Tag Archives: pi

“π Earth”: Astronomers discover Earth-sized planet that takes 3.14 days to orbit its star

Every new exoplanet discovery is remarkable in its own way, and if that planet happens to be Earth-sized, it’s even more special. If it’s connected to a famous constant (Pi), it’s basically an astronomy party.

Pi, the ratio of a circle’s circumference to its diameter, isn’t exactly 3.14. In fact, it’s 3.141592653589793238… and goes on forever. But for most people, 3.14 is a good enough approximation — and for the astronomers looking for this new planet, the similarity was too striking.

“The planet moves like clockwork,” says Prajwal Niraula, a graduate student in MIT’s Department of Earth, Atmospheric and Planetary Sciences (EAPS), who is the lead author of a paper published today in the Astronomical Journal, titled: “π Earth: a 3.14-day Earth-sized Planet from K2’s Kitchen Served Warm by the SPECULOOS Team.”

“Everyone needs a bit of fun these days,” says co-author Julien de Wit, of both the paper title and the discovery of the pi planet itself.

The planet is called K2-315b but already, astronomers have nicknamed it π Earth. It’s the 315th planetary system discovered with data from K2, the successor of the Kepler telescope — just one shy from another coincidence that would have made it number 314.

The first signs of the planet were reported in 2017, but it was only confirmed more recently. π Earth has approximately 95% of Earth’s mass, making it essentially Earth-sized, and orbits a star that’s 5 times smaller than the Sun.

However, there’s virtually no chance of life as we know it on the planet. For starters, the planet orbits very close to its star, and astronomers estimate that it heats up to around 450 Kelvin (177 degrees Celsius, or 350 degrees Fahrenheit). As mentioned, the planet also circles its star every 3.14 days — so a ‘year’ on the planet is little more than three days, which means it moves at a blistering speed of 81 kilometers per second, or about 181,000 miles per hour (compared to 30 km/s at the Earth’s equator).

However, the planet is interesting in itself, more than being a mathematical curiosity.

“This would be too hot to be habitable in the common understanding of the phrase,” says Niraula, who adds that the excitement around this particular planet, aside from its associations with the mathematical constant pi, is that it may prove a promising candidate for studying the characteristics of its atmosphere.

“We now know we can mine and extract planets from archival data, and hopefully there will be no planets left behind, especially these really important ones that have a high impact,” says de Wit, who is an assistant professor in EAPS, and a member of MIT’s Kavli Institute for Astrophysics and Space Research.

The researchers are also interested in a follow-up study on the Pi planet with the upcoming James Webb Space Telescope (JWST), to see potential details of the planet’s atmosphere. For now, they are combing through other telescope datasets for signs of Earth-like planets — Pi or non-Pi.

The study has been published in the Astronomical Journal.

Pi.

What is Pi (π) and what is it good for?

If you have a straight line but want a circle, you’re going to need some Pi.

Pi.

Image via Max Pixel.

I’m talking about the number, not the delicious baked good. It’s usually represented using the lowercase Greek letter for ‘p’, ‘π’, and probably is the best known mathematical constant today. Here’s why:

The root of the circle

Pi is the ratio of a circle’s diameter to its circumference. No matter the size of a circle, its diameter will always be roughly 3.14 times shorter than its circumference — without fail. This ratio, π, is one of the cornerstones upon which modern geometry was built.

Bear in mind that (uppercase) ∏ is not the same as (lowercase) π in mathematics.

For simplicity’s sake, it’s often boiled down to just two digits, 3.14, or the ratio 22/7. In all its glory, however, pi is impossible to wrap your head around. It’s is an irrational number, meaning a fraction simply can’t convey its exact value. Irrational numbers include a value or a component that cannot be measured against ‘normal’ numbers. For context, there’s an infinite number of irrational numbers between 1.1 and 1.100(…)001. They’re the numbers between the numbers.

There is no unit of measurement small enough in rational numbers that can be used to fully express the value of irrational ones. They’re like apples and oranges — both fruits, but very different.

Real numbers.

Apart from the fact that it implies there are numbers which are neither rational or irrational (there aren’t), this Euler diagram does a good job of showcasing the apples/oranges relationship between the two groups.
Image credits Damien Karras.

