The 17 Equations That Changed The Course Of History

Mathematics is all around us, and it has shaped our understanding of the world in countless ways.

In 2013, mathematician and science author Ian Stewart published a book on 17 Equations That Changed The World. We recently came across this convenient table on Dr. Paul Coxon’s twitter account by mathematics tutor and blogger Larry Phillips that summarizes the equations. (Our explanation of each is below):

 

Here is a little bit more about these wonderful equations that have shaped mathematics and human history:

1) The Pythagorean Theorem: This theorem is foundational to our understanding of geometry. It describes the relationship between the sides of a right triangle on a flat plane: square the lengths of the short sides, a and b, add those together, and you get the square of the length of the long side, c.

This relationship, in some ways, actually distinguishes our normal, flat, Euclidean geometry from curved, non-Euclidean geometry. For example, a right triangle drawn on the surface of a sphere need not follow the Pythagorean theorem.

2) Logarithms: Logarithms are the inverses, or opposites, of exponential functions. A logarithm for a particular base tells you what power you need to raise that base to to get a number. For example, the base 10 logarithm of 1 is log(1) = 0, since 1 = 100; log(10) = 1, since 10 = 101; and log(100) = 2, since 100 = 102.

The equation in the graphic, log(ab) = log(a) + log(b), shows one of the most useful applications of logarithms: they turn multiplication into addition.

Until the development of the digital computer, this was the most common way to quickly multiply together large numbers, greatly speeding up calculations in physics, astronomy, and engineering.

3) Calculus: The formula given here is the definition of the derivative in calculus. The derivative measures the rate at which a quantity is changing. For example, we can think of velocity, or speed, as being the derivative of position — if you are walking at 3 miles per hour, then every hour, you have changed your position by 3 miles.

Naturally, much of science is interested in understanding how things change, and the derivative and the integral — the other foundation of calculus — sit at the heart of how mathematicians and scientists understand change.

 
Isaac Newton

4) Law of Gravity: Newton’s law of gravitation describes the force of gravity between two objects, F, in terms of a universal constant, G, the masses of the two objects, m1 and m2, and the distance between the objects, r. Newton’s law is a remarkable piece of scientific history — it explains, almost perfectly, why the planets move in the way they do. Also remarkable is its universal nature — this is not just how gravity works on Earth, or in our solar system, but anywhere in the universe.

Newton’s gravity held up very well for two hundred years, and it was not until Einstein’s theory of general relativity that it would be replaced.

5) The square root of -1: Mathematicians have always been expanding the idea of what numbers actually are, going from natural numbers, to negative numbers, to fractions, to the real numbers. The square root of -1, usually written i, completes this process, giving rise to the complex numbers.

Mathematically, the complex numbers are supremely elegant. Algebra works perfectly the way we want it to — any equation has a complex number solution, a situation that is not true for the real numbers : x2 + 4 = 0 has no real number solution, but it does have a complex solution: the square root of -2. Calculus can be extended to the complex numbers, and by doing so, we find some amazing symmetries and properties of these numbers. Those properties make the complex numbers essential in electronics and signal processing.

 A cube.

6) Euler’s Polyhedra Formula: Polyhedra are the three-dimensional versions of polygons, like the cube to the right. The corners of a polyhedron are called its vertices, the lines connecting the vertices are its edges, and the polygons covering it are its faces.

A cube has 8 vertices, 12 edges, and 6 faces. If I add the vertices and faces together, and subtract the edges, I get 8 + 6 – 12 = 2.

Euler’s formula states that, as long as your polyhedron is somewhat well behaved, if you add the vertices and faces together, and subtract the edges, you will always get 2. This will be true whether your polyhedron has 4, 8, 12, 20, or any number of faces.

Euler’s observation was one of the first examples of what is now called a topological invariant — some number or property shared by a class of shapes that are similar to each other. The entire class of “well-behaved” polyhedra will have V + F – E = 2. This observation, along with with Euler’s solution to the Bridges of Konigsburg problem, paved the way to the development of topology, a branch of maths essential to modern physics.

 

The normal distribution.

7) Normal distribution: The normal probability distribution, which has the familiar bell curve graph to the left, is ubiquitous in statistics.

The normal curve is used in physics, biology, and the social sciences to model various properties. One of the reasons the normal curve shows up so often is that it describes the behaviour of large groups of independent processes.

