Frenet–Serret Formula ✍️

It explains how a curve reveals its hidden geometry by tracking the way it bends and twists through space. Imagine tracing the path of a roller coaster, a winding river, or the spiral of a DNA strand. At every point along the path, the curve is constantly changing direction, and the Frenet–Serret formulas provide a precise way to describe that change.

They do this by attaching a moving frame of three special directions to each point on the curve. The first points forward along the path, showing where the curve is heading. The second points inward, toward the direction of bending. The third stands perpendicular to both, capturing how the curve twists out of its plane. Together, they form a local coordinate system that travels with the curve itself.

As you move along the curve, these three directions rotate and evolve. The formulas measure this evolution using two key quantities: curvature and torsion. Curvature tells how sharply the path bends, while torsion tells how strongly it twists into three dimensions. If curvature vanishes, the path becomes straight; if torsion vanishes, the curve lies flat in a plane.

Mathematicians and physicists use the Frenet–Serret formulas to study motion, design smooth paths in engineering, understand particle trajectories, and analyze natural shapes. They transform a simple line into a rich geometric story, revealing exactly how space is being navigated at every step.

#FrenetSerretFormula #DifferentialGeometry #Geometry #Mathematics #Math #PureMathematics #AppliedMathematics #MathematicalPhysics #Physics #STEM #ScienceEducation #MathEducation #Curvature #Torsion #SpaceCurves #VectorCalculus #Calculus #LinearAlgebra #GeometricAnalysis

RE: https://mathstodon.xyz/@johncarlosbaez/116662332807347504

In memory of Ivan Todorov (1933–2025).
His publications:
<https://scholar.google.com/citations?user=wZiNNXoAAAAJ>
(INRNE is Institute for Nuclear Research and Nuclear Energy.)

After expanding to show all 433 articles, "octonion[ic]" appears in five titles.
The most cited of the latter is a 2018 paper with Svetla Drenska,
"Octonions, exceptional Jordan algebra and the role of the group F₄ in particle physics".

#Algebra
#MathematicalPhysics
#Octonions
#ParticlePhysics
#TheoreticalPhysics

Today my paper "Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds" got published in the Journal of Geometry and Physics!

The paper defines: A globally hyperbolic spacetime $(X,g)$ is observer-refocusing if there exists a point $p$ and a timelike curve $\gamma$ in $X$ so that all lightrays emitted from $p$ intersect $\gamma$. The paper proves that spacetimes $(X,g)$ with $\dim(X)\geq 3$ which are observer-refocusing with respect to a compact timelike curve have compact Cauchy surfaces with finite fundamental group. This extends known results on strongly refocusing spacetimes, which are spacetimes with points $p,q$ so that all lightrays through $p$ go through $q$. Further, observer-refocusing spacetimes of dimension at least $3$ with an analytic metric are strongly refocusing.

These results lead to immediate corollaries in Riemannian geometry: Let $(M,h)$ be a connected, complete Riemannian manifold and let $x \in M$. We call $(M,h)$ a $Z^x$ manifold if all geodesics starting at $x$ return to $x$. We show that if $\dim(M) \geq 2$ and if the return time of unit-speed geodesics starting at $x$ is uniformly bounded, then $M$ is compact with finite fundamental group. Further, if the metric of a $Z^x$ manifold is analytic, then all unit-speed geodesics starting at $x$ return to $x$ at a common time. This resolves the question "Are all $Z^x$ manifolds $Y^x_l$ manifolds for some $l>0$?" posed in Besse's book "Manifolds all of whose geodesics are closed" affirmatively for analytic manifolds.

https://doi.org/10.1016/j.geomphys.2026.105834
https://arxiv.org/abs/2503.23565

#DifferentialGeometry #MathematicalPhysics #GeneralRelativity

“Looking back, my path has not been linear, and I changed direction more than once; however, what has stayed constant is curiosity, even when the topics and the places were changing. This is also what I like most about mathematics: there is room for many different trajectories, as long as you keep following questions that genuinely interest you.” - Mikaela Iacobelli

➡️ https://hermathsstory.eu/mikaela-iacobelli

#Academia #PhD #AssociateProfessor #MathematicalPhysics #PDEs #KineticTheory #MathsJourney #WomenInSTEM #WomenInMaths #HerMathsStory

Urs Schreiber:
> It took the editors at "Quantum Topology" 111 weeks to conclude they are "not interested" in our arxiv.org/abs/2309.07245 .

http://ncatlab.org/schreiber/show/Model+Structure+on+K-Linear+Infinity-Local+Systems

The Twitter post:
https://x.com/UrsSchreiber/status/2011394201674711188

#Math #MathematicalPhysics #HoTT #CategoryTheory

The Hidden Math of Ocean Waves

https://fed.brid.gy/r/https://www.wired.com/story/the-hidden-math-of-ocean-waves-crashes-into-view/

Postdoc Fellow in Bhamla Lab
Bhamla Lab at Georgia Institute of Technology

See the full job description on jobRxiv: https://jobrxiv.org/job/bhamla-lab-at-georgia-institute-of-technology-27778-postdoc-fellow-in-bhamla-lab/

#biologicalphysics #biomechanics #frugalscience #mathematicalphysics #Raman #ScienceJobs #hiring #research
https://jobrxiv.org/job/bhamla-lab-at-georgia-institute-of-technology-27778-postdoc-fellow-in-bhamla-lab/?fsp_sid=5072

The ‘10 Martini’ Proof Connects Quantum Mechanics With Infinitely Intricate Mathematical Structures

https://fed.brid.gy/r/https://www.wired.com/story/the-10-martini-proof-connects-quantum-mechanics-with-infinitely-intricate-mathematical-structures/