Category Archives: Mathematics & My Art

Change a four sided continuous structure into a single surface Mobius torus, or Mobioid.

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Drawing of Doug Eglen continuous metal sculpture with sides in four colours. Mick Burton, continuous line artist.

In my last post, on 3 June 2025, I said that I would like to draw Doug Eglen’s 3/8″ square forged metal sculpture. 

Continuous Lines in forged metal, by Doug Eglen.

Here it is, but I have used separate colours for each of the four sides.  You can only see Red, Green and Blue because the metal remains flat overall and you cannot see the Black I allocated for the underside.

I have tried to reflect the 3D quality of the sculpture, with its Celtic over and under style, highlights and shadow.

When I first saw the structure, I wanted to understand what would happen if I applied twists to the sculpture.  I thought that there should be one or more types of twist which would turn the culture from being four sided with four surfaces into four sided with one surface.  I knew that the Mobius Strip has two sides before the half twist and one continuous surface afterwards and thought that the single surface should also be possible for Doug’s structure.

Another thing that I wanted do was to try and clarify my idea that his structure reflected the shapes of two Boomerangs.  I wondered what twists I would need to do this.

The poker which I made at the age of 12 at school appeared to have a half twist but it is difficult to visualise how this changed the position of the sides.

Brass handle with half twist on my poker, made at school when I was 12. Mick Burton, continuous line artist.

I had to use a pencil to draw along one surface to confirm that it changed the top side into the bottom side.  This meant that I could use two half twists near the centre of a new drawing of the sculpture to produce two boomerangs which had individual colours.

Using two twists to produce Black and Red Boomerangs on drawing of Doug Eglen’s metal sculpture. Mick Burton, continuous line artist.

Black has now appeared at the top for half the time, so we have four colours on view.  Another outcome is that the two sides, which are Green and Blue, swapped from one side to the other at the twists which has resulted in blue almost disappearing.  This is due to the close photographic angle by Doug showing most of the insides of each boomerang shape and I coloured them almost equally Green and Blue in my first drawing.  Green stayed the same within the Red boomerang here and Blue changed to Green in the new Black one.

I have mentioned the Mobius effect.  The Boomerang drawing has not produced a change in the number of surfaces but just altered their positions. Sides still equal surfaces.  This is because I have used two half twists.  If I had only used one half twist then the number of surfaces overall would become two surfaces, one colour for top and bottom and another for both sides.

It seemed to me that employing one quarter twist would produce a Mobius single surface throughout, as each time a surface comes round again it deflects a quarter and on the fourth approach it is back on top. Maybe a three quarter twist would also achieve this, but I found this more difficult to visualise.  First approach deflects to side 4, Second to side 3, Third to side 2 and fourth to side 1 again.

Anyway, I have drawn simply a single quarter twist as an addition to Doug’s sculpture.

Drawing of Doug Eglen continuous metal sculpture, but with added quarter twist. Mick Burton, continuous line artist.

I have started with black and the result is All Black.  This drawing depicts a 4 sided metal continuous sculpture which has only one surface.  It is a complete Mobius result.  Visually, you can see that it is a quarter twist and our knowledge of the general sculpture helps us to realise that there is in fact one surface.

Yes, a Boomerang drawing and an “All Black” drawing suggest that I have Australian and New Zealand interests.  My mother, maiden name Brenda Mace, was born in a pub in Bedale, North Yorkshire.  Nearly 100 years earlier six brothers were born in the same pub and four of them went to gold fields in both Australia and then New Zealand in the early 1860’s.  They were cricketers as well and Christopher Mace played for Victoria against the first team to visit from England and two years later John and Harry joined with him, in Otago New Zealand, against the first English team to play there.

Back to the drawings.  I looked on the Internet for images of general structures which included a quarter twist and failed to find any.  There were examples of the pure circular twist, including 4 sided, but these are difficult to visualise as there is no obvious start point. 

Mobius Strip Structure of Rectangle Geometric Shape. Issuu website.

