Do You Love Me?

"You broke my heart,
'Cause I couldn't dance.
You didn't even want me around.
And now I'm back,
to let you know,
I can really shake 'em down.”

As a teenager growing up in central Montana, dances were a very important social event. Whether at the high school, the local teen club, or the Eagles, these were opportunities to make it of break it with the local gals. And, as a teenager, the local gals were very high on my list of priorities.

But, sadly, I couldn’t dance. Not that that really mattered much. The key was getting up the gumption to just ask a girl to dance. And be prepared for rejection. Oh, they did say yes now and then. And those were good times.

The music was always part of it. Even if you couldn’t get a dance partner, you could still groove to the music. That’s right kids of today, we would GROOVE to the music. Sometimes it was just records and we took off our shoes to save the gym floor: sock hop. Other times it was local bands. Especially John Uribe and the Sultans. Lot’s of memories that I’m sure many from my time and place share.

Those six little spoken lines that I started this essay and the song that comes after were written by Motown CEO Berry Gordy, Jr. There is some controversy whether he wrote the song for The Temptations or originally for the group that had it as their one big hit. That would be The Contours.

Now I admit I probably heard it first covered by some NW band such as the The Wailers, Kingsmen, or Paul Revere and the Raiders. The song has been covered umpteen times and was even resurrected when it was used for the soundtrack of “Dirty Dancing.”

The Contours struggled throughout their career and this was there only really big hit, their third try. Funny about how they got into the business. A possible lesson for anyone on dealing with rejection.

Joe Billingslea and Billy Gordon founded a singing group called The Blenders in their native Detroit, Michigan in 1959. They completed the group with Billy Hoggs and Billy Rollins, who had responded to an ad placed in the local newspaper by Billingslea. The group soon added Leroy Fair (in place of Billy Rollins) and bass singer Hubert Johnson and changed the name to The Contours.

In the fall of 1960, the group auditioned for Berry Gordy's Motown Records. Gordy turned the act down, prompting the group to pay a visit to the home of Johnson's cousin, R&B star and Gordy associate Jackie Wilson. Wilson in turn got The Contours a second audition with Gordy, at which they sang the same songs they had at the first audition, the same way, and were signed to a seven-year contract. Get that, second audition, same as the first, but this time they got the job! Or was that Jackie Wilson’s doing?

The 1962 release of their third song with its shouted lead vocals from Billy Gordon and references to several popular dances such as the Mash Potato, hit #1 on Billboard's R&B chart and crossed over to #3 on the Hot 100 in 1962. It sold over one million copies, and was awarded a gold disc.

Founding member Billingslea, now 77 years old, continues to perform with his group, The Contours with Joe Billingslea, which is among the acts featured in a DVD released by Motown in January 2007 called “Motown: The Early Years,” featuring its appearances on the Public Broadcasting System specials. In March 2010, The Contours were inducted into the Doo-Wop Hall of Fame. The band has released 15 singles, many of which charted although never higher than #16, although the ’88 rerelease of “Do You Love” me went to #11.

They have ten albums including a 2011 release and you’ll find their work on several Motown compilation albums. There have been various ventures by individual band members and a few law suites over use of the name, but then lawyers were always an essential part of rock and roll.

Le’ me see you dance.
He can’t dance.
You can’t dance.
Watch me now!

Super Bowl 50

Fifty Years of Super Bowls. So how are the teams doing? Who are the perennial champions and who are the flashes in the pan? I just love statistics. So I sought out these numbers to get it all straight. It didn’t take a lot of searching. As usual, Wikipedia has it all ‘splained out.

Since the inaugural game on January 15, 1967 there have been 50 contests, with this year’s to be decided on Sunday, February 7. (Notice how the season has lengthened and the big game coming later in the year. That’s one kind of progress.)

So which teams have been there the most? Won the most? Lost the most? Never even been to the big dance? Well here’s how it maps out. There are four teams that, over the last 50 years, (which represents 100 opportunities since there are two teams in each Super Bowl,) have played in the final showdown the most.