8) Wave Equation: This is a differential equation, or an equation that describes how a property is changing through time in terms of that property’s derivative, as above. The wave equation describes the behaviour of waves — a vibrating guitar string, ripples in a pond after a stone is thrown, or light coming out of an incandescent bulb. The wave equation was an early differential equation, and the techniques developed to solve the equation opened the door to understanding other differential equations as well.

9) Fourier Transform: The Fourier transform is essential to understanding more complex wave structures, like human speech. Given a complicated, messy wave function like a recording of a person talking, the Fourier transform allows us to break the messy function into a combination of a number of simple waves, greatly simplifying analysis.

The Fourier transform is at the heart of modern signal processing and analysis, and data compression.

10) Navier-Stokes Equations: Like the wave equation, this is a differential equation. The Navier-Stokes equations describes the behaviour of flowing fluids — water moving through a pipe, air flow over an aeroplane wing, or smoke rising from a cigarette. While we have approximate solutions of the Navier-Stokes equations that allow computers to simulate fluid motion fairly well, it is still an open question (with a million dollar prize) whether it is possible to construct mathematically exact solutions to the equations.

11) Maxwell’s Equations: This set of four differential equations describes the behaviour of and relationship between electricity (E) and magnetism (H).

Maxwell’s equations are to classical electromagnetism as Newton’s laws of motion and law of universal gravitation are to classical mechanics — they are the foundation of our explanation of how electromagnetism works on a day to day scale. As we will see, however, modern physics relies on a quantum mechanical explanation of electromagnetism, and it is now clear that these elegant equations are just an approximation that works well on human scales.

12) Second Law of Thermodynamics: This states that, in a closed system, entropy (S) is always steady or increasing. Thermodynamic entropy is, roughly speaking, a measure of how disordered a system is. A system that starts out in an ordered, uneven state — say, a hot region next to a cold region — will always tend to even out, with heat flowing from the hot area to the cold area until evenly distributed.

The second law of thermodynamics is one of the few cases in physics where time matters in this way. Most physical processes are reversible — we can run the equations backwards without messing things up. The second law, however, only runs in this direction. If we put an ice cube in a cup of hot coffee, we always see the ice cube melt, and never see the coffee freeze.

 

Albert Einstein

13) Relativity: Einstein radically altered the course of physics with his theories of special and general relativity. The classic equation E = mc2 states that matter and energy are equivalent to each other. Special relativity brought in ideas like the speed of light being a universal speed limit and the passage of time being different for people moving at different speeds.

General relativity describes gravity as a curving and folding of space and time themselves, and was the first major change to our understanding of gravity since Newton’s law. General relativity is essential to our understanding of the origins, structure, and ultimate fate of the universe.

14) Schrodinger’s Equation: This is the main equation in quantum mechanics. As general relativity explains our universe at its largest scales, this equation governs the behaviour of atoms and subatomic particles.

Modern quantum mechanics and general relativity are the two most successful scientific theories in history — all of the experimental observations we have made to date are entirely consistent with their predictions. Quantum mechanics is also necessary for most modern technology — nuclear power, semiconductor-based computers, and lasers are all built around quantum phenomena.

15) Information Theory: The equation given here is for Shannon information entropy. As with the thermodynamic entropy given above, this is a measure of disorder. In this case, it measures the information content of a message — a book, a JPEG picture sent on the internet, or anything that can be represented symbolically. The Shannon entropy of a message represents a lower bound on how much that message can be compressed without losing some of its content.

Shannon’s entropy measure launched the mathematical study of information, and his results are central to how we communicate over networks today.

16) Chaos Theory: This equation is May’s logistic map. It describes a process evolving through time — xt+1, the level of some quantity x in the next time period — is given by the formula on the right, and it depends on xt, the level of x right now. k is a chosen constant. For certain values of k, the map shows chaotic behaviour: if we start at some particular initial value of x, the process will evolve one way, but if we start at another initial value, even one very very close to the first value, the process will evolve a completely different way.

We see chaotic behaviour — behaviour sensitive to initial conditions — like this in many areas. Weather is a classic example — a small change in atmospheric conditions on one day can lead to completely different weather systems a few days later, most commonly captured in the idea of a butterfly flapping its wings on one continent causing a hurricane on another continent.

