1. Fun fact: By rotating the surface around the center, you should also be able to find a curve with a zero normal curvature, i.e. a straight line. Can you find it? 2. Technically, there is a specific way to find out the normal curvature with respect to a certain direction, that is to create a perpendicular plane along that direction, and the normal curvature at that point is exactly the curvature of the point on the line of intersection between the plane and the surface. 3. This is possible due to the extreme value theorem. Since the surface is smooth, rotating it to consider every direction is a continuous process so the normal curvature must attain a maximum and a minimum each at least one. A significant consequence of Theorema Egregium is that a ball cannot be completely laid flat since the Gaussian curvature of a ball is positive while a flat plane is zero. In particular, it is impossible to create a flat map of the Earth without any stretching, so the world map must be distorted to compensate for the Gaussian curvature. More interestingly, although there is no areapreserving map, there are still maps which preserve angles, such as maps on the Mercator projection (figure 4). These maps are crucial especially for marine navigation to ensure that one can reach the destination by heading to the angle shown on the map. How to Eat a Pizza Decently? Finally, we go back to the Gaussian hold of pizza. A pizza slice has a zero Gaussian curvature everywhere since the dough was rolled out flat before baking. By bending the crust to fold the pizza slice sideways and not in any other direction, we preserve a zero normal curvature everywhere of the pizza slice to maintain its zero Gaussian curvature (refer to figure 2(b) again), in a way that keeps the pizza “straight” so that the tip of the pizza can safely arrive at our lip. Congratulations! Now you know not only how to eat a pizza elegantly, but also the reason behind the technique. This problem regarding the curvature of surfaces is in the field of differential geometry, which Figure 4 The world map on Mercator projection, which preserve angles but not area. Regions at high latitudes are largely distorted. Photo credit: Daniel R. Strebe [1] deals with the geometries of smooth shapes and involves mathematical techniques from calculus and linear algebra. Now try it yourself, the Gaussian hold for pizzas! 薄餅慘劇 如果你現在要吃一塊薄餅，你會怎樣做？通常事情會 這樣發生： 自薄餅誕生多個世紀以來，這一直都是我們吃薄餅的 悲慘宿命。然而，你只要從批邊將薄餅沿長邊對折，就能 使薄餅保持筆直，然後輕鬆地吃掉薄餅。姑且讓筆者稱這 個方法為「高斯握法」，靈感來自數學家卡爾．弗里德里希． 高斯（Carl Friedrich Gauss）！ 從數學角度看，這個秘技與一個叫曲率的概念有關。曲 率的概念對我們來說並不陌生，它是衡量「彎曲」程度的 一個值。對於一維曲線，曲線在某一點上越為彎曲，其曲率 就越大。從數字上量化的話，一個半徑為 r 的圓在每一點 上的曲率均為1/r，因此大圓的「彎曲程度」比小圓要小。 那很自然地，直線上任何點的曲率皆為零，即是沒有彎曲。 此處要注意的是描述某一點的曲率是較為可取的做法，因 為曲線在不同點上可以有不同的曲率。 然而，在我們原本的問題中，我們處理的是一個表面 （嚴格來說，因為薄餅有厚度，所以並不是一個表面，但 我們可以暫且將其視為一個表面）。那麼，曲率的概念如 1) 你握住薄餅邊緣。 2) 薄餅從你手中垂下，來回搖晃。 3) 配料掉到桌上，你無奈地坐在那裡， 看著桌子變成一團糟！
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