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

  • Accolades

Physics Professor Nadya Mason has been selected a University Scholar by the Office of the Vice President for Academic Affairs at the University of Illinois at Urbana-Champaign. The highest honor bestowed on faculty by the university, this award recognizes faculty who have made significant contributions in their fields of research and teaching, in line with the university’s reputation for leading-edge innovation and excellence. Mason is one of five faculty members on the Urbana campus to be named to this honor in this selection round.

  • Accolades

The 347 scientists from around the globe who worked together to capture the first-ever image of a black hole have won the 2020 Breakthrough Prize in Fundamental Physics. The Breakthrough Prize Foundation announced today that members of the international Event Horizon Telescope (EHT) Collaboration were selected for this honor, with the citation: “For the first image of a supermassive black hole, taken by means of an Earth-sized alliance of telescopes.”

The $3-million prize will be shared equally among all contributing scientists named on the six EHT papers published on April 10, 2019, in conjunction with the public unveiling of the black hole image—the first direct evidence of a supermassive black hole and its shadow.In addition to Gammie, U of I’s EHT research team members named Breakthrough Prize laureates include current physics graduate students George Wong and Ben Prather; current astronomy graduate student Andrew Nadolski; former astronomy graduate students Ben Ryan (now an R&D scientist at Los Alamos National Laboratory), Dimitrios Psaltis (now an astronomy professor at The University of Arizona), and Hotaka Shiokawa (now a data scientist at Rakuten Institute of Technology); and former physics postdoctoral researchers Monika Moscibrodzka (now an astrophysics professor at Radboud University, the Netherlands) and Roman Gold (now a postdoctoral researcher at Goethe University Frankfurt am Main, Germany). Each of these scientists will receive upwards of $8,600 in prize money.

  • Accolades

Humankind’s first image of a monster black hole, the tangled proteins that cause destructive neurodegenerative diseases, and the pathways telling our brains we’re in pain are among the discoveries honored by this year’s Breakthrough Prizes.

With $3 million accompanying each major prize, the Breakthrough Prizes are the most lucrative in science. Bankrolled by Silicon Valley titans, including Yuri Milner and Mark Zuckerberg, the prizes honor cutting-edge achievements in life sciences, physics, and mathematics. And unlike some other well-known science prizes, it’s not unusual for the Breakthrough team to honor teams rather than a few select individuals.

  • Accolades

The Department of Physics at the University of Illinois at Urbana-Champaign has established the Gary Kelly Staff Excellence Award to recognize staff members who demonstrate exceptional job performance and dedication in support of departmental goals. The annual award will be presented for the first time at the 2019 Fall Physics Reception to up to two outstanding staff members; the award comprises a commemorative plaque and $2,000 for each recipient. The Gary Kelly Staff Excellence Award is funded by a generous estate gift from Illinois Physics alumnus Gary Kelly (MS, 1974).  

 

Illinois Physics Associate Head for Administration Jennifer Jorstad comments, “This award was created to recognize the hard work that the civil service and academic professional staff contribute to the mission of the department. Many of our staff go above and beyond their regular job responsibilities in order to serve the department. Those strong performers deserve to be recognized.”