17) Black-Scholes Equation: Another differential equation, Black-Scholes describes how finance experts and traders find prices for derivatives. Derivatives — financial products based on some underlying asset, like a stock — are a major part of the modern financial system.

The Black-Scholes equation allows financial professionals to calculate the value of these financial products, based on the properties of the derivative and the underlying asset.
 Here are some traders in the S&P 500 options pit at the Chicago Board Options Exchange. You won’t find a single person here that hasn’t heard about the Black-Scholes equation.

ORIGINAL: Business Insider
Andy Kiersz
Mar 13 2014

Is Free Energy Slowly Being Unveiled?

ORIGINAL: Activist Post
August 19, 2013

Ere many generations pass, our machinery will be driven by a power obtainable at any point of the universe. Throughout space there is energy. -- Nikola Tesla, 1892

DARPA's RINGS - A renewable electromagnetic power generator being tested at the International Space Station

When Tesla invented the first wireless communication device, he also discovered a form of free energy radiating throughout the whole universe. He planned on displaying wireless electric power with his Wardenclyffe Tower until it was sabotaged by financier J.P. Morgan.

Tesla conceded that his World Power System project was "retarded by laws of nature. The world was not prepared for it. It was too far ahead of time. But the same laws will prevail in the end and make it a triumphal success."

Perhaps the the world is now prepared for this technology as small applications using Tesla's discoveries are finally being revealed to the public.

In a 1900 magazine article, The Problem of Increasing Human Energy, Tesla discusses a machine that can gather heat from the ambient air and other forms of harvesting energy from the natural world.

Just a short 112 years later, in March of last year, scientists in Hong Kong built a graphene battery that turns ambient heat into electric current. This technology was picked up by UCLA researchers who claimed this same discovery as their own, seen in the video below:

More interesting Tesla technology came out earlier this year when a German university student invented a device that harvests electromagnetic waves to charge a battery.

Two more significant achievements have been announced in this month. One was seemingly being rolled out to acclimate the public to this coming technology, while the other quieter story is one where similar technology is already being used to power space propulsion.

Researchers from Washington University unveiled a wireless communication device called Ambient Backscatter that requires no battery.

They describe their device as:

Ambient Backscatter transforms existing wireless signals into both a source of power and a communication medium. It enables two battery-free devices to communicate by backscattering existing wireless signals. Backscatter communication is orders of magnitude more power-efficient than traditional radio communication. Further, since it leverages the ambient RF signals that are already around us, it does not require a dedicated power infrastructure. They included an informative video explaining the technology and its possible uses:

Now try imagine this type of self-harnessing power technology used on a larger scale. It's currently being tested on the International Space Station.

On August 12th, the University of Maryland announced their success in powering the propulsion of satellites and the space station with a renewable electromagnetic power source. The project, sponsored by DARPA and NASA, is called RINGS (Resonant Inductive Near-field Generation System).

Besides testing electromagnetic propulsion, they also intend to use RINGS to demonstrate wireless power transfer (WPT). "WPT may offer a means to wirelessly transfer power between spacecraft and in turn power a fleet of smaller vessels or satellites."

According to their press release:

Add caption

New electromagnetic propulsion technology being tested by the University of Maryland's Space Power and Propulsion Laboratory (SPPL) on the International Space Station could revolutionize the capabilities of satellites and future spacecraft by reducing reliance on propellants and extending the lifecycle of satellites through the use of a renewable power source.

Because a finite propellant payload is often the limiting factor on the number of times a satellite can be moved or repositioned in space, a new propulsion method that uses a renewable, onboard electromagnetic power source and does not rely on propellants could exponentially extend a satellite's useful life span and provide greater scientific return on investment.

Associate Professor of Aerospace Engineering Ray Sedwick and his research team have been developing technology that could enable electromagnetic formation flight (EMFF), which uses locally generated electromagnetic forces to position satellites or spacecraft without relying on propellants. Their research project is titled Resonant Inductive Near-field Generation System, or RINGS.

RINGS was sent to the International Space Station on August 3 as part of a payload launched on Japan’s HTV-4 Cargo Ship from the Tanegashima Space Center. The project is scheduled for four test sessions on the research station. Astronauts will unpack the equipment, integrate it into the test environment and run diagnostics. From there, RINGS will undergo three science research sessions where data will be collected and transmitted back to the ground for analysis.

