Pi Are Round!

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).

I’ve written about pi itself quite a few times (see Pi Day 2015 and Pi Day 2016). It’s come up a few other times for being transcendental and (we think) normal.

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?

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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:

The expected distribution for each digit is 10% — the actual frequencies (of the first 10 million digits) are very, very close. [click for big]

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:

Area = \pi \times radius^2

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:

(\pi (\frac{10}{2})^2) \times 1 = 78.54\\\\(\pi (\frac{8}{2})^2) \times 2 = 100.53

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:

Two 8″ pizzas are bigger than one 10″, and two 10″ pizzas are way bigger than one 12″. (A 16″ is 200 square inches of pizza!) [click for big]

(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.)

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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.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

15 responses to “Pi Are Round!

  • SelfAwarePatterns

    I’m a mathematical empiricist, but I think we both discover and invent math. We discover the underlying relationships, but we invent the notations, concepts, and methodologies we use to work with those relationships. But like any invention, the underlying reality puts constraints on what we can invent.

    I read about that quantum experiment. I’m skeptical. (Of course, I’m always skeptical.) Partially because I had trouble finding a coherent description of what exactly they did, and because the most detailed article I could find talked about possible loopholes.

    If our consciousness does construct our reality, then mine is a bastard, at least this week.

    • Wyrd Smythe

      We do absolutely invent the language of math. It’s those underlying relationships and patterns that fascinate me and which (to me) seem ontological. A thrown object follows a parabola, essentially x2, which is a beautifully simple pattern given the obviousness of squaring a distance (to make, whachamacallits,… “squares” 🙂 ).

      I mentioned studying the mathematics of rotation, and that’s another place where the language we invent seems to describe an elegant underlying pattern.

      I’m with you on the skepticism about that quantum experiment, and I noticed the same lack of details and hand-waving. Apparently the idea has been known for quite a while, but only recently have laser optics gotten to the point of allowing them to actually perform it.

      A lot of my skepticism comes from either not buying into, or not understanding (which is more likely), recent discussions about observers observing experimenters and various states of entanglement and superposition among them all. It apparently leads to paradoxical results, but I have a hard time when it comes to the reality of people being in superposition or entanglement. I wonder if they’ve gotten “Lost in Math.”

      This seems related to me, although they do have an actual experiment they performed. I probably just don’t get it.

      Unless you subscribe to solipsism, other minds are participating in reality and wrecking your week. It’s not all on you! 🙂

      • SelfAwarePatterns

        I decided to actually look at the paper for that quantum experiment. https://arxiv.org/pdf/1902.05080.pdf
        Entanglement seems to feature heavily, which (in my probably confused understanding) seems to make this an expansion of the test of Bell’s inequality theorem. The final paragraph in the discussion section at the end makes what is being discussed in the popular articles one possible interpretation. The results appear to be compatible with MWI, Bohmian mechanics, the relational interpretation, and “QBism”, in addition to the explicit consciousness-causes-collapse theories. It does seem to rule out, or at least challenge, objective collapse theories. (The more epistemic versions of Copenhagen are also consistent.)

        A large part of my week has been my body giving me grief (bad cold, tooth problem, etc). If other minds are causing this, then they are bastards. 😡

      • Wyrd Smythe

        My condolences! A cold is bad enough, but tooth problems are miserable.

        Thanks for the details on the paper. They don’t surprise me. When it comes to the popular press handling science topics, it’s almost always wrong in some regard.

      • SelfAwarePatterns

        Thanks, but it wasn’t just that. I also found out that an old friend had passed away back in December, that all of our project funding at work got yanked, and just today on returning to work discovered that both my boss and boss’s boss are “transitioning out”.

        On that paper, I also didn’t notice the first time on the news story that this hasn’t been through peer review yet. It’s usually seen as a red flag in science when someone goes public before that’s done. (Although admittedly, it’s happening a lot more these days on less controversial studies.)

      • Wyrd Smythe

        Wow, what is it about this week? I have a friend who’s having a “week from hell” and I know someone else who’s also dealing with some serious crap lately. My condolences all the more; hope shit gets better!

        Yeah, releasing work to the public before peer review has gotten to be a whole thing, hasn’t it. Need to be sure to beat others to the punch, but there have been some serious OOPS! lately (cough-BICEP2-cough). And of course we all remember faster-than-light neutrinos. 😀

        (Although, in fairness, the OPERA folks were clear they didn’t understand the apparent results, were pretty sure it was an error, but couldn’t find it. They were almost looking for help at that point, although, if it were real, then it was a pretty big deal. It was everyone else that really ran with that ball. Hundreds of papers explaining why it was real. 😀 )

      • SelfAwarePatterns

        I had forgotten that BICEP2 was released pre-review. Yeah, those guys were so worried about being scooped that they decided to take the chance, and got burned.

        I think the OPERA folks were being a bit disingenuous. They presented it as an error they wanted help on, but if it hadn’t turned out to be an error, well. But really, who in the science world goes public asking for methodological help? They could have brought in outside experts for review. Issuing a press release was a calculated move on someone’s part.

      • Wyrd Smythe

        Ha, yes, mos def! That’s why I said they were “almost” asking for help. They were scared to death it might be real! 😀

  • Wyrd Smythe

    (I bet no one noticed this was posted at 3:14 AM! 😀 )

    • SelfAwarePatterns

      I sure didn’t, but I do see where it showed up in InoReader at exactly 3:14! Well done. Did you schedule it or stay up until 3:14 to hit the button? 🙂

      • Wyrd Smythe

        Scheduled it.

        I was just setting it up to post on the day (I had other plans yesterday), but you have to pick a time, and I was trying to decide between 7:00 AM and 7:30 AM when I realized 3:14 was better. 😀

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