Necessary Facts: pi

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Saturday, March 14, 2015

Happy Pi Day of the Century

Every March 14, we celebrate Pi Day. 3.14 is a pretty good approximation, better than 1 part in 1000. This year, being 2015, we acknowledge 3.1415 in significant digits. You can carry it out further at 9:26:53 AM and PM: 3.141592653.

Also this year, you can celebrate for another second to 9:26:54 because 3.1415926535… continues to 8979… So, 3.1415926536 is an acceptable estimate.

You can have a pretty good celebration next year, as 3.14159 rounds to 3.1416 , which is a better approximation than 3.1415.

The first 50 decimal digits of pi are 3.14159265358979323846264338327950288419716939937510 (Wikipedia here).

Pi Day dot org gives many suggestions for circular festivities – and the first million digits of pi here. And they sell a clock, with a circumference measured out in radians or fractions of pi.

The formula for the circumference of a circle is pi times the measure of the diameter: pi times d. We usually learn C = 2 pi r first because twice the radius (2r) is the diameter, and we usually draw a circle with a compass by measuring the radius.

Australian Math Teacher

So, half way around a circle (180 degrees) measures pi. All the way around (360 degrees) is 2 pi. 90 degrees is pi/2 … 45 degrees is pi/4… 30 degrees is pi/6 or one-twelfth of a circle or 1 o’clock.

We use the lowercase Greek letter pi because it begins the Greek words “perimeter” (“measure around” perimetron) and “periphery” (“carry around” perifero). It is something of a modernization because Euclid, Archimedes, and the other ancients only had what we now call capital letters. The lowercase letters were invented in the Middle Ages.

(The Middle Ages followed the Dark Ages, which is where software applications management and marketing teams live permanently. Blogspot has no Symbol font; neither does WordPress. We live in an age of ignorance. Seeking a personal Renaissance, I have a year to figure out the CSS and code my own or find a plug-in, sort of like when the West learned Arabic numerals.)

Also on Necessary Facts

Happy Pi Day 2013

Pi in the Sky over Austin (2014)

Patterns in Pi

Science in the Middle Ages

Saturday, February 28, 2015

The Unit Circle

Truth to tell, I was disappointed to find a really nice write-up on Wikipedia. I was hoping that this was still arcane knowledge. As far as I can tell, it is not commonly taught in trigonometry classes. We use the sine and cosine for problems with vectors; and that seems to be about it. In point of fact, these constructions are the root and rock of computational trigonometry. If you draw well and measure carefully, you can get two decimal places, or fractions to eighths.

Words have meaning.
If you ever suffered from a respiratory allergy,
then you know that it affects the "bay" the "sinus" cavities.
What we call the sine of an angle is the half-sine,
the semi-bay.
Similarly, our "tangent"
is the measure of the semi-tangent,
ignoring the reflection
below the part that we care about.

The trigonometric identities
come from the Pythagorean theorem.

In Feynman's Lost Lecture, the professor allows that we do our maths with algebra and calculus. We no longer rely on geometric constructions. Feynman had to create his own derivations for Newton's Laws simply because he could not follow Newton's easy claims about conic sections. Newton used geometry to create the calculus. However, calculus is such a powerful tool that we stopped learning the geometry that Newton knew.

You can write out the algebraic statements
but a picture is worth a thousand words.

In Cosmos, Carl Sagan tells that Pythagoras fled from Samos because he could not tolerate the tyrant Polycrates, whom Sagan denigrates for having "started out as something like a caterer." (Bold though he was, Sagan shared the anti-capitalist mentality.) But in The Ancient Engineers, de Camp tells us that engineers working for Polycrates bored through a mountain, starting at opposite ends, met in the middle, and were not off by a foot in a mile. Lacking the positional notation of Arabic numerals, of course, they did all their calculations with geometry.

Descartes is credited with uniting algebra and geometry.
His work reflects the seemingly intractable
analytic-synthetic dichotomy of philosophers.
In truth, it should have erased the distinction
between the logically consistent and the empirically verifiable.
See the works of Gregory Browne on this blog here and here.

Long ago, at Curious Books in East Lansing, I found a old manual for apprentice carpenters that showed sines, cosines, and tangents to 32nds of an inch for triangles of given measurement. Many times I have regretted not buying it then. Even before that, when Laurel and I were first married, her father asked us what calculus was all about. We told him. He asked, "Do you mean like this?" and took out a foot-sized caliper and showed us how he measured stair cases. Rise over Run. Ain't no doubt.

PREVIOUSLY ON NECESSARY FACTS
Pi in the Sky over Austin (2014)
Patent Nonsense
World Peace Through Massive Retaliation
Circumference
Anthropocene: A Bad Name for a Good Thing

Thursday, March 13, 2014

Pi in the Sky over Austin

Coming home at 6:30 PM, I looked up and saw a strange cloud formation... it resolved... 3 141592...

Five sky-writing aircraft from AirSign, a Florida aerial advertising company, will write out π, the Greek letter pi, the ratio of the circumference of a circle to its diameter. The calculation produces an unending series of digits, beginning with 3.141592. In a press release, the company said that its pilots will produce “several hundred characters” of the numerical sequence. Tomorrow, the company explained, is March 14, or 3.14, national "Pi Day." Building on that, the company says the planes will begin their 90-minute writing process at 6:28 p.m. because “6:28 is pi times two, a number some believe is the truer reflection of the power of the circle. (The Austin-American Statesman here.)

