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I just discovered Barry Martin’s Hopalong Orbits Visualizer — an excellent abstract visualization, which is rendered in 3D using Hopalong Attractor algorithm, WebGL and Mrdoob’s three.js project. Here is a demo video: Hop to the source website using your desktop browser (with WebGL and Javascript support) and enjoy the magic. Hopalong Attractor Algorithm Hopalong Attractor…
In this study note, I have discussed and explained about real sequences — a very important topic in Real Analysis and Calculus. I will cover the topic, definitions and examples under the following headings: Real Sequence: A sequence of real numbers A sequence of real numbers (or a real sequence) is defined as a function…
Drop a ball through a tunnel drilled straight through Earth. It won’t fall forever—it’ll oscillate back and forth, popping out on the other side in exactly 42 minutes. Every. Single. Time. Here’s the physics behind one of the most elegant thought experiments in classical mechanics. The Setup: A Tunnel Straight Through Earth Imagine drilling a…
The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were correct to limited extent but still stun the mathematicians. The prime generating equation by Euler is a very specific binomial equation on prime numbers and yields more primes than any…
Ensembles in Chemical Sciences and Statistical Physics are a collection of a number of macroscopically identical but essentially independent systems. The System, here, is defined by the collection of a large number of particles and the term macroscopically identical means that each of the systems constituting an ensemble satisfies the same macroscopic conditions, like Volume,…
This paper presents a statistical analysis of 20 years of meteorological data (1989-2008) from the Pantnagar weather station in Uttarakhand, India. The research was conducted under the INSPIRE-SHE Scholarship Program, Department of Science and Technology, Government of India. The goal: extract meaningful patterns from atmospheric data using regression analysis, correlation studies, and time-series methods. Abstract…
This introductory guide to Object Oriented Ontology is an on-going collection of central theses and surrounding debates. If you’d like to submit a text, blog, or discussion thread that you think is particularly useful to the would-be OOO scholar you can post a link in the comments or email us at gaurav@gauravtiwari.org or use this…
How do you prove a number is irrational? You can’t just look at a number and declare it irrational. You can’t check infinitely many decimal places. You need a proof. And proofs of irrationality are some of the most elegant arguments in all of mathematics. The basic strategy is always the same: proof by contradiction.…
The area of a disk with radius \( R \) is \( \pi R^2 \). You probably memorized this in middle school. But do you know why it’s true? Most people don’t. They just accept it. And that’s a shame, because the derivation is beautiful. It shows you how calculus and geometry connect. It shows…
The triangle inequality is one of those results that sounds obvious until you try to prove it rigorously. Here’s the geometric intuition: no side of a triangle can be longer than the other two sides combined. If it were, you couldn’t close the triangle. Simple enough. But this same principle extends to real numbers, complex…
In 1911, a 23-year-old clerk from Madras posed a problem in the Journal of the Indian Mathematical Society that stumped professional mathematicians for months. The answer? A simple “3.” But getting there required a formula so elegant it’s still studied today. Welcome to the world of Ramanujan’s nested radicals. What Are Nested Radicals? A nested…
I was very pleased on reading this news that Government of India has decided to celebrate the upcoming year 2012 as the National Mathematical Year. This is 125th birth anniversary of math-wizard Srinivasa Ramanujan (1887-1920). He is one of the greatest mathematicians India ever produced. Well this is ‘not’ the main reason for appointing 2012…
