My love for pizza is not over yet. A while ago I have read an article about new and different ways mathematicians have found to cut pizza and I had to check it out. To my surprise I got to read something called Pizza Theorem in Geometry!!! My enthusiasm grew exponential at that moment.
It turns up that this is a proper mathematical problem and that it is very interesting. In elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a disk in a certain way. Sounds really straight forward, nothing major. The proper theory sounds like this:
Let p be an interior point of the disk, and let n be a number that is divisible by four and greater than or equal to eight. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n/2 − 1 times by an angle of 2π/n radians, and slicing the disk on each of the resulting n/2 lines. Number the sectors consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states that:
- The sum of the areas of the odd numbered sectors equals the sum of the areas of the even numbered sectors (Upton 1968).
The theorem is more related to circular geometry and the proofs have a lot of algebra manipulation of area of a circle and area sector. The theorem has a couple of generalizations and interesting proofs during the years. It all started as a challenged problem in 1968 and it evolved to proper proofs and generalizations. Moreover, mathematicians have also defined what a crust is:
The crust may be interpreted as either the perimeter of the disk or the area between the boundary of the disk and a smaller circle having the same center, with the cut-point lying in the latter’s interior.
So hopefully I made you more interested in the mathematics behind this. Check the Wikipedia Page for this, also check some of these if you want to find out more about the proof for this theorem. Also, check the following video for more:
Hope this made you a little more curious in this subject. If you want to read my previous post on this topic check Cutting Pizza ~part 1~. And because a post about pizza without an image of a pizza is nothing… Bonn Appetite!
Don’t forget that maths is everywhere! Enjoy! ~LThMath~