This pure circular twist is more understandable than most and the Issuu website shows how they built one which you can walk round in.  I am not sure if this includes walking upside down at one stage!

https://issuu.com/vsvu/docs/prof_is_1000/s/16586493

I feel that a twist anchored into an actual general structure, which has some straight and flat sections, is important.  It is easier to identify the type of twist and how the effect of the twist radiates through the whole structure in a more meaningful way.

Some other reference sites mention continuous surfaces, with more than the single edge and surface of a Mobius Strip, as Morbioids.  They compare their structure with the Torus (when it has parallel lines drawn around it which can be regarded as equivalent to edges which can produce the Mobius effect).

There are specific explanations of degree of twist, the number of sides, leading to the number of surfaces.  These confirm my assumptions about a quarter (90 degree) twist, or three quarter (270 degree) twist resulting in a single surface for the square structure.  A half (180 degree) twist has two surfaces.  Others explain 5 or 6 sides and there is a formula for n surfaces according to twist and sides.

A useful link that works is headed “Name for a 3 sided Mobius Strip?”  https://reddit.com/r/topology/comments/1bfdu7m/

SamwiseGanges    said that he was going to call them Mobius prisms.  When he referred to square Mobius prisms, he confirmed my assumptions about the effect of their twists.

AceThe Aro   said that Dr. Cye Waldman called them Mobioid’s in 2017 and you can click onto his moving images.  You can also click on Ace’s own slideshow which runs through all the different twist and number of sides possibilities.

I would like to refer to my drawing of Doug Eglen’s structure, with a quarter twist, as an “Anchored mobius torus”, or “Anchored Mobioid”.

Doug may consider loosening the join on his double boomerang ironwork and resetting it with a quarter (90 degree) twist.  That would produce a real single surface anchored Mobioid.  

 

 

Continuous Lines in forged metal, by Doug Eglen.

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Continuous Line knots in tempered metal forged by Doug Eglen. Photo by Doug sent to Mick Burton, April 2025.

Doug Eglen contacted me in March 2025 to say that he had recently started having an interest in knots and unknots and came across my painting of Haken’s Gordian knot, which you can see in my earlier posts of 9 May 2018 and 5 June 2015

Continuous Line Artist view of Haken’s Gordian Knot.

How do you construct Haken’s Gordian Knot?

He asked for my permission to paint his own version of the Gordian knot, based upon my painting.  He intended to exhibit his painting, along with the new metal knots which he was producing in his forge.  Doug has an exhibit case at Purdue University, Indiana, in the Math department library.  I agreed, and answered his questions about assumptions he had made about the construction of my painting. 

Doug later sent me a copy of his painting.  This is impressive, as are his metal knots, and you can see them on his website   

https://http://www.deglen.org/math-art

I particularly liked Doug’s photo of his 3/8″ square metal, in what I would call a sort of “double boomerang” shape without any twists.

Doug Eglen square 3/8″ metal without twists in a continuous line, copied to Mick Burton.

The alternate over and under style, like Celtic knotwork, produces great highlight graduations as well as suggestions of colour that Doug achieves with his firing treatments. I like the dark shade of the sides of the metal compared with the shadow of the piece.  I would like to draw this.

At school, when I was 12, I produced a metal poker with brass handle which has a half twist.  I have had no further experience of metal work.

Brass handle with half twist on my poker, made at school when I was 12. Mick Burton, continuous line artist.

I did do some wire bending to produce a single continuous line wire horse in 1967.  At first I placed it underneath a piece of clear glass covering the dining table, to hold it down.  Then I secured it to a wooden board through a centre page magazine photo of a horse grazing in a meadow.

Continuous line Horse in a length of garden wire done in 1967. Mick Burton.

Next, I went sculptural and used cardboard strips to take the Horse a stage further in 1970 when I lived in Nottingham.

Continuous line Horse using strips of cardboard 1970. Mick Burton.

I showed it to someone at work who’s Dad owned a company which produced steel castings. He said he would ask his Dad about the possibilities of doing a metal horse.  The answer was a “Yes”, but it would cost me £3,000 !  I now know someone who’s cousin does 3D printing, so maybe I can get a more reasonable quote now.