On top of that list of four are the Pittsburg Steelers. They’ve played in 8 Super Bowls and won 6 for a 75% winning record. They were there in the 70s, the 90s, and 4 times in the 21st century. They were, are, and continue to be a contender with their last appearance in 2010.

Next in order is the Dallas Cowboys. They also played in 8 big games and won 5 keeping their record over .500. All these successes were in the 70s and the 90s, with no games since 1995.

The New England Patriots are 4 and 4 in their appearances and had an opportunity to jump to 9 Super Bowl appearances before the Denver Broncos beat them in this year’s AFC final. They have played in Super Bowls in ’85 and ’96, and have been most successful in the 21st century, including a win last year.

Last in the list of top teams are the Denver Broncos. This year is their 8th appearance, but their record is a rather dismal 2 wins out of 7 attempts. It remains to be seen if they will improve that record this year. This also means they hold the record for losing the most Super Bowls, a stat that sits at 5 losses pending the outcome of this week’s game. Only challenged in this numerical category by the Vikings and Buffalo Bills, each with four losses amongst four attempts — the Bills with the terrible result of losing four in a row from ’90 to ’93.

Certainly the Broncos hold many records for poor performance with large losses and blowouts in many of their appearances. On the positive side, they’ve been a constant participant with Super Bowl appearances in the 70s, 80s, 90s, and an upcoming second appearance in this century. Not many other teams have been such a regular, with Denver appearing in 4 of the 6 decades of play. The Patriots have also been in 4 decades, but their first time was in ’85.

The San Francisco 49ers have 6 appearances based largely on a long run in the 90s. Five teams have 5 overall appearances including the Packers, Giants, Raiders, Red Skins, and Dolphins. Best of that group is the four wins for Green Bay and New York. Green Bay may have the longest range of successes with a win in the first Super Bowl as well as a win in 2010. The Dolphin’s successes were all back in the 70s and 80s, before most of today’s fans were even born.

The Colts join the Vikes and the Bills in playing in four games, but better their records with two wins. The Rams and the Seahawks have been three times, each with one win. The St. Louis Rams lost in ’79, and the L.A. version won one and lost one in this century. The Seahawks success has been quite recent.

Six teams have been there twice, including the Ravens, Chiefs, Bears, Eagles, Bengals, and this year’s Panthers. The Baltimore Ravens have bested this group with two wins, while the Eagles, Bengals, and Panthers have yet to win one.

With only a single appearance we have seven: the Jets, Buccaneers, Saints, Chargers, Falcons, Titans, and Cardinals. Only the first three have managed a win.

So who has never been to the Super Bowl of the current teams? Well this sad lot includes the Cleveland Browns, a team the Broncos kept out of the final game twice back in the 80s. Also with no trips are the Detroit Lions, Jaguars, and Houston Texans, that last team being relatively new to the NFL formed in 2002 and the Jags have only been around since ’95.

Although the Broncos have not done well with their visits, this year’s contest as yet undecided, they are in the elite group of teams with repetitive wins. After winning a Super Bowl, the following season is very difficult for many reasons. Only seven teams have ever won back-to-back with the Steelers repeating that conquest in both IX and X and XIII and XIV in the 70s when they pretty much owned the NFL. Other teams with two in a row include Green Bay with the first two Bowls (and a win of the NFL Championship prior to the Super Bowl), the Dolphins, San Francisco, and the Broncos and the Patriots. No team has ever won three in a row, and few have even been there three times consecutively except for the Dolphins who lost the first of three in the early 70s and those poor Bills who played four-in-a-row without a single victory in the 90s.

So there you have the stats for the upcoming game. Denver continues success getting to the final contest, and in less than a week we’ll know if they added a win to their success rate, or add another poor performance and blow-out to a pretty dismal showing in the big game. Still, as a Denver fan since before they started playing in the final show down, I’m just super excited they get another chance. They have been a consistent power house in football since the 70s, and that is a record few other teams or cities can match.

