Sarah and I were talking and the subject of pizza came up. It turns out she's a big fan - particularly of Hawaiian pizza. Rather than getting mired in the cultural war of pizza types, let's go for the basics - the mathematics of pizza. You'll only need a piece of paper, a ball (Sarah recommends a volleyball), and a marker.

Take a piece of paper and lay it flat on something. It's flat with zero curvature everywhere. if you draw a triangle on it and sum the angles, you get 180°. A circle will have a circumference of 2Pi times the radius - 2πr - and an area of πr^{2}. All is well. Now fold it along an axis - say the long one if it's a rectangular piece. You've introduced some curvature. Now grab it by a corner and let it flop. Again there's curvature, but notice which one is stronger and save that thought.

Here's where the math comes in. Exploring what curvature means, Carl Friedrich Gauss wondered if there was anything intrinsic about the curvature of a surface. The sheet of paper - what is the relation between its flat shape and when you fold it? His approach was to consider the paths you can take along the surface. Imagine you're tiny and walking on the sheet of paper. You can take any path you want. When it's folded state there are a lot of paths that are concave (he called these negative curvature) and others, along the axis of the fold, with no curvature. Gauss defined the curvature at any point on the surface as the product of the two most extreme curvatures running through that point. In this case the red one that runs along the axis has zero curvature and the most curved green path has some negative value. The total curvature for that point is zero times a negative number or zero. The beautiful part is the curvature for all points on the paper are zero if it's flat or folded in such a way that doesn't stretch or tear it.

Now find a volleyball or some other sphere. The surface is convex everywhere and the curvature is always positive in any direction. The Gaussian curvature for any point - the product of the curvature of the two extreme paths - is always a positive number times positive number or positive. (for now just consider the sign of the curvature - the actual measure is a bit beyond this post). This is why you can't perfectly wrap a sheet of paper - or pizza - around a ball. The paper always has zero curvature and the ball is always positive. This also explains why can't make an accurate flat map of the Earth and have to rely on projections with distortions.

Many surfaces are more complicated. For fun explore points on bagel or bananas, but for now back to the pizza. If you let it flop it still has zero curvature, but no strength. Putting a fold down an axis gives it rigidity. You find this throughout nature and engineering. An empty Coke can should hold your weight if you put it on the ground upright and carefully stand on it. Lay it on its side and will crush flat under your weight. The fold in a surface of zero curvature gives a lot of rigidity in one direction. Next go somewhere that is easy to clean and try to break an egg by squeezing it in your hand. The positive curvature everywhere makes it surprisingly strong. If the shell has an imperfection or if you tap it with the edge of a knife or fork while squeezing, you'll have a mess. The positive curvature was distorted and the surface failed.

Assuming you didn't make a mess, go back to the ball. Draw an equator and then mark a North pole. Drawing an arc at a right angle to the equator, you'll always intersect the North pole - it's a line of longitude. Next pick another point on the equator and make another arc at a right angle. You've made a spherical triangle. Sum up the angles. If you started with the points close together the sum will be just over 180° and if you go all the way around it will be nearly 540°. Our old rules of trig don't work on this non-flat surface.^{1} A neat thing you can do is measure the shape of the Universe in a way that is similar to making these trigonometric measurements. So far it looks extremely flat and that's a deep result about the structure and evolution of the Universe.

So get some pizza to celebrate the Remarkable Theorem its explanation of why we hold pizza the way we do.

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^{1} If you draw and measure some circles on the sphere you'll notice the circumference is less than 2πr and the area is less than πr^{2}. If you try it on a surface with negative curvature - a saddle shape or the inner part of a bagel for example - the circumference is greater than 2πr and the area is greater than πr^{2}.

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