Kjeller, August 2002.

Knots in art
Piotr Pieranski


Scientists tend to think, when a good idea comes to their mind, that they are the first to have it.
How often they are wrong!
Not only because they do not take into consideration that their colleague could have had a similar idea before.
But also because they do not at all take into account the possibility that
an ordinary man could be the first, many, many years ago.
Ordinary people do not publish, but, fortunately, what they do is of interest for artists.
Artists record the life and work of ordinary people.
This happens quite often, in particular when the ordinary man happens to be a beautiful, young girl.
I  am searching the archives for convincing evidence of my claim, and I think I have already found some.
It is the aim of the essay to present results of my search. It is up to you to decide, if I'm right.
Obviously, I focused my attention on the science of knots.
This is more or less the content of Part I.
Having it ready I discovered that there is a lot of other information gathered as a side product of my main research.
Thus I decided to extend the work and step by steps six more parts were born.
Part II deals with the vital question if looking for the ideal knot, thus, first of all, untangling the knots is safe and easy.
Part III is just a warning what a wrong approach to the untangling activity can do to a man.
In Part IV I am trying to present the beauty of knots, in particular those, in their ideal conformations.
Part V tries to answer the touchy question where from the mathematicians take their brilliant ideas.
Part VI presents my discoveries concerning the very beginnings of the knot theory
and some chilling or funny stories from its history.
I put a lot of effort to illustrate well my discoveries.
Fortunately, as you will see, prophetic Polish painters provided me with a lot of a ready for use materials.
Final remarks, acknowledgments and excuses <---- are here

Part I

Who was the father of the ideal knot problem?

As I have discovered, we should rather ask "Who was the mother of the ideal knot problem?"
Why? After an extensive study, I come to the conclusion,
that  the first person to think of the knot tightening problem
could have been a young female Polish shepherd met by Jozef Chelmonski.
Have a look at the picture below. The gossamer knot is hardly visible, but it is there!

Gossamer with a knot
(based on Gossamer by Jozef Chelmonski)

A similar problem, although in rather different circumstances,
was also considered by an anonymous lady known by Wojciech Gerson,
a Polish painter living at the end of the XIX century.
Her quite original idea was to use the viscous drag of the flowing water
to tighten and untangle knots.

The Rest with the Entangled Trefoil
(based on The Rest by Wojciech Gerson)

As it often happens, the same ideas were born, independently, in many minds.
There exists a convincing evidence, that also a lady known by Amadeo Modigliani was
making some experiments with untangling knots. Her position in the knot tightening debate
was, apparently, completely different.


Reclining nude untangling a knot
(based on: Reclining Nude from the Back by Amadeo Modigliani) 

One of the most intriguing questions that every mathematician
asks him/her-self in his/her childhood is:
 Where do the ideal knots come from?
Well. The question is touchy in the case of humans. You know: awkward facts of life.
Fortunately, in the case of knots the situation is simple and clear.
One of the pictures by Chełomoński is devoted to it.
It shows Jun O. making his first scientific discovery:
Theorem I.1. Ideal knots are delivered by storks.
Seeing is believing. No proof is needed.

New arrival
(based on Storks by Jozef Chelmonski)

Part II

Is bringing a knot to its ideal conformation safe and easy?

As we know, tightening a knot may bring it to the ideal conformation. It was probably
very late evening, that a girl observed discretely by Georges de La Tour managed to
tighten a quite complex achiral knot to its ideal conformation.
She seems to be satisfied with her work.
It took her not more than a while to arrive at the final conformation.
(Have a look at the oil lamp. Only a bit of the oil is missing.) )

Magdalen with a knot
(based on: Magdalen with the Smoking Flame by Georges de La Tour)

Needless to say, not all endeavors to untangle knots and bring them to the ideal conformations
 are successful. The knot untangling business is not a joke. Look at the poor boy
portrayed by Caravaggio. Being bitten by an entangled knot is not a pleasant experience!

Boy bitten by a knot
(based on: Boy bitten by a lizard by Caravaggio)

One of the most spectacular failures was recorded by the Norwegian painter Edvard Munch.
He spent a part of his life in Paris and Berlin. It could be in Berlin, I guess, that the tragedy
depicted in one of his most famous pictures took place. The fellow shown on the left failed completely.
That is why he is hiding his face. The lady has had already her first try.
Apparently: unsuccessful. She is depressed but not hopeless.
Which was the end of the knot untangling party?
Gloomy, I guess.

