Pi Cycles

Pi: Places where Pi shows up:

Finding Pi Outside of the Circumference

The number, which we celebrate on Tuesday, shows up more places than you might think.

Photo illustration by Natalie Matthews-Ramo. Photos by iStock/Thinkstock.

Photo illustration by Natalie Matthews-Ramo. Photos by iStock/Thinkstock.

This story originally appeared on the Conversation and is reprinted here with permission.


Happy Pi Day, where we celebrate the world’s most famous number. The exact value of π=3.14159… has fascinated people since ancient times, and mathematicians have computed trillions of digits. But why do we care? Would it actually matter if somebody got the 11,137,423,895,285th digit wrong?

Probably not. The world would keep on turning (with a circumference of 2πr). What matters about π isn’t so much the actual value as the idea and the fact that π seems to crop up in lots of unexpected places.

Let’s start with the expected places. If a circle has radius r, then the circumference is 2πr. So if a circle has radius of one foot, and you walk around the circle in one-foot steps, then it will take you 2π = 6.28319… steps to go all the way around. Six steps isn’t nearly enough, and after seven you will have overshot. And since the value of π is irrational, no multiple of the circumference will be an even number of steps. No matter how many times you take a one-foot step, you’ll never come back exactly to your starting point.

Calculating the area of a circle with wedges. Jim.belk
Calculating the area of a circle with wedges.


From the circumference of a circle we get the area. Cut a pizza into an even number of slices, alternately colored yellow and blue. Lay all the blue slices pointing up, and all the yellow slices pointing down. Since each color accounts for half the circumference of the circle, the result is approximately a strip of height r and width πr, or area πr2. The more slices we have, the better the approximation is, so the exact area must be exactly πr2.

You don’t just get π in circular motion. You get π in any oscillation. When a mass bobs on a spring, or a pendulum swings back and forth, the position behaves just like one coordinate of a particle going around a circle.

If your maximum displacement is one meter and your maximum speed is 1 meter/second, it’s just like going around a circle of radius one meter at 1 meter/second, and your period of oscillation will be exactly 2π seconds.

The area of the space under the normal-distribution curve is the square root of pi.
The area of the space under the normal-distribution curve is the square root of pi.

Autopilot,CC BY-SA

Pi also crops up in probability. The function f(x)=e-x², where e=2.71828… is Euler’s number, describes the most common probability distribution seen in the real world, governing everything from SAT scores to locations of darts thrown at a target. The area under this curve is exactly the square root of π.

How did π get into it?! The two-dimensional function f(x)f(y) stays the same if you rotate the coordinate axes. Round things relate to circles, and circles involve π.

Another place we see π is in the calendar. A normal 365-day year is just over 10,000,000π seconds. Does that have something to do with the Earth going around the sun in a nearly circular orbit? Actually, no. It’s just coincidence, thanks to our arbitrarily dividing each day into 24 hours, each hour into 60 minutes, and each minute into 60 seconds.

What’s not coincidence is how the length of the day varies with the seasons. If you plot the hours of daylight as a function of the date, starting at next week’s equinox, you get the same sine curve that describes the position of a pendulum or one coordinate of circular motion.

More examples of π come up in calculus, especially in infinite series like
1 – (13) + (15) – (17) + (19) + ⋯ = π/4
12 + (12)2 + (13)2 + (14)2 + (15)2 + ⋯ = π2/6

(The first comes from the Taylor series of the arctangent of 1, and the second from the Fourier series of a sawtooth function.)

A graph of the exponential function y=e^x.
A graph of the exponential function y=e^x.

Peter John Acklam,CC BY-SA

Also from calculus comes Euler’s mysterious equation e + 1 = 0 relating the five most important numbers in mathematics: 0, 1, i, π, and e, where i is the (imaginary!) square root of -1.

At first this looks like nonsense. How can you possibly take a number like e to an imaginary power?! Stay with me. The rate of change of the exponential function f(x)=ex is equal to the value of the function itself. To the left of the figure, where the function is small, it’s barely changing. To the right, where the function is big, it’s changing rapidly. Likewise, the rate of change of any function of the form f(x)=eax is proportional to eax.

