Inner polygon (4): an irrelevant question about the mountain travel worm.

Let's take a look at a seemingly unrelated question before continuing to explore the problem of polygon inner graphics. Start from a point on the horizontal line, draw a line segment above the horizontal line, and return to the horizontal line (for example ). Think of this broken line as a mountain. We divide the whole mountain into two parts based on the highest peak. Now, let us assume that there is a fall-in-love mountain, one standing at the far left side of the mountain foot (point 0) and one standing at the far right side of the mountain foot (point 0 ). The two men will depart from the foot of the mountain and reach the top of the mountain at the same altitude. Regardless of the shape of the mountain, can this romantic idea always be realized?

Note: During the mountain climbing process, the mountain can go back to take care of each other. For example, for the hill shown in the figure, two people can follow the methods below to Achieve Synchronous mountain climbing. The routes for the two people are:

0 → 1 → 2 → 3 → 4 → 5 → 6 → 5 → 4 → 3 → 2 → 1 → 2 → 3 → 4 → 5 → 6 → 7
0' → 1' → 2' → 3' → 2' → 3' → 4' → 5' → 6' → 5' → 6' → 7' → 8' → 9 '→ 8' → 9' → 10' → 11'

In fact, no matter what the line segment is, such a mountain climbing method can always be achieved. Just like the rectangular Partitioning Problem introduced earlier in this blog, this problem also involves multiple proof methods, such as mathematical induction proof, construction proof, and Graph Theory Model proof. Next we will provide a proof of topology, which is the most ingenious proof I have ever seen.

Take the left-side Mountain as the horizontal axis and the right-side Mountain as the vertical axis. Place all the points at the same height on the plane to form a continuous curve. Next, we only need to explain that this curve connects the points in the lower left corner and in the upper right corner. To this end, we only need to explain that any continuous curve from the top left to the bottom right corner will surely be at the same time with this curve. This is obvious, because when a person walks from the top of the hill to the foot of the hill, and another person walks from the foot of the hill to the top of the hill, they will inevitably encounter at the same height ".