ROBOTICS
ROBOTICS
Pi is useful for all kinds of calculations involving the volume and surface area of spheres, as well as for determining the rotations of circular objects such as wheels. That's why pi is important for robotics scientists who work with planetary bodies and spacecraft.
Fast Facts:
The number pi, the ratio of circumference to diameter of a circle, is celebrated every year on March 14, or 3/14
NASA/JPL scientists and engineers use pi frequently in calculations
Take the JPL Education Pi Day challenge
If you like numbers, you will love March 14, 2015. When written as a numerical date, it's 3/14/15, corresponding to the first five digits of pi (3.1415) -- a once-in-a-century coincidence! Pi Day, which would have been the 136th birthday of Albert Einstein, is a great excuse to eat pie, and to appreciate how important the number pi is to math and science.
Pi is the ratio of circumference to diameter of a circle. Any time you want to find out the distance around a circle when you have the distance across it, you will need this formula.
Despite its frequent appearance in math and science, you can't write pi as a simple fraction or calculate it by dividing two integers (...3, -2, -1, 0, 1, 2, 3...). For this reason, pi is said to be "irrational." Pi's digits extend infinitely and without any pattern, adding to its intrigue and mystery.
Pi is useful for all kinds of calculations involving the volume and surface area of spheres, as well as for determining the rotations of circular objects such as wheels. That's why pi is important for scientists who work with planetary bodies and the spacecraft that visit them.
At NASA's Jet Propulsion Laboratory, Pasadena, California, pi makes a frequent appearance. It's a staple for Marc Rayman, chief engineer and mission director for NASA's Dawn spacecraft. Dawn went into orbit around dwarf planet Ceres on March 6. Rayman uses a formula involving pi to calculate the length of time it takes the spacecraft to orbit Ceres at any given altitude. You can also use pi to think about Earth's rotation.
"On Pi Day, I will think about the nature of a day, as Earth's rotation on its axis carries me on a circle 21,000 miles (34,000 kilometers) in circumference, which I calculated using pi and my latitude," Rayman said.
Steve Vance, a planetary chemist and astrobiologist at JPL, also frequently uses pi. Lately, he has been using pi in his calculations of how much hydrogen might be available for chemical processes, and possibly biology, in the ocean beneath the surface of Jupiter's moon Europa.
"To calculate the hydrogen produced in a given unit area, we divide by Europa's surface area, which is the area of a sphere with a radius of 970 miles (1,561 kilometers)," Vance said.
Luisa Rebull, a research scientist at NASA's Spitzer Science Center at the California Institute of Technology, Pasadena, also considers pi to be important in astronomy. When calculating the distance between stars in a projection of the sky, scientists use a special kind of geometry called spherical trigonometry. That's an extension of the geometry you probably learned in middle school, but it takes place on a sphere rather than a flat plane.
"In order to do these calculations, we need to use formulae, the derivation of which uses pi," she said. "So, this is pi in the sky!"
Make sure to note when the date and time spell out the first 10 digits of pi: 3.141592653. On 3/14/15 at 9:26:53 a.m., it is literally the most perfectly "pi" time of the century -- so grab a slice of your favorite pie, and celebrate math!
Pi represents the ratio of the circumference of a circle to its diameter
Pi is a number that relates a circle's circumference to its diameter. Pi is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. That's because pi is what mathematicians call an "infinite decimal" — after the decimal point, the digits go on forever and ever.
Students are usually introduced to the number pi as having an approximate value of 3.14 or 3.14159. Though it is an irrational number, some people use rational expressions, such as 22/7 or 333/106, to estimate pi. (These rational expressions are accurate only to a couple of decimal places.)
Mathematicians and math enthusiasts are interested in calculating pi to as many digits as possible. The record for reciting the most digits of pi belongs to Suresh Kumar Sharma of India, who recited pi to 70,030 decimal places in 2015, according to the Pi World Ranking List
. Meanwhile, some computer programs have calculated the value of pi to an astounding 62.8 trillion digits, Live Science previously reported. Calculations like these are often unveiled on Pi Day, a pseudo-holiday that occurs every year on March 14 (3/14).
By definition, pi is the ratio of the circumference of a circle to its diameter. In other words, pi equals the circumference divided by the diameter (π = c/d). Conversely, the circumference of a circle is equal to pi times the diameter (c = πd). No matter how large or small a circle is, pi will always work out to be the same number. Pi (π) is the 16th letter of the Greek alphabet and is used to represent the widely known mathematical constant.