(based on: Ashes by Edvard Munch )

Problems of the Norwegian couple are not strange to me.
The particularly nasty knot they recklessly decided to play with has a bad reputation.
You will find it in one of the most famous Polish pictures by Jan Matejko.
It shows the royal joker, Stanczyk, thinking about the problems of Poland.
To be more precise: thinking about the relation of Poland with its neighbors.
The relation is the famous Polish Gordian Knot.

Stanczyk thinks about the Polish Gordian Knot
(based on: Stanczyk w czasie balu na dworze królowej Bony wobec straconego Smolenska, Jan Matejko)

What is a depressing problem to some people, can be a source of joy to others.
It's just the matter of an appropriate approach to difficulties.
A splendid illustration of this truth can be found in paintings of Axentowicz.
Have a look at the young lady portrayed by the gifted Polish artist.
Being young and beautiful, the lady does not get depressed when she finds the Polish Gordian Knot too difficult to untangle.
She treats the sinister knot as an intriguing piece of jewelry. How charming is she checking if it matches her carnation.
Don't you agree?
(I think, I'm slightly in love with the girl. And I feel she likes my knot. The power of art is incredible. Just try.)

Spring looking at a knotted jewel.
(based on Spring by Teodor Axentowicz)

Part III

What a wrong approach to ideal knots can do to a man.

Contemporary artists are also interested in the ideal knots problem.
Obviously, their view is completely different. It is so, because the world around them is different:
more aggressive, dangerous, terrifying. People involved in the knot untangling business are often perverted.
A good example of what a wrong approach to the untangling activity may do to a man has been
illustrated by Starowieyski. His "Serial knot untangler" shows a monster ready to use his
brutal skill on a completely helpless knot. To me it is awesome.
Certainly, not all means leading to the ideal goal are permitted.

Serial knot untangler
(based on: Nieuchwytny morderca by Franciszek Starowieyski)

That untangling knot can be connected with perversion was known already to ancient Greeks.
As I have found out, the activity for which Oedipus gained such a terrible reputation was but a cover
to something even more terrifying. Not only he was untangling knots by himself using non-Reidemeister moves,
 but he also was ordering it to his own son! Henry Fuseli revealed this horrifying truth.
Just look at the poor juvenile. How hideous this order must have been to him.
Non-Reidemeister moves! Gosh!

Oedipus ordering his son to use non-Reidemeister moves
(based on: Oedipus Cursing his Son, Polyneices, Henry Fuseli)

Entangled knots are not good for human minds.
Among the artists who knew this simple truth was Edgar Degas.
He seems to be the first to discover the cause of the blue mood ubiquitous among the absinth drinkers:
nastily entangled, knotted molecules of higher cyclic alcohols.
Today we know: mind is not able to untangle them.
On the contrary, the naughty molecules easily entangle the mind.
(Try. With caution!)

Knots drinker
(based on : Absinth drinker by Edgar Degas)

Part IV

The beauty of ideal knots

Let me change the mood.
From the gloomy one to more optimistic.
Untangling knots is neither easy nor safe, but the result - ideal knots - are worth the risk: they are pretty!
Some artists knew this before the scientists started to think about them.
There are many wonderful examples of  ideal knots depicted in the most precious pieces of art.
Being Polish, I went through the art galleries trying to find ideal knots in pictures painted
by Polish artists, in particular those, whom I like most: Wyspianski, Axentowicz, Zmurko.
I think I have found some interesting pieces.
Let me start with Zmurko. He is less known, but as I find, his knowledge of ideal knots was deep.
One of his paintings can be seen as a proof of the existence of ideal knots. I like the proof.
In contrast to the proof by Cantarella, Kusner and Sullivan, Zmurko's proof is full of warmth and feelings.
(CKS, sorry, but this is true.)

A cigarette, a fan and an ideal trefoil
(based on: A fan and a cigarette by Franciszek Zmurko)

It seems that many Polish artists who spent some time in Paris were aware of the existence of ideal knots.
Certainly, Wyspianski knew they existed. More. He was apparently aware that the path leading to the ideal
conformation could be blocked by the misleadingly beautiful, local minima.
Looking at his Girl with a knot you will certainly recognize the Gordian Unknot.
Neither I, nor anybody else was able to provide a formal proof of its existence.
Wyspianski's approach is different: "The proof of the pudding is in eating".
He simply portrayed it.

Girl with the Gordian Unknot
(based on: Girl with a flowerpot by Stanislaw Wyspianski)

As we are with Wyspianski, my favorite painter, let me tell you something more about him.
His intuitive knowledge of the knot theory must have been a good one.
In one of his paintings, we find a clear image of a nicely tightened, big achiral knot.

Girl with an achiral knot
(based on: Girl in a hat by Stanislaw Wyspianski)

In another, I recognize a toy in form of a cable knot.

