Happy Pi Day! Order some pizza and use pi to make sure you get the most pie possible! I made a handy chart that may change how you order pizza.
Or not. It’s something I heard about early in the year that caused a minor tweet storm (I’m not on the Twitter, so never saw nothing, which I’m fine with). It centered around how it was often better to order two smaller pizzas than one large one (depending on pricing and assuming your goal is the most pizza possible per peso).
Since pi is involved in this pizza pie probe, I thought it would make a fun topic for Pi Day (not to mention March Mathness).
Which might seem a strange thing to say about a weird number, like pi, but, in mathematics, a normal number is one whose digits are perfectly distributed; no digit appears more than any other digit.
When we talk about how pi (or e or other transcendental numbers) ‘contain every finite number sequence within their digits,’ this depends on the digit sequence being normal.
There is no constructive proof that any number we know of (including pi and e) is normal, but mathematicians believe it to be the case. It has not been proven not to be the case. (There are numbers explicitly constructed to be normal.)
This seems to correspond with how, in the real world, there are no perfect circles, so pi as a precise value cannot exist physically. It exists only as an abstraction of the circles we can observe.
It raises the question whether we invent or discover math. What does mathematical infinity mean in terms of how math reflects reality? How real are idealizations?
I just read about an experiment that, as interpreted, seemed to demonstrate that objective reality doesn’t exist, that it’s possible for two observers to make contradictory observations.
Since high school, I’ve wondered about idealism. Is it possible consciousness literally shapes reality?
We assume our internal model is reasonably accurate, given we can only see visible light, and our other senses are equally limited. Our empirical reality seems solid and consistent.
But what if, at the quantum level, that’s just not true? What if it is literally the case you find what you expect to find?
Could consciousness be a fundamental force?
But I digress. Back to math!
In the 2016 post, I wrote about downloading ten-million digits of pi so I could see just how “normal” it was given that many digits.
It turned out to be extremely evenly distributed (although not perfect). I showed the results in some tables, but visual data is always more fun:Like I said: extremely evenly distributed.
Above, I mentioned a supposed tweet storm (of excitement, apparently, not criticism or disdain).
It all revolved around the well-known formula for the area of a circle:
The infamous “pi R squared” (except, of course, they usually aren’t)!
The question is whether you sometimes get more pizza area if you order two smaller pizzas rather than a single large one. As an example:
So clearly two 8″ pizzas gives you more pie than a single 10″ pizza. Even a 12″ pizza only gives you 113.1 square inches — not a lot more.
Here’s a handy chart:(The curves are just part of a parabola that comes from the r2 part of the equation. Pi is just a constant and doesn’t really matter to the curve.)
The next step is to factor in the price to give us the price per square inch.
Usually, the smaller the unit size, the higher the price per whatever (ounce, inch, etc). That turns out to be true with pizza.
Assume a simple (low-end) price structure of (Domino’s cheese pizza):
- 10″ (78.540 sq. in.) — $5.99
- 12″ (113.097 sq. in.) — $7.99
- 14″ (153.938 sq. in.) — $9.99
Then we can just divide the price by the area to get the price per square inch of pizza. In this case, it’s:
- 10″ — 7.63¢ per inch2
- 12″ — 7.06¢ per inch2
- 14″ — 6.49¢ per inch2
Considering a better pizza, their Wisconsin Six Cheese commonly prices out like this:
- 10″ (78.540 sq. in.) — $11.99
- 12″ (113.097 sq. in.) — $13.99
- 14″ (153.938 sq. in.) — $15.99
- 16″ (201.062 sq. in.) — $17.99
And, of course, you still pay less per square inch when you buy a bigger pizza:
- 10″ — 15.27¢ per inch2
- 12″ — 12.37¢ per inch2
- 14″ — 10.39¢ per inch2
- 16″ — 8.95¢ per inch2
In fact, rather dramatically! (Which is why it’s always a good idea to “do the math” — others are doing it for the express purpose of taking advantage of you!)
So if price is the primary concern, larger pizzas are the way to go. If getting the most possible pizza is the goal, ordering multiples is a good option.
As a final word about pi, a big part of its fascination is that infinite string of (possibly normal) digits, but in terms of real-world use, we just don’t need that many digits.
Consider the diameter of the Earth, 7917.6 miles. (The diameter varies, so this is an average value, which means precise results are kinda silly.)
If we (considering the date, 3.14) use just two decimal digits of pi, we get a circumference of 24,861.264 miles, which isn’t horribly inaccurate, although it turns out to be off by miles (never mind the decimal fraction, which is pure fiction).
If we use the easily remembered “zipcode” pi, 3.14159, we get a circumference of 24,873.852 984, which is accurate to tenths of a mile.
Twice as many digits, 3.1415926535, and we get 24,873.873 993 351 60, which is accurate to one-hundred-thousandth of a mile (a bit over half an inch). In fact, it’s pretty close down to one-millionth of a mile.
What’s the “real” value? It depends on how many digits we use, but at some point the accuracy becomes sub-atomic. You might notice that precision follows the number of digits; use 20 digits and the first 20 digits of the answer will be accurate.
If we go with the mystic number 42 and use 3.14159 26535 89793 23846 26433 83279 50288 41971 69 (yeah, I know, right?), the the Earth’s circumference is: 24,873.873 994 062 546 944 851 825 251 453 792 035 919 505 274 4 (which is beyond ridiculous).
So, generally speaking, the zipcode of pi (14159) should be enough for most occasions. If you can remember the extended zipcode (2653), all the better. Anything more is mostly a party trick.
Pi Day is nice, but let’s have twice the pizza pi in about three months, on Tau Day!
Stay transcendental, my friends.