Introduction This article is written by way of a reference for my longstanding PF colleague and prolific poster of challenging Maths/geometry problems @chwala . The particular post this article will discuss is an example of one such problem which he raised on Physics Forums. https://www.physicsforums.com/threads/find-angle-adb-in-this-isoceles-triangle-given-some-extra-information.1063874/ The Problem We take the liberty of including the original...

The Relativator (revisited) This is an update of my 2006 post (reconstructed in 2014) Relativator: The circular slide-rule for physicists. This is a circular slide-rule for doing relativistic calculations for elementary particle physics that I learned about from – an article by Elizabeth Wade ( “Artifact: Relativator”, Symmetry (FNAL/SLAC), 01/01/06, https://www.symmetrymagazine.org/article/december-2005january-2006/artifact-relativator  https://www.symmetrymagazine.org/sites/default/files/legacy/pdfs/200512/artifact_relativator.pdf ), which is...

Introduction: “Convenient Notation” In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred  to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta.  The Kronecker delta is simply the indexed components of the identity operator in matrix algebra: [tex]\delta^j_k =\left\{\begin{array}{lcl}1&\text{ if }...

Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated: Before we end, let’s now briefly talk about the birth of quantum information science, a pivotal shift that began in the 1990s. This is the era when researchers...

Abstract This article will build on the hints about treating the complex numbers as a branched surface, briefly described and pictured in section 4.2 of https://www.physicsforums.com/insights/views-on-complex-numbers/#The-Radish. Using a particular set of conventions, all the problems described in https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/  can be removed, and the rules described there as applying only to reals generalized to complex numbers. A...

Abstract Fermat’s Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book...

The Concept A vector space is an additively written abelian group together with a field that operates on it. Intuitive picture vs abstract definition Vector spaces are often described as a set of arrows, i.e. a line segment with a direction that can be added, stretched, or compressed. That’s where the term linear to describe...

Introduction The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement that the order of two consecutive operations does not matter: ##(1+2)+3=1+(2+3).## That’s it. The concept of a group is so simple that I still wonder why...

Introduction Overview That there is no tidal bulge is the key premise of this article. Upper-level oceanography undergraduates and above know this. Yet the tidal bulge is still used to portray why the Moon is receding the Earth. If there is no tidal bulge, some other explanation is in order. That other explanation uses gravitation...

Abstract Topology as a branch of mathematics is a bracket that encompasses many different parts of mathematics. It is sometimes even difficult to see what all these branches have to do with each other or why they are all called topology. This article aims to shed light on this question and briefly summarize the content...

Stochastic calculus has come a long way since Robert Brown described the motion of pollen through a microscope in 1827. It’s now a key player in data science, quant finance, and mathematical biology. This article is drawn from notes I wrote for an undergraduate statistical physics course a few months ago. There won’t be any...

Comment on PBS Space Time video PBS Space Time produces some very good videos on the foundations of quantum mechanics (QM), so let me comment on their video What If Physics IS NOT Describing Reality to provide (crucial) missing information. This comment pertains only to the first 9 min of the video, i.e., it has...

Abstract It took until the last century for physicists and mathematicians in the Netherlands to question the Euclidean concept of dimension as length, width, and height. Luitzen Egbertus Jan Brouwer published a ground-breaking paper On the Natural Concept of Dimension (Amsterdam, [2]) in 1913 about the mathematical definition of dimension picking up a thought from...

Abstract Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted to gather the many different views that can be found elsewhere – Euler’s and Gauß’s perspectives, i.e. various historical views in the light of the traditionally parallel...

The “Addition of Velocities” formula (more correctly, the “Composition of Velocities” formula) in Special Relativity [tex]\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c} }{1 + \frac{v_{AB}}{c} \frac{v_{BC}}{c}}[/tex] is a non-intuitive result that arises from a “hyperbolic-tangent of a sum”-identity in Minkowski spacetime geometry, with its use of hyperbolic trigonometry. However, I claim it is difficult to obtain this by looking at...

The concept of quantum superposition (or superposition for short) is very counterintuitive, as Schr##\ddot{\text{o}}##dinger noted in 1935 writing [1], “One can even set up quite ridiculous cases.” To make his point, he assumed a cat was closed out of sight in a box with a radioactive material that would decay with 50% probability within an...

The slinky drop is a rather simple experiment. In its most basic form, it requires only a popular toy for children, a stable hand, and a keen eye. For a better view, using a modern smartphone to capture a video of the experiment also helps to capture the falling slinky. Apart from the commonly quoted...

Direct answer: Cosmologists classify the universe as open, flat, or closed based on spatial curvature. Current cosmic microwave background (CMB) measurements indicate the universe is very close to flat, but small curvature remains observationally allowed. Whether the universe is ultimately finite or infinite cannot be settled by theory alone, since all three curvature types are...

Introduction The figure on the right shows a “double-double” Atwood machine with three ideal pulleys and four masses.  All pulleys are released from rest simultaneously.  Which of the choices below describes the angular motion of the top pulley P after some time has elapsed and why? It rotates clockwise with increasing angular speed. It rotates...

Introduction This elaborates some of the claims in my insights article on digital audio. The Sinc Function Upsampling and Zero Insertion The first link in my insights article has a section on filters, but I will detail the most critical case. The bit depth is assumed to be so large for all practical purposes that...