Saturday, March 14, 2015

It's Pi-Day!

This being year 15 in the Twenty-First Century, most of us in the general public became aware today, for the first time ever, that based on a tradition inaugurated by the San Francisco Exploratorium, a museum, in 1988, there is such a thing as Pi-Day every March 14th. That is because that famous number, begins 3.14159926…. This year only, the number of the year itself, ignoring those thousands, is an echo of π.

This led me ponder how π came to be discovered. I opened my Excel and started to play with ratios. To the toying mind, the ratio is relatively easily discovered. If you already know the circumference of a circle and its diameter, dividing the first by the last produces that magic number. Back before such wonders as Excel, the early geometer could use string and two sticks to draw a pretty good circle in sand—and then use more string to measure circumference and the diameter. Matching those strings will yield π more or less—in round numbers 3—but the circumference string will be ever-so-slightly longer. Early on the ratio calculated was 22/7th, which yield 3.1428… These days we know π out to 13.3 trillion digits—but using 3 or 22/7 will get there for pie-baking purposes. If we have a diameter or 4 inches, the formula

     Circumference = π * d

will yield 12.566 inches. If we substitute plain 3, the result will be 12 inches; if we use 22/7, the result will be 12.571 inches.

The value of π, however, was in calculating the actual Area of the circle. This was initially done by overlaying a circle with a polygon—from squares to up to 96-sided figures (Archimedes) and then calculating the area by calculating the area of the polygon—which is at least straight-forward. Eventually, by examining results so discovered, not least the ratio of those areas to the radius of the circle, yielded the understanding that if the area is divided by radius squared, we get the same number as we get by dividing the circumference by the diameter. Therefore the formula for the Area is….

     Area = π * r2

Worth noting here is that π required measurement before it came to be revealed. Circumference was easy, but area required major efforts to approximate the circle with shapes that have definitely measurable angles—as shown in the graphic from Wikipedia (link):



Herewith, finally, some ratios of a circle’s measurements just for the fun of it.

Some Ratios
r
d
C
A
C/(r+d)
A/((r+d)/2)
C/r
C/d
A/r
A/d
A/r^2
1
2
6.283
3.142
2.1
2.1
6.283
3.141593
3.142
1.571
3.141593
2
4
12.566
12.566
2.1
4.2
6.283
3.141593
6.283
3.142
3.141593
3
6
18.850
28.274
2.1
6.3
6.283
3.141593
9.425
4.712
3.141593
4
8
25.133
50.265
2.1
8.4
6.283
3.141593
12.566
6.283
3.141593
5
10
31.416
78.540
2.1
10.5
6.283
3.141593
15.708
7.854
3.141593
6
12
37.699
113.097
2.1
12.6
6.283
3.141593
18.850
9.425
3.141593
Formulas:
Symbols:
     Circumference = pi * d
r = radius
C = Circumference
     Area =  pi * (r*r)
d = diameter
A = Area

Much fun in an entirely open-ended way—to play with these ratios. That the numbers don’t come out clean, and that even after trillions of digits π does not yield a repeating series of decimals, makes me think of God’s sense of humor.

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