# 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.

# “Vortex” by David Kilpatrick. Single Continuous Line and Alternate Overdraw colouring.

“Vortex” by David Kilpatrick, artist from Atherton, Australia.   Single Continuous Line using the Alternate Overdraw method to allocate colours.   March 2017.   Mick Burton blog.

I have been exchanging ideas with David Kilpatrick recently and he has agreed to let me put some of his pictures in my blog.  “Vortex” stands out to me, as I have been a fan of Vorticism for many years.  He has used Alternate Overdraw to allocate colours in sequence and it has worked well.

David’s design gives the impression of a sheet of plastic, coloured green on one side and red on the other, and each twist showing the other side.  With overlaps you get darker greens or darker reds.  Four internal areas let the background shine through.  The whole thing is very natural, including David’s own style of patchy colour radiating outwards.

Next is David’s “Knight’s Tour” which he is still working on.

“Knights Tour” by David Kilpatrick, artist from Atherton, Australia.   Single Continuous Line based upon moves of a knight and using Alternate Overdraw to allocate colour sequence.   April 2017.  Mick Burton blog.

I did a Single Continuous Line “Knight’s Moves on a Chessboard” in 1973 (see Gallery 1965-74) with the intention of colouring it, but never tackled it properly.  One of the problems was the number of fiddly small areas.  It led to my “Knight’s Tour Fragments” instead (see my previous post on 16.2.2017).

But now we have GIMP!  David said that he used this to move the lines about on his “Knight’s Tour”.  I googled GIMP and it means “GNU Image Manipulation Program”.  Some areas are still fairly small but he has produced a vibrant structure.

David says that these are trial colours (I presume from GIMP) and he intends to work out an improved scale of colours in his own style.

However, the colours shown already demonstrate the natural balance inherent in the Alternate Overdraw colour allocation.  The composition suggests to me an island with yellow “beaches” as well as reds within opposite “volcanic” zones.

There is a choice regarding background, which would naturally be the same colour as the light blue internal areas and result in a surrounding “sea”, or it could be left white as shown above.

I look forward to seeing the final version, which I am sure will be another splendid example of Vorticism.

Another picture that caught my eye was his “The Pram” which is based on a magic rectangle.

“The Pram” by artist David Kilpatrick, from Atherton, Australia.   Based on a Magic Rectangle. 2015.    Mick Burton blog.

This pram picture has lots of line ends in it and makes me want to attempt one myself using a Continuous Line animal.  Such a design would make you want to connect up so many loose ends.  My Spherical pictures already do this to an extent, as I take a line out of the picture at one side and bring it back in at the corresponding opposite side.

I think that David chose the positions of the displaced squares in a sort of random way.  Maybe I would want to be confident that I could move them around, in the way you could on the movable squares game of my childhood, and get back to the actual original picture.

You can see much more of the art of David Kilpatrick on

https://www.redbubble.com/people/fnqkid

# Twisting, Overlapping, Envelope Elephant. Continuous Line Drawing colouring.

“Fluorescephant”, the original version of “Elephant Grass” which is at the top of this continuous line blog, was my first successful Colour Sequence painting.  The sequence ran from yellow through greens to blues in steps of colour and tones which gave a natural three dimensional effect and dynamism.  Part of this was the overlapping nature of continuous lines which was reflected by the successive darker colouring.

The painting was accepted for the International Amateur Artist exhibition, in Warwick Square London, in February 1973 and then a month later in the National Society annual open exhibition in the Mall Galleries.

Fluorescephant. Continuous line drawing with colour sequence. National Society Open Exhibition, Mall Gallery, London, 1973. Mick Burton.

I was never totally happy with the colouring.  I thought that there was an extra natural effect, on top of the overlapping, which I was missing.  When I started my art again in 2012, after a gap of nearly 40 years, I once more tried to sort this out.  I realised that I could enhance the twisting of the design and highlight gaps where the outside would show through.

Here is the result, “Twisting, Overlapping, Envelope Elephant”.  Imagine that the continuous lines are describing a sheet of plastic, which is coloured Blue on the front and Red on the back.  Each time a twist occurs, against the outside background, then I colour it Red.  When the overlaps build up, the shades of the blue front go darker blue, and the shades of the twisted areas become darker red.  Where the blue front and the red back occasionally overlap, then I use violet to reflect the mix.

This 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.

You can see considerable areas of background colour within the animal showing through. This looks natural within the form of the elephant.

The blue areas, including darker blue overlaps, are the same as the blue areas in the “Fluorescephant”, so it is good to keep a large part of the original colour sequence in this change of style.