Sleeping Mietek with a cable knot
(based on: Sleeping Mietek by Stanislaw Wyspianski)

To end with something special, let me introduce to you another remarkable Polish painter, Jozef Mehoffer.
In Friburg, Switzerland, you may find his wonderful stained glass pictures. Here, I would like to make you
acquainted with one of the most unusual piece of his work, a dream-like picture of a garden.
It would be strange, if there were no ideal knots in it. There are two, both trefoils.

Strange garden with trefoil knots
(based on: Strange garden by Jozef Mehoffer)

Part V

Where does the inspiration of the knot theorists come from?

The number of scientists involved in studies connected with the theory of knots grows at an exponential rate.
Why? From where do they take their most brilliant ideas?
These are the questions which I am asking myself quite often, in particular when I am reading
papers written by my mathematically oriented colleagues.
Due to the apparent time worm-holes Polish painters were able to provide answers to the questions
before the questions have been posed.
One of the less known Zmurko's paintings reveals the truth. The pleasant truth, I would say.
(More and more I am tending to think that maybe I should have become a mathematician, not a physicist.)

Inspiration of Eric R.
(based on Faust illumination by Franciszek Zmurko)

The method used by Eric R. reminds me "Joseph Balsamo", the novel by Alexandre Dumas (father),
which I was secretly reading in my youth.
That mathematicians are able to convince brilliant girls to share their secret thoughts with them is wildly known.
Another story of this kind has been recorded by Axentowicz..
The victim is different, the predator is different, the method is different.
but the goal the same - a brilliant idea.

Inspiration of Rob K.
(based on Reading girl by Teodor Axentowicz)

Some scientist are too shy to do, what Eric R. or Rob K. do, to find brilliant ideas.
It does not mean, of course, that their minds are free from temptations.
And temptation, as we all know, can be easily transformed into reptation.
Once more it has been Axentowicz to reveal the truth.

Temptation of Tetsuo D.
(based on Redhead by Teodor Axentowicz) 

Not always the circumstances at which mathematicians get their best ideas are so dramatic.
Sometimes, the ideas appear all af a sudden when nobody really expects them, e.g. during a picnic.
It was a sunny Sunday afternoon. (
I cannot find out at which AMS spring meeting it was.)
Jon S. and Greg B. were enjoying an afternoon tea, when a very good idea came to their mind.
They left for a while the company to discuss it in private..
This very moment has been recorded by Gierymski.
The subject of the discussion will remain unknown. Peccato!

Jon S. and Greg B. discussing a new idea
(based on In the garden house by Aleksander Gierymski) 

Good ideas can be fished for also in tranquility, far from the hectic life of modern society.
I find an example of this approach in one of the paintings by Siemiradzki.
He shows Ben L. after two days in raw spent at the keyboard.
Ben seems to be innocently watching the fishing children, but it is only appearances, appearances...
Can you see the ideal knot resting behind the tree? In a while Ben will grab it.

I know he will. Have a look at his home pages. The poor knot is there!

Ben L. fishing for ideal knots
(based on Fishing by Artur Siemiradzki)

Part VI

Knot X Files
(Stories from the history of the knot theory)

Searching the archives of old paintings we find traces of the events which had an essential influence on the development.of the knot theory..
One of such events was recorded for posteriority by Jan Matejko, a great Polish painter of whom you will hear more.
The painting I am presenting below shows Dale R., the prophet of the knot theory,
at the memorable day when he received from the Royal Press (Publish or Perish)
the author copy of his famous treatise "Knots and Links".
Having received the copy he decided to hand it to his protector, the King of Poland (not visible in the painting).
As the legend says, the King read the book and awarded Dale with the title of the Royal Knotter.

Dale R. handing his treatise on knots to the King of Poland
(based on Blind Wit Stwosz with his granddaughter by Jan Matejko)

Knot theory, as a part of topology is dated for a bit more than 200 years.
Is this right?
My visit to Las Vegas convinced me that this dating may be completely, absolutely, totally wrong! Why?
Just have a look by yourself at the picture I took at the entrance to one of the biggest casinos.
Don't you think the sphinx could have been a knot theorist?
I think he was.

The first knot theorist
  (photograph taken by PP)

Knot theory was born in sweat and pain.
The birth of the physics of knots took place in radically different circumstances.
That this was the case one can see looking at another painting by Axentowicz.
No sweat. No pain. Just a friendly look.
And a knot of a mutual understanding.
Maybe more.