History
Pi has been known for nearly 4,000 years and was discovered by the ancient Babylonians. A tablet dating to somewhere between 1900 B.C. and 1680 B.C. found pi to be 3.125, according to the Exploratorium in San Francisco
People in ancient Egypt were making similar discoveries, as evidenced by the Rhind papyrus of 1650 B.C. In this document, the Egyptians calculated the area of a circle by a formula giving pi an approximate value of 3.1605. There is even a biblical verse where it appears pi was approximated, according to a correspondence published in 1999 in the journal Nature
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. — I Kings 7:23 (King James Version)
One of the first calculations of pi was carried out by Greek mathematician Archimedes of Syracuse (287 B.C. to 212 B.C.), according to the Exploratorium
Archimedes used the Pythagorean theorem to find the areas of two polygons. Archimedes approximated the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed. The polygons, as Archimedes mapped them, gave the upper and lower bounds for the area of a circle, and he approximated pi at between 3 1/7 and 3 10/71.
Earlier, Chinese mathematician and astronomer Zu Chongzhi (429 B.C. to 501 B.C.) had calculated pi using a similar method, finding the value to be 355/113. Unfortunately, Zu's book of writing has been lost, so little is known about his work or methods.
British mathematician William Jones was the first to begin using the symbol π to represent pi, in 1706.
While no Mars landing is exactly the same, they do share one thing in common: parachutes. Slowing down a rover or lander as it drops through the thin Martian atmosphere is imperative if engineers hope to slow the spacecraft enough to give the descent rockets time for a soft landing. NASA’s engineers take all sorts of things into consideration when designing a parachute: the mass and velocity of the spacecraft, the elevation of the landing site and the density of the atmosphere, just to name a few. Pi helps engineers determine how big the parachute needs to be in order to generate the drag needed to slow down.
NASA’s Cassini spacecraft spent 13 years orbiting Saturn, discovering seas and jets of water ice on its moons, and observing its majestic rings. Twice during the mission, engineers used a technique called a pi transfer to alter the spacecraft’s orbit. With a precisely steered flyby of Saturn’s largest moon, Titan, Cassini’s orbit was flipped 180 degrees to the opposite side of the planet. (In radians, 180 degrees is equal to pi, hence the name pi transfer.) With the lighting conditions also flipped 180 degrees, from Cassini's perspective, the spacecraft was able to see Saturn and Titan in a whole new light.
Just like Earth’s ancient explorers, when spacecraft visit other planets and worlds, they make a map. Even spacecraft that orbit familiar places, like Earth, make maps of processes scientists want to understand, such as how water flows around the globe. Spacecraft make maps by taking images as they orbit – like in the animation above, which shows the Juno spacecraft mapping Jupiter. Their cameras often have rectangular fields of view that capture images in “bands” on the surface of a planet. Scientists use pi in the formula for surface area to figure out how many images it will take to map the entire planet or body.
Engineers use pi to help estimate the amount of uncertainty in the position where a Mars lander or rover will touch down. Many aspects of landing on Mars are uncertain: winds, air density, the initial speed and position of the spacecraft when approaching Mars from Earth. Even the exact position of Mars itself is not perfectly known. Before a Mars landing, most of these uncertainties can be modeled using mathematical distributions that include pi in the calculations. When simulated together, the result is potentially miles of position uncertainty surrounding the targeted landing spot. Engineers take this uncertainty into consideration and are careful about where they aim! For example, they can aim close, but not too close to a mountain – like they did with the Curiosity Mars rover, which landed near Mount Sharp.
Scientists use pi to search for exoplanets, which are planets that orbit stars other than our own Sun. Powerful ground- and space-based telescopes track how much light is emitted by distant stars. When a planet passes in front of its star, the telescope sees a dip in the amount of light emitted. Knowing the percentage of this decrease and the formula for the area of a circle, scientists can deduce the planet’s size.
When scientists discover new exoplanets, one of the things they want to know is whether these worlds could support life as we know it. These “potentially habitable” worlds orbit within what’s known as the habitable zone of their parent stars – a location that’s a safe distance from the star, not too close, where water would turn to gas, and not too far, where it would become ice. Scientists use pi to locate the inner and outer edges of the habitable zone around a given star. And they use pi, along with Kepler’s third law, to calculate how long it takes the exoplanet to make one full orbit of its star, which reveals the planet’s location and whether it’s in the habitable zone.
