## Quibbles and Bits

# The Volume Of A Pizza

Numbers, and the mathematics that describe them, can help you with many interesting things, including the volume of a pizza.

There are some wonderful surprises hidden in plain old numbers. Things that can delight you because you can’t imagine how such simple things could be possible. To many, the temptation is to read things into them that can’t possibly be true. Other times, even professional mathematicians just sit back and shake their heads in amazement. A perfect example of the latter would be the Mandelbrot Set, an extraordinary pattern generator based upon a single, absurdly simple mathematical equation. Mathematicians continue to study the Mandelbrot Set, and are always coming up with complicated new analyses to explain just one single feature, but none come close to shedding light on the extraordinary level of infinitely repeating – yet always subtly varying – swirling patterns for which the Set is justifiably famous.

The video below zooms deeply into a random part of the Mandelbrot Set. Try to watch it full screen. The entire video is 16 minutes long, but by the 2’45” mark we have zoomed in so far that the complete Mandelbrot Set would occupy the size of the entire observable universe. By the end of the video, the size of the complete Set would so large as to be beyond any meaningful ability to describe it. Yet every last micro detail you see in the entire video is generated by just one trivial equation. You can enjoy it in phenomenal video resolutions as well…up to 2160p60.

Here’s a much simpler piece of mathematical trivia. The square of a Prime number is always a multiple of 24, plus one. Think about that for a moment. Can that be true? Is life really as simple as that? A few moments spent on a calculator readily shows that it holds good for every Prime number your calculator can handle. But what on earth is so special about 24? Why on earth should Prime numbers have that intriguing property? Mathematics holds the simple answer in its hands.

Here it is. If *P* is a Prime number, and *P ^{2}* is 1 plus a multiple of 24, then it follows that

*P*would need to be a multiple of 24

^{2}–1*.*We can simplify this by observing that:

* **P ^{2} – 1 = (P–1)(P+1)*

Therefore, our question instead becomes: Is the product of *(P–1)* and *(P+1)* always a multiple of 24?

- First, we observe that
*(P–1)*,*P*, and*(P+1)*form a run of three consecutive numbers. Therefore, one of them must be a multiple of 3. Obviously that can’t be*P*, since it is Prime. So one of either*(P–1)*or*(P+1)*must be a multiple of 3. - Next, since
*P*is Prime, it must be odd, so both*(P–1)*and*(P+1)*must both be even. With any two consecutive even numbers, one must be a factor of 2 and the other a factor of 4.

So *(P–1)* and *(P+1)* between them must include the factors 2, 3, and 4, whose product is 24. Therefore, the product of *(P–1)* and *(P+1)* is always a multiple of 24.

Less intriguing, but just plain old cool, is an observation involving factorials. The factorial of a number (it only applies to integers) is obtained by taking the number and multiplying it by every integer less than itself. So, the factorial of 4 (*denoted by placing an exclamation point after the number*) is *4!* = 4*x*3*x*2*x*1, which is 24. (*There’s that number again!*). Factorials arise most often when calculating probabilities and combinations. For example, the number of ways to order a deck of cards is *52!* which is a seriously huge number. Therefore, if you perform a perfect shuffle on a deck of cards, it is virtually guaranteed that the exact card sequence you’ll get will have never previously occurred in the history of the universe, and furthermore will probably never occur ever again! And my cool observation regarding factorials is this…the number of seconds in 6 weeks is exactly *10!*

Let’s check that one out. There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 7 days in a week. So, the number of seconds in 6 weeks is:

60 *x* 60 *x* 24 *x* 7 *x* 6

I can expand one of the 60’s into (10 *x* 3 *x* 2) and the other into (5 *x* 4 *x* 3). I can also expand the 24 into (3 *x* 8). So, the number of seconds in a week becomes:

(10 *x* 3 *x* 2) *x* (5 *x* 4 *x* 3) *x* (3 *x* 8) *x* 7 *x* 6

You’ll see there are three 3’s in there, so all I have to do is take two of them and multiply them together, which turns them into a 9:

(10 *x* 3 *x* 2) *x* (5 *x* 4) *x* (9 *x* 8) *x* 7 *x* 6

This contains one each of all the numbers from one to ten, multiplied together, which is 10 factorial. Therefore, there are *10!* seconds in 6 weeks.

OK, so that was also a little trivial. But here’s something you are flat out not going to believe. What do you get if you add up all of the positive integers? Infinity, right? We’re talking about:

*1 + 2 + 3 + 4 + 5 +6 + 7 + 8 + + +*…

…and so, every number you add to the tally only makes the result exponentially bigger, all the way to infinity. But what would you think if I told you that the answer is actually ** minus one twelfth**? Yep, all the positive integers add up together to

*‑1/12*. A totally absurd proposition, I agree, but quite surprisingly, this answer forms the basis for some of the most important problems in advanced theoretical physics, including String Theory. It was first noted by a famous Indian mathematical savant, Srinivasa Ramanujan, in 1913. I should observe that many mathematicians will point out, quite rightly, that you are dealing with infinities here, and that while the result may be perfectly correct in one context, it may equally be totally incorrect in another. But I thought it was worth throwing in there, even though I’m not going to offer up the proof (since it is a bit too elaborate, although not in fact all that challenging).

