# Continuous Line Artist view of Haken’s Gordian Knot.

Depth of lines in black and white, in Haken’s Gordian Knot. Mick Burton, single continuous line drawing 2015.

Here is an update on posts which I did in May and June 2015 regarding the above Knot and the interest these posts have since generated.

As a Continuous Line Artist I have looked at many angles of what my lines may mean and what they can do.

One such examination was triggered by Haken’s Gordian Knot, a complicated looking knot which is really an unknot in disguise – a simple circle of string (ends glued together) making a closed line, which I saw in a book called “Professor Stewart’s Cabinet of Mathematical Curiosities”.   The drawing above is my version of Ian Agol’s illustration of the Haken Knot (see it in my post of 31 May 2015).  I used dark and light shades to emphasize the Overs and Unders shown for the line.

The reason that I was so interested was that it reminded me of my “Twisting, Overlapping, Envelope Elephant” (see below).

This single continuous line drawing is coloured to represent a “Twisting, Overlapping, Envelope Elephant”, which is Blue on one side and Red on the other. Mick Burton, 2013.

How this elephant line works is explained in my post of 31 May 2015.  In essence, you need to imagine that the composition is made up of a flexible plastic sheet which is Blue on the front and Red on the back.  Each time there is a twist, on an outer edge in the drawing, you see the other colour.

In the Gordian Knot, I spotted that there is a narrow loop starting on the outside (lower left on first illustration above)  which seemed to lead into the structure, with its two strands twisting as it went, each time in a clockwise direction.  I followed the two twisting lines throughout the drawing until they ended in a final loop on the outside (left higher).  I counted 36 clockwise twists and one anticlockwise.  My thoughts are explained in full in my post of 2 June 2015.

To aid the explanation I completed a painted version, where I used the same Blue and Red colours, as for the above elephant, to emphasize the twists.

Twisting, overlapping colouring of Haken’s Gordian Knot. Mick Burton, single continuous line drawing painting 2015.

Note that the colours in the Elephant define two sides of a surface, but in the Unknot the colours are illustrating the twist of two lines travelling together.  The twin lines go through other loops continually so there are no real surfaces.

After completing the above two posts, I decided that I would try and find out more about the Knot and came across a question posed by mathematician Timothy Gowers, in January 2011, on the MathOverflow website.  He had asked for examples of very hard unknots and after many answers he had arrived at Haken’s “Gordian Knot”.  He described the difficulties he was having.  Timothy said that he would love to put a picture of the process on the website and asked for suggestions.

As I had already done two pictures before I read his post I decided to respond.  The work that I did on this is detailed in my post of 5 June 2015 entitled “How do you construct Haken’s Gordian Knot?”.

My response duly appeared on the MathOverflow website in early 2015, but within a day or two it had been taken down and a notice appeared stating that only mathematicians of a certain status should post on the site.

That’s fine as my only maths qualification is General Certificate of Education at school.  At Harrogate Technical College I was thrown out of Shorthand and, with only three months to go to GCE exams they put me in for Maths and Art.  I owe many thanks to the Shorthand teacher, who thought my only skill was picking locks when someone forgot their locker key.  Also I have never had any discussion face to face with a mathematician about my art or my maths.

Following this setback I decided to set it all down in my Blog, in the three posts up to 5 June 2015.

Although I have not actually talked directly to a mathematician, I did correspond with Robin Wilson and Fred Holroyd at the Open University in the mid 1970’s about my ideas on the Four Colour Map Theorem.  I set out my ideas briefly in my post of 18 August 2015 “Four Colour Theorem continuous line overdraw”.

When Fred Holroyd responded to my write up, he used my own expressions and definitions which was very impressive.  He said that I had proved a connected problem, only proved in the world as recently as 16 years previously.   When I asked Robin Wilson about the announcement from a mathematician who said that he had proved the Four Colour Theorem, Robin said not to worry as he thought that this one was unlikely to be validated.  He said that he would prefer that my theory could be proved as it was elegant and also that they could use it.

The theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken, involving running one of the biggest computers for over 1000 hours.  After this I decided to go onto other things, leaving my art and maths behind for almost 40 years.

Yes, its the very same Wolfgang Haken, who devised the Gordian Knot!

Ok, lets move on.  In February 2016 I received an e-mail from Noboru Ito, a mathematician in Japan, saying that he had read my article of 5 June 2015 “How do you construct Haken’s Gordian Knot?” and it was very helpful.  He would like to add it to the reference of his new book “Knot Projections”.

Of course I agreed and he later confirmed that he had referenced my work to the preface of his book.

Here is a picture of my copy of his book which was published in December 2016.

“Knot Projections” by Noboru Ito, published December 2016 by CRC Press, a Chapman & Hall Book.

Additionally, in November 2017 I received an e-mail from Tomasz Mrowka, a mathematician at the Massachusetts Institute of Technology.  He said that he was interested in acquiring a copy of my Twisting, Overlapping colouring of Haken’s unknot.  “It’s really quite striking and I would love to hang it in my office”.

I was delighted to send him a photo which he could enlarge and frame.

# How do you construct Haken’s Gordian Knot?

After completing my drawings of Haken’s Gordian Knot, which I covered in my previous continuous line blog post, I decided that I needed to find out more about how this unknot was created.  It is one thing me portraying the route of the two strands running through a completed structure, but possible something very different if I construct it from scratch.

A Google search for Haken’s Gordian Knot took me to a page of MathOverflow website, where a question that appeared “Are there any very hard unknots?” posed by mathematician Timothy Gowers, in January 2011.  In an update after many answers he said that he had arrived at Haken’s “Gordian Knot”.

Haken’s Gordian Knot, from Ian Agol. A simple circle of string (an Unknot) formed into a complicated continuous line.

Timothy said that, after studying the knot for some time, “It is clear that Haken started by taking a loop, pulling it until it formed something close to two parallel strands, twisting those strands several times, and then threading the ends in and out of the resulting twists”. This approach is something like the suggestions I made in my last post on the basis of my Twisting, Overlapping, Envelope painting of the Haken Knot.

Twisting, overlapping colouring of Haken’s Gordian Knot. Mick Burton single continuous line drawing painting.

Timothy then added that “The thing that is slightly mysterious is that both ends are “locked” “.  When I started to build the structure from scratch I began to realise what “locked” may mean.

Constructing Haken’s Gordian Knot. Stages 1 & 2. Mick Burton.

After leaving the looped end at the start, the ongoing route first meets its earlier self at Stage 2.  However instead of the ongoing route going through the earlier one, the initial loop goes back through the later one. This must be what is meant by the first “lock”.

Constructing Haken’s Gordian Knot. Stages 3 to 7. Mick Burton

Continuing, things were as expected up to Stage 7.  I now realised that the route could be simplified to one line, as the Twists were not affecting progress but the feed through points were crucial.  I switched to drawing the route by using a simple line (to represent the twin twisting strands) and showed Feed Through points as Red Arrows.

Haken’s Gordian Knot, Simplified Route showing Feed Points. Mick Burton.

You can see that after point “C”, where the reverse Feed occurs, there are 12 expected Feed Through points until you arrive at point “E”.  Here instead of Feeding through an earlier part of the route, Haken indicates that you are expected to Feed through the End Loop at “E” which is too soon. This must be the other “Lock”.

At this stage, of course, I had no idea what to do.  Timothy did not seem to be using a lot of paper like me, but a “twisted bunch of string” and a small unknot diagram.  So I found some string, but was at a loss to make much sense of anything using that.  Timothy, however, was disappointed that it was so easy with his string initially, but delighted when it became more difficult !

What I did realise about the sections of route lying beyond point “E”, which I have coloured Green, is that they all lie beneath the rest of the structure.

This would allow the Green Area to be constructed separately before you sort of sweep it underneath as a final phase.  When I say “separately” I can only assume that you would need to do all this first, feed the result through the final loop and encapsulate the result.  You would then take this bundle to the start and use it to spearhead the building of the structure, leaving the loop at the other end of the two stranded string back at the start.

