Five interesting topology transformation Problems

If you like the space imagination challenge of the last time, you will certainly like the intuitive topology book of V. V. prasolov. The first chapter of the book contains five very classic "topology transformations" Puzzles, which we will share with you here. Note the game rules: we assume that all objects are made of rubber and can be stretched, squashed, and bent at will, but do not allow any operation to change the essential structure of the image, such as cutting and sticking.

1. Can I consecutively change the left image to the right image?

2. Can I consecutively change the left image to the right image?

3. There is a circle on the surface of the three-dimensional figure shown in the left. Can this circle be changed to the position shown in the right image through continuous transformation?

4. Create a hole in the surface of a tire. Can I flip the interior surface of the tire to the outside through continuous transformation?

5. Can I consecutively change the left image to the right image?

1. Can I change the left to the right continuously?

The answer is yes, as shown in:

This means that if the human body can be deformed as freely as a rubber person, then we can leave our fingers open after making two rings with both fingers and forefoot, returns the ring. Algorithmic and computer methods for three-manifolds draw a very beautiful book:

What's more interesting is that if you only have a watch on your wrist, the above solution will not work:

2. Can I change the left to the right continuously?

The answer is yes, as shown in:

3. There is a circle on the surface of the three-dimensional figure shown in the left. Can this circle be changed to the position shown in the right image through continuous transformation?

The answer is yes, as shown in:

4. Create a hole in the surface of a tire. Can I flip the interior surface of the tire to the outside through continuous transformation?

The answer is yes. First, make a continuous transformation as shown in. As you can see, a tire with holes on the surface is essentially two circles of paper sticking together! However, note that paper circle 1 and paper circle 2 are not in the same position: one is a white surface (that is, the internal surface of the first tire) outside, one is the shadow surface (that is, the outer surface of the first tire. Now, we use the paper sphere 2 as the original paper sphere 1, and the paper sphere 1 as the original paper sphere 2, turning them back into the tire shape, and turning the internal and external surfaces of the tires upside down.

Interestingly, after turning the interior surface of the tire out, the "warp" and "weft" (let's say that) on the tire will also be reversed:

Wikipedia has an extremely handsome animation that shows the process of turning the inner surface of a ring to the outside. This animation is very addictive, but it takes 10 minutes to watch it!

5. Can I change the left to the right continuously?

The answer is yes. First, make a continuous transformation as shown in, so it becomes the figure (A) in question 1 ). Using the method of Question 1, we can change the shape we want.

 

From: http://www.matrix67.com/blog/archives/5140