My Favorite Pieces of Pi

March 12, 2015

Sprial 1

Over the past couple of years I’ve dedicated a number of posts to the way I’ve been looking at Pi.  In recognition of the upcoming Pi Day, I’m reposting some of my favorite images from these past posts.

Though not very beautiful the following image, which I did in as an Excel document, is one of my favorites, as it visually illustrates different common estimations of Pi, and how they compare to each other.

Pi- comparisons of approximations

Pi- comparisons of approximations

From these images, it looks to me that there’s a  difference, though not much,  between 3.14 and 3.141592 though the difference between 3.1416 and beyond is indistinguishable to mere mortals.

I wrote a post about a cut-and-paste pi project. It’s illustrated with 19 photos. The link to this post embedded in the caption below the photo.

What I like most about this project is that it lets you hold a piece of .14 in your hand.

I also made some coloring pages, for pi, which I collected in a Pi Coloring Book-style pages pdf.

pi big fraction

This fractional representation of Pi is the most accurate estimation of pi with a denominator under 10,000. Here’s the way I colored it in:

The most accurate fractoin representing pi with a denominator less than 30,000

The most accurate fractoin representing pi with a denominator less than 30,000

Here are a couple more of my Pi coloring pages:

PI symbol

Pi

 

pi 22 7

a useful estimation of pi

 

Then, I decided to try my hand at creating a one-page explanation of  Pi :

Circumference wrapped by by 3.14 Diameters, or 2(pi)r

Circumference wrapped by by 3.14 Diameters, or 2(pi)r

This past summer my cousin was visiting. He needed to figure out the size of the rim on a wheel of his bicycle, and was having trouble with the measurement because it was circular. You know where this story is going, right? I suggested that he measure the diameter then multiply by 3.14. He seemed politely interested, but I’m not sure if think he actually considered trying it. After all, it sounds like hocus-pocus.

Happy Pi Day!

 

Hen

I have one last thing that I want to cackle about before finishing with pi: exactly what number should we use for pi?

Pi is an irrational number, but it’s just not practical to use an infinite amount of decimal places when using pi in an equation, right? So, how many decimal places should we use? This depends, in part, on how accurate you want your answer to be.

Okay, so pi equals about 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510….. but it’s just silly to think about all these numbers in terms of measurement because most of us simply do not have the means  to measure anything this accurately. So, what number should we use? To answer this I used Microsoft Excel to help me show you a picture (!) of a comparison of approximations of pi. (I’m really excited about this.)

Pi- comparisons of approximations

Pi- comparisons of approximations

Ta-Dah! Now, take a look at this! It looks like using 3 as an approximation of pi  can really be off the mark. Going out one digit is better, and it’s close to the value that the Babylonians came up with (3 1/8)  two thousand years ago, but it’s still not so good. Now, 3.14 looks like it’s close to the rest of the representations of pi, which become increasing indistinguishable from each other, and it’s a value that I’m satisfied with. But for anyone who is not so easily satisfied we get to zoom in on this picture for a closer look.

Zooming in on PI

Zooming in on PI

I have to tell you that when I created these images I didn’t know how the various approximations of pi would look compared to each other.  You can see here that 3.14 looks like it’s less than 3.141, and 3.141 looks smaller than 3.1415 and maybe 3.1415 looks smaller than 3.14159 but after that, even though there are differences between the  subsequent values of pi, the differences are so small that, to the senses and  for the purposes of us mere mortals, these differences are indistinguishable from each other.

So there you have it. A real look at pi.  So, what’s so important about this number anyway? There are books on this subject, but, briefly, pi shows up predominantly in two places. One is standardized tests. Every student who takes standardized tests, which include the SATs and ACTs will absolutely 100% be asked a question that will test their understanding of pi. Is this a practical reason to understand it? I dunno, but that’s the way it is. Will you ever use it for any practical reason? The only time I have ever used it was when I needed to roll out a clay coil that was long enough to become the foot of a bowl I was making. The bottom of my bowl was about 6 inches across so I knew that my coil needed to be nearly 19 inches long plus about another inch for overlap.

