# 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

# Nessie the cockapoo visits Gledhow Valley

Nessie the cockapoo arrives with a favourite toy. How did she know my favourite colour range. Mick Burton, continuous line artist.

Nessie the cockapoo has come to stay for a week whilst Helen and Janet are in California.  She arrived waving one of her favourite toys, which just happens to have a range of colours similar to those in a recent painting of mine.

“Knight’s Tour Fragments”, acrylic on canvas. Exhibited at Harrogate and Nidderdale Art Club Exhibition in November 2016. Mick Burton, continuous line artist.

Nessie is two and a half and lives in a village in Worcestershire in a house almost surrounded by common land.  Strangely, there are no cats in the village and no squirrels (although several years ago one appeared in the garden the day we arrived for a visit, and it was suggested that it had been a stowaway in our car).  Hens roam free in the garden – so where are the foxes?  No greater spotted woodpeckers, they are all green.

Nessie’s favourite spot in our house is by the French Windows at the back.  She watches the birds and squirrels endlessly, and it is good to lie down on the job.

Nessie, the cockapoo, watching birds and squirrels. Why not take it easy?   Mick Burton, continuous line artist.

But Nessie is not used to seeing cats, and we have plenty of those.  Suddenly we hear barking and scraping at the window.

Nessie spots a cat and all hell breaks loose.  Mick Burton, continuous line artist.

Hopefully one of the foxes will turn up whilst Nessie is here.  We often see one or more during the day, and we even had one on the garage roof marking its territory.  Here is a photo of one in the garden in late January 2017.

A Gledhow Valley fox in the garden in January 2017.  Mick Burton, continuous line artist.

Nessie eats sensibly and feels that there my be more nurishment in the cardboard box than in the breakfast cereals themselves.

Nessie tucking in to a cardboard box which had contained breakfast cereals.  Mick Burton, continuous line artist.

Of course the highlight of each day for Nessie is the walk through Gledhow Valley Woods to the lake.

Nessie, the cockapoo, can’t wait to go to Gledhow Valley Woods, and the lake, with Joan and me.  Mick Burton, continuous line artist.

We are used to seeing the odd rat scamper across the path by the lake, as well as seeing how well they swim.  One rat dashing across suddenly realised that Nessie was passing and took off, missing Nessie’s nose by a whisker.  I am not good at taking photos of flying rats, so here is one nearby wondering what is going on.

A rat peeping from behind a tree on the banks of Gledhow Valley Lake.  Mick Burton, continuous line artist.

Twenty ducks who were sitting on the bank and the path fly off when they see Nessie, and Joan has brought some oats to feed to the Swan.  There is only one swan left at the lake just now and it is still in its first year.

Young swan, now living alone on Gledhow Valley Lake.  Mick Burton, continuous line artist.

We have been concerned for some months about the swans, particularly since the water level dropped after a digger cleared rubbish from the dam end. Large areas of silt have been on view where the swans nest.  Here are the adults and one youngster in late January 2017.

Family of swans on Gledhow Valley Lake. Photo taken in late January 2017 before the adults abandoned the lake.  I hope the swans did not have to pull bread slices from this wrapper themselves.   Mick Burton, continuous line artist.

At the time of this photo, showing the two adults and the above young swan in late January 2017, the second youngster had been ostracized and was sitting in a corner of the lake.  When we were litter picking this Sunday on the monthly action day with Friends of Gledhow Valley Woods, they told us that soon after the photo foxes killed this young bird and then the adults left the lake.  One adult was found wandering in the Harehills area and the RSPCA took it to Roundhay Park lake.  A lady told us that the other adult was walking past her house in Oakwood, presumably heading for Roundhay Park lake too.  So we hope that things work out well for the adults at Roundhay and our young  survivor here in Gledhow.

Nessie spots a cat she has not seen before. Henry adopts defensive mode.  Mick Burton, continuous line artist, Leeds.

On the way home from the lake, Nessie confronted a cat.  This is Henry and he stood sideways and seemed to double in size.  Nessie was on her lead, which was probable just as well for Nessie.