Haken’s Gordian Knot, Prior action for the Green Route, before starting main structure. Mick Burton.

Timothy said that he would love to put a picture of the process on the website and asked for suggestions.

Even though I am an artist and not a mathematition, I had already done two pictures of Haken’s knot before I found the MathOverflow website and was fascinated by the production process of the knot and so did some extra diagrams of my own.

I will ask if my drawings match Timothy’s thoughts in any way.

# Haken’s Gordian Knot and the Twisting, Overlapping, Envelope Elephant.

I constantly look for Continuous Lines in many fields of art, history, mathematics – anywhere, as I just do not know where they are going to crop up.  Currently I am casting an eye on Islamic Art and Celtic art and am developing ideas on those.

Recently I glanced through a book called “Professor Stewart’s Cabinet of Mathematical Curiosities” and came across Haken’s Gordian Knot, a really complicated looking knot which is really an unknot in disguise – a simple circle of string (ends glued together) making a closed line. Here it is.

Haken’s Gordian Knot, from Ian Agol. A simple circle of string (an Unknot) formed into a complicated continuous line.

When I looked at the Knot, it reminded me of my “Twisting, Overlapping, Envelope Elephant” continuous line in that it has a lot of twists. I realised straight away that a narrow loop on the outside (left lower) seemed to lead into the structure with its two strands twisting as it went, each time in a clockwise direction.  I followed the two twisting lines throughout the drawing until they ended in a loop on the outside (left higher).

I wanted to draw and paint this knot. My first drawing was of the line on its own. The depth of some of the lines reminded me of one of my earliest paintings “Leeds Inner Ring Road Starts Here”, which was based upon a sign board which appeared near Miles Bookshop in 1967 informing us of the route the new road would carve through the City. This was several years before Spagetti Junction was built near Birmingham. My picture had lines swirling all over at various heights in one continuous line.

Leeds Inner Ring Road Starts Here. Use of varying thickness of single continuous line drawing, overs and unders. Pre dates Spagetty Junction near Birmingham. Mick Burton, 1967.

My first picture of the Gordian Knot, in black and white, concentrated on the heights of the lines following the overs and unders shown by Haken.

Depth of lines in black and white, in Haken’s Gordian Knot. Mick Burton, single continuous line drawing.

But my main aim now was to use blue and red to show the twisting nature of the pair of lines running between the starting loop and the end loop.  This was intended to allow the viewer to more easily follow the loop and the twists throughout the structure.

Twisting, overlapping colouring of Haken’s Gordian Knot. Mick Burton single continuous line drawing.

Just like viewing my “Twisting, Overlapping, Envelope Elephant”, from my previous post, imagine that you have a strip of plastic which is blue on the front and when you twist it over it is painted red on the back.  Where blues cross each other you have darker blues, and correspondingly with reds.  Where blue crosses red you have violet.  I show the strips feeding through each other, like ghosts through a wall.  There are some darks and lights in there as well.  Most usefully, the background shines through to help make the strips stand out.

You can now get more of a feel for what is going on.  I counted 35 clockwise twists and two anti-clockwise (numbers 19 and 26).  Continued twists in the same direction tie in the ongoing loop, when it feeds through the two strands of its earlier route at least 12 times.

This is a preparatory painting, in acrylic but on two sheets of copy paper sellotaped together.  When I exhibit these pictures they will be hung as portrait, rather than the landscape shown here for comparison with Haken (as you will note from where my signature is).  I think they look a bit like the head of the Queen in portrait mode !

Having got this far, I realised that I should find out more about the Haken knot (or unknot), beyond Professor Stewart’s brief introduction.  How did Haken construct the knot and why?

Please see my next post, on this continuous line blog, to see how I got on.

# Winding Number Theory and Continuous Line Drawing

Whirlpool in Space, or Petrol polluted Puddle. Spherical single continuous line drawing with repeat colour sequence with Rainbow colours. Mick Burton, Continuous Line, 1971.

