720° untangles 360° tangles

Despite growing up in three dimensions, as a kid I did not recognize one of 3D’s deep and subtle properties: full rotations tangle, but double rotations untangle!

Following physicist Paul Dirac, twist a belt one full turn about its length. The 360° single twist cannot be undone without changing the belt buckles’ orientations, although the twist can be changed from clockwise to counterclockwise. Now twist the belt two full turns about its length. Amazingly, the 720° double twist can be undone without changing the belt buckles’ orientations. The double twist is the true identity.

The animations below show belts twisting concretely in 3D space and abstractly in a 3D projection of the 4D quaternion unit sphere. Points on the sphere represent all 3D rotations (twice). Each blue dot represents a belt cross section rotation and is located by a radius vector. Suppressing rotations about the 1‑direction, the radius vector’s projection onto the 2-3-plane is the section’s rotation axis and twice its co-latitude is the section’s rotation angle.

The sphere’s north and south poles represent the same orientation, 0° ≡ 720° and 360°, but different orientation-entanglements. Identifying the north and south poles as the same orientation allows closed loops on the quaternion sphere to represent both 360° and 720° belt twists, but only the latter can smoothly contract to the north pole identity rotation. This multiple connectivity is reminiscent of a torus (or donut with hole), where toroidal loops (around-the-hole) are contractible but poloidal (through-the-hole) loops are not, rather than the simple connectivity of a sphere, where all loops are contractible.

Elastic belt with a 360° twist. Blue dots on quaternion sphere projection represent belt cross section rotations. Blue curve connecting dots can not be smoothly contracted to the untwisted state represented by the north pole, but without changing the orientation of the belt’s ends, the twist can be changed from clockwise to counterclockwise as indicated.

Elastic belt with a 360° twist. Blue dots on quaternion sphere projection represent belt cross section rotations. Blue curve connecting dots can not be smoothly contracted to the untwisted state represented by the north pole, but without changing the orientation of the belt’s ends, the twist can be changed from clockwise to counterclockwise as indicated.

Elastic belt with a 720° twist. Blue dots on quaternion sphere projection represent belt cross section rotations. Blue curve connecting dots can be smoothly contracted to the untwisted state represented by the north pole, so without changing the orientation of the belt’s ends, the twist can be be undone as indicated.

Elastic belt with a 720° twist. Blue dots on quaternion sphere projection represent belt cross section rotations. Blue curve connecting dots can be smoothly contracted to the untwisted state represented by the north pole, so without changing the orientation of the belt’s ends, the twist can be be undone as indicated.

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Wooster physics reunion in Eugene, Oregon!

I recently returned from a refreshing and productive leave at the University of Oregon in Eugene.  I received my own Ph D in the field of quantum optics at Oregon, and my leave was a great opportunity to continue work with prior colleagues.  However, I am excited to be back and interacting daily with Wooster students again!

Five Wooster alumni are currently enrolled in U of Oregon’s Physics Ph D program.  This summer I managed to gather four of these physicists together in one place–one of my favorite old haunts from my graduate school days, McMenamins Cafe.

Pictured here from left to right are Deepika Sundaraman ’14, myself, Tzula Propp ’15, Nicu Istrate ’15, and Andrew Blaikie ’13.  Not present–but with us in spirit–is Amanda Steinhebel, who had recently left Oregon to spend a year on site at the world’s largest particle accelerator (the Large Hadron Collider) at CERN in Switzerland.  All attendees (plus Amanda) received this year’s Wooster physics club t-shirt!

Later in the summer, I attended a the fall conference of the Oregon Center for Optical, Molecular, and Quantum Science, where Andrew Blaikie ’13 presented his fascinating research on a nanoscale device employing graphene to make extremely sensitive measurements (see Andrew in action below).

As for the research I worked on in Oregon while on leave–that is a different blog post for a different day!