Because it can’t properly be conveyed through a fraction, it follows that pi also has an infinite string of decimals. Currently, we’ve calculated pi down to roughly 22.4 trillion digits. Well, I say ‘we’, but it was actually our computers that did it.

Truth be told, we don’t actually need that many digits. They’re very nice to have if you’re NASA and people live or die by how accurate your calculations are — but for us laymen, 3.14 generally does the trick. It’s good enough because it’s just about at the limit of how accurately we can measure things around us. We simply don’t need that much precision in day-to-day activity.

Go around the house, pick up anything round, and run a length of string along its circumference. Unwind it and measure it with a ruler. Measure the circle’s diameter with the same ruler, use this value to divide the circumference, and you’ll get roughly 3.14 each and every time. In other words, if you cut some string in several pieces, each equal to the diameter in length, you’d need 3.14 of those strips to cover the circumference.

Because this simplification is so widely-used, we celebrate Pi day on March 14 (3/14) every year.

If you do happen to need a more-detailed value for Pi, here it is up to 100 million decimal places.

What’s it for?

Pi is used in all manner of formulas. For example, it can be used to calculate a circle’s circumference (π times diameter), or its area: A=πr2 — how I keep this formula lodged in my neurons is using the “all pies are square” trick. It’s also used in calculating various elements of the sphere, such as its volume (3/4πr3) or surface area (4πr²).

But it also shows up in a lot of engineering and computational problems. Weirdly enough, pi can be used to obtain the finite sum of an infinite series. For example, if you add up the inverse of all natural squares — 1/12+1/22+1/32+….+1/n2 — you get π2/6.

Most branches of science stumble into pi in their calculations at one point or another. Computer scientists use it to gauge how fast or powerful a computer is, and how reliable its software, by having the device crunch numbers and calculate pi. It’s very useful for determining both circular velocities (how fast something is spinning) as well as voltage across coils and capacitors. Pi can be used to describe the motion of waves on a beach, the way light moves through space, the motion of planets, or to track population dynamics if you’re into statistics.

Another place pi pops up (that you wouldn’t suspect) is in the value of the gravitational constant. This shows how fast an object will accelerate towards the ground as it’s falling. Its most widely-accepted value is 9.8 m/s2. The square root of that value is 3.1305-ish, which is close to the value of pi. That’s actually because the original definition of a meter involved a pendulum that took 1 second to swing either way. Wired has a more comprehensive explanation here.

Pi also underpins modern global positioning systems (GPS) since the Earth is a sphere. So give a little mental thanks to mathematics the next time you’re drunkenly thumbing your phone to hail an Uber.

Who discovered pi?

Domenico Fetti Archimedes.

“Archimedes Thoughtful” by Domenico Fetti, currently at the Gemäldegalerie Alte Meister in Dresden, Germany. Archimedes calculated one of the most accurate values for Pi during the Antiquity.

Pi is not a newcomer to the mathematical stage by any means. We refer to it using the letter ‘π’ from the ancient Greek word ‘περίμετρος’ — perimetros — which means ‘periphery’ or ‘circumference’. It was introduced by William Jones in 1706 and further popularized by Leonhard Euler. The notation was likely adopted in recognition of the efforts of one great ancient mathematician: Archimedes.

Archimedes put a lot of effort into refining the value of pi. He was also the first to use it to calculate the sum of an infinite number of elements over 2,200 years ago, and it’s still in use today.

But he wasn’t the first to realize the importance of pi(e). In his book A History of Pi, professor Petr Beckmann writes that “the Babylonians and the Egyptians (at least) were aware of the existence and significance of the constant π” as far back as 4,000 years ago. They likely only had rough estimations of its exact value (maths was still a new ‘tech’ back then) but they were in the right ballpark.

“The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for pi, which is a closer approximation,” writes Exploratorium in a look at the history of pi.

They add that ancient Egyptian mathematicians also settled on a quite-ok-for-the-time value of 3.1605, as revealed by the Rhind Papyrus. Chinese and Indian mathematicians also approximated the value of pi down to seven or five digits, respectively, by the 5th century AD.