While mathematics is arguably the ultimate precise science, the one with the least possible room for ambiguities and dispute, it has nonetheless had to deal with ambiguities and disputes since time immemorial. Back in about 520BC, the school of Pythagoras believed, and taught, that everything could be explained using numbers. And by numbers, they meant integers. Of course, they knew that quantities existed between adjacent whole numbers, but they insisted that these could all be represented as fractions, or ratios between pairs of integers. However, problem began to emerge when they established the famous theory of Pythagoras, that for a right-angled triangle, the lengths of the three sides were governed by the relationship:

* **A ^{2} + B^{2} = C^{2}*

The question was, if A = 1 and B = 1 then what is C? It is a quantity which when multiplied by itself gives the result 2, and which we call √2. A scholar named Hippasus of Metapontum was reputedly the first to recognize a fundamental problem which this forced them to face. Clearly, the answer was a number between 1 and 2, and so (according to Pythagoras) had to be a ratio between two integers:

** ****√2 = M/N**

M and N were both simplified by cancelling out any common factors. Therefore, it is clear that at least one of M and N must be odd (*otherwise there would be a common factor of 2 that could be cancelled out*). However, if we then consider that ** M^{2}/N^{2} = 2**, this would require that both M and N must both be even. This obvious contradiction means that there are no such integers M and N that could satisfy the criterion. There was no possible ‘rational’ number that could represent the quantity √2. For this apparent heresy, Hippasus was thrown from a boat and drowned.

Some corners of mathematics receive what appears to be a disproportionate amount of detailed attention, and at the head of that line is undoubtedly **π**. People have devoted remarkable energies to evaluating π to extraordinary degrees of precision. When I was at university in the 1970’s (studying Physics), one of the professors in our math department claimed to have been the first to calculate π to a million decimal places. The result, printed on computer paper, occupied a wall in his office. Today, the record stands at 22,459,157,718,361 decimal places, and represents not so much the limits of capability, as the limits of patience, combined with the desire for a certain ‘coolness factor’ (*that number of decimal places was carefully chosen for its quirky significance, appreciated only by mathematicians*). Sufficient computer paper does not exist to print it out!

The algorithm currently used to enumerate π, Alexander Yee’s “y-cruncher”, was originally developed as a tool to torture-test CPUs. It is in the public domain, and has been the only show on the road since 2010. It runs on readily available, although carefully specialized, PC hardware. Here is a question for you to ponder…how much hard disk space would you need to store a 22,459,157,718,361-digit number? BTW, aside from “y-cruncher” there is an algorithm available which can quickly and efficiently calculate just the *N*th digit of π, for ** any** value of

*N*, if such a thing is of value to anyone. Seems incredible to me, but there you are. I guess you could assign part of your evening to calculating the 22,459,157,718,362

^{nd}digit, if you were of a mind to do so.

People have studied π to assess whether there are any unusual features in the distribution of digits in π, and to a remarkable degree the distribution is indeed totally random. Not only that, but auto-correlation tests show that second- to fifth-order distribution features are also all totally random. This has also been extended to representations of π in bases other than 10, with the same result. That the digits of π pass every test thrown at them for randomness puzzles some people. There is a philosophical point in play here…the dichotomy between what appears to be a truly random process, yet one which arises from a fundamentally structured quantity. But there is also a lunatic-fringe element who are determined to uncover a hidden message from some kind of higher power. Good luck with that…but if they do discover such a message, I’ll be careful to re-designate them as prophets.

Oh, and the volume of a pizza? Well, if the radius of the pizza is ** Z**, and its thickness is

**, then its volume is given by:**

*A**Pi*Z*Z*A*

Because for some obtuse reason each reply on Copper magazine is restricted to a number of characters complex topics that require lengthy responses require multiple entries. I have no idea why.

I have lived in a world of mathematics pretty much all of my life. My mother was a mathematician with a master’s degree from Columbia University, my father the quality control manager of a high tech company out on Long Island that built exotic electronics for the military and NASA. My mother supervised my math education. I don’t know if I would have made it through first year calculus without some tutoring help from her. I was always a good math student but when I was 14 years old I devoured Euclidian plane geometry. I didn’t memorize proofs, I implicitly understood the concepts behind them and could regenerate them instantly at will. Going from two dimensions to three was not too terrible a problem. Cartesian coordinates easy, spherical coordinates a little more difficult but not terribly so. Cylindrical coordinates harder but very useful for fluid dynamics such as flow though pipes and air ducts as the velocity and flow profiles are parabolas with the flow at the boundary zero. If I could turn a problem into a geometry problem I could usually solve it. For example once when I returned from a cruise I asked my office mate, a crackerjack mechanical engineer how far you could see to the horizon on a ship. He said figure it out. And I did. All you need to know is trigonometry and the fact that the earth is 8000 miles in diameter.

That was the way my brain works best, visualization. It was therefore very frustrating to me that my mind could not visualize more than three dimensions. In my twenties I invented two tricks to visualize phenomena in more than three dimensions. In one, the problem of acoustics I visualized it in six dimensions. In another, a problem in physics I visualized it in four using a different trick. These tools allowed me to understand problems that seem incomprehensible to a three dimensional mind that only has experiences in three dimensions. I am quite certain that there are more than three dimensions and the proof is quantum entanglement. While two objects may be distant physically in the three dimensions we experience they are in close proximity in a dimension we don’t experience. This is the only plausible explanation I can think of for how entanglement could work.

During this period I realized that the thousands of equations I had learned in so many fields told me what would happen, they addressed only that but told me nothing about how and why they happen. It was little comfort to realize that no one else understood it either. For example an equation will tell you what the force is between two charged particles based on their distance and the magnitude of their charge but they tell you nothing about how they interact or why like particles repel and opposite particles attract. They also don’t explain why protons on top of each other in the nuclei of atoms except hydrogen don’t fly apart. All physicists can do is put a name on it called the strong force but that explains nothing. The standard model still has a lot of explaining to do about things it does not address.

If you’re interested I’ll tell you how I solved the acoustics problem.