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 P2 is 1 plus a multiple of 24, then it follows that P2–1 would need to be a multiple of 24. We can simplify this by observing that:

 P2 – 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! = 4x3x2x1, 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:

 A2 + B2 = C2

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 M2/N2 = 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 Nth 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 22,459,157,718,362nd 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 A, then its volume is given by:

Pi*Z*Z*A