Blogs, Modeling

Pizza Division

I had my wisdom tooth pulled today.  I have just gotten home from the procedure and things went very well. While both my wife and mother-in-law questioned the wisdom of such a decision, I decided the first thing I wanted to eat was pizza. I think it was a great idea and I thoroughly enjoyed it.

If you happen to live in the Richmond area, I would definitely suggest the meat feast from Vinny’s. The meat is incredible and well proportioned on a nice New York style crust. I’ll refrain from having my own debate on whether New York, Chicago or other types of pizza are best, because that isn’t really the topic of this post. However, feel free to do so in the comments.

While slowly eating the pizza so that I wouldn’t get any on the right side of my mouth, I began to think about was how can you ensure a quality distribution of a pizza? There are many ways to interpret this, so let’s look at a few.

Firstly, if there is one person eating the pizza, then they do whatever they want. If there are two people eating the pizza, the classic solution is to let one cut the pizza in half and let the other person choose which slice they want. There have been further techniques created for division among three or more people as well. Francis Su suggests a technique in which you suggest all three pieces letting the people choose. You can then adjust where to cut until each person chooses a different piece, thus resulting in an envy free distribution.

The problem I wanted to consider was significantly easier, as I just wanted to find equal portions of the pizza. A precise division into eight equal slices was know to Euclid using only a straight-edge and compass. The process involved choosing any three point on the edge of the pizza. You could then construct the circumcenter of the triangle, giving you the center of the pizza. This allowed you to create a diameter which gave you an angle you could divide in half twice with the compass and straight-edge in order to get an equal eight slices. However, cutting a pizza into, say 18, exact slices using only a straight and compass is impossible (see Geometric Problems of Antiquity)  While methods are possible using a trisectorix or conic sections, but these problems gave geometers work for more than 2000 years leading to many disoveries, areas of mathematics and more.

Again, however, I have just gotten a tooth pulled and between the pain medication and lingering pain I was aimed at an even easier problem. That is, since pizza come in sizes given by the diameter, could I determine the size of a annular slice of pizza, that is a circle with an inner circle missing, that would account for 1/8 the area of a 16 inch pizza. If I could do so in general, I could use a slice number to give a fair comparison between pizzas with different diameters.

Now that I’ve gotten to my question, I want to point out that, assuming a circular pizza with A=πr2, the total area of a 16 inch diameter pizza is π(8)2=64π. Therefore, 1/8 of this would be 8π square inches, so the radius of the first piece would be √8≈2.828 inches, that is a 2√8≈5.657 inch pizza. The next slice would have the same area, so both of these together would have radius πr2=16π, so the radius would be 4. Hence the second slice would be the annulus with inner radius √8≈2.828 inches and outer radius of 4 inches. That is we could say a 8 inch diameter pizza is a 2 slice pizza with respect to a 16 inch pizza. If we generalize these results, we get that the nth slice would have outer area 8nπ inches and an inner area 8(n-1)π inches giving an outer radius of √(8n) inches and an inner radius of √(8(n-1)) inches.  Therefore, if we wanted compare circular pizzas linearly, we could use the following table for pizzas up to 20 slices or 25.3 inches.

1 Slice 5.66 inch 11 Slice 18.76 inch
2 Slice 8.00 inch 12 Slice 19.60 inch
3 Slice 9.80 inch 13 Slice 20.40 inch
4 Slice 11.31 inch 14 Slice 21.17 inch
5 Slice 12.65 inch 15 Slice 21.91 inch
6 Slice 13.86 inch 16 Slice 22.63 inch
7 Slice 14.97 inch 17 Slice 23.32 inch
8 Slice 16.00 inch 18 Slice 24.00 inch
9 Slice 16.97 inch 19 Slice 24.66 inch
10 Slice 17.89 inch 20 Slice 25.30 inch

While conventions are hard to change, and I doubt pizzerias will start giving the size of pizzas in using slices as units, it is still a useful idea if you are trying to determine the relative costs of pizzas based on their size.

If you found this blog interesting or helpful, please like this post and let me know in the comments.  If you would like to discuss the best pizza or type of pizza, pricing options at different sizes, also feel welcome to leave your comments below.

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