This painting was one of two pictures hung at the Chelsea Painters Open Exhibition, held at the Chenil Galleries, Chelsea,1971.  I walked along the Kings Road looking for the gallery (I had submitted through and agency) and it was great to see the Whirlpool first in a window by the entrance.

The design was triggered by a browse through mathematics books in the library and coming across Winding Number Theory.  This used a continuous line and every time the line was drawn winding in a counter-clockwise direction a level was added and if you wound back in a clockwise direction a level was taken off.

So far, in my colour sequence numbering based upon Alternate Overdraw, I had not had a sequence of colours greater than about nine.  If I used the Winding Number method and continued to wind around counter-clockwise many times I could have a long colour sequence.  I went a bit mad with the Whirlpool, which was done on a spherical basis (allowing drawing out of one side of the paper and back in at the opposite side), and has a sequence of 20.

I did not simply go from light to dark over the whole 20 (the steps in shade between each colour would have been too small), but oscillated up and down with a smaller range of colours similar to a rainbow.  I had seen the rainbow effect produced by sunlight on petrol spilt on a puddle.

Alternate overdraw with colour sequence numbers. Continuous Line, Mick Burton.

To illustrate the difference between my Alternate Overdraw and the Winding Number, I start with the Alternate Overdraw in Red on the left.  As shown in earlier posts, each channel of areas between overdraws has two numbers alternately.  You move naturally up to a higher channel or down to a lower channel, through the red line and continue the numbering.

As before, you start with “0” on the outside and 0 can also appear within the drawing.

When we come to Winding Number allocation, we can use the same basic drawing with little arrows showing the direction it has been drawn.  It can be either direction of course, but I have chosen one which will match the result above.

Winding number allocation. Continuous Line, Mick Burton.

Starting at “0” for the outside, if we cross to another area through a counter-clockwise border for that area, it will be a level higher.  If we cross through a clockwise line we reach a lower area.

Here the + areas are of course higher levels and the (-) areas are lower.

The numbering matches the Alternate Overdraw illustration above.

Now, just to show the initial thrust of the drawing for the Whirlpool, with many levels, the next illustration shows a line spiralling from the centre outwards and I have shown just 13 winds.

Part of initial Winding Number spiral for the Whirlpool in Space painting.    Mick Burton, single continuous line drawing.

Then I have drawn a line from the centre of the spiral directly back to the bottom of the picture as the first phase of breaking up the spiral so that lots more areas start to  appear.  The next such phase is a meandering line near the top of the picture.

As I have said, this is a shorter version than the one usd for the main Whirlpool in Space picture above.

Next I have decided that the direction of  the line will be counter-clockwise starting from the bottom of the spiral, so that the levels go up towards the centre.  The numbers have then been added.

On the original painting, which is 24″ x 20″, I used alternate overdraw to allocate the colours.  This was because I was used to using that method and I think it is better for an artist anyway.  The Winding Number theory was simply the inspiration for producing a sequence of 20 or more.

Now, to go off at a tangent, I am interested in other artists in the family who keep coming to light.  As part of my research into my mother’s family, the Mace’s from Bedale (from 1825) and much earlier from Cambridge, I have come across a book by Thomas Mace called “Musick’s Monument” published originally in 1676.  It was published again about six years ago.  Amongst his own illustrations within the book is this one of a spiral depicting his idea of God’s world.

“Mysterious Centre of All Mysterie…” in Musick’s Monument, by Thomas Mace, 1676. Continuous Line, Mick Burton.

Thomas was a famous composer of the 17th century, and craftsman who made lutes and viols, whose main job was a chorister at Trinity College, Cambridge.

To add to this “sort of” co-incidence, my Uncle Harry Mace from Bedale, North Yorkshire, was a joiner and builder.  When he retired he started to make old style instruments, such as viols, and sold them to a music shop in Leeds.  I am sure he did not know about Thomas of Cambridge.

Thomas was not too impressed with a relatively new instrument in his time, the violin.