Have the Broncos replaced the Cowboys as “America’s Team”? Some would say so. Me, I’m waiting for the numbers and the final tally. There have been 50 of these final contests during my lifetime, and each one has been an exciting visit by the home teams. They haven’t all been good games or close games or entertaining games, but they’ve all been “super.” Just ask any team that hasn’t been there … recently or at all.

Of course it will soon start all over … the season, the playoffs, Super Bowl 51 (LI) … so there is still a chance that your team will be added to this roll of honor. Will the Patriots up their score to 9 appearances? Will the Broncos be back? What about Green Bay? The Bears? Cleveland? Well, we’ll have to wait another year for that. Right now I’m getting ready for SUNDAY!

Oliver Heaviside

I’ve written about the brilliance of James Clerk Maxwell, the man who developed a complete set of equations that describe electro-magnetic emissions that we commonly call “radio waves.” His work was one of the key stepping stones to our modern understanding of the universe including the constant speed of light and Einstein’s equations of relativity.

But these advances, key as they are, were more like tiny steps with many minds slowly unraveling the mysteries of creation. Maxwell’s breakthrough was a giant leap, but it was based on the previous insights of others including Michael Faraday, Johann Carl Friedrich Gauss (one of the three greatest mathematicians of all time in addition to a damn good physicist), and André-Marie Ampère. Throw in some important work by Hendrik Antoon Lorentz and Oliver Heaviside and you arrive at the refined version: the four vector equations now commonly known on college campuses as Maxwell’s equations. (I’m sure you’ve seen the tee shirt.)

The Lorentz Force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. The first derivation of the Lorentz force is commonly attributed to Oliver Heaviside in 1889, although other historians suggest an earlier origin in an 1865 paper by James Clerk Maxwell. Hendrik Lorentz derived it a few years after Heaviside.

May 18 is the birthday of Heaviside, who was born in 1850 in London. Heaviside's uncle was Charles Wheatstone, who co-invented one of the first practical telegraph systems. Wheatstone supported his nephew's education and his career in the growing field of telegraphy. By the age of 22, Heaviside had published a paper on the best way to use a Wheatstone bridge.

In 1873, when Heaviside came across James Clerk Maxwell's A Treatise on Electricity and Magnetism, he recognized its revolutionary importance and resolved to study the mathematics needed to understand it. Heaviside went beyond merely understanding the work. Using vector notation, he recast Maxwell's original 20 equations into the four familiar equations that we learn today.

He was a self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations (later found to be equivalent to Laplace transforms), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science for years to come.

He applied the equations to investigate such problems as the behavior of charges moving in electromagnetic fields and to predict the existence of an ionized layer in Earth's upper atmosphere that allows the transmission of radio signals around the planet's curved surface. He also invented and patented the coaxial cable for transmitting radio-frequency electromagnetic signals.

Perhaps he is best known among the general public for an upper layer of the Earth’s atmosphere that carries his name. The Kennelly–Heaviside layer, named after Arthur E. Kennelly and Heaviside, also known as the E region or simply the Heaviside layer, is a layer of ionized gas occurring between roughly 56–93 miles above the ground — one of several layers in the Earth's ionosphere. It reflects medium-frequency radio waves, and because of this reflection, radio waves can be propagated beyond the horizon.

This atmospheric phenomenon has appeared in more than just scientific texts. The Heaviside layer is used as a metaphor for heaven in Andrew Lloyd Webber's musical Cats. This reference is based on a quotation found in a letter written by T. S. Eliot, whose book Old Possum's Book of Practical Cats forms the basis of the musical. In the musical, one cat is chosen each year by Old Deuteronomy to go to the Heaviside Layer and begin a new life. In the song "The Journey to the Heaviside Layer,” it is stated that the Heaviside Layer is "past the Russell Hotel" and "past the Jellicle moon,” indicating that it is very far away and difficult to access. (“Jellicle” is a kind of cat, commonly nocturnal black-and-white cats.)

