For example,

(1+2+3)^2 = 36 = 1^3 +2^3 + 3^3.More generally,

(1 + 2 + \ldots + n)^2 = 1^3 + 2^3 + \cdots n^3or

\left( \sum n \right)^2 = \sum n^3.The accompanying animation illustrates the identity, where the cubes can be rearranged into either a square or a sequence of composite cubes of the same total volume.

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Assuming the **Taylor** series expansion

where z = x + i y \in \mathbb{C} with x,y \in \mathbb{R} and i = \sqrt{-1}, the derivative condition

f^\prime(z) = f^{(1)}(z) = \sum_{n=1}^\infty \frac{f^{(n)}(0)}{(n-1)!} z^{n-1} = \sum_{n=0}^\infty \frac{f^{(n+1)}(0)}{n!} z^n \equiv f(z) = f^{(0)}(z)coupled with the normalization condition

f^{(n+1)}(0) = f^{(n)}(0) = f^{(n-1)}(0) = \cdots = f^{(1)}(0) =f^{(0)}(0) = f(0) \equiv 1implies

f(z) = \sum_{n=0}^\infty \frac{z^n}{n!} \equiv \exp z.Numerically plot this expression to discover** two jewels**. As in the figure, the exponential function maps the imaginary axis to the unit circle f(i\mathbb{R}) \rightarrow \mathbb{S}, with negative real parts forcing complex numbers inside (red) and positive real parts forcing complex numbers outside (blue). The function is exponential on the real axis with **e-folding time** 1, so f(1)=e, but periodic on the imaginary axis with **period** \tau, so f(z+i\tau)=f(z). Specifically, as it maps 1 to e, it maps horizontal strips of height \tau = 2\pi onto the entire complex plane, where

and

e = 2.71828182845904523536028747135266\ldotsare associated with **Archimedes** and **Euler**.

Last fall, as part of his senior thesis, Yuchen Gan ’21 and I used computer simulations to generalize this famous result to uniform spinning planets, where **Coriolis** and **centrifugal** effects force the tunnels into arcs curving away from the center and intersecting the surface in multiple places. We discovered many families of **periodic** tunnel networks that connect multiple surface locations even at non-equatorial latitudes, as in the animation. Such tunnels could ideally provide energy-free communication and transportation for the planets’ inhabitants.

But in January, in a wonderful **aha!** moment, we were surprised and delighted by a dramatic perspective change: the motion of an object or passenger (a “terranaut”) freely falling through the tunnel system is both spiky concave arcs with respect to the planet *and* a smooth convex **ellipse** with respect to inertial space! We subsequently proved mathematically that the inertial motion is that of a two-dimensional harmonic oscillator, and the ellipses are **centered** (not **focused**) on the planet.

Download a higher-resolution QuickTime MOV version of the animation with or without the red elliptic trace.

]]>*in* the surface and the curvature *of* the surface itself. Geographic tongue promises to be a new exemplar of reaction–diffusion on curved surfaces. Our computer simulations connect reaction–diffusion processes on curved surfaces with moving patterns on the human body.

The car-sized **Perseverance** **rover** hangs just above the surface, suspended by a **bridle** of three 7.5-meter nylon **tethers** from its powered descent stage, seconds before touchdown. The curly electrical **umbilical** that transported the 1s and 0s encoding the photo also dangles from the descent stage to the rover as the rockets blast **regolith** outward. Moments later **pyrotechnics** severed the tethers and umbilical, and the descent stage flew away to crash at a safe distance, leaving the rover six-wheels-on-**Mars**.

The six stars of TCY 7037-89-I orbit each other in three binary pairs, as in the schematic. The primaries are slightly larger and hotter than our sun and the secondaries are about a half as large and a third as hot. All stars eclipse each other as seen from Earth, and a** neural network** helped identify them from the TESS light curves.

In 1941, a Columbia University chemistry graduate student published one of the most famous short stories of science fiction’s **Golden Age**. **Isaac Asimov**‘s “Nightfall” imagines a planet orbiting six suns with a civilization evolved in perpetual daylight. Only once every 2000 years does an undiscovered moon eclipse a sun when it is alone in the sky plunging the civilization into darkness — and revealing tens of thousands of newly visible stars!

Extending previous work with Reba Glaser (SUNY Geneseo)’19 and Nate Smith ’18, we wrote computer simulations to document the distinct recovery of **reaction–diffusion** wavefronts disrupted by a variety of obstacles. Curvature dependent wavefront velocities ultimately restore the wavefronts, with perturbations that decay as power-law functions of time. But concave, spiral, and fractal obstacles can sustain wavefronts locally for long times. Soft obstacles with variable diffusivity, either intrinsically or due to light sensitivity, can enforce one-way propagation and, appropriately configured, can locally and indefinitely sustain incident wavefronts, creating **clocks** or repeaters, beating hearts for these excitable systems.

My eyes were glued to NASA-TV last weekend as I followed the flight of the **SpaceX Dragon** “Resilience” to the **International Space Station**. Ferrying a diverse **Crew 1** of Mike Hopkins, Victor Glover, Shannon Walker, and Soichi Noguchi for a six-month stay on the ISS, this was the first commercial **FAA certified** human spaceflight.

Only later did I learn of Penrose’s important contributions to **General Relativity**, including topological arguments to demonstrate the inevitably of **gravitational collapse** leading to astrophysical **black holes**. And earlier this month I was excited to hear that Penrose — in his 90th year — shares the **2020 Nobel Prize** in physics!

So I got out my Penrose tiles (thanks Woody) and assembled a small pattern. It’s not easy, but a combination of **local** edge and vertex rules (or a **global** inflation rule) can extend the aperiodic pattern to infinity.