# 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.

# Identikit and Key Features for Continuous Line Drawing

Identikit image of suspect for the murder of Elsie Batten, London, March 1961.

Edwin Bush, recognised and arrested as a result of the Identikit image for the murder of Elsie Batten.

I use Key Features as part of my Continuous Line Drawing of Animals and People. Faces have always been of interest to me and when young I loved to do caricatures of people. I was in the police for a time and of course identification in its many forms was also fascinating.
In 1960 I left Harrogate Secondary Technical School and became a Police Cadet with the West Riding Police at the Wakefield Headquarters. My first post was in the Prosecutions Department, opening the mail (which required signing the Official Secrets Act) and making the tea. One of the officers preparing cases for Quarter Sessions and the Assizes was Inspector George Oldfield, who nearly 20 years later was Assistant Chief Constable Crime and in charge of the Yorkshire Ripper investigation.
My next posting in early 1971 was in “Modus Operandi” the Criminal Records Office for the North of England. Each criminal’s record sheet had at least one photograph of him attached, and in one case there were eight or nine photos ranging from a man’s early teens up to his seventies. Of course he had changed vastly over this time, but I noticed that his ears were identical in all the photos.
One day there was a bit of commotion in the large office with detectives collected in a corner arguing and laughing. I went to see what it was all about and they were looking a two images of suspects for a robbery. They explained to me that the images were produced by selecting parts of a face printed on cellophane sheets and placing them on top of each other to build up a face based on witness descriptions. It was called an Identikit set and was new in this country after being started in the USA. There were accompanying descriptions of a large man and a small man. The joke was that it was Laurel and Hardy. It was not given much chance of being successful by some of the crowd. A detective walking past peeped at what we were looking at and said, “Oh, I know them, but I think they are in prison”. Everyone laughed. Five minutes later he returned and said that he had looked up the two records and they had come out of prison a week before. They were not out for long!
I then found out about a recent case of murder in London which had been solved by the use of the Identikit. It was following the murder of Elsie Batten, an antique shop owner in London in March 1961. The detective investigating had two witness descriptions and produced two images on his new Identikit set. A constable saw a man resembling one of the images and arrested him. It was Edwin Bush, who admitted to the murder. He was executed at Pentonville Prison in July 1961. The image and a photo of Bush are shown above.
A few months later I was working at Horsforth Police station, near Leeds, when a detective I had worked with in the Records Office came in and said that he was going round all the police stations handing out Identikit sets and explaining how they should be used. So I had a go and it was great fun. Then the Inspector came in and he chatted to the detective for 20 minutes and said that he was not convinced that the Identikit would be much good. I had been watching as the detective had surreptitiously created an image of the Inspector whilst they were talking, and it was spot on, to everyone’s amusement!

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Of course Identikit was not directly successful in all cases. In the late 1970’s The Yorkshire Ripper committed at least 13 murders and there were several Identikit images available to assist George Oldfield as he tried to solve them with his large team of officers. There were many complications and George was side tracked by the tape recordings from a man with a Geordie accent who said that he was the killer (most of my Geordie friends were interviewed). When Peter Sutcliffe was captured in Sheffield it was realised that several of the Identikit images were good likenesses to him.