(Photo by Postdlf, via Wikipedia)

Some people fear alien spaceships. Alas, spaceships exist as much as pink flying elephants. The term “spaceship” is a lie in one word.

Other people fear that some evil energy may appear and suddenly take over the world. No, this will not happen. Energy is conserved. Nature follows rules.

Other people sell shares in companies that extract energy from the vacuum. But the vacuum is defined as the state of lowest possible energy. Such energy extraction is impossible. These people are lying.

Still other people say that nature has secrets, that there are mysterious phenomena in nature, or that there are hidden powers. Instead, a bit of physics will make you understand that this is not the case. Many cults spread such fears, all based on lies. Physics, your own experience, and your own reason will reject these fears and lies.

Physics does so by explaining that every change in nature is due to motion. Therefore, every change in nature has conserved quantities. In particular, in nature, things cannot appear or disappear suddenly. Not even lightning. Such fears are unfounded.

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But physics also helps to push curiosity to the limit. One learns what matter is made of, how light moves, how computers work, how lasers function, how atoms and molecules are constructed, and how stars shine and move. One explores and learns to explain chemistry, biology, machines, and the universe, with its size and its history.

Physics also tells us that everything that moves in nature is made of smallest countable constituents. This applies to people, animals, stars, black holes, and even to the curvature of empty space.

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Physics also tells us how we can change the world. When we dream about the future of medicine, of agriculture, of communication, of computing, of transport, of space travel, or of serving others in any other way, it makes sense to check what is possible and what is not by using all knowledge about how change and motion occur in nature. Physics is the science of motion.

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For a simple start, enjoy the free physics textbook in 5 volumes, downloadable at motionmountain.net, written to be entertaining and motivating on every page, for readers between 16 and 106 years of age.

(Photo by John Samuel, via Wikimedia)

Nature has a smallest length which arises when combining the maximum speed c, the maximum force coming from black hole limits, and the smallest action ħ:

However, in complete contrast, every equation in physics you ever learned assumes that points can be as close as desired. Every equation in physics either assumes that G vanishes or that ħ vanishes.

If we assume that G vanishes and assume ħ differs from zero, we call the description ‘quantum theory’ or ‘particle physics’ and get its equations.

If we assume that ħ vanishes and G differs from zero, we call the description ‘general relativity’ and get its equations.

If, however, we assume that neither vanishes, we get the minimum length, and equations are not possible.

Well, there is one way out. We could assume that the speed of light c is infinite. In other words, believing in a single equation is like believing in speeds faster than light.

This is a dream. Alas, nature does not allow it. Only fantasy does.

In technical terms, there are no trans-Planckian effects in nature. In simple words: believing in a single equation describing all of nature is science fiction.

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References:

The maximum force: motionmountain.net/maximumforce.html

The smallest action: motionmountain.net/actionquantum.html

The minimum length: motionmountain.net/minimumlength.html

(Image of an earthquake damage by Martin Luff, via Wikimedia)

Nature also has a limit to jerk. In American English, it has become common to (humorously) use the term jerk, which is usually abbreviated as j, for the time derivative of acceleration:

Like all quantities, it has a Planck-scale limit. The limit is

Like all higher time derivatives of acceleration, jerk has no practical use in the description of nature. Except one.

The internet gives a platform to many strange opinions. This includes the mistaken opinion that physical systems - or nature - can exceed the Planck limits. This is wrong.

Sometimes, the Planck mass is cited as an argument. But the Planck mass only limits elementary particle mass. And indeed, all elementary particles have smaller mass values than the limit.

In reality, there is no experiment whatsoever that allows trans-Planckian values to be achieved, to be measured or even to play a role in nature. Trans-Planckian quantities simply do not exist in nature. They are as real as pink flying elephants.

In fact, all measured jerk values are much smaller than the limit. Earthquakes can reach 300 m/s^3 - and such then lead to effects as those shown in the photograph above. Macroscopic jerk values are rarely larger than 10^5 m/s^3. Such values are found in crash tests and mechanical failures. For particles in accelerators, values in excess of 10^20 m/s^3 are achieved. But all these values are much smaller than the theoretical limit given above.

In other words, whenever you encounter a jerk, recall that every jerk is limited. This realization will greatly increase your quality of life.

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A literature reference on the jerk limit is Physical Review D 104 (2021) 124079.

Imagine a rope of constant radius that is infinitely flexible. Tie a tight knot into it. By how much did the ends of the rope come closer?

In other terms, find a formula to calculate the length of a tight knot in a rope.