But I’m just “Ham”-ing it up with that fact.

Happy birthday Oliver.

Pi -- Part Four

In 1706, William Jones — a self-taught mathematician — published his seminal work, Synopsis palmariorum matheseos, roughly translated as “A summary of achievements in mathematics.”

It is a work of great historical interest because it is where the symbol “π” appears for the first time in scientific literature to denote the ratio of a circle’s Circumference to its Diameter. This is where the association of the Greek letter and the important ratio of circle measurements began.

Jones realized that the decimal 3.141592 … never ends and that it cannot be expressed precisely. “The exact proportion between the diameter and the circumference can never be expressed in numbers,” he wrote. That was why he recognized that it needed its own symbol to represent it.

It is thought that he chose π either because it is first letter of the word for periphery (περιφέρεια) or because it is the first letter of the word for perimeter (περίμετρος). (Or because of both).

The symbol π was popularized in 1737 by the Swiss mathematician Leonhard Euler (1707–83), but it wasn’t until as late as 1934 that the symbol was adopted universally. By now, π is instantly recognized by school pupils worldwide, but few know that its history can be traced back to a small village in the heart of Anglesey, an island off the north west coast of Wales.

William Jones was born in 1674 on a small holding close to the village of Capel Coch in the parish of Llanfihangel Tre’r Beirdd, north of the county town of Llangefni in the middle of the island.

When he was still a small child the family moved a few miles further north to the village of Llanbabo. He attended the charity school at nearby Llanfechell, where his early mathematical skills were drawn to the attention of the local squire and landowner, who arranged for Jones to go to London, where he was given a position as a merchant’s accountant. He later sailed to the West Indies, an experience that began his interest in navigation.

When he reached the age of 20, Jones was appointed to a post on a warship to give lessons in mathematics to the crew. Based on that experience, he published his first book in 1702 on the mathematics of navigation as a practical guide for sailing. On his return to Britain he began to teach mathematics in London, possibly starting by holding classes in coffee shops for a small fee. Shortly afterwards he published Synopsis palmariorum matheseos, a book written in English, despite the Latin title.

William Jones became friendly with Sir Thomas Parker, later the Earl of Macclesfield, and tutored the young George Parker, who was to become the second Earl. He later lived at the family home, Shirburn Castle, near Oxford, where he developed close links with the family. Through his numerous connections William Jones amassed at Shirburn an incomparable library of books on science and mathematics. He also maintained links with Wales, particularly through the Morrises of Anglesey, a family of literary brothers renowned for their cultural influences and activities who, although a generation younger than William, came from the same part of Anglesey and had strong London-based connections.

In the wake of publishing his Synopsis, William Jones was noticed by two of Britain’s foremost mathematicians: Edmund Halley (who had a comet named after him) and Sir Isaac Newton. He was elected a Fellow of the Royal Society (FRS) in 1711 and was vice-president of the society during part of Sir Isaac Newton’s presidency. William Jones became an important and influential member of the scientific establishment. He also copied, edited and published many of Newton’s manuscripts. In 1712 he was appointed a member of a committee established by the Royal Society to determine whether the Englishman, Isaac Newton, or the German, Gottfried Wilhelm Leibniz, should be accorded the accolade of having invented the calculus — one of the jewels in the crown of contemporary mathematics. Not surprisingly, considering the circumstances, the committee adjudged in favor of Newton.

In his will William Jones bequeathed his library of roughly 15,000 books together with some 50,000 manuscript pages, many in Newton’s hand, to the third Earl of Macclesfield. Some 350 of these books and manuscripts were written in Welsh, and this portion of the original library was safeguarded in about 1900 to form the Shirburn Collection at the National Library of Wales in Aberystwyth.

And so concludes my short expedition seeking the story of π. Whether these mathematical relationships are invented or discovered, we do owe the simple Greek letter all school children learn early in their mathematical careers to the Welshman of some fame.