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Luna’s Convex Orbit

Luna orbits Earth and Earth orbits Sol (where Luna is Earth’s moon and Sol is Earth’s star, the sun). As a kid, I thought Luna’s solar orbit formed a loopy spirograph pattern. Instead, Luna’s orbit is convex!

Neglecting the eccentricity and tilts of the orbits and the incommensurability of the periods, Luna’s counterclockwise path around Sol is roughly a dodecagon, as in the idealized animation, where the distance between Earth and Luna varies from cartoon large to proportionally correct. Colors code the path’s curvature, red for counterclockwise-rotating velocity vectors and cyan for clockwise-rotating velocity vectors, with magnitudes proportional to the saturation, so white is straight. Due to the convexity of both Earth and Luna’s orbits, some planetary scientists consider Earth and Luna to be a double planet.

Luna's solar orbit is almost a convex rounded dodecagon. Red-white-cyan colors code curvature.

Luna’s solar orbit is almost a convex rounded dodecagon. Red-white-cyan colors code curvature.

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1+2+3+… = -1/12?

In quantum electrodynamics, the bare charge of an electron is infinite, but the renormalized dressed charge is finite. The bare electron shields itself by polarizing the virtual electron-positron pairs of the nearby quantum vacuum to reduce its coupling at large distances to

\infty - \infty \stackrel{I}{=} \sqrt{\frac{1}{137}}

in natural units, where the “I” decorating the equals sign denotes an informal relationship. Renormalization techniques help physicists make sense of divergent sums and integrals used to calculate quantities like running coupling “constants” that vary with interaction energy and set the strengths of the fundamental forces.

Corresponding sum and integral diverge together

Corresponding sum and integral diverge together

For a simple example, the sum of natural numbers

\sum_{n=0}^N n = 1+2+3+\cdots+N= \frac{N}{2}(N+1)

diverges as N\rightarrow \infty. Similarly, the analogous integral of positive real numbers

\int_{0}^X \hspace{-0.8em}dx\ x =\frac{X^2}{2}

diverges as X\rightarrow \infty. To cancel these corresponding discrete and continuous divergences, introduce an exponential convergence factor and formally subtract the integral from the sum

\sum_{n=0}^\infty n -\int_{0}^\infty \hspace{-0.8em} dx\, x =\color{red}\lim_{\lambda \rightarrow 0}\color{black}\left( \sum_{n=0}^\infty n\, \color{red}e^{-\lambda n}\color{black}-\int_{0}^\infty \hspace{-0.8em} dx\, x\, \color{red}e^{-\lambda x} \color{black}\right).

To evaluate the cancellation, differentiate the integral with respect to the convergence parameter \lambda>0 to get

\mathcal{I}=\int_{0}^\infty \hspace{-0.8em}dx\ x\, \color{red}e^{-\lambda x}\color{black} =-\frac{d}{d\lambda}\int_{0}^\infty \hspace{-0.8em}dx\ e^{-\lambda x} =-\frac{d}{d\lambda}\left(\frac{e^{-\lambda x}}{-\lambda}\right)\bigg|_{0}^\infty=-\frac{d}{d\lambda}\left(\frac{1}{\lambda}\right)=\frac{1}{\lambda^2}.

Next differentiate the sum with respect to \lambda

\mathcal{S}=\sum_{n=0}^\infty n\,\color{red} e^{-\lambda n}\color{black} = -\frac{d}{d\lambda} \sum_{n=0}^\infty e^{-\lambda n} = -\frac{d}{d\lambda} \sum_{n=0}^\infty \left(e^{-\lambda}\right)^n

and sum the resulting geometric series to find

\mathcal{S}= -\frac{d}{d\lambda} \left( \frac{1}{1-e^{-\lambda}} \right)= \frac{e^{-\lambda}}{\left(1-e^{-\lambda}\right)^2}\color{red}\frac{e^{2\lambda}} {e^{2\lambda}}\color{black}= \frac{e^{\lambda}}{\left(e^{\lambda}-1\right)^2}\color{red}= \frac{e^{\lambda}}{\left(1-e^{\lambda}\right)^2}\color{black},

which is thus an even function of \lambda. Replace the exponentials by their power series expansions