Mathematicians call this the ropelength problem. It is unsolved for every non-trivial knot! The problem is unsolved for open knots (as in the figure) or for closed knots (as mathematicians like to define them). Only computer approximations to the ropelength exist. A formula does not. Clearly, different knots have different ropelengths:

In simple words, nobody in the world has a clue for the solution of the ropelength problem, despite the efforts of several mathematicians who have spent their lives on the problem.

The ropelength problem is fun

(1) A famous knot theorist humorously called the lack of solution a “shame” for the whole field of knot theory, and even a shame for mathematics as a whole.

(2) A few days ago, in early February 2026, I posted the problem on social media as a challenge for Ai systems. One big and proud mathematics Ai project - which I do not want to name here - wrote me back and stated that the problem was “ill-defined” and asked me whether I could provide a “reference”. Now, if an Ai system is even unable to find a wikipedia article, will it be able to solve the mathematical problem described in that article? I doubt it.

(3) Other Ai projects might succeed. One of them even liked my social media post. It will be fun to see what happens.

Mathematics has many beautiful and simple open problems. The ropelength problem might well come out at the top.

The ropelength problem is interesting

Apart from testing the abilities of mathematical Ai systems, the solution to the ropelength problem would be of interest because solving similar geometrical problems would solve the main open issues in fundamental physics.

Is it artificial or absent intelligence?

The ropelength problem is a typical research problem that cannot be solved by reading the literature. Some new insight is necessary. Just ask your favorite Ai system: “Can you provide formulae for the ropelength of the open and the closed overhand knot?”

Is Ai absent innovation?

This specific question is highlighted by the ropelength problem. It is not required that mathematical Ai systems be like Ramanujan. (Though that would be fascinating, of course.) But can Ai systems produce any innovation at all?

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References

The image is from P. Pieranski, S. Przybyl, A Stasiak, Tight open knots, The European Physical Journal E 6 (2001) 123-128, doi 10.1007/s101890170012, and is also found on ResearchGate.

The wikipedia article on ropelength is here: https://en.wikipedia.org/wiki/Ropelength. Some papers follow.

On the Minimum Ropelength of Knots and Links: https://arxiv.org/abs/math/0103224.

Tight open knots: https://www.researchgate.net/publication/225672330_Tight_open_knots.

The shapes of physical trefoil knots: https://www.sciencedirect.com/science/article/pii/S2352431621000043.

More questions that check the abilities of Ai systems are here: https://www.motionmountain.net/tiny.html#Ai. At present, they all disappoint - deeply.

https://www.motionmountain.net/simplifiedchinese.html

Enjoy.

With a big thank you to Liu Jiahui.

Equivalently, nature has a maximum power, given by c⁵/4G - around 10^52 W. No engine and no lamp in nature can be stronger. Interestingly, the stongest known lamps in the universe are binary black hole mergers, as shown in this computed image found on wikipedia:

See here for a computed animation of this incredible phenomenon: wikipedia/commons/a/a4/BBH_gravitational_lensing_of_gw150914.webm. When two black holes merge, they radiate extremely intense gravitational waves, which can be detected on Earth. The merger itself takes only a fraction of a second. (You might want to ponder why the highest energy sources emit gravitational waves, and why they cannot be light sources. It is a fun puzzle.)

In July 2025, a number of papers on the most powerful energy source ever found were published, the black hole merger GW231123 (see the full list of papers about it here).

The power limit c⁵/4G corresponds to emitting 50700 solar masses in a single second. The measured peak luminosity value for that black hole merger is around 150 solar masses per second. Even though this is a huge amount, it is still far from the upper limit. Therefore the power limit and thus the force limit are not exceeded. But why is this of interest?

The force limit c⁴/4G allows to deduce general relativity. This parallels the speed limit c, which allows deducing special relativity. It also parallels Planck’s quantum of action ℏ, which allows deducing quantum theory. And it parallels the smallest system entropy k ln 2, which allows deducing thermodynamics. These are the four basic Planck limits.

In other words, the force and power limits are tests of general relativity. But there is more. Maximum force is also built into the strand tangle model, the only unified description of general relativity and quantum theory that explains the gauge interactions, the elementary particles, and the fundamental constants. In fact, maximum force was the starting point of the strand tangle model.

Therefore, any test of maximum force is also a test of the strand tangle model. More precisely, the strand tangle model implies and predicts that no trans-Planckian effect can be observed. In particular, no larger force than the Planck limit c⁴/4G can be observed. If any larger force or any other trans-Planckian effect would ever be found, the strand tangle model would be immediately falsified. So far, this has not happened. On the contrary, so far, all observations confirm the strand tangle model.

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The second volume is available in Persian. See motionmountain.net/persian.html

All volumes are available in Naški, or Serbo-Croatian. See motionmountain.net/naski.html

A Chinese translation is in work at present (spring 2025). This is good news for many young people.