The Welch term for “beauty,” is “harddwch.” And there’s beauty in that too, although I have no idea how to pronounce it. And so ends our journey through mathematics. May you find enlightenment and beauty in the story I’ve told.

Pi -- Part Three

But there’s still more to π. After all, other famous irrational numbers, like e (the base of natural logarithms) and the square root of two, bridge different areas of mathematics, and they, too, have never-ending, seemingly random sequences of digits.

What distinguishes π from all other numbers is its connection to cycles. For those of us interested in the applications of mathematics to the real world, this makes π indispensable. Whenever we think about rhythms — processes that repeat periodically, with a fixed tempo, like a pulsing heart or a planet orbiting the sun — we inevitably encounter π. There it is in the formula for a Fourier series:

A Fourier series is a way to represent a wave-like function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The Discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series.

The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1. Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem applied, among other uses, in the encoding of CDs, DVDs, and modern "digital" television transmission. The study of Fourier series is a branch of Fourier Analysis.

The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Harmonic analysis is among the most successful and applicable branches of modern mathematics. It is an indispensable tool in subjects ranging from number theory to partial differential equations and numerical analysis. All that starts with a simple circle and the relationship given the name of the most famous Greek letter.

Combining the trigonometric functions of sine and cosine, the Fourier series, among other interesting facts, embodies the concept that all complex waveforms from music to the strange waves produced in computer circuits are just made up of an appropriate combination of basic “sine waves” from simple harmonic motion, spinning in a circle, and you know about circles and pi, they go together like apple pie and ice cream. Yummy … and so ubiquitous it fits everything from the rotation of our galaxy to spinning atoms and all the sizes between.

The Fourier series is an all-encompassing representation of any process, x(t), that repeats every T units of time. The building blocks of the formula are pi and the sine and cosine functions from trigonometry. Through the Fourier series, pi appears in the math that describes the gentle breathing of a baby and the circadian rhythms of sleep as well as the wakefulness that govern our bodies. When structural engineers need to design buildings to withstand earthquakes, pi always shows up in their calculations. Pi is inescapable because cycles are the temporal cousins of circles; they are to time as circles are to space. Pi is at the heart of both.

For this reason, π is intimately associated with waves, from the ebb and flow of the ocean’s tides to the electromagnetic waves that let us communicate wirelessly. At a deeper level, π appears in both the statement of Heisenberg’s uncertainty principle and the Schrödinger wave equation, which capture the fundamental behavior of atoms and subatomic particles in quantum mechanics. In short, π is woven into our descriptions of the innermost workings of the universe.

How’s that for an inclusive statement? All from this ratio of the simplest measurements of a circle. This number that seems to have no end … either in digits or in uses. Now that is beautiful!

Pi — Part Four

Pi -- Part Two

Pi is irrational, meaning it cannot be expressed as the ratio of two whole numbers. There is no way to write it down exactly: Its decimal expansion continues endlessly without ever settling into a repeating pattern. No less an authority than Pythagoras repudiated the existence of such numbers, declaring them incompatible with an intelligently designed universe.

I was taught in school the simple fraction 22/7 as an approximation for pi. It only has 3 digits, yet it is a bit closer to the correct value than the apparently equivalent 3.14 in digits.

Twenty-two-sevenths works out to 3.142857… compared to the actual pi of 3.14159…. Another approximation from history is 256/81= 3.16049… and 339/108 = 3.1888…. The Chinese used 3927/1250 = 3.1416 exactly … about the closest of any of these attempts. Whether these ancients thought these were approximations or simply the best they could come up with isn’t clear.

The ancient Greeks had a puzzle that was never solved. It is called “squaring the circle.” That means to create a square using just the geometric tools of a compass and a straight edge that has the same area as a given circle. Modern mathematicians know that this is impossible since we now understand that pi is transcendental which means it that is not algebraic — that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent transcendental numbers are π and e, the base of natural logarithms. Further, it can be proven that any shape created with only a compass and straight edge are the algebraic numbers and doesn't include a value like pi.