The relationship between an angle, its sine, cosine and a circle.
The relationship between an angle, its sine, cosine and a circle.

345Kai, CC BY-SA

We can then define f(x)= eix to be a complex function whose rate of change is i times the function itself, and whose value at 0 is 1. This turns out to be a combination of the trigonometric functions that describe circular motion, namely cos(x) + i sin(x). Since going a distance π takes you halfway around the unit circle, cos(π)=-1 and sin(π)=0, so e=-1.

Finally, some people prefer to work with τ=2π=6.28… instead of π. Since going a distance 2π takes you all the way around the circle, they would write that e = +1. If you find that confusing, take a few months to think about it. Then you can celebrate June 28 by baking two pies.


Martin Armstrong, Pi, and the Markets

I believe there is one man who has cornered the cycles-theory market, and that man is Martin Armstrong. I do not know of any other trader who has done as much study and work on economic cycles as he. http://www.armstrongeconomics.com



Here is an article by a former PEI employee: http://www.sandspring.com/_private/MartysDateFeb2007a.pdf

Pi & Disease: Here is an article that I sent Marty which he posted on his blog: https://www.armstrongeconomics.com/uncategorized/measles-51-6-year-cycle/

“Something about the measles vaccine, that I found really startling in my recent research, is that the vaccine failures that we’re seeing today were actually all predicted in our medical literature. There was a Dr. David Levy, in 1984, who predicted the problems that were to come, and that was during the period where there was just the single vaccine. But then, there’s a Dr. Heffernan, from 2009, who did his own predictions based on very sophisticated mathematical analyses during a two-vaccine period, and what he said, verbatim, is, “we predict that after a long disease-free period, the introduction of infection will lead to far larger epidemics than that predicted by standard models.” And even more compelling, he said that, “large-scale epidemics can arise with the first substantial epidemic not arising until 52 years after the vaccination program has begun,” well, guess what year 52 years is? 2015. So, now, can you see why the CDC is staying up late at night and having panic attacks?”

52 is very close to the 51.6 Pi cycle.

Pi Numbers


8.6 years is 3139 days, round that to 314, a pi.

4.3 is half 8.6.

8.6 + 4.3 = 12.9 (about 13).

Pi & Earthquakes

Mexico City had a 8.0 earthquake on Sept 19, 1985. 32 years (pi) later it experienced a 7.1 earthquake on Sept 19, 2017.


Pi & Relationships

13 years often mark turning points in relationships.

Pi & Religion

“Marty-love your analysis based on 4.3 and 8.6 year cycles. You state that this is all over everywhere and in all markets. Correct, and it is even seen in the Book of Daniel (Bible). Towards the end of Daniel, we see the angel mentioning about 1290 weeks. Well, 1290 (just the number) is an exact multiple of 8.6 – 150 times.

Of more interest is that 1290 is exactly 300 times 4.3! This is YOUR 300 year number and I guess we can say that there is no new thing under the sun!

Best regards,


REPLY: Fascinating. Another has pointed out that 430 is an important number from the Bible. As the story goes, God told Abraham in a vision that his descendants through Isaac would end up as slaves in a foreign country. God would, however, release them from this bondage after 400 years (Genesis 15:12 – 16). Exactly 430 years later to the very day, on the same night, this prophecy was fulfilled as the children of Israel left Egypt on the 15th day of the first month (Exodus 12:40 – 41). So I was told that this number has worked to the very day before.” https://www.armstrongeconomics.com/uncategorized/the-8-6-seems-to-be-in-religious-texts-as-well/



I find all religions interesting for there seems to be kernels of wisdom in all traditions and beliefs. Is this 8.6 frequency just the perfect cycle?


Collapse of the Roman Silver Monetary System 

The waterfall event occurred in about 8.6 years, during the reign of Gallienus.


Solar Pi Cycles


Weather and Pi


Extra Reading on Numerology

The Alphanumeric Conspiracy