Further work, most notably that of Archimedes, helped refine this value. He used the Pythagorean Theorem to measure the area of a circle via the areas of inscribed and circumscribed regular polygons. If you slept during math class, that’s the polygon inside the circle and the one that contains the circle, respectively. It was an elegant method, but it did have its limits — since the areas of those two polygons aren’t exactly the same as the surface area of the circle, what Archimedes got was an interval that contained pi. He was aware of this limitation. His calculations revealed that pi must fall between 3 1/7 and 3 10/71 — which is between 3.14285 and 3.14085. Today we know that the five-digit value of pi is 3.14159, so that result isn’t at all bad for a guy without a proper pen to write it down with.

The first method to calculate the exact value of pi came up during the 14th century, with the development of the Madhava-Leibniz series. By the time the 20th-century swang by, pi was known down to about 500 digits.

Image: University of Rochester

Classic formula for ‘pi’ connects pure math and quantum mechanics like a ‘magic trick’

Image: University of Rochester

Image: University of Rochester

Pi or  π is the ratio between a circle’s circumference and diameter. It doesn’t matter how big or small the circle is – the ratio stays the same, and the constant has proven to be indispensable for mathematicians. Along the ages, computing pi – an irrational number, hence its decimal representation never ends (supercomputers managed to calculate trillions of digits for pi) – has proven to be both an entertaining and head banging quest for mathematicians. The first calculation of pi was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes bracketed a circle with polygons, which allowed him to break the circle into squares. Using Pythagoras’ theorem and the perimeter of a square, both known to Archimedes, the Greek savant used this trick with 96 sided polygons to correctly estimate Pi to about two digits (3.14), proving 3.1408 < Pi < 3.1428.

Until the advent of calculus and computing infinite series, not that many digits were added to the ones found by Archimedes for more than a 1,500 years. One major breakthrough was made in 1655 when the English mathematician  derived a formula for pi as the product of an infinite series of ratios. Oddly enough, but not that surprising considering the prevalence of pi in nature, researchers from the University of Rochester reached the same formula while they were computing the quantum mechanical energy stats of hydrogen.

In quantum mechanics, a technique called the variational approach can be used to approximate the energy states of quantum systems, like molecules, that can’t be solved exactly. Carl Hagen, a particle physicist at the University of Rochester, made a habit out of teaching the technique to  his students by applying it to hydrogen. The thing about the hydrogen atom is that its energy levels can be computed directly using the quantum calculations developed by Danish physicist Niels Bohr in the early twentieth century. By applying the variational approach and then comparing the result to the exact solution, students could calculate the error in the approximation. But after Hagen himself started solving the problem, he noticed a peculiar trend: the error of the variational approach was about 15 percent for the ground state of hydrogen, 10 percent for the first excited state, and kept getting smaller as the excited states grew larger.

Two pages from the book “Arithmetica Infinitorum,” by John Wallis.

Two pages from the book “Arithmetica Infinitorum,” by John Wallis.

Hagen needed some backup, so he recruited mathematician Tamar Friedmann to help out. The two found that the ratio yielded—effectively—the Wallis formula for π.

Specifically, the calculation of Friedmann and Hagen resulted in an expression involving special mathematical functions called gamma functions leading to the formula

pi-Figure1

which can be reduced to the classic Wallis formula.

pi-Figure2

“We didn’t just find pi,” said Friedmann, a visiting assistant professor of mathematics and a research associate of high energy physics, and co-author of a paper published this week in the Journal of Mathematical Physics. “We found the classic seventeenth century Wallis formula for pi, making us the first to derive it from physics, in general, and quantum mechanics, in particular.”

“The value of pi has taken on a mythical status, in part, because it’s impossible to write it down with 100 percent accuracy,” said Friedmann, “It cannot even be accurately expressed as a ratio of integers, and is, instead, best represented as a formula.”

It’s amazing to see pi pop up in such a natural way, with no circles involved what so ever. And how elegant to find this connection by reaching the same results as a XVIIth century mathematician.

“This derivation of pi is a surprise of the familiar, much like a magician’s trick,” said Moshe Machover of King’s College London, who was not involved in the study. “A child who sees a trick done for the first time may be only surprised. But an adult, who has seen numerous tricks over the years, experiences both surprise and familiarity.”

“Nature had kept this secret for the last 80 years,” Friedmann said. “I’m glad we revealed it.”