So the ancient search for a method to “square a circle” is now known to be impossible. Sometimes in math and science it is as important to know what is impossible as it is to know how to calculate something.

It’s fair to ask: Why do mathematicians care so much about π? Is it some kind of weird circle fixation? Hardly. The beauty of π, in part, is that it puts infinity within reach. Even young children get this. The digits of π never end and never show a pattern. They go on forever, seemingly at random — except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of π.

And yet π, being the ratio of a circle’s Circumference to its Diameter, is manifested all around us. For instance, the meandering length of a gently sloping river between source and mouth approaches, on average, π times its straight-line distance. Pi reminds us that the universe is what it is, that it doesn’t subscribe to our ideas of mathematical convenience.

Pi also opens a window into a more uncharted universe, the one consisting of transcendental numbers, which exclude such common irrationals as square and cube roots. Pi is one of the few transcendentals we ever encounter. One may suspect that such numbers would be quite rare, but actually, the opposite is true. Out of the totality of numbers, almost all are transcendental. Pi reveals how limited human knowledge is, how there exist teeming realms we might never explore.

A short explanation of rational and irrational numbers. Start with the integers … the whole numbers. (We’ll ignore zero and negative values.) Rational numbers are any number that can be formed from a ratio of integers or whole numbers. That is, a fraction of integers like 22/7. The ancient Greeks thought that was all there is. That’s why they called them “rational.” Then some guy figured out that the square root of 2 can’t be rational. He proved that, if it was a ratio or fraction of whole numbers then the numbers were not even, nor were they odd. Since all integers must be one or the other, there could not be such a ratio. That made such an impact on the Pythagoreans (a club that the guy was in and famous for their theorem or formula) threw him out of the boat … literally … they drowned him … or so the story goes.

Makes a good story whether true or not. One aspect of irrational numbers is that they can not be represented by a finite decimal expansion such as 1/2 = 0.5 or a repeating decimal such as 1/3 = 0.3333… (or 1/11 = 0.090909…). Transcendental numbers are even more complicated. They can’t even be represented by any root of a regular polynomial equation … but I promised to keep this simple, so I’ll stop there.

The combination of utility and mystery makes π a perfect symbol for all of mathematics. Surely the ancients, had they understood π better, would have worshipped it, just as they did the moon and the sun. They would have praised pi’s immutability: Pi = 3.14159... is one of the few absolutes that remain, unchangeable in a world of temporary existence.

Or is it absolute? The ratio of circumference to diameter might not be as fixed as we think. To understand why, imagine a circle drawn on the surface of a sphere. Its diameter, as measured along the bulging surface, will be greater than if the same circle is traced out on a flat sheet of paper. This observation might have been of only academic interest except for our inability, so far, to definitively determine whether the geometry of our universe is flat. If there is even a little curvature, then the value of π, as defined by this ratio, is not what we think. Thanks to Einstein we now have an absolute speed of light, but π might not be fixed.

Yes π, on cue, reminds us that it is an abstraction, like all else in mathematics. The perfect flat circle is impossible to realize in practice. An area calculated using π will never exactly match the same area measured physically. This is to be expected whenever we approximate reality using the idealizations of math.

To this day, some think that 22/7 or 3.1416 are the exact value of pi. Perhaps they fear the unknown of the unknowable and are as traumatized as Pythagoras by the idea of a non-fractional universe. The Indiana State General Assembly once proposed establishing pi as something like 3.2 to make commerce easier, but the law didn’t pass because Professor C. A. Waldo of Purdue University talked them out of it.

Maybe life would be simpler if pi = 3.2, but I would argue that life could not exist if π has such a simple value. There is no reason to be afraid. I’ve learned that it’s only when we try to stretch our minds around mathematics’ enigmas that true understanding can set in.

That’s the beauty of it!

