The first is the most common question: Is 0.999 ... equal to 1? In fact, according to the current definition of real numbers, these two numbers are strictly equal, and the limit is 0.9999 ... which is not equal to 1. Strict proof can be proved by Dydykin division, and the general proof of 1/3 is not rigorous, because
1+ 1=2 in arithmetic is not an axiom, and it can be strictly proved according to Piano's axiom.
Koch curve: limited area, infinite perimeter.
Torricelli trumpet: limited volume, unlimited surface area.
Fixed point theorem: knead a map of the world into a ball and throw it on the ground at will. The vertical projection of a place on the map must coincide with the real place in space.
E is irrational, π is irrational, so are e+π, e-π, e*π and e/π rational or irrational? No one knows such a simple question.
Countless: Cai Tingheng, it sounds a little incredible. Cai Ting constant is a definite number, but it has been proved theoretically that you will never find it.
The rootless solution of quintic equation is frustrating and puzzling, but it is a fact.
Going up and down the mountain: climbing the same mountain, the climbing speed is 3m/s, the downhill speed is 5m/s, and the average speed is not 4m/s ... It's a bit counter-common sense, but a simple calculation will tell.
Harmonic series is divergent!
Rubber band and ant problem: an ant climbs from one end of the rational elastic rope to the other at a speed of 1cm per second. At the same time, the elastic rope is stretched evenly at the speed of 10cm per second. Can ants climb to the finish line? What happens if you stretch evenly at a speed of 100cm per second?
Cycloid length: Cycloid length is equal to four times the diameter of the circle, which is closely related to the circle. How to treat an "unreasonable" line, the length is not only irrelevant to π, but also a beautiful integer multiple! A line rolled out of a circle has nothing to do with π, which is really incomprehensible.
There are infinitely many regular polygons, but what about regular polyhedrons? A little surprised, there are only five. In fact, this is not difficult to prove. Just use euler theorem.
The most meaningful number: the Greyhound number (not now, of course, but his structure is the easiest to understand, and other structures such as Tree(3) are difficult to understand). The first layer of this number is far beyond human imagination, and you can't even say the number of digits of this number (write N digits at will). . . . . . (For example, the digit of 1234567890 is 10, and the digit of 10 is 2, and the digit of 2 is 1).
About dimension: the definition of spatial dimension in mathematics is different from that in physics. There are too many definitions of spatial dimension in mathematics.