\mathcal{S}=\frac{1+\lambda+\lambda^2/2!+\lambda^3/3!+\cdots}{\left(1+\lambda+\lambda^2/2!+\lambda^3/3!+\cdots - 1 \right)^2}=\frac{1+\lambda+\lambda^2/2+\cdots}{\lambda^2\left(1+\lambda/2+\lambda^2/6+\cdots \right)^2},

expand the denominator square

\begin{array}{ccc}\left(1+\lambda/2+\lambda^2/6+\cdots\right)^2=&+1\hspace{1.2em}+\lambda/2\hspace{0.7em}+\lambda^2/6\hspace{0.2em}\phantom{+\cdots}\\ \rule{0pt}{1.3em}&+\lambda/2\hspace{0.43em}+\lambda^2/4\hspace{0.37em}\color{red}+\lambda^3/12\color{black}\phantom{+\cdots}\\ \rule{0pt}{1.3em}&+\lambda^2/6\color{red}+\lambda^3/12+\lambda^4/36+\cdots&\color{black} = 1+\lambda+7\lambda^2/12+\cdots\end{array}

and use long division

\begin{array}{r}1-\lambda^2/12+\cdots\\ 1+\lambda+7\lambda^2/12+\cdots \,{\overline{\smash{\big)}\,1+\lambda+\phantom{9}\lambda^2/2\phantom{9} +\cdots}}\\ \underline{1+\lambda+7\lambda^2/12+\cdots }\\0-\lambda^2/12+\cdots\\ \underline{-\lambda^2/12+\cdots}\\0+\cdots \end{array}

to show

\mathcal{S} = \frac{1}{\lambda^2} \left(1-\frac{\lambda^2}{12}+\mathcal{O}[\lambda^4]\right) =\frac{1}{\lambda^2}-\frac{1}{12}+\mathcal{O}[\lambda^2].

For finite \lambda, the difference

\mathcal{S}-\mathcal{I}=\sum_{n=0}^\infty n\, e^{-\lambda n}-\int_{0}^\infty \hspace{-0.8em} dx\, x\,e^{-\lambda x} = \frac{1}{\lambda^2}-\frac{1}{12}+\mathcal{O}[\lambda^2]-\frac{1}{\lambda^2}= -\frac{1}{12}+\mathcal{O}[\lambda^2],

and for vanishing \lambda

\color{red}\lim_{\lambda \rightarrow 0}\color{black}\left( \sum_{n=0}^\infty n\, \color{red}e^{-\lambda n}\color{black}-\int_{0}^\infty \hspace{-0.8em} dx\, x\, \color{red}e^{-\lambda x} \color{black}\right) = -\frac{1}{12}.

Thus, removing nonphysical infinity in a controlled way exposes a finite component

\infty - \infty \stackrel{I}{=} -\frac{1}{12}

of the divergent natural numbers sum. To celebrate this hidden “golden nugget”, write

1+2+3+\cdots \stackrel{R}{=} -\frac{1}{12},

where”R” denotes regularized, renormalized, and remainder. “R” also denotes Ramanujan, who discovered this association without any formal mathematical training, and Riemann, whose famous zeta function \zeta[s] provides an alternate path to it (though Euler noticed it earlier).

In brief, use the convergence factor to tame infinity by isolating the diverging, remaining, and vanishing parts of the natural numbers sum

\sum_{n=1}^\infty n=\color{red}\lim_{\lambda \rightarrow 0}\color{black}\sum_{n=0}^\infty n\color{red}\, e^{-\lambda n}\color{black}=\color{red}\underbrace{\ +\frac{1}{\lambda^2}\ \ }_\text{Diverge}\color{black}\underbrace{\ -\frac{1\vphantom{\lambda^2}}{12}\ \ }_\text{Remain}\color{red} \underbrace{\vphantom{-\frac{1\vphantom{\lambda^2}}{12}}+\mathcal{O}[\lambda^2]}_\text{Vanish}\color{black}\ \stackrel{R}{=}-\frac{1}{12}.