Pi — Part Three

Pi -- Part One

I talk and write about the “beauty of math” all the time, but it is very hard to get the idea across. Those that are familiar with advanced math — and it doesn’t have to be all that “advanced” — get it. They understand what I’m saying. But those that haven’t really studied the queen of the sciences just don’t realize what I’m talking about. It isn’t like a beautiful painting or a powerful sunset or even a lovely women. No, the beauty of math isn’t in the eye, but in the brain of the beholder. Sure, there are some neat graphics and drawings with a mathematical basis, but the beauty I’m referring to is an “inner beauty.” It is the purest thought stuff and the excitation is directly in the aesthetics part of the human mind and soul. Philosophers may argue about it. There may be traditions and periods and literature describing beauty. There can be schools and branches and lists and theories. But that is a cold attempt at capturing the internal joy that true beauty brings to one.

So it seems the question is, does a relatively simple mathematical example portray this beauty that the learned mathematician speaks so highly of? Is there some “simple” math that also is deep in this special quality that aestheticians or epistemologists (or metaphysicians) so elegantly long for? I think the answer is yes. This is a little bit of math that was introduced to most students in the 6th grade level have conquered this principle and have a good understanding of … at the very least … some basic facts about pi (π).

And what are those basic facts? I’m thinking about the ratio of the Circumference of a circle (the distance around a circle) to the Diameter of a circle (the length of a line across a circle that passes through its center). These sixth graders know that it is symbolized by the Greek letter π called “pi” that rhymes with Apple Pie. It is approximately equal to 3.14 or 22/7. And that is another fact they typically have conquered: there is no precise value for pi. It is an irrational number, although sixth graders may not use the term “irrational.” But they mostly know you can’t write the number down. It just goes on, and on, and on: 3.1415926535897932385 … That’s the value to 20 places, but there are more, and they don’t repeat, they appear to be random and non-ending … because they are.

Many mathematicians celebrate Pi Day each year on March 14 or 3/14. This year (2015) was especially interesting pi day because the year plus the hour, minutes, and seconds, extended the number to more decimals than will occur for over a 1,000 years. March 14 is also Albert Einstein's birthday!

Of course, these same sixth graders will likely use pi in various equations and may even physically measure a few circles just to confirm, as best simple measurements can confirm, that the number is in the neighborhood of a 3.14. The formula for the area of a circle will be introduced and concepts such as “squaring” take on practical mathematical usefulness. But there is so much more. What else could we teach these sixth graders, or perhaps someone with a bit more math education like a High School graduate? Or we could just state that cake are square … not pie … old joke!

From the basic definition that the Circumference equal pi times the Diameter, we can write this as an equation: C = π D. Further, we know that the radius, r, is half of the Diameter, so this would be: C = 2πr.

The next thing we learned in the sixth grade math class was how to calculate the area of a circle. That too used this magic number pi. I still recall how it was explained to me so long ago at Garfield Elementary school. You take the circle and divide it up into sections (called segments). You then rearrange the segments into this nearly rectangular shape.

The shape on the right is a parallelogram and the formula for the area of a parallelogram is A = b x h, or base times height. The base is half of the circumference, since the other half makes up the top. If the total circumference is 2πr, then half of the circumference would be πr. Since height of each segment is the radius, then height would be r. So the area of the circle is πr times r which equals πr².

When I first saw this example I had a problem. It isn’t exactly a parallelogram. The top and the bottom are “wavy” and the wave part reduces the area from a “real” parallelogram. So the formula seemed bogus. Then I thought about what the teacher had said about there being an infinite number of numbers in pi and that got me thinking about infinity. What if, instead of cutting the circle into 8 segments, you cut it into 16 and made the parallelogram out of the 16. The waviness would be less and the formula would seem more accurate.

Keep up dividing the circle into smaller and smaller pieces and building the parallelogram. As the number of segments goes to infinity (I now know the correct description is “approaches” infinity), the waviness would disappear and the formula would be exactly right. If you make the parallelogram out of the the tiniest slices imaginable, then the curvature would disappear. And there you have it … as a sixth grade student I’d just invented the Calculus. Well, sort of.

