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n-Ball Volumes
The way a ball’s volume varies with dimension is both fascinating and practically important (in statistical physics, data science, and elsewhere). Here I offer an accessible derivation. Recursion Let be the volume of a ball of radius in dimension . Note that , , , and . The volume of a unit -ball and so…
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Orbital Starship
Up early this morning to watch Starship flight 14. Eighteen years to the day since SpaceX‘s Falcon first achieved orbit, today so did Starship, and it deployed 26 large third-generation Starlink satellites. Superheavy Starship is the world’s largest launch vehicle, by far, and is designed to be completely reusable. Much work remains to perfect it,…
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Navier-Stokes Singularity
Yesterday, OpenAI announced the solution to the Millennium-Prize Navier-Stokes problem: an initially smooth fluid can develop a singularity in finite time. More precisely, in a fluid modeled by the Navier-Stokes equations (Newton’s second law for incompressible fluids), from zero initial speeds a smoothly varying localized force can create a vortex that combines inward spiraling with…
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Roman vs. Hubble
Up early this morning for the successful launch and deployment of the Roman Space Telescope, whose development I have followed for 15 years. Considered a successor to the Hubble Space Telescope, Roman’s 2.4-meter primary mirror is the same size as Hubble’s but is shaped differently, and its shorter focal length provides a much larger field-of-view.…
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First Photo From Mars
50 years ago was summer 1976, post Apollo, but pre-Space Shuttle. On July 20th, the seventh anniversary of the Apollo 11 Moon landing, the Viking 1 spacecraft made the first successful landing on Mars. (5 years earlier, the Mars 3 spacecraft ceased communicating just 110 seconds after landing.) My family owned and operated a small…
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Constructing Pentagons
A highlight of Euclid’s Elements is the exact construction of a regular pentagon using a straight edge and a compass. But I’ve noticed recently that some of the pentagon constructions on YouTube are merely approximations – and are not advertised or understood as such. Furthermore, many of these constructions require non-collapsible compasses, which can “carry”…
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Negative Resistance
I recently coauthored an article in the International Journal of Bifurcation and Chaos that reports efficiently performing logic with a simple nonlinear electronic device that exhibits negative resistance, something I had previously theorized. At the heart of the device is a single p-n junction, which is formed by a silicon semiconductor crystal doped with impurity…
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Stokes’ Drag Law
Introductory physics often assumes without proof that the drag force on an object is proportional to its velocity, at least for smooth or laminar flow. In particular, a sphere of radius falling slowly with velocity in air of viscosity experiences a drag force which was first derived by George Stokes in 1851. Here is a…
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1+2+3+… = -1/12 Revisited
Why is minus-one-twelfth associated with the sum of the natural numbers? It’s the constant term in the series expansion of the corresponding smoothed or regularized sum! Introduce a decay factor, and in the limit of vanishing decay, the finite-non-zero part of the resulting sum is minus-one-twelfth, as one can quickly verify in Mathematica using (for…
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Pythagorean Animations
Known for thousands of years, hundreds of proofs of the Pythagorean theorem have been published, including one by U.S. President James Garfield. Here I animate three of my favorites. Each shows that the sum of the squares of the lengths of the legs of a right triangle equals the square of the length of its…
Author: John F. Lindner
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Nice post, John! Thanks for writing these. I always enjoy them.
David Smith’s ein stein is an awesome shape!
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Thanks, Mark! I enjoy reading your posts as well.