Of course, this is all “small beer” compared to Doug Eglen’s metal workings.

Alien Creatures on Train Tracks Puzzle Continuous Line.

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Alien Creature continuous line from Train Tracks Puzzle.    Puzzle Madness, Large, 1.5.2022.  Mick Burton.

This is an update from my post of 24 December 2020 when I explained how I had started Train Tracks Puzzles during Covid lockdowns.    Red Alert, Continuous Line Detected on Train Tracks Puzzle.

When I try to solve a Train Tracks Puzzle, I draw out the grid freehand and copy in the cross references of number of tracks in vertical or horizontal rows as well as the given pieces of track.  A big part of the process is marking the squares, which will not have track pieces, with a circle or cloud shape.  I draw in the track I have decided upon with a line between given pieces of track.

Most people will complete the puzzle on their phone or laptop from start to finish, but I just like drawing things, rather than hitting wrong buttons on keypads most of the time.  Here is my initial drawing.

Initial drawing of Train Tracks Puzzle 1.5.2022. Mick Burton, continuous line artist.

I coloured all “given pieces of track” yellow and the interior of the Alien red in Sharpie pen and did lighter background with coloured pencils, leaving the cloud shapes white.

Here is another Alien from two days later.

Green Alien based upon Train Tracks Puzzle.    Large 3.5.2022 Puzzle Madness.   Mick Burton, continuous line artist.

I mentioned in my post of 24.12.2020 that I was at position 272 out of the 863 people listed.  Also that I had scored 17.925 points compared to the top rated Stirlingkincaid with 2,766,965. This was nearly twice the points of anyone else and I wondered if he ever slept.

I got to about position 130 and for a time just tried to keep treading water at that level.  The number of participants had more than doubled since I started.  Here is the summary a few months ago, at about the time I ceased to do the puzzles.  You can find the Train Tracks puzzles on     https://puzzlemadness.co.uk      .

All time Train Tracks table on Puzzle Madness in June 2022. Top three all over 4 million points and Mick Burton on 240 thousand.

GA had drawn level with Stirlingkincaid on 4,221,000 but both trailed Stevo by 665.  Fourth place Antique was only on 2,708,990.  Do any of them ever sleep?

I have kept quite a lot of my initial drawings and may do some more Aliens.

 

 

Winding Number Man, continuous line drawing.

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Winding Number Man. Continuous line drawing with colour sequence. Mick Burton.

In 1972, at the same time as I was experimenting with winding number lines such as in Petrol Polluted Puddle (see my post of 24.11.2014),       Winding Number Theory and Continuous Line Drawing

I drew Winding Number Man, which involved looping around in the same direction from start to finish.

Winding Number Man. Continuous line with alternate overdraw. Mick Burton.

If I had done this in a concentrated area, like the close spiral I used in Petrol Polluted Puddle, I would have had a long single series of overlapping colours.  However, as I progressed around the head, body, legs and arms of the man I avoided too many overlaps.  As I gained new overlaps, previous ones fell away.

The longest sequence of colours is six, whereas with PPP the real sequence is 19.  I could not cope with one sequence of colours that long, with only slight changes between each one, so I used a repeat rainbow sequence which provided the puddle effect I wanted.

The shorter single sequence of colours on the Winding Line Man gives him the form and density I required.

All along I had in mind something similar to the Michelin Man who advertised the tyres.  Here is a recent representation.

Michelin Man logo. Creative Review.

Strangely, I was remined to get on with this blog when watching the new Shetland TV series, where Detective Sergeant  “Tosh” McIntosh was trapped inside a caravan which was about to explode.  I paused the TV in the middle of the explosion and the freeze frame flame looked a bit like the Michelin Man.

Shetland explosion looks like the Michelin Man. End of Episode 3, Series 7.

Watch Episode 4 to see what happened to Tosh.

Continuous Line Artist view of Haken’s Gordian Knot in “Unknot Hall of Fame”.