Henry the marmalade cat from Gledhow Valley.  Dogs beware.   Mick Burton, continuous line artist.

Anna and Emma, the children next door, went to the woods with Nessie today and had been looking forward to it for days.  Nessie gets on well with everyone.

She has enjoyed her holiday in Gledhow Valley and we are taking her back to the land of green woodpeckers.

# Four Colour Theorem continuous line overdraw.

Continuous lines overdrawn on Skydiver formation design, using Four Colour Theory method. Mick Burton

My recent post about the formation design used by the record breaking skydivers included a continuous line overdraw of their design (modified slightly be me to complete links which would have been present with more skydivers).  I said that I would explain how the overdraw (above) was completed.

The structure is made up of circles which have 3 way junctions throughout (3 handed in the case of skydivers ! ).  This can be regarded a map and so I will apply my Four Colour Theorem continuous line overdraw which I devised in the early 1970’s.

I was trying to prove the Four Colour Theorem, which states that no more than four colours are required to colour all the regions of a map.  My basic idea was that drawing a single continuous overdraw throughout a map would split it into two chains of alternate regions, which would demonstrate that only 4 colours were required.  If more than one continuous overdraw resulted then there were still only two types of chains of alternate regions.

As you will probably know, this theorem has many complexities which I will not attempt to cover here.  In the mid 1970’s I corresponded with two mathematicians at the Open University about my approach, Robin Wilson and Fred Holroyd, who were both very helpful and encouraging.  The theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken, running one of the biggest computers for over 1000 hours.  I soon decided that it was time to go onto other things!  However, my journey had been fascinating with numerous amazing findings which have been so useful in my art.

I can keep to relatively simple methods for my pictures.

Here is the design, used above, with my initial overdraws shown in red.

Assumed formation design used by Skydivers, with initial overdraws. Mick Burton four colour overdraw.

On final completion of the overdraws, every junction should have two of its three legs overdrawn and so the start decision (1) above overdraws two legs and this means that the third leg, which I call a “spar”, links to another junction where the other two legs must be overdrawn.

We then carry on making decisions which trigger other overdrawn lines across spars.  Usually there is a “knock on” effect where new overdraws connect with already overdrawn lines which then trigger more overdraws.

If we go wrong and a junction is triggered which has all three legs overdrawn, or none, we have to go back and change earlier decisions in a controlled process.  I usually photocopy the overdraws completed, every two or three stages, so that going back is not too time consuming.

Here is the situation after decision (3).  Decision (2) in blue had only triggered two overdraw sections but decision (3), in green, has triggered ten sections to be overdrawn in green.

Four Colour Overdraw decision 3 triggers 10 further overdraws, in green. Mick Burton, continuous line artist.

Here is the completed overdraw.  It can be seen that some decisions still only trigger one or two overdraws, but decisions 5 and 7 triggered 13 and 12 overdraws respectively.  There are 80 junctions in the design and it took 11 decisions to complete the overdraws.

completed Four Colour Theorem overdraw, on design based upon Skydivers formation design. Mick Burton, continuous line artist.

The completed overdraw has several continuous overdraws.  I tried other variations but had to accept that this design cannot be overdrawn with a Single Continuous overdraw.  This is due to the design having basically only two full rings of circles, which means that some tips of petals cannot be included in a continuous overdraw.

Continuous lines overdrawn on Skydiver formation design, using Four Colour Theory method. Mick Burton

This situation can be overcome by adding links between the tips of the petals to produce that extra ring of areas.  Here is the expanded design and the stages of overdraw.  I managed to complete the Single Continuous overdraw in one sequence without having to go back to change any decisions.

Increased size design with successful Single Line Overdraw using Four Colour Theorem method. Overdraw decisions shown. Mick Burton.

Of course it looks better with one solid colour overdraw and no decision numbers.

Skydiver formation design with links between out petals completed, overdrawn with a Single Continuous Line using Four Colour Theorem method. Mick Burton.

I have said that the method of overdraw was developed with Four Colours in mind, and so you could use one pair of colours alternately within the above overdraw and another pair of colours on the outside of the overdraw (which can include the background).

