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

There have been accusations for years that the Major League ball is “juiced,” thus accounting for the increasing power numbers.

MLB officials have categorically denied that, and last year, commissioned a study of the baseball and how it’s produced.

In the landmark 85-page independent report replete with color graphs, algorithms and hypotheses, a group of 10 highly-rated professors and scientists chaired by Alan Nathan determined that the ball is not livelier or “juiced.” Nathan is a professor emeritus of physics from the University of Illinois at Urbana Champaign.

The surge in home runs “seems, instead, to have arisen from a decrease in the ball’s drag properties, which cause it to carry further than previously, given the same set of initial conditions – exit velocity, launch and spray angle, and spin. So, there is indirect evidence that the ball has changed, but we don’t yet know how,” wrote Leonard Mlodinow, in the report’s eight-page executive summary.

  • In the Media

Growing up in Trinidad and Tobago, Kandice Tanner went to a school where she was one of only a dozen girls among 1200 pupils. She had switched from an all-girl school to avoid the distractions of socializing and to take the more advanced math classes offered at the boys’ school. “Being submerged in an all-male environment early on was beneficial to me,” Tanner says. “I felt comfortable with guys, and more important, I knew I could hold my own in a male-dominated environment.”

  • Research
  • Condensed Matter Physics

Illinois Physics Professor Philip Phillips and Math Professor Gabriele La Nave have theorized a new kind of electromagnetism far beyond anything conceivable in classical electromagnetism today, a conjecture that would upend our current understanding of the physical world, from the propagation of light to the quantization of charge. Their revolutionary new theory, which Phillips has dubbed “fractional electromagnetism,” would also solve an intriguing problem that has baffled physicists for decades, elucidating emergent behavior in the “strange metal” of the cuprate superconductors.

This research is published in an upcoming colloquium paper in Reviews of Modern Physics (arXiv:1904.01023v1).

  • Accolades
  • Student News

The BPS Art of Science Image Contest took place again this year, during the 63rd Annual Meeting in Baltimore. The image that won first place was submitted by Angela Barragan, PhD Candidate at the Beckman Institute UIUC. Barragan took some time to provide information about the image and the science it represents.