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

Peter Prevos, has included me in his “Unknot Hall of Fame”, within an article on his website about “The Art and Magic of the Trivial Knot”, which also explains many technical aspects of the trivial knot and how magicians have incorporated those ideas. There are designs, in the Hall of Fame, by Goeritz, Thistlewaite, Ochiai and Haken as well as art by Vanuatu and myself and reference to James Sienna. Have a look on – https://horizonofreason.com/science/unknot-gallery/

I had already done my painting when I saw a post on Mathoverflow website “Are there any very hard unknots?” by mathematician Timothy Gowers – https://mathoverflow.net/questions/53471/are-there-any=very-hard-unknots

I responded with posts on my website on in June 2015 and a later update in May 2018.

Noboru Ito, mathematician now at the of University of Tokyo, contacted me in February 2016 about his near completed book “Knot Projections” and my article is referenced in the Preface. “It was very helpful”.

Tomasz Mrowka, mathematician at the Massachusetts Institute of Technology, asked in November 2017 for a copy of my painting, as “it’s really quite striking and would love to hang it in my office”.

David Eppstein, computer scientist and mathematician at university of California, Irvine, featured my painting on his website in November 2018 “Mick Burton, an artist known for drawings that use a single continuous line to create the impression of complex and naturalistic shapes, looks at knot theory, self-overlapping curves, and the visualization of Seifert surfaces.” I had to look in Noboru Ito’s book to check out Seifert surfaces !

In essence my painting of Haken’s Gordian Knot is another example of me finding a well known structure which I can apply my continuous line knowledge and experience to. The way that nature can work in these structures often surprises me.

This is separate to my ongoing art work of producing single continuous lines and colouring based upon interesting subjects – which can be animals, landscapes, portraits, still life and abstracts.

Other examples of looking at structures have been –

Four colour theory maps, where my overdraw method could divide a map up into two streams of alternate colours, hence the four colours. I corresponded with mathematicians Robin Wilson and Fred Holroyd in the mid 1970’s. See my post in August 2015 on Skydiver patterns and my Four Colour Theory.

The artist Escher’s favourite tile at the Alhambra in Spain, which I realised had two continuous lines running through it. I saw that the artist could have made in into a single line with two small alterations. See my posts of April 2015.

Knights moves on a chess board starting and finishing at the same square and landing once on all the other squares. See my 1974 picture in the Gallery 1965-1974.

I am always on the look out for new structures which are suitable.

Red Alert, Continuous Line Detected on Train Tracks Puzzle.

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I started doing Train Tracks puzzles in the Daily Mail a few months ago and then moved onto Medium puzzles (dimensions up to 10 x10) on puzzlemadness.co.uk and a month ago tried Large difficulty (dimensions up to 12 x 12).

You start off with a grid which states the number of cells which occur vertically or horizontally and they give you some bits of track initially, including start and end track at the edge.

Train tracks from puzzlemadness.co.uk Large difficulty 12.12.2020. Mick Burton, continuous line artist.

There are many attractive elements to this sort of puzzle, including the possibility of solving them totally without trial and error.  The first thing to do is to add initial offshoots for all these start tracks.  Next look for any rows which already meet the number of cells containing track, such as the right hand vertical which has the required two including the offshoot.  This allows you to allocate spaces to the remaining 10 cells. 

Being an artist, I know the value of space in a picture and it is particularly important here.  Then you have to consider the various types of track and on you go.  Constantly checking and rechecking is the key as you add pieces.  Bear in mind that the aim is to end up with one route from start to finish, avoiding dead ends, and use that to your advantage.  Finding dead ends is also useful as you can allocate spaces.

It is best to start off with smaller easier Train Tracks puzzles to get used to the process.

I attempt my puzzles on paper where I draw the grid and enter the numbers and given track pieces.  My fingers are too wide and clumsy to do much prodding on my mobile phone and if I complete the puzzle I then tap in the answer.  Here is my initial drawing of the above puzzle.

Initial attempt at the rail track puzzle (large difficulty) of 12.12.2020. Mick Burton, continuous line artist.

