Mar. 13, 2015

Kitchen Math: How to Eat π Pies

by Eugenia Cheng

Click to enlarge images
Tune in to Science Friday on March 14, 2015, for a Pi Day Celebration.
Every child knows how to make their pie last: Eat half, then eat half of what’s left, then eat half of that…and so on. The part that’s left over will get smaller and smaller until eventually you can’t even see it any more, and the total amount of pie eaten will get very, very close to one, although it will take forever.
What if you want to eat not one pie, but π pies? Here’s how you do it. Start with four whole pies on plates. Remove a third from each of them. Then, out of each third, cut a slice the size of a fifth of a whole pie. Put those fifths back with the pies on the plates. 
So far, you have this much pie on the plates:

Now keep going. Remove a slice the size of a seventh of a whole pie from each of the pies on the plates. Then cut a sliver the size of a ninth of a whole pie from each of those slices, and put the slivers back with the pies on the plates. Then remove an eleventh. Put back a thirteenth. Remove a fifteenth. Put back a seventeenth. The process you’re carrying out is this:


The amount of pie left on the plates will eventually have to stabilize, because at each stage you move a little less than at the previous stage, so your total amount of pie will have to approach something—and that thing happens to be the irrational number π. Now, enjoy your pi(e)!
It’s a fascinating fact about numbers that you can add up an infinite sequence of rational numbers (fractions where the top and bottom are whole numbers) and get an irrational number (a number that can’t be expressed as a fraction). This doesn’t sound like it makes any sense, because every time you add a fraction on, you get another fraction…but when you start adding up infinite numbers of things, weird stuff can start happening.
It takes several chapters of an advanced calculus book to get to the point of understanding why the process described above really gives pi. But if you’re into baking, there’s an oddly easier way to get an approximation for pi using kitchen math. Many bakers know that you can use a recipe for a 9-inch round cake to make an 8-inch square cake instead. Considering the formula πr^2 (used to calculate the area of a circle), this baker’s trick suggests that π4.5^2 =8×8=64.
If we use this to approximate pi, we get:


...which is closer than the approximation we got above using fancy math, and definitely good enough for cake! 
About the Author
Dr. Eugenia Cheng is tenured in the School of Mathematics and Statistics at the University of Sheffield, UK, and is Scientist in Residence at the School of the Art Institute of Chicago. Her first book, How to Bake π, will be published by Basic Books in May. Follow her @DrEugeniaCheng


About Eugenia Cheng

Dr. Eugenia Cheng is Scientist in Residence at the School of the Art Institute of Chicago. Follow her @DrEugeniaCheng

The views expressed are those of the author and are not necessarily those of Science Friday.
EVAL ERROR: Entity: line 1: parser error : Space required after the Public Identifier

Entity: line 1: parser error : SystemLiteral " or ' expected

Entity: line 1: parser error : SYSTEM or PUBLIC, the URI is missing


line 1: package tmpevalpackage;
line 2: sub doEval { 
line 3: 	my($parent);
line 4: 	
line 5: 	if($LayoutManager::url_resolved_values{'SEGMENT.nickname'}) {
line 6: 		$parent = $LayoutManager::url_resolved_values{'SEGMENT.nickname'};
line 7: 	}
line 8: 	elsif($LayoutManager::url_resolved_values{'VIDEO.nickname'}) {
line 9: 		$parent = $LayoutManager::url_resolved_values{'VIDEO.nickname'};
line 10: 	}
line 11: 	elsif($LayoutManager::url_resolved_values{'DOCUMENT.nickname'}) {
line 12: 		$parent = $LayoutManager::url_resolved_values{'DOCUMENT.nickname'}
line 13: 	}
line 14: 	
line 15: 	if($parent) {
line 16: 		my(@books) = &Database::SelectClause('BOOK',"parent = $parent");
line 17: 		if(!@books) {
line 18: 			$parent = '';
line 19: 		}
line 20: 	}
line 21: 	
line 22: 	if(!$parent) {
line 23: 		my(@sel) = &Database::SelectClause('GLOBAL','record all ""');
line 24: 		if(@sel) {
line 25: 			$parent = 'GLOBAL.' . $sel[0];
line 26: 		}
line 27: 			$main::ENV{'reading_header'} = "FEATURED READING";
line 28: 	}
line 29: 	
line 30: 	 = '';
line 31: 	
line 32: 	if($parent) {
line 33: 		my(@books) = &Database::SelectClause('BOOK',"parent = $parent");
line 34: 		0 = 0;
line 35: 		my $dots;
line 36: 		foreach(@books) {
line 37: 			my(%data);
line 38: 			&Database::GetRow($_,'BOOK',\%data);
line 39: 			my($status,$title,$author,$url,$image,$width,$height) = &SciFri::Schema::getAmazonItem($data{'isbn'});
line 40: 			if($data{'title'}) {
line 41: 				$title = $data{'title'};
line 42: 			}
line 43: 			if($data{'author'}) {
line 44: 				$author = $data{'author'};
line 45: 			}
line 46: 			if($status eq 'ok') {
line 47: 				 .= "<div class=\"box-2x1-item box-slide\" data-href=\"$url\">";
line 48: 				 .= "	<div class=\"box-2x1-item-photo\">";
line 49: 				 .= "		<div class=\"image-wrapper\" data-jsclass=\"imageWrapper\" data-align=\"right\">";
line 50: 				 .= "			<img src=\"$image\" data-width=\"$width\" data-height=\"$height\">";
line 51: 				 .= "		</div>";
line 52: 				 .= "	</div>";
line 53: 				 .= "	<h4>$title</h4>";
line 54: 				if($author) {
line 55: 					 .= "	<p>by $author</p>";
line 56: 				}
line 57: 				 .= "	<div class=\"box-2x1-more-button\"><a href=\"$url\"><img src=\"/images/v1/icon_text_more_white.png\" width=47 height=15 border=0></a></div>";
line 58: 				 .= "</div>";
line 59: 				++0;
line 60: 			}
line 61: 		}
line 62: 	}
line 63: 	if($parent eq "GLOBAL.1") { $main::ENV{'reading_header'} = "FEATURED READING"; }
line 64: 	else { $main::ENV{'reading_header'} = "RELATED READING"; }
line 65:  };
line 66: &doEval();
line 67: 1;

Science Friday® is produced by the Science Friday Initiative, a 501(c)(3) nonprofit organization.

Science Friday® and SciFri® are registered service marks of Science Friday, Inc. Site design by Pentagram; engineering by Mediapolis.