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Round all about, A long day on Venus

Q: What do you mean by the circumference of the circle? Who first found the formula for the circumference of a circle? When? (Raja, Chennai, India)

A: The circumference of a circle is the length of its boundary (also called the "perimeter"). The blue line of the second figure illustrates a circumference.

Circles in nature — armillaria mushrooms clustered about a round stump [Courtesy of Scott]

An astounding thing about circles is — if you divide the distance around it (the circumference) by the distance across it (the diameter) — you always get the same constant number, no matter how big or small the circle is. We denote that number by the Greek letter π (pi, which sounds like "pie") because it’s the first letter in the Greek word "peripherei," meaning perimeter or circumference. Pi, a non-repeating decimal expression that never ends, is 3.14 (to the first approximation — we have computed over 1 trillion digits of pi and found no pattern.)

So, expressing the words of the previous paragraph with an equation gives

π = C/d,

where C is the circumference and d is the diameter of the circle. This gives the formula for the circumference:

C = π d

People have known that "dividing the circumference by the diameter equals pi" for so long that it’s untraceable and likewise, the formula for the circumference.

The problem, though, is — what is the value of pi? People, through the ages, sneaked up on its value by measurements. They measured the diameters of circles and (by curling a string or a rope around the outside and then measuring the stretched out string) the circumferences. Then, they determined pi by dividing the circumference by the diameter.

The Bible, for one, implies its value is 3. The "pi" bible passage lists specifications for Solomon’s temple (built around 950 BC) and says,

And he made a molten sea, ten cubits from one brim to another: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7,23).

Here, the text describes a large brass casting, with 10-cubit diameter and 30-cubit circumference. Thus, π is 3. A cubit originally was the length of the forearm from the tip of the middle finger to the elbow. Anyway, the circumference-pi formula was known in 950 BC.

Before that even, the Babylonians knew that pi was about 3 1/8 (3.125), a more accurate figure.

In about 1650 BC, an Egyptian manuscript, the Rhind Papyrus, gives the value (accurate to within less than one percent): π = 4(8/9)² = 3.16.

Thus far through the history of pi, though, people approximated pi strictly through physical measurements. In about 250 BC, Archimedes was the first to give a rigorous scientific estimate of pi: a value between (3 and 10/71) and (3 and 10/70). He proved mathematically that the first three digits of pi are 3.14.

A polygon inscribed inside the circle and another circumscribed outside — together enclosing the circle. [Drawing by the author.]

His geometric argument rested on the fact that a circle can be squeezed ever closer by polygons inscribed inside and circumscribed outside. See figure. He knew that the area of a circle is its radius squared times pi. He could calculate the area of the polygons. So, he just increased the number of polygons until he got the accuracy he wanted: pi lies between (3 and 10/71) and (3 and 10/70). That took 96 polygons. It also involved a fantastic imagination and incredible powers of calculation (without the help of decimal notation, trigonometry, or algebra). Clever.

Here’s an extra challenge from mathematician Ilan Vardi: "To test your deeper understanding, prove that pi exists. 50% credit if you can explain what the question means." Vardi’s proof and discussion.

By the way, mathematician John O’Connor, of the University of St Andrews, Scotland, gives this mnemonic for the first 24 digits of pi is the following scheme: each successive digit of pi is the number of letters in the corresponding word of this sentence — How (3) I (1) want (4) a (1) drink (5), alcoholic (9) of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard...

Thus, π = 3.14159265358979323846264...

Further Reading:

Pi  (A surprise)

University of St Andrews, Scotland: A history of pi by JJ O’Connor and EF Robertson

Proof that pi exists by Ilan Vardi

Wikipedia: The Rhind Papyrus

Google: define pi

Wikipedia: the letter pi

Readers' comments:

I wanted to point out that in Greek, Ð does not sound like "pie" but like "pea" or "pee." The prounciation "pie" is likely due to the obvious conflict it would have in mathematics with the Latin character P that shares the same pronunciation.

Brian, Ridgecrest, California

Q: How long does it take Venus to orbit the sun? (Matthew, Bromley, England)

A: She orbits the Sun in 225 Earth days — at a distance that’s about 72% of Earth’s orbit. Venus moves about 18% faster along her orbit and so her year is shorter than ours.

More intriguing, though, is Venus’ day. She rotates about her axis incredibly slowly — 243 of our days (as compared with our 1-day rotation). Venus’ day is longer than her year! We don’t know why her day is so long.

Venus landscape, the 10,000-foot (3000-m) high volcano, Gula Mons, appears on the horizon. [NASA/JSC]

Also, another strange thing — Venus rotates about her axis in the opposite direction that she orbits the Sun. Again, we don’t know why. Perhaps — eons ago when the solar system was first forming and chaotic bodies, large and small, circled Sol — she was hit by a gigantic body moving in a direction opposite to her earlier forward rotation. The collision might have stopped her forward spin and started her spinning slowly backwards from then on.

"The thick and largely featureless Venusian atmosphere hides the surface from view so it always looks pretty much the same through a telescope, at least in visible light," emails astronomer Robert Massey of the Royal Observatory Greenwich in London.

"So, early astronomers basically guessed the length of the day but the estimates were wide-ranging. In 1961 scientists at radio observatories in Jodrell Bank (England), Goldstone (California) and Evpatoriia (the then Soviet Union) used radar to measure the rotation rate. Today's accepted value comes from this and later observations from space probes like the Magellan orbiter that surveyed Venus in the early 1990s."

Further Reading:

Royal Observatory Greenwich: Venus

Nine-planets.org: Venus

StarrySkies.com: A day and a year on Venus

University of Mississippi: Venus by Luca Bombella

Tuorla Observatory, Finland: Discovery of the first quasi satellite of Venus

University of Michigan: Windows to the universe — Our solar system

(Answered Oct. 21, 2005)