Hopalong Orbits Visualizer: Stunning WebGL Experiment

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. https://www.youtube.com/watch?v=YjuexKGHLTM Hop to the source website using your desktop browser (with WebGl and Javascript support) and enjoy the magic. PS: Hopalong Attractor Algorithm Hopalong Attractor predicts the locus of points in 2D using this algorithm (x, y) -> (y - sign(x)*sqrt(abs(b*x - c)), a -x ) That is, $ x= y- \mathrm{sign}{(x)} \cdot \sqrt{…

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Photo Archives from ‘Only Gaurav’

'Only Gaurav' was a photo-blog, I operated parallel to MY DIGITAL NOTEBOOK. Both sites worked well together until February 2012. After February 2012, I got busy in college stuffs and my blogging life took a huge leap of one year. MY DIGITAL NOTEBOOK was able to survive this gap and now it is continuously getting updated. On the other hand, 'Only Gaurav' wasn't that lucky. It was neglected due to several reasons. Starting from January 2013, I'm regularly using  Flickr…

Real Sequences

Sequence of real numbers A sequence of real numbers (or a real sequence) is defined as a function $ f: \mathbb{N} \to \mathbb{R}$ , where $ \mathbb{N}$ is the set of natural numbers and $ \mathbb{R}$ is the set of real numbers. Thus, $ f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a function which produces a sequence of real numbers $ r_n$ . It's customary to write a sequence as form of functions in brackets, e.g.; $…

Smart Fallacies: i=1, 1= 2 and 1= 3

This mathematical fallacy is due to a simple assumption, that $ -1=\dfrac{-1}{1}=\dfrac{1}{-1}$ . Proceeding with $ \dfrac{-1}{1}=\dfrac{1}{-1}$ and taking square-roots of both sides, we get: $ \dfrac{\sqrt{-1}}{\sqrt{1}}=\dfrac{\sqrt{1}}{\sqrt{-1}}$ Now, as the Euler's constant $ i= \sqrt{-1}$ and $ \sqrt{1}=1$ , we can have $ \dfrac{i}{1}=\dfrac{1}{i} \ldots \{1 \}$ $ \Rightarrow i^2=1 \ldots \{2 \}$ . This is complete contradiction to the fact that $ i^2=-1$ . Again, as $ \dfrac{i}{1}=\dfrac{1}{i}$ or, $ i^2=1$ or, $ i^2+2=1+2$ or, $ -1+2=3$ $ 1=3…

Set Theory, Functions and Real Number System

SETS

In mathematics, Set is a well defined collection of distinct objects. The theory of Set as a mathematical discipline rose up with George Cantor, German mathematician, when he was working on some problems in Trigonometric series and series of real numbers, after he recognized the importance of some distinct collections and intervals.

Cantor defined the set as a ‘plurality conceived as a unity’ (many in one; in other words, mentally putting together a number of things and assigning them into one box).

Mathematically, a Set $ S$ is ‘any collection’ of definite, distinguishable objects of our universe, conceived as a whole. The objects (or things) are called the elements or members of the set $ S$ . Some sets which are often pronounced in real life are, words like ”bunch”, ”herd”, ”flock” etc. The set is a different entity from any of its members.

For example, a flock of birds (set) is not just only a single bird (member of the set). ‘Flock’ is just a mathematical concept with no material existence but ‘Bird’ or ‘birds’ are real.

Representing sets

Sets are represented in two main ways:

Four way valid expression

People really like to twist the numbers and digits bringing fun into life. For example, someone asks, "how much is two and two?" : the answer should be four according to basic (decimal based) arithmetic. But the same  with base three (in ternary number system) equals to 11. Two and Two also equals to Twenty Two. Similarly there are many ways you can add them and get different results. Dmitri A. Borgmann, the German recreationalist, puzzler and father of logology, noticed the following expression…