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Section4Revolutionary 8

We turn our attention now to the scientific revolution. While \(8\) entered this enlightened time admired by bunny lovers and chocolate aficionados with a penchant for gathering in groups of five, it had not yet joined the ranks of the truly great numbers. But that was about to change, in large part due to its collaboration with two giants of the age: Isaac Newton and Leonhard Euler.

No one embodies the scientific revolution more than Sir Isaac Newton. Newton was born on Christmas Day in 1642, two months after his father died. He survived a premature birth and the plague, no small feat in mid 17th century England. Indeed, in 1665 Cambridge colleges closed their doors due to an outbreak of plague. Sent home, where he had no access to the internet, Newton experienced his anni mirabilis, 20 miraculous months of intense creativity during which he formulated the key ideas of all his major discoveries: gravitation, optics, and, central to our story today, the calculus.

In calculus one investigates change, typically of some process over smaller and smaller intervals. As such, very small, nebulous positive quantities such as \(\Delta x\) and \(\Delta y\) came to steal the spotlight from many larger numbers such as three. How did \(8\) manage in this age of infinitesimals? By standing on the shoulders of many tiny numbers, as we shall see.

Newton was an astute observer of the natural world, although he occasionally sat beneath over-ripe apples with little regard for his personal safety. For instance, in his youth he put forth a serious effort to quantify his happiness. He even developed a unit of measure for happiness: the warmfuzzy. Newton ran controlled experiments involving bread puddings from which he established a relationship between the rate at which his happiness changed and the number of bread puddings he had consumed. Although the original notes have not survived to the present day, this very paragraph points to the following result of Newton's observations:

\begin{equation*} H^\prime = 6p(2-p), \end{equation*}

where \(H^\prime\) represents the rate at which his happiness level is changing (in units of warmfuzzies per bread pudding), and \(p\) is the number of bread puddings consumed on a given day.

As Newton eats his bread pudding his happiness level will rise as long as \(H^\prime\) is positive. According to the model, this will be the case as long as \(0 \lt p \lt 2\text{.}\) In fact, his happiness will be increasing fastest when \(p = 1\) (at which point it is increasing at a whopping rate of 6 warmfuzzies per bread pudding!), which no doubt makes it difficult not to have a second bread pudding. His happiness will continue to rise until \(p = 2\text{,}\) after which \(H^\prime\) is negative. If he were to keep on eating bread puddings after having two, his happiness level would begin to fall toward what he called grumpiness.

So, Newton's brain told him what his gut already knew: He maximizes the happiness gained from bread pudding consumption by eating exactly two of them. The natural question, of course, and the one that I dare say motivated the development of integral calculus, is this: How much happiness does he actually gain by consuming two bread puddings?

As calculus students know today, the net change in happiness level corresponds to the integral

\begin{equation*} \int_0^2 6p(2-p)~dp \end{equation*}

which evaluates to \(8\) warmfuzzies.

So it came to pass that in the pursuit of happiness Newton found \(8\text{.}\) (As an aside, some etymologists believe the phrase “No, thanks. I just ate” has its origin in this saying of Newton's: “No more! I'm up eight!” This may explain why eight is one of the few numbers that is also a verb, phonetically.)

Leonhard Euler was a second giant of mathematical thought during the scientific revolution. A native of Basle, Switzerland, Euler was born in 1707, twenty years before Newton's death. He spent much of his working life in St. Petersburg and Berlin. He generated an immense quantity of fundamental, ground breaking work in mathematics.

Pertinent to our story, Euler is responsible for many common symbols used in mathematics to this day, including the symbols for the numbers \(\pi\) and \(e\text{.}\) To honor \(8\text{,}\) which already had the strongest of symbols (as we shall see in Section 6), Euler invented a new field of mathematics called graph theory.

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Figure4.1The seven bridges of Königsberg.

Graph theory was born from the solution to a puzzle that came to Euler's attention in 1736. In Königsberg (now Kaliningrad, Russia) one found a picturesque scene on the Pregel River: seven bridges joining four different land masses, as pictured in Figure 4.1. The puzzle that Euler answered was this: Can one walk in Königbserg in such a way as to traverse each bridge exactly once, following these rules: (1) you can only access the islands by the bridges; and (2) there is no backtracking on bridges (once you begin to walk on a bridge you must cross to the other side). There was no requirement that you end your walk where you started it. Such a walk, if it exists, is now called an Eulerian path.

Euler not only solved the puzzle (the answer is “no”), but also, being a mathematician, provided a method by which one could very quickly tell whether such a walk was possible for any number of bridges and any number of land masses. Thus, graph theory was born and Euler provided its first theorem. But why did Euler choose this particular puzzle with which to launch this new field? Scratching below the surface in a way eerily similar to the work Robert Langdon regularly did in The Da Vinci Code, we find that his choice was motivated by the desire to offer a tribute to the number \(8\text{.}\)

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Figure4.2If we had an eighth bridge of Königsberg.

I invite the reader to check that if you were to add an eighth bridge such a walk becomes possible. Such is the power of \(8\) that this eighth bridge may be added anywhere, connecting any two land masses, and wherever it is placed, an Eulerian path now exists! Here, I've added an eighth bridge randomly to the scene in Figure 4.2. Check that now a path traversing each bridge exactly once is possible.

The power and versatility of \(8\) displayed in the Bridges of Königsberg puzzle has been captured in song that can still be heard today, ringing in the great halls of Basle during Octoberfest, as long as you are drinking with this author:

Oh bridges, oh bridges, you puzzle me so
I cross one and cross two, which way do I go
to traverse each one of them exactly once?
I can't seem to do it, don't think me a dunce!
Oh Pregel, Oh Pregel, I dare not go in!
My love doth, my love doth, find it a great sin
to call on her family in fine clothes sodden
but what choice do I have with bridges seven?
Oh Euler, Oh Euler, I think I've a plan
if you could please plop down just one other span
Plop it down here or there, and you'll hear me cry
“I cross all \(8\) bridges while keeping me dry!”
Oh Bridge \(8\text{,}\) Oh Bridge \(8\text{,}\) we drink to you this beer
you'd stretch out to give us a solution clear,
to the puzzle that inspired graph theory great
in an effort to celebrate big number \(8\text{!}\)