Knots in art
by
Piotr Pieranski
Introduction
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Ordinary people do not publish,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
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)
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.
Tangle
(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)
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)
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)
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)
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!
Yes! 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.
Amen.
