如何不失禮 The Pizza Tragedy If you were to eat a pizza slice right now, how would you do it? Typically, it goes as such: For centuries since its invention, this is how pizza is tragically eaten. However, if you bend the crust to vertically fold the pizza into half, it can be held straight and easily eaten. I call this the “Gaussian hold,” inspired by the mathematician Carl Friedrich Gauss! The secret behind this trick – in the mathematical point of view – relates to a concept called curvature. The idea of curvature is not that foreign to us; it is simply a value measuring how “curved” or “bent” something is. For a one-dimensional curve, the more curved it is at a certain point, the greater its curvature. To give a numerical sense, a circle of radius r has curvature 1/r everywhere, so a large circle is “less curved” than a small circle. It is also natural to say that any point on a straight line has no curvature, i.e. zero curvature. Note that it is more accurate to describe the curvature at a certain point, because a curve can have different curvatures at different points. Nevertheless, in our original question, we are dealing with a surface (well, strictly speaking, a pizza is not exactly a surface because it has thickness, but we can view it as one). So how does the concept of 1) You grab a slice of pizza by its crust. 2) The pizza droops and dangles on your hand. 3) The topping falls on the table, and you sit in despair as you struggle with the mess! curvature relate to the laws of bending surfaces? The answer relies on a remarkable result related to a specific kind of curvature called the Gaussian curvature. Enter the Gaussian Curvature Let’s discuss how we describe the curvature of a surface. Think about the surface of a three-dimensional object such as a piece of paper, a ball, a cylinder, a pringle chip, or even a donut. For a certain point on the surface, notice that you will get a different curvature depending on the orientation of the surface you are considering. To get a more concrete example, take a look at the surface in figure 1 and find the curvature at the center (origin) of the graph. Depending on which orientation of the surface we take (red or blue), we have a different curvature with the red curve going upward on both sides, while the blue curve going downward on both sides. In this case, at the center point, we can say the red curve has a positive curvature, whereas the blue curve has a negative curvature (the definition of positive and negative curvature depends on the convention we use). Note that even though the curves span across the surface, we are considering only the curvature at the center point. Figure 1 A surface with both a positive (red) and a negative (blue) normal curvature at the center (origin) (footnote 1). a Pizza Slice How to Eat
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