The birth of the physics of knots
(based on Redhead by Teodor Axentowicz)

Getting a proper theoretical insight into the solution of some topological problems is not a trivial task.
Often one has to fight for it risking physical and mental abilities.
No wonder such fights attracted the interest of painters.
The painting presented below shows Jozek P., known also as Zawisza the Black, in the memorable moment
of winning his fierce battle to understand functioning of a nasty topological puzzle known as Chinese Rings.
He pierced the puzzle with the sharp edge of his brilliant mind.
Obviously, it is not easy to paint piercing a problem with a mind, thus Matejko made it in a symbolic manner.
(Notice the young face in the left bottom corner of the painting. It's the face of Tomek, the faithful squire of Zawisza.
They say that it was him to help Zawisza to win the battle.)

Jozek P. piercing the puzzle of Chinese Rings
(based on The Grunwald Battle by  Jan Matejko)

Let us move for a while to realm of scultpure.
We all admire the ancient sculptures discovered by archeologists.
Egiptian sphynx are among the most impressive ones.
But let me draw your attention to something even more impressive: the knotted cubes.
They were discovered in the middle of a desert by the team lead by Alexander G.
Nobody knows which their function was.
My personal guess is that they had no function but simply presented the state of mind of the sculpturer
after a few days of  thinking about the sense of his work.
Think yourself about the problem
and check the state of your own mind.

Alexander G. resting after the discovery of the knotted cube
(Based on a POV-ray scene created by Gilles Babin. Thanks Gil!)

Which is the use of the knot theory?
I'm very often asked the question.
(By people, who are not able to tie properly their shoelaces.
I happen to know some. They claim that finding an ideal knot is a problem of logistics.)
Gosh! Have a look at historical paintings by Grottger. There you will find the answer,
Let me explain. Polish national sport is plotting uprisings.
It is important, since it is only due to the sport that we managed to survive as a nation.
To win an uprising you need weapons.

The picture, to which I am drawing your attention, shows Polish mathematicians preparing lethal knots.
You may recognize the face of one of the fellows who hammer the knot.
Yes. It is our Canadian ally, Rob S.. the famous knot plotter.
(Thanks Rob! Great job! It's not your guilt that the uprising was lost.)

Rob S. hammering a knot
(based on Hammering scythes by Artur Grottger)

Strange enough, most applications of the knot theory are connected with rather violent events.
To support my claim let me present another historical masterpiece by Matejko.
This time he reconstructed a really gloomy story.
One of our best queens, Bona, was poisoned. She had lots of enemies.
Why? She was Italian and it was her to introduce la verdurra into our traditional kitchen.
As the legend says, the royal knot theorist, Jon S. was trying to save her life with the extraction from an ideal knot.
(Click on the image to see it better.)

And what?...I am sorry to say - it did not work.
I think he used too simple knot. He used a mere trefoil.
Perko claims that 10.161 or 10.162 would be serve better.

 Jon S. trying to save Queen Bona
(based on Poisoning of Queen Bona by Jan Matejko)

Question "Why does the trefoil knot weaken ropes more than the figure eight knot?" bothered human race for centuries.
Some people lost their lives trying to find the cause of this crucial law of nature.
In vain. The puzzle remained unsolved.
It needed the brilliance of  Giovanni D., his Italian fantasy, to arrange a proper experiment and find out the truth.
Once more let us visit the gallery of Matejko's work.
The picture below shows Giovanni D. at the moment of the discovery.
Notice the bunch of spaghetti at his feet. Crucial experiments were done before.
What you see is but the moment of illumination: CURVATURE!

Giovanni D. discovers the truth
(based on: Copernicus or the dialog with the God by Jan Matejko)

Matejko, Matejko... Not too much of the Matejko? NO!
He deserves it. It were his paintings that stimulated our imagination where we were children.
Stories told by Matejko's brush get deep into the memory of every Polish child.
For instance, the story of the alchemist Greg B. vel Sedziwoj.
He was an alchemist. But he was different from other alchemists.
They were looking for the philosophical stone. He was looking for the philosophical knot.
He succeeded. The picture shows Sedziwoj showing the philosophical knot to the King Sigmund III.
(I think, it's 4.1. Ideal conformation.)


  Greg B. presenting the philosophical knot to King Sigmund III
(based on Sedziwoj by Jan Matejko)

It would be an unforgivable mistake, if you were not introduced to the art of Jacek Malczewski.
His painting is highly symbolic. Things and events you see in his pictures have hidden meanings.
Looking recently at one of the pictures I realized how prophetic Malczewski was.
The mysterious Coronation of the Swiss King of Knots now reveals its sense.
I think I recognize the face of the king. Yes. No doubt. It's Andrzej S.
He accepts the crown with dignity and modesty.
One cannot oppose the will of gods.

