1. In archaeology, there is a story about counting ostrich eggs.
Uleni? The archaeological team led by Friedman found an ancient city site, the earliest known Egyptian temple, a brewery, a potter's house burned by a nearby kiln fire, and the only known elephant tomb in ancient Egypt when investigating the ancient Egyptian site in Niken (better known as its Greek name "Shirakopolis").
There, archaeologists saw the broken ostrich eggshells unearthed in HK6 area. The complete ostrich egg was originally used as a cornerstone and placed in the foundation of the new building. After thousands of years, these ostrich eggs have long been fragmented, so the first question is "How many eggs are there?"
Typical ostrich egg fragments in Syracuse.
The plan to piece together eggshells (known as the "egg head plan") proved to be too time-consuming. Therefore, archaeologists hope to divide the total area of eggshell fragments by the surface area of an ordinary ostrich egg to get an estimate. This is where mathematics comes in handy. What is the surface area of ostrich eggs? Or by extension, what is the surface area of an egg?
Arthur, who is in charge of this research? Moore realized that the shape of an egg is very similar to putting two semi-ellipsoids together. If you can find the surface area of an ellipsoid, you can divide it by 2, and then add up the surface areas of the two parts. Little friends can do their own calculations ~
2. How to tell the direction of the bicycle?
The picture below shows the wheel prints left by the cyclist. Does he travel from left to right or from right to left?
To infer the direction of a bicycle, we must first determine which wheel tracks are left by the front wheel and which are left by the rear wheel.
If the wheel print is straight, the direction of the wheel is the same as that of the wheel print. However, if the wheel footprint is curved, the driving direction of the wheel will be consistent with the tangent direction of each point on the wheel footprint. A tangent is a straight line that has only one contact point with a curve. For the sake of understanding, please carefully observe the wheel footprints left by the unicycle in the picture below. The wheel directions of point A, point B and point C are consistent with the tangent direction I marked.
A bicycle has two wheels. The front wheel can point in any direction, but the rear wheel has no freedom in the driving direction-it must always point in the direction of the front wheel.
So no matter where the rear wheel is, the front wheel is in the tangent direction of the rear wheel footprint, and the distance from the rear wheel must be equal to the length of the bicycle body. In other words, all tangents of the rear wheel print must intersect with the front wheel print, and the distance between the tangent point and the intersection point must be equal to the body length.
Now, please look at point D on the thick line below. The tangent of this point does not intersect the two wheel tracks. Therefore, we can draw the conclusion that the curve where point D is located is not the rear wheel print, but the front wheel print.
Now, we can determine the direction of the bike. We know which wheel tracks were left by the rear wheel. The above discussion also tells us that the length of bicycle body will intersect with the front wheel footprint at a certain point along any tangent line on the rear wheel footprint. So we take two points, e and f, on the rear wheel print, draw the tangents of the rear wheel print respectively, and observe the intersection of these two tangents with the front wheel print. As can be seen from the figure below, the lengths of the two line segments on the left side of point E and point F are equal, while the lengths of the two line segments on the right side are not equal. The distance between the wheels will not change during the running, so it can be seen that the direction of the bicycle is from right to left, which is too simple!
(Content transferred from Mathematical Jingwei)