Are you starting to see the beauty of mathematics? How you can play games in your mind and do impossible things with thought. Einstein called these little mental exercises (in German, of course) gedanken experiments.

Early mathematicians realized pi’s usefulness in calculating areas, which is why they spent so much effort trying to dig its digits out. Archimedes used 96-sided polygons to painstakingly approximate the circle and showed that pi lay between 223/71 and 22/7.

By calculating the area of the polygon drawn within the circle you get the lower bound for pi. The polygon that circumscribes the circle is a bit larger and gives the upper bound. If you increase the size of the polygon … the number of sides … you get values closer and closer to the actual value of pi. But this method is very tedious and better methods were soon found.

By the time Madhava (in India, around 1400) calculated pi to over 10 decimal places using his groundbreaking infinite series (which regrettably bears Leibniz’s name), it was already more than accurate enough to address all practical applications. Pursuing pi further had essentially become a mathematical challenge.

The equation we now call “Leibniz’s Equation” is very interesting for its simplicity. It is an infinite equation, which means it has an infinite number of terms. After all, if an equation existed that didn’t have an infinite number of terms then that would imply pi has an exact or “algebraic” value, and we know it does not.

This interesting equation is simply the sum and difference of all the odd numbers written as fractions. Actually it is the formula for one quarter of pi, so you have to multiply the result by 4.

One, minus a third, plus a fifth, minus a seventh, etc. is the calculation. Here, in the terse symbols of math is what I just said.

Here it is in summation notation.

Leibniz's formula converges extremely slowly: it exhibits sublinear convergence. Calculating π to 10 correct decimal places using direct summation of the series requires about five billion terms. There are much better formulas that converge on the correct answer with less terms. These “better” equations are now used to calculate pi. Two mathematicians, Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo, calculated pi to 206,158,430,000 decimal places in 1999. And that’s not the record.

Still, imagine the power and beauty of mathematics. This simple yet difficult to write down fully number that comes from the simple forms of geometry and measuring a circle also shows up in an infinite series of the odd numbers. How could this be? But wait, even more amazing “coincidences” (or are they) will appear as we dig into the simple, yet complex number pi. This is part of the beauty of mathematics I keep talking about. Are you starting to “grok” it?

With the advent of computers, pi offered a proving ground for successively faster models. But eventually, breathless headlines about newly cracked digits became less compelling, and the big players moved on. Recent records (currently in the trillions of digits) have mostly been set on custom-built personal computers. The history of pi illustrates how far computing has progressed, and how much we now take it for granted.

So what use have all those digits been put to? Statistical tests have suggested that not only are they random, but that any string of them occurs just as often as any other of the same length. This implies that, if you coded this monograph, or any other article or book, as a numerical string, you could find it somewhere in the decimal expansion of pi. One could argue that all knowledge of man, both current and future discoveries, is hidden somewhere in the digits of pi. Don't be too amazed, that's just the concept of probability extended to the infinite.

Of course, that’s relatively useless, since you don’t know where to find the material you want … exactly where in the decimal expansion of pi would you look. An apt metaphor for an age when we are being asphyxiated by mushrooming clouds of information.

But pi’s infinite randomness can also be seen more as richness. What amazes me, then, is the possibility that such profusion can come from a rule so simple: Circumference divided by Diameter. This is characteristic of mathematics, whereby elementary formulas can give rise to surprisingly varied phenomena. For instance, the humble quadratic can be used to model everything from the growth of bacterial populations to the manifestation of chaos. Pi makes me wonder if our universe’s complexity emerges from similarly simple mathematical building blocks. That is a basic belief of science encoded in Occam's Razor, the philosophical concept that the simplest explanation is the correct one. Without that belief, science would probably be impossible. Did you know that Science involves "faith"? Yes it does. That's rather beautiful in itself.

Oh, I’ve got a lot more to say about pi. There is so much more wonder and mystery in this simple concept known since the beginning of civilization. I’ll let my humble readers digest this morsel before I pour on more syrup and add some butter. So, until the next installment, happy trails.

Pi — Part Two