Pi is not as constant as you think!

Yulia Maximenko
3/12/2015

 

A lot of people are celebrating Pi Day on March 14, 2015 at 9:26 am. (This makes sense, of course, only if you put the month before the day when you write your date.) But let us not forget that π (the ratio of a circle's circumference to its diameter) is not actually constant in non-Euclidean geometry. And since we live on a two-dimensional spherical surface, this might actually make a difference for circles much smaller than we would intuitively might have guessed. But first, let's do some simple geometry: Imagine a sphere of radius R. We define a circle on the surface of that sphere as we would define a circle anywhere: a geometrical shape consisting of points equally distant from a selected point. On a 2D spherical surface those circles look like this:

      Fig. 1. 3D sphere with circles on its surface.

Note that radius r is measured along the curved line on the surface of the sphere from a point also on that surface. Now, if we actually calculate the circumference of one of the circles of radius r, it would be L=2πR sin(r/R)=2πR sin(α/2), where α is the flat angle from the center of the sphere to the circle on its surface (see Fig. 1). So, π', the varying ratio of the circumference to the diameter, would be
π'=L/D=π sin (α/2)/(α/2)=π sin (r/R)/(r/R).

      Fig. 2. Plot of π vs. circle radius on Earth.

So, for α changing from 0 to π (assuming that from π to 2π the picture would be symmetrical), π' would be changing from π to 2 in the limiting case of the circle on the equator (Fig. 2).

Now, some fun facts: for a circle of radius 1000 miles, the value of "π" would be around 3.10867! For a 50 mile radius, "π" would be 3.14151. And even the engineers who built the Large Hadron Collider should have worried about the value of "π", since for a circular structure 2.7 miles in radius (which is the case for the LHC) "π" would be 3.141592415! So, we strongly encourage all high energy physicists and their sympathizers to celebrate Pi Day two minutes earlier than the rest of the world to honor our non-Euclidean geometry! As for the community of general relativity... we encourage them to redo all the calculations in a non-minkowskian metric for a non-massless Earth to know exactly when they should celebrate Pi Day. Also, advocates of the Indiana Pi Bill who root for legally making π equal to 3.2 should probably reconsider and change it to a value smaller than 3.1415926, since no circle on Earth would give them their desired result! Though if the surface of our planet was a saddle, that would be a completely different matter...

As a bonus, we suggest another interesting geometrical observation: If you have a rope around the Earth hovering h=1 foot off the ground, can you guess how much longer the rope would have to be (than the circumference of the Earth)? That's right, 2π=6.28 feet longer than the 25018 mile-long circumference of the Earth. If this small number seems counterintuitive, you can check it easily:
L' - L=2π(R+h) - 2πR = 2πh=6.28 feet.

Happy Pi Day!

Yulia Maximenko

Recent News

  • In the Media
  • Student News
  • Atomic Molecular and Optical Physics
  • Quantum Information Science

When it comes to furthering our overall understanding of the physical world, ultracold quantum gases are awfully promising. As the famous physicist Richard Feynman argued, to fully understand nature, we need quantum means of simulation and computation. Ultracold atomic systems have, in the last 30 years, proven to be amazing quantum simulators. The number of applications for these systems as such simulators is nothing short of overwhelming, ranging from engineering artificial crystals to providing new platforms for quantum computing. In its brief history, ultracold atomic experimental research has enhanced physicists’ understanding of a truly vast array of important phenomena.

  • Research
  • Condensed Matter Physics

A Majorana particle is a fermion that is its own anti-particle. Majorana particles were postulated to exist by Ettore Majorana in a now famous paper written in 1937. However, such particles have not  been discovered in nature to date.  The possible realization of Majorana particles in condensed matter systems has generated much excitement and revived interest in observing these particles, especially because the condensed matter realization may be useful for topological quantum computation. A new paper by Illinois Physics Professor Vidya Madhavan and collaborators recently published in Science shows the first evidence for propagating 1D Majorana modes realized at 1D domain walls in a superconductor  FeSexTe1−x

  • In the Media

Albert Einstein was right again. More than 100 years ago, his calculations suggested that when too much energy or matter is concentrated in one place, it will collapse in on itself and turn into a dark vortex of nothingness. Physicists found evidence to support Einstein’s black hole concept, but they’d never observed one directly. In 2017, 200-plus scientists affiliated with more than 60 institutions set out to change that, using eight global radio observatories to chart the sky for 10 days. In April they released their findings, which included an image of a dark circle surrounded by a fiery doughnut (the galaxy Messier 87), 55 million light years away and 6.5 billion times more massive than our sun. “We have seen what we thought was unseeable,” said Shep Doeleman, leader of what came to be known as the Event Horizon Telescope team. The team’s name refers to the edge of a black hole, the point beyond which light and matter cannot escape. In some ways, the first picture of a black hole is also the first picture of nothing.

Institute for Condensed Matter Theory in the Department of Physics at the University of Illinois at Urbana-Champaign has recently received a five-year grant of over $1 million from the Gordon and Betty Moore Foundation. The grant is part of the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems (EPiQS) Initiative, which strives to catalyze major discoveries in the field of quantum materials—solids and engineered structures characterized by novel quantum phases of matter and exotic cooperative behaviors of electrons. This is the second 5-year EPiQS grant awarded to the ICMT by the Moore Foundation. The two awards establish an EPiQS Theory Center at the Institute for Condensed Matter Theory.