Scientists use pi to study earthquakes and, soon, marsquakes! NASA’s InSight Mars lander is equipped with an instrument for measuring seismic activity on the Red Planet, which will tell us more about what’s going on inside the planet. During a marsquake, surface waves – a type of seismic wave – travel outward from the epicenter in all directions on Mars. By timing the arrival of these surface waves at the InSight lander and using pi, scientists can determine what time the marsquake occurred.
Sending messages to distant spacecraft and receiving them requires a network of massive antennas stationed around the globe so that, as the world turns, we never lose contact. Together, these antennas make up NASA’s Deep Space Network, or DSN. The engineers who communicate with spacecraft through the DSN use pi in the math equations needed to send messages and process those that are sent back. It’s a pretty important task considering the messages are used to do things like land rovers on Mars and get images from a spacecraft flying closely by Pluto for the first time.
There are no joysticks or steering wheels on Mars rovers. Instead, rovers receive commands from operators on Earth that tell them when and how to drive, take pictures, turn their wheels and use their robotic arms. Some of these functions are measured in degrees and others in radians (slices of a circle), so pi is regularly used to convert between the two.
Engineers use pi to put spacecraft into orbit around other planets. To do this, they have to slow down the spacecraft just enough and at exactly the right time for it to get pulled into orbit by the planet’s gravity. Engineers determine how much that gravity will tug on the spacecraft, how fast the spacecraft is going and the details of the new orbit. Using those numbers, along with pi, they can compute exactly how much they need to put on the brakes – which for a spacecraft, means firing its forward-facing thrusters at just the right moment.
One of the jobs of comet and asteroid hunters, like those at NASA’s Center for Near-Earth Objects, is to determine how quickly an object is rotating. From their observations of the object, scientists can estimate how long it takes the object to make one complete rotation on its axis. Then, unit conversion is used to find the object’s angular velocity, which is often measured in radians per second. (You can think of radians as slices of a circle, or better yet, slices of pi.)
Rover wheels have distinct designs on them that leave patterns on the ground as they turn. These patterns serve as visual markers that help operators while driving the Mars rovers remotely from Earth. Pi is used to calculate how far the rover should travel with each wheel rotation. By measuring the distance from one wheel mark to another, rover drivers can determine if the wheels are slipping or if they’ve driven the expected distance.
Scientists studying extreme environments, such as those on comets and the moons of Jupiter and Saturn, want to learn how processes unfold on their surfaces. In the case of icy environments, one way to do that is by using lasers in the laboratory to explode ice samples and then studying the chemical reaction that takes place. Scientists use pi to calculate the beam width of the laser and understand how much energy is hitting their ice sample.
Just like cars, spacecraft require fuel to get where they’re going and to maneuver throughout their journey. But in space, there’s no refueling along the way. Determining how much fuel a spacecraft will need and how much it has used is a delicate task. Engineers use pi to compute how much fuel is available in spacecraft tanks, which are commonly spherical, and how quickly that fuel travels through their cylindrical fuel lines. Even donut-shaped (toroidal) tanks, which can hold a lot of propellant but take up much less space, require the use of pi.
While studying the surfaces of other worlds and even Earth, scientists use pi to determine the size of features on the surface. To size up circular shapes, such as craters, the math is simple, while unusual shapes, like Pluto’s “heart,” require trigonometry or calculus.
Craters can tell scientists a lot about the surfaces of planets, moons and other bodies. Just by determining how circular a given crater is – using pi and the crater’s perimeter and area – planetary geologists can reveal clues about how the crater was formed and the surface that was impacted.
How do scientists find out what other planets and asteroids are made of if they can’t visit them in person? Using pi, of course. Planetary scientists use pi to determine the volume of rocky planets or asteroids. Volume, together with the object’s mass, tells them its density. And because planetary materials like rock, ice and metal have known densities, scientists can make informed guesses about what the planet or asteroid might be made of based on the object’s density.
One of the ways that scientists study what’s happening inside the thick swirling clouds on gas giant planets, like Jupiter and Saturn, is by sending spacecraft that can analyze the chemical makeup of these worlds. Scientists then use pi in combination with the spacecraft sensor data to estimate the volume of materials in the planet’s atmosphere. For example, in 1995, the Galileo spacecraft dropped a probe into Jupiter and detected unusually low levels of helium in the upper atmosphere. After studying the data, scientists hypothesized that helium could be raining out of the upper level of Jupiter’s atmosphere and pi held the key to how much. Today, the Juno spacecraft, which arrived at Jupiter in 2016, is helping scientists get an even better picture of what’s going on inside the planet.