They circled twice continuing the expansion.

If you want a quick approximation, 355\113 = 3.141592... which is better than 22/7 = 3.1428... and it has an order and symmetry that makes it easy to remember: 3 55 over 11 3: all the numbers are odd, 1-3-5: 355113 starts and ends with 3. It's a piece of cake, well, okay, a piece of pie...

PREVIOUSLY ON NECESSARY FACTS
Happy Pi Day 2013
Patterns in Pi

Thursday, March 14, 2013

Happy Pi Day 2013

That we celebrate Pi Day is curious. Granted, e = 2.71828… does not lend itself well to February 71 or the 27^th day of the 18^th month, whereas March 14 exists and finds millions of kids in school mathematics classes. Yet our fascination with Pi – versus the speed of light, Planck’s Constant, G, or anything else – may speak to a basic belief that reality is not just easy to understand: it is fun.

You can make a pretty good compass from a length of string and a thumbtack. In ancient Egypt, geometers were called “arpedonapti”, those who knot ropes. (Garden of Archimedes at the University of Firenze (Florence) Italy in English here.) You can make a right angle by knotting any convenient length of string into a 3-4-5 triangle. With a known right angle, much else can be drawn and thereby computed. This was important to the Egyptians because the annual flooding of the Nile erased boundary lines.

The compass is more reliable, precise, and accurate than a knotted rope. The inventor that instrument has been lost to time. We can call him "Daedalus" for convenience. Clearly, the architect’s compass (bow compass or divider) had to have been a specific and sophisticated invention. The earliest known is from the 6^th century BCE. Wikipedia on the Drafting Compass. Wikipedia on the Divider Caliper. The invention of formal geometry is credited to Thales of Miletus. Pythagoras of Samos was a near contemporary. This was also the time when philosophy was supplanting religion for explanations, when coinage was replacing commodities as stores of value, and when hereditary rulers were replaced first by tyrants and then oligarchies and democracies. Those were heady times

By the European Middle Ages, the importance of builders (masons) made the compass a symbol for God’s work and order within the universe.

William Blake's Ancient of Days
from Wikimedia commons

The opinion that a benevolent God placed us within His creation was different from the classical view that life is unpredictable because even the gods cannot change the work of the Fates, three blind sisters who spin, measure, and cut the threads of our lives. In our postmodernist era, we have fallen back to that pagan view, perhaps best stated by Albert Einstein. While he denied that God plays dice, he felt that the universe is stranger than we can imagine. That may well be true – and it may also explain some aspect of our fixation on pi. It is at once understandable and unpredictable.

Yet, paradoxically, formulas to generate pi are known. (See the presentation on “Machin’s Formula” by Dr. Milan Milanović here.) Interestingly, you can find the nth digit of pi without knowing the preceding numbers. (See "Fun Facts" from the mathematics department of Harvey Mudd College here. )

One Million Digits of Pi: http://www.piday.org/million/

Four Million Digits of Pi: http://zenwerx.com/projects/pi-digits/pi/

Previously on Necessary Facts:

Patterns in Pi

“Slow Down and Think” by Daniel Kahneman reviewed here.

She's Such a Geek!
Visualizing Complex Data

Monday, November 5, 2012

Patterns in Pi

Five "heads" in a row! The next toss is likely a "tails." The gambler's fallacy rests on the erroneous assmption that small subsets will correlate highly with the universal set. Thinking Fast and Slow by Daniel Kahneman (reviewed on Necessary Facts here) demolishes this confusion of "intuitive" for "calculative" reasoning.

The following appear in pi to 100,000 places, (here). These numerical facts are highly suggestive -- and totally irrelevant.
The following runs appear:
00000
11111
22222
33333
55555 (twice)
66666
99999 (three)
999999
(But 44444, 77777, and 88888 do not appear at all.   President Obama insists that this is unjust and Gov. Romney insists that these are the ones that create jobs.)

The following single digits appear. (Should the government pass laws to equalize these disparities?)

0 125 of 10004
1 167 0f 10143
2 131 of 9908
3 81 of 10028
4 92 of 9970
5 29 of 10028
6 138 of 10030
7 133 of 10028
8 25 of 9978
9 94 of 9910

The following strings appear:
02468 (four times)
13579 (twice)
12345
23456
34567 - 3 times
56789 - 2 times
43210 - once and none others like 98765

31415 - pi itself
31416 - pi rounded
271828 - e the base of natural logarithms

000001 once but none of the others from 000002, …, 000009)

1776 - 15 times
1492 - 5 times

2997 (the speed of light in a vacuum in meters per second) 9 times
29978 (once)
66260 - Planck's constant (662608) once
66261 - Planck's constant once

17321 - the square root of 3, once
24495 - the square root of  6 once
26458 - the square root of  7 once
33166 - the square root of  11 (twice) (but none of the others below 10)
20500 (The White House ZIP code, once.)

Does order exist, or do we project order? Or is the truth stated entirely differently from either of those?

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