Coronation of the Andrzej S.
(based on Tobiasz i Parki by Jacek Malczewski)

Well, Andrzej S.  is maybe the Swiss King of Knots, but it does not mean he was the fellow working of knots in Helvetia.
Certainly not. Above I have already mentioned another one: Giovanni D. the discoverer of the destructive force of curvature.
Here comes another one: John M. John M. the Global, to be precise.
 He portrait was painted by Leon Wyczolkowski.  It shows  John the Global in the memorable day
when he could proudly present to the whole world  the precious jewel
manufactured by one of his apprentices, Oscar G. (his happy face is shown on the left).
It took months to manufacture the incredible piece of the jewelry art:
each of the infinitely thin golden spokes emanating from the points of the trefoil frame represents the diameter of a disc that realizes rhoG at that point.
If you would like to know, what rhoG is, visit the home page of  John's manufacture or the home page of his skilful apprentice Oscar.
You will not be disappointed.

John M. with the Constant Global Curvature Trefoil
(based on Ludwik Rydygier by Leon Wyczolkowski )

As mentioned above, Malczewski's paintings are full of symbols.
Their meaning was a mystery for his friends. But today all becomes simple and clear.
Have a look at the picture below.
What, for goodness sake, are the balls released by the mysterious gentleman?!
First of all, these are not balls, but holes. Magnetic holes.
The fellow who releases them into this world is Arne S. A Norwegian.
(The Norwegians were always doing strange things!)
Which is the use of them? This you will become clear soon.

Arne S. releasing magnetic holes
(based on U studni by Jacek Malczewski)

Thus. Which is the use of Arne's holes?
This is demonstrated by his former Ph.D. student Geir H.
The holes can be used to plait wonderful braids.
And you know, how ladies are fond of braids.
Look at the girl. She really seems to be interested by what Geir is doing for her.
You may try to do the same for your girls, but mind your feet!
Looks easy. Is not.

Geir H. plaiting a braid
(based on Polonia by Jacek Malczewski)

Sorry, but I cannot help returning to Matejko.
Why? Because in Part  IV I recklessly mentioned the CKS proof.
I expressed a bitter opinion that the proof lacks warmth and feeling.
One can understand why is that so.
The work on it was full of violent clashes between the three giants of the knot theory.
Presenting a proof it is not dating a girl, although results may be similar.
The sharp eye of Matejko managed to grab the very moment in which
Jason C. shows his version of the proof to Rob K. and John S.
You may ask, why he put on his armor, why pulled out the sword.
Look at the faces of the other two fellows.
What I see is sarcasm and irony.
It is not easy to be a mathematician. Maybe it is fortunate that I am only a physicist?
(By the way. Do you know who the other two fellows are?)

Jason C. presenting his proof
(based on Wladyslaw Lokietek by Jan Matejko)

A work on a proof of a non-trivial theorem is a fight, sweat (even booze).
But, when the proof is ready - GLORY! The author enters the limelight.
His fellow mathematicians are truly happy. No sign of envy. No more ironic or sarcastic remarks.
Let me present the last, (cross fingers - the last!) of the Matejko's paintings.
It shows the triumph of Jorge C.
He spent days and nights trying to prove that the knot shown in one of the flags does not exist.
 Matejko portrayed him in the happy moment at which, blessed by his adviser (on the left), Jorge C. delivers his
proof to the editorial office of Journal of Knot Theory and Its Ramifications.
You may ask why Jorge C. wears this out-of-fashion, heavy duty helmet.
Well. When working with physical knots you should really take precautions.
Remember! I know something about it!

Jorge C. delivering his proof to JKTR
(based on: Constitution of May 3 by Jan Matejko)

To be continued ... until somebody protests...


I hope you enjoyed this short course on the presence of knots in art..
If my comments made you smile from time to time, I am happy.
It was my intention just to entertain you, nothing else.
If, by the way, you memorized some Polish names, I am pleased.
It seems to me that some of them are just worth remembering.
Names of Polish mathematicians are well known in the whole world.
What about Polish painters?
Think about it.


I wish to thank all my friends who drew my attention to the pieces of art which I used to illustrate my considerations.
Since what I am doing is neither serious nor profitable I did not ask the owners of the pictures for permission to use them.
The essay is aimed to entertain my friends, mostly mathematicians, and will never be published.
However, trying to be be fair, I linked all of the used pictures to the pages from which they were taken.
I hope this will satisfy the owners of the copyrights.
Most of the pictures were taken from
the Gallery of Polish Paintings Zascianek.
Its www pages are among those, which I visit most often.
Click in there! Have a look!
The picture by Starowieyski was taken from the Polish Posters.

Back to the main page