When I loaded this on my phone, I expected that as I tapped in the last piece the completed puzzle would disappear to be replaced by congratulations across the screen, for completing a route from start to finish.  Instead I saw the Red Alert.  It is not normally an offence to produce a continuous line in this blog.

I am good at mending this sort of thing of course and here is the final result – there is a X (space indicator) so that you see the complete shape before the last piece goes in causing the whole thing to disappear.

Correct completion of Train Tracks puzzle, with just the last bit to go in. Mick Burton, continuous line artist.

I am interested in various stand alone structures which have an environmental feel to them, where all the different elements can produce a surprising result.  

As it has been Lockdown etc,  I have completed 94 in about 10 weeks scoring 17,925 points, which put me at position 272 out of 863 listed.  Top is Stirlingkincaid with 2,766,965 !

On the monthly list I am 91st with 7,650 points.  Stirlingkincaid has 228,640 – does this person ever sleep?

Personally, I will probably move on now, looking for more structures which I can unlock with my continuous line knowledge.  Also, I need to finish my current work about Drawing Prime Numbers.

Colour Sequence Application to Continuous Line Drawings by Mick Burton – demonstration continued.

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img_5140 - copy

Clyde the Elephant, single continuous line with colour sequence by Mick Burton.

This is the continuation post covering my demonstration and workshop at Harrogate and Nidderdale Art Club on 6 December 2018.

Here is a reminder of my marker pen attempt at a continuous line elephant.

IMG_4950

Demonstration of a Single Continuous Line Elephant, initial drawing, at Harrogate and Nidderdale Art Club by Mick Burton, on 6 December 2018.

At home later I followed the line/s around and found that there was more than one line and I needed to do one or two diversions to correct that.  As the pattern at the front of the neck has a sort of square which I needed to get rid of I used that region to also turn the drawing into a single line throughout.  With a bit of general smoothing of arcs all round I arrived at the following revised elephant.

img_20190108_0002

Revised single continuous line elephant.    Mick Burton, Leeds Artist.

The next stage was to apply my Colour Sequence to the lines, which I completed in the last few days.  The result is shown at the top of this post.

To explain the process I use, and how it works, I will briefly go through the illustrations which I used later on in the Demonstration at Harrogate and Nidderdale Art Club.

We start by drawing a winding line in an anticlockwise direction.

anticlock 1 line

Stage 1. Single line drawn anticlockwise.   Mick Burton explains colour sequence.

Then, starting on an outside section of line, overdraw in red alternate sections of line.  This results in three different continuous line sections bounded by a red line.

anticlock 2 alt overdraw

Stage 2. Overdraw in red missing alternate sections.   Mick Burton explains colour sequence.

We can now number all the areas to indicate where the colours in the sequence go.  Call the outside 0 and number through the sections to 5 in the middle.  You will see that each channel between red lines has alternately numbered areas.

anticlock 3 number alloc

Stage 3. Number the areas in sequence from the outside (being 0) to the middle (being 5). Mick Burton explains colour sequence.

I have already decided on a sequence of colours to use, running from light tones to darker and from yellow to red.  First apply yellow and gold alternately throughout the outer corridor.

anticlock 4 first alt colours - copy

Stage 4. Paint alternate colours within the outer corridor. Mick Burton explains Colour Sequence.

Paint in the next two colours from the sequence – orange (which looks reddish here) and red – alternately in the inner corridor.  You can see how the colours are lining up in natural sequence of tone and colour.

anticlock 5 second alt colours - copy

Stage 5. Paint second set of alternate colours (orange, which looks reddish here, and red) in the next corridor.   Mick Burton explains colour sequence.

Lastly, for our anticlockwise line we paint the central area (which has its own red line surrounding it).  The result is a simple set of sequences running from the outside to the middle.

anticlock 6 full colours

Stage 6. The last colour in the sequence (dark red) is added in the centre. Mick Burton explains colour sequence.