 

I think that the next time pi shows up in a meaningful way is in Calculus class.  There’s a really cool thing that pi does with calculus formulas:

Egg and Egg Cup

If you ever wondered what calculus does, well one of the things it does is provide a formula using pi which can calculate the surface area of an egg-shell, which of course, is not a perfect circle. And other use of Pi in calculus is that it is used in yet another formula which can calculate the volume of an egg cup, which, again, is not a regular sort of shape. Now. just think, if you can figure out the surface area of an egg you can figure out how much material you will need if you ever want to build irregularly shaped buildings like Santiago Calatrava’s.

Curvilinear Architecture by Santiago Calatrava

Curvilinear Architecture by Santiago Calatrava

For more big eggs like this take a look at http://artmagonline.wordpress.com/2011/12/15/on-calatrava

And if you want to know how much liquid it takes to fill up your bathtub, your swimming pool or your oil tanker, calculus and pi (paired with the right calculus formula) comes in handy again.

Now, it’s a good thing that Chinese New Year is coming up, because I’m feeling inclined to abandon pi and start writing about accordion style bindings…

Addendum: I just came across a great post on pi in the Scientific American Blog. The articles reports that NASA uses 15 or 15 digits in their Pi calculations, in the comments reader suggested that the cube root of 31 is  an excellent pi approximation. Read for yourself http://blogs.scientificamerican.com/observations/2012/07/21/how-much-pi-do-you-need/

cut and paste pi (1)

Click the MORE…tag to reveal this whole post on how to do a cut-n-paste pi

Here’s a hands-on, cut-n-paste  paper project that, hopefully,  puts pi in your hands. This post is 19 photographs long, so, in deference to anyone to whom  seeing that circumference divided by diameter equals 3.14 is inconsequential, I have hidden the bulk of this post under the MORE tag button in this post. If  you want to see the whole project, please click the more button, and I will see you on the other side…. Read the rest of this entry »

I  love circles. Learning about circles formally, however, was so obscure and opaque that I’ve put in some time trying to imagine visuals to clear up my confusion.

Here’s a picture I came up with to visualize  the circumference/diameter relationship:

Okay, so does this make sense to anyone out there? Please let me know.

The Diameter of a circle equals 3.14 diameters

Pi

I put this together because I like to  know what things look like.  Telling me that circumference divided by diameter equals 3.14159…, or pi,  conjures up absolutely no pictures  at all.  I’ve often confused radius with diameter, which is completely understandable considering that after being taught the circumference/diameter thing, forever afterwards all the formulas use radius so, you know how it goes? My wires got crossed and all of a sudden radius and diameter seemed somehow synonymous.

For those of you who know what I mean, but are still unsure about the difference between radius and diameter, I offer this:  Diameter comes from the Greek: meter means measure, and dia means across, so diameter is the length that measures all the way across the circle. Now, think of radius like a ray from the sun: it starts in the center and travels out, so it’s half of a diameter.

By the way, I so much like the way that the image above visually explains the Pi relationship that I’ve put together a hands-on cut-and-paste activity to really drive the point home. It’s 19 photographs long.  I think it will be a few days before I put it up.

Bubble Marbling and Crayon Batik Book Covers

Bubble Marbling and Crayon Batik Book Covers

And, just in case you’re wondering, this is still a bookmaking and paper works blog…see, I use circles in bookmaking, too!

ADDENDUM: 12/2016 Below are some links to some of my other Pi Pages

Pieces of Pi: Colored in examples of some  Coloring Pages for Pi to help with remembering the different ways pi is expressed when doing  calculations that need to yield a specific rational number.

Pi Post #3 : A hands-on, cut and paste pi project

Pi Post #4 :Contains a cool graphic comparing different approximations of pi

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