center, as in Fig. B. A focused beam pulls a sphere behind the focus toward the focus, as in Fig. C. The push in Fig. A stabilized by the nudge in Fig. B balances the pull in Fig. C to hold the sphere just behind the focus, as in Fig. D: a Nobel Prize hiding in plain sight.

A single Nobel Prize cannot be awarded to more than three people and cannot be awarded posthumously. Arthur Ashkin developed optical tweezers in the 1970s and 1980s. Over 20 years ago his collaborator Steven Chu shared the Nobel Prize with Claude Cohen-Tannoudji and William Phillips for trapping atoms. This year at age 96 Ashkin receives the Nobel Prize for his optical tweezing pioneering.

A falling cat’s twisting returns its shape to normal but rotates its body to land feet down. Earth’s spin returns a Foucault pendulum to its initial position in one day but rotates its oscillation plane. Parallel parking cyclically rotates a car’s front wheels but shifts the car sideways. These are examples of nonholonomic motions or mechanical anholonomies.

Hwan Bae, Norah Ali, and I just published a featured article in the journal Chaos on another famous anholonomy, Hannay’s hoop, which involves a bead sliding frictionlessly on a horizontal noncircular hoop: A slow cyclic rotation restores the hoop to its original state but unavoidably shifts the moving bead by an angle that depends on the hoop’s geometry. Rotating a noncircular hoop indelibly imprints its geometry on the bead’s motion.

In the limit of slow rotation and fast beads, the shift is called Hannay’s angle (and is analogous to Berry’s phase in quantum mechanics). We mathematically generalized the shift to any speed, fast or slow, and were able to observe it in a simple experiment involving wet ice cylinders sliding in 3D printed channels.

Norton’s Dome is a fascinating counterexample in classical mechanics: A frictionless mass balanced at the dome’s top can remain there forever — but can also spontaneously slide down!

The Shape

Norton’s dome has a cubed square root profile: if the downward height is z at dome arc length s, then

If max height z_m = 2/3 when max arc length s_m = 1, then

By the Pythagorean theorem, ds^2 = dx^2 + dz^2, and so

Integrate to find the horizontal coordinate in terms of the arc length

and inversely

Eliminate the arc length to find the profile

and inversely

If the arc length 0\le s \le 1, then the rectangular coordinates -2/3 \le x,y \le 2/3 and 0\le z \le 2/3.

The IVP

For sliding frictionlessly on the dome, form the total energy as the sum of the kinetic and potential energies

for unit mass and gravity. Differentiate with respect to time to get

and divide by \dot s to get the equation of motion

Alternately, form the Lagrangian as the difference in the kinetic and potential energies

and insert into the Euler-Lagrange equation

to again get

The acceleration along the dome increases as the square root of the arc length. Combine the equation of motion with the initial conditions s[0]=0 and \dot{s}[0]=0, corresponding to a mass at rest on top of the dome, to form an initial value problem (IVP).

The Solutions

Surprisingly, two solutions exist: infinite rest

and spontaneous sliding

To check the latter, differentiate to get

How can a physical system have two evolutions?

The Resolution

If an IVP’s equations are continuous, solutions exist; if the first derivatives are also continuous, the solution is unique. In this case, the square root \sqrt{s} in the equation of motion means that the first derivative is not continuous at the origin, so solutions exist but are not unique.

The peak of Norton’s Dome is continuous, and its slope is continuous, but the slope of its slope is discontinuous. The apex of a cone is more extreme, as its slope is discontinuous.