Quantumaniac is where it’s at - and by ‘it’ I mean awesome.
Over here I post a ton of physics / math / general interesting science related posts. I try to be as informative as possible, all while posting fascinating things that hopefully enlighten us both a little to the mysteries of our truly wondrous universe(s?). Plus, how would you know if the blog exists or not unless you observe it? Boom, just pulled the Schrödinger’s cat card. Now you have to check it out - trust me, it said so in an equation somewhere.
Catching Elephant is a theme by Andy Taylor
Symmetree
Why Does 0.999… = 1?
Consider the real number that is represented by a zero and a decimal point, followed by a never-ending string of nines:
0.99999…
It may come as a surprise when you first learn the fact that this real number is actually EQUAL to the integer 1. A common argument that is often given to show this is as follows. If S = 0.999…, then 10*S = 9.999… so by subtracting the first equation from the second, we get
9*S = 9.000…
and therefore S=1. Here’s another argument. The number 0.1111… = 1/9, so if we multiply both sides by 9, we obtain 0.9999…=1.
Klein Quartic
Happy Face Math - Charlie Smith
Zeno’s Dichotomy Paradox
Zeno (ca. 490 B.C. - ca. 430 B.C.) created a number of famous paradoxes, some which baffle students (and cause great amusement for teachers) to this day. One of this most famous, the dichotomy paradox - seems to destroy the basic concept of motion and destination.
Zeno’s dichotomy paradox simply states that in order for an object to move to a certain destination, it must reach certain ‘goals’ along the way. Let’s use the example of a train. In order for a train to get from New York to Washington D.C., it must travel half of the distance. Once that is covered, it will reach halfway to Washington D.C. again (now 1/4th of the full distance away), and will then reach another halfway - the train will inevitably reach an infinite amount of halfway points, without ever reaching Washington D.C. Another way to think of it is in order for the train to even reach the halfway point, it must travel 1/4th of the full distance, then 1/8th, then 1/16th - and it will never actually arrive.
Although intuitively this cannot be true, mathematically it is not easily disproven at first. However, one can set up a sum to simplify this nicely. Let’s assume that each ‘halfway point,’ is its own unique interval, and there are an infinite amount of them. So, we can set up a sum by setting 1/2 to the nth power:
We find that the sum is equal to 1 - or the full distance. For our less mathematically inclined readers, think of it this way: bring your finger about halfway from your face to the computer screen. Now take that distance and move it halfway to the screen again. If you continued this process, although you would technically be moving halfway each time, and there would always be another halfway point to get to - the distance becomes trivial. In this same way, the sum will eventually reach a point where it is equal to 0.99999999999999999999 etc., and 1 will suffice. Nice try Zeno - better luck next time.
The Poincaré Conjecture
Imaginestretching a rubber band around the surface of an apple, then shrinking it down slowly. This shrinking could occur without tearing the rubber band or breaking the apple - and the band would never have to leave the surface. However, if this rubber band were to be stretched across, say, a tire - there is no way to shrink to a point without breaking one or the other. The surface of such an apple is “simply connected,” but the tire is not. Henri Poincaré (shown below), during the early twentieth century - knew that two dimensional spheres had this ‘connected’ property - and he asked if the same applied for three dimensional spheres.
The conjecture turned out to be immensely difficult to prove. After more than a century, Grigori Perelman finally devised a solution. In 2006, Perelman was awarded the Fields Medal for this contribution, but he decided to turn it down, stating that:
“I’m not interested in money or fame, I don’t want to be on display like an animal in a zoo.”
Newton v. Leibniz - The Calculus Controversy
In Latin, the word ‘calculus’ means ‘pebble,’ meaning that small stones were used to calculate things. Calculus is essentially the study of change, and the pebbles represent small, gradual changes that can produce impressive results. The origin of calculus is not the work of a single man, not even the work of the two men pictured above - but like most major discoveries, a gradual build of overlapping discoveries, something very similar to calculus itself. The question over the creation of the branch of mathematics has become one of the fiercest rivalries in modern history - that between Isaac Newton and Gottfried Leibniz.
In 1666 (and perhaps earlier), when Newton was 23 - he had begun work on what he called “the method of fluxions and fluents,” effectively what we know as calculus. Newton’s discovery of calculus was mainly a result of practical use - he needed a method to solve problems in physics and geometry, and calculus was what resulted. On the other hand, Leibniz had become fascinated by the tangent line problem and began to study calculus around 1675.
The ideas of the two men were similar, although it is unlikely that either of them knew the specifics of the other’s work. The two men spoke in letters often, and discussed mathematics - and although the Royal Society found Leibniz effectively guilty of plagiarism later, this was not likely the case. Both men came to similar discoveries in different ways - Leibniz came to integration first, while Newton began his work with derivatives.
Although Newton discovered the principles of calculus first - he did not publish them until many years after Leibniz did. Leibniz published his first paper employing calculus in 1684, but Newton did not publish his fluxion notation form of calculus until 1693, and a complete version was not available until 1704! Nonetheless, Newton still came to the discovery first - and although both men are officially credited, Newton is the one that most people remember.
However, Newton doesn’t deserve all the credit here. The famous dy/dx notation that calculus students have come to love and hate was developed by Leibniz. Although Newton may have come to the discovery first, Leibniz attacked the problems with far better notation - and we have naturally adopted it. Instead of Leibniz’s dx/dt (shown below) notation for derivatives, Newton preferred ‘dot’ notation:
However, this dot notation can become confusing, especially when used for higher order derivatives, so it has been generally dismissed - except for hardcore Newton fanatics who insist on using his notation. Newton did not even have a standard notation for integration, but frequently switched; but Leibniz used the recognizable integration symbol:
This has developed into a fantastic controversy over the years - and has become as much of a moral question as it is scientific. Many Leibniz advocates belief that Newton doesn’t deserve full credit because he didn’t publish his findings first - while many others believe that Newton came to the discovery first, so the credit is his. Personally, I have to place myself on the side of Newton - although Leibniz’s notation is wonderful, Newton discovered the principles first.
Which side are you on?
Sierpinski Valentine
The Simplest Perfect Squared Square
Twenty one different squares - all in one square. Covering a 112 in. x 112 in. sized area, these varying square sizes comprise the smallest perfect square. It has been proven that a square cannot be divided and tiled with fewer than twenty one distinct square inside.
You can thank Bob Mackay for this ingenious piece.
Using Calculus to find the Identity of Batman
