Category: Mathematics

  • 3D Ein Stein

    John F. Lindner

    In yet another recent exciting advance in mathematical tiling, last month Ioannis Tsiokos and the reasoning model GPT-6 Astra announced the discovery of a 3D aperiodic mono tile, an ein stein (or “one-stone”), a single shape that forces a non-periodic tiling of space, and reported formal verification by the Lean 4 theorem-prover software. Independently Felix…

  • n-Ball Volumes

    John F. Lindner

    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…

  • Navier-Stokes Singularity

    John F. Lindner

    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…

  • Roman vs. Hubble

    John F. Lindner

    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.…

  • Constructing Pentagons

    John F. Lindner

    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”…

  • Stokes’ Drag Law

    John F. Lindner

    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…

  • 1+2+3+… = -1/12 Revisited

    John F. Lindner

    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…

  • Pythagorean Animations

    John F. Lindner

    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…

  • The Mathematics of Wonder

    John F. Lindner

    Since childhood I have been fascinated by M. C. Escher‘s extraordinary graphics. Escher once wrote, “I never feel quite at home among my artist colleagues; what they are striving for, first and foremost is “beauty” … I guess the thing I mainly strive after is wonder … .” In 1945 Escher produced a lithograph called…

  • e is Transcendental

    John F. Lindner

    Introduction The Euler-Napier-Bernoulli constant is not just irrational, it is transcendental, as first proved by Charles Hermite in 1873. Inspired by the work of Mathologer (Burkard Polster with Marty Ross), here I offer an elementary proof of ‘s transcendence. As warmup, I first present a well-known proof of its irrationality while hinting at the proof…

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