As you will have realised, each loop going over earlier parts of the drawing adds a level, like overlapping shadows or leaves on a tree looking darker as they overlap.  The direction of darker tones of colour in the sequence reflects this.

In more complex drawings, however, the sequences of colours can change direction.  To show this we need to have a different single continuous line.

Start drawing your line with two loops from the lower left in an anticlockwise direction as before.  When you reach the upper left change to doing three loops in a clockwise direction and then go back to the start by a line running underneath.

clock 1 line

Stage 7. Start drawing your line from the lower left in an anticlockwise direction doing loops and when you reach the higher left change to clockwise loops running back to the right. Then finish clockwise running underneath to the start. Mick Burton explains colour sequence.

Here is confirmation of the directions of the line, anticlockwise and clockwise, and how they change and run back over earlier lines.  We now have a more complex drawing for colouring.

clock 2 directions of line

Stage 8. Here is the completed single line with the directions shown – red for anticlockwise and blue for clockwise. Mick Burton explains colour sequence.

By applying alternate overdraw in red we split the drawing into corridors which look a bit more complicated than the simple anticlockwise drawing we did earlier.

clock 3 alt overdraw

Stage 9. Alternate overdraw in red splits the new drawing up into corridors for colouring. Mick Burton explains colour sequence.

When we number the areas, starting at 0 on the outside as before, we have plus numbers at the top of the drawing but minus numbers appear in the lower corridor.  When we follow the natural sequence of numbers downwards from 2 through 1 and 0 we hit -1 and -2.

clock 4 number alloc

Stage 10. Numbering from 0 on the outside as before we get minus numbers as well as plus. Mick Burton explains colour sequence.

After I had been doing my colour sequence for a few years I found out that mathematicians call this mix of anti and clock directions Winding Number Theory.  When you continue with loops in an anticlockwise direction you are adding levels of overlap and when you change to clockwise you start reducing levels.

We can now apply alternate colours yellow and red to the upper channel.

clock 5 first alt colours

Stage 11. First set of alternate colours in the upper channel on the complex drawing. Mick Burton explains colour sequence.

Then we can complete the positive colour direction.

clock 6 last upward colour

Stage 12.  Completing the plus direction colours by adding dark red.   Mick Burton, continuous line artist.

Now looking at the lower colours, in the clockwise section of the drawing we add the final two colours alternately.

clock 7 downward colours

Stage 13.  Complete colour sequence on single continuous line drawn in both anticlockwise and clockwise directions. Mick Burton, Leeds artist.

So that is the basis of how I do my colour sequence.  

For my elephant, it is more complicated and I show below my sketch after doing the alternate overdraws to create the corridors of alternate colours and then numbered the colours throughout.

img_20190108_0001

Single continuous line elephant showing alternate overdrawn lines in red and colour numbering. The key to the colour sequence and numbering is shown. Mick Burton, continuous line artist.

I have shown the key to the colour sequence and numbering in the top right corner.  The colours can be employed in the opposite direction, of course, but with all my drawings the choice of which direction of sequence to adopt is not too difficult.  The darker colours fall lower down or on the main body of the animal and the more delicate red, orange and yellow mostly on the face. 

I only use red once, and that is on the eye.  This really reflects the greater detail on a face which extends the colour range.  Several of my colour sequence animals have the eye coloured by an end of range colour only used once in the drawing, eg. Iguana, Harriet the Hen and Olympic Lion.

The completed elephant, at the top of the post, has a story behind it.  I did the initial drawing in my demonstration to Harrogate and Nidderdale Art Club on 6 December 2018, which is the day my first grandson, Lucas, was born in Glasgow, son of Kate and Mark. 

I have decided to call the elephant Clyde after the famous Glasgow river.  Lucas can have a picture on his wall which is exactly as old as he is.    

Continuous Line Artist view of Haken’s Gordian Knot.

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Depth of lines in black and white on Haken Gordian Knot. Mick Burton, continuous line.

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).

Twisting, overlapping, envelope elephant. Continuous line.

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 Gordian Knot. Mick Burton, continuous line.

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

“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.