Discard the first, retain the second, and let go the third.

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A Better Table

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.)

Click for a larger version of my latest periodic table design

Click for a larger version of my latest periodic table design

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Math Grenade

I just bought a new calculator. New to me, that is, but older than me.

Inspired by the 1600s Gottfried Leibniz stepped cylinder and the 1800s Thomas de Colmar arithmometer, the Curta mechanical calculator design was developed by Curt Herzstark while imprisoned in the Buchenwald concentration camp during World War II. Curta calculators were manufactured between 1947 to 1972 in Liechtenstein until electronic calculators became widely available.

A true number cruncher, the Curta has been nicknamed the pepper grinder because of its crank — or the math grenade due to its resemblance to some hand grenades. My Curta II has about 719 metal parts yet fits comfortably in my hand. I remember seeing advertisements for Curta calculators in Scientific American magazine. Now collector’s items, the one I just bought costs an order of magnitude more.

“The world’s first, last, and only hand-held mechanical calculator”, the Curta is an elegant marvel of mechanical engineering, the culmination of generations of work to mechanize arithmetic. And as I can now attest, it is a joy to hold and use.

After 2 slides and 2 cranks, a Liebniz wheel computes 5 + 3 = 8

A Liebniz wheel at the heart of the Curta computes 5 + 3 = 8 with 2 slides and 2 cranks

Curta ad from my childhood

Curta ad from my childhood

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Optical Tweezers

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.

Microsphere in a light beam: Fig. A push stabilized by Fig B nudge balances Fig. C pull to trap sphere

Microsphere in a light beam: Fig. A push stabilized by Fig B nudge balances Fig. C pull to trap sphere

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.

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Dr. Rendezvous

Edwin Aldrin earned his PhD from MIT in 1963 with a thesis titled, “Line-of-sight guidance techniques for manned orbital rendezvous”. Just three years later in 1966, Aldrin was the pilot of Gemini XII, the last flight of the Gemini program, a critical precursor to the Apollo moon program. Aldrin and his commander James Lovell were attempting to rendezvous and dock with an Agena target vehicle when their onboard radar failed. Aldrin used a handheld sextant to repeatedly measure the angles between Gemini XII and the Agena ahead and above them to help Lovell successfully complete the rendezvous and docking.

Aldrin would join Neil Armstrong and Michael Collins on the historic flight of Apollo 11 in 1969 — and later formally change his first name from “Edwin” to his other nickname “Buzz”.

When Gemini XII's radar failed, the man who wrote the book on rendezvous was onboard to help

When Gemini XII’s radar failed, the man who wrote the book on spacecraft rendezvous was onboard to help

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Saturnday

Ancient cultures everywhere observed seven “wanderers” move against the apparently fixed stars of the night sky: our star the sun, our natural satellite the moon, and the brightest planets Mars, Mercury, Jupiter, Venus, and Saturn. In many languages, these wanderers became the basis for the names of the seven days of the week; for example, in English & French the days are:

Sunday & dimanche
Monday & lundi
Tuesday & mardi
Wednesday & mercredi
Thursday & jeudi
Friday & vendredi
Saturday & samedi

In many languages, the days of the week are named after the classic "planets"

In many languages, the days of the week are named after the classic “planets”

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Anholonomy

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.

Spacetime diagrams of a bead sliding on spinning hoop in the hoop (left) and lab (right) frames. Red spheres are experiment, cyan tubes are simulation, yellow tubes numerically extrapolates bead sliding in the absence of hoop spinning, and black arcs are the generalized Hannay’s shift.

Spacetime diagrams of a bead sliding on spinning hoop in the hoop (left) and lab (right) frames. Red spheres are experiment, cyan tubes are simulation, yellow tubes numerically extrapolates bead sliding in the absence of hoop spinning, and black arcs are the generalized Hannay’s shift.

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