RINGS is composed of two units, each of which contains a specially fabricated coil of aluminum wire that supports an oscillating current of up to 18 amps and is housed within a protective polycarbonate shell. Microcontrollers ensure that the currents oscillate either in-phase or out-of-phase to produce attracting, repelling and even shearing forces. While aluminum wire was chosen for its low density in this research prototype, eventual systems would employ superconducting wires to significantly increase range and performance.

In the spring of 2013, RINGS was tested for the first time in a microgravity environment on NASA's reduced gravity aircraft. UMD graduate students Allison Porter and Dustin Alinger were on hand to oversee the testing. RINGS achieved the first and only successful demonstration of EMFF in full six degrees of freedom to date.

"While reduced gravity flights can only provide short, 15-20 second tests at a time, the cumulative test time over the four-day campaign provided extremely valuable data that will allow us to really get the most from the test sessions that we’ll have on the International Space Station," said Sedwick.

In addition to EMFF, the RINGS project is also being used to test a second technology demonstrating wireless power transfer (WPT). WPT may offer a means to wirelessly transfer power between spacecraft and in turn power a fleet of smaller vessels or satellites. Having the power to support multiple satellites, and using EMFF as a propellant-less means to reposition those same satellites, provides the flexibility to perform formation control maneuvers such as on-orbit assembly or creating synthetic aperture arrays. A synthetic aperture array uses a network of smaller antennas to function collectively as one large antenna. Larger antennas are capable of producing higher resolution images and better quality data. And these are just some of the recent examples of the unveiling of this technology which seemed to have been largely under wraps until now.

Is the age of free energy upon us? Will some of this technology escape and go open source?

Things we can't see

ORIGINAL: Truth Theory

11 November 2012

When you think about it, there is a great deal out there that we can’t see.

Our eyes only respond to a very narrow range of electromagnetic radiation. The following diagram demonstrates just how narrow our range of vision compared to the overall electromagnetic spectrum.

Image Source: http://9-4fordham.wikispaces.com/Electro+Magnetic+Spectrum+and+light

So we can’t see anything that generates or reflects wavelengths equal to or longer than infrared, as the following image demonstrates. Even the Hubble Space Telescope can’t see the distant infrared galaxy that the Spitzer Space Telescope can see with its infrared sensors.

Image Source

And we can’t see anything that generates or reflects wavelengths equal to or shorter than ultraviolet, as the following image from NASA demonstrates. Only instruments with special sensors that can detect ultraviolet or x-rays can see some of the objects in the sky.

Image Source

Of course, we can’t see things that are smaller in size than about 40 microns, which includes germs and molecules.

We can’t see things that are camouflaged by technology, such as the Mercedes in the following picture.

Image Source

Sometimes, it isn’t our eyes that can’t sense something that is right in front of us, but rather, our brain. We actually stare at our noses all day long but don’t notice because our brains effectively subtract it out from our perception, given that we don’t really need it. Our brains also fill in the imagery that is missing from the blind spot that we all have due to the optic nerve in our retinas.

In addition to these limitations of static perception, there are significant limitations to how we perceive motion. It actually does not take much in terms of speed to render something invisible to our perception.

Clearly, we can’t see something zip by as fast as a bullet, which might typically move at speeds of 700 mph or more. And yet, a plane moving at 700 mph is easy to see from a distance. Our limitations of motion perception are a function of the speed of the object and the size of the image that it casts upon your retina; e.g. for a given speed, the further away something is, the larger it has to be to register in our conscious perception. This is because our perception of reality refreshes no more than 13-15 times per second, or every 77 ms. So, if something is moving so fast that it passes by our frame of perception in less than 77 ms or so, or it is so small that it doesn’t make a significant impression in our conscious perception within that time period, we simply won’t be aware of its existence.

It makes one wonder what kinds of things may be in our presence, but moving too quickly to be observed. Some researchers have captured objects on high-speed cameras, for which there appears to be no natural explanation. For example, there is this strange object captured on official NBC video at an NFL football game in 2011: Whether these objects have mundane explanations or might be hints of something a little more exotic, one thing is for certain: our eye cannot capture them. They are effectively invisible to us, yet exist in our reality.

This article originally appeared at TheUniverseSolved.

Sources:

Walking Times

Image Credit