I have found another interesting result in that if you use strong colours inside the overdraw, as it is the main image, and neutral colours outside (or even leave the outside blank) then the gaps between the “petals” show good use of space.  Here is the design simply coloured in strong red inside the overdraw, which creates a good contrast as the background seeps in.

Solid colour within single continuous overdraw, with Four Colour method, showing good use of space. Mick Burton.

The chains of areas produced by the continuous overdraws can be coloured, not just in two pairs of colours to demonstrate Four Colours, but with a colour sequence or a mixture of sequence, alternate colours or even one colour.  In the last picture I have used colour sequence on main chains of areas related to the central space and, as a contrast,  light grey on the chains connected to the outside of the design.

Star Burst. Four Colour Theorem applied to a map of shell shapes wound round from the centre. Rainbow sequence of colours. Mick Burton, 1971

This is one of the first paintings that I produced after discovering my Four Colour Theorem overdraw in 1971. I called the picture “Star Burst”, one of my first planetary pictures.

# 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.

# Escher Islamic Mosaic Change to One Continuous Line. STAGE 5.

Escher painting 1922 of Islamic Mosaic tile at the Alhambra. WikiArt. Continuous line study by Mick Burton.

In my post of 4 April 2015, Continuous Lines in Escher Islamic Mosaic painting, STAGE 1, I mentioned that the original Islamic artist had deliberately created two Continuous Lines, when he could have just as easily created one, because he wanted to retain overall symmetry of design and border connections.

I stated that I had examined the design and worked out how to make a change to the border connections of lines to create one continuous line throughout the design, and this is how it’s done.

Here is the chart from STAGE 2 again, which shows the Main Continuous Line in Red and the Minor Continuous Line in Blue and the colours are also shown as the connections loop outside the Border.  The change has to be done without changing the Alternate Overdraw in the main design and this is done by linking a Red Overdraw with a Blue Overdraw at the same time as linking two not overdrawn lines.

Minor Continuous Line, Alternate Overdraws in Red and Blue. Mick Burton Escher Mosaic study.

We need a crossover on the Border involving a Red loop and a Blue loop.  If we part them at that junction and re-join the Red with the Blue, and then join both not overdrawn ends as well, we have united the Main and Minor continuous lines.  See Below.

Joining of Main Red Continuous Line to Minor Blue, leaving both non Alternate Overdraw lines joined at the former junction. Mick Burton Escher Mosaic study.

To show how this change is reflected in the Border, here is a before and after “Spot the difference” comparison which I have drawn.

Change of Border on Escher Mosaic to enable one continuous line. Mick Burton study.

As you can see, the difference between having two continuous lines and one is just a couple of flicks of a pen. Obviously, the artist would have known there was a one continuous line option and that he could have done it without losing any design or colouring options.

Presumably, the artists were required to retain overall symmetry above all else, including in the Border.  Eric Broug has also informed me that continuous line drawing is very rare in Islamic geometric design.

I think that the Artist chose two continuous lines in the Mural Mosaic to demonstrate that he was only one step from having one line, and he made sure that the Border was drawn so that this change opportunity (which occurs on each of the four sides)  was as simple as possible.  He is saying “I could easily have drawn One Continuous Line ! “

After completing my research into the Escher painting, and explaining the one continuous line alternative, I realised that I needed to draw the single continuous line myself.  Here it is.

One Continuous Line Drawing, including Border signals, based on Escher Islamic Mosaic. Mick Burton, March 2015.

This completes my five STAGES of explaining my thoughts on Escher’s terrific painting, in 1922, of the Islamic Mural Mosaic in the Alhambra.  I hope you found this abstract continuous line it to be interesting and stimulating.

Finally, I would like to thank Margaret Graves, Assistant Professor of Islamic Art and Architecture at Indiana University, for her encouragement and guidance after I completed my research.

# Why “Continuous Line” ? What is the point of it ?

You may ask why I am so bothered about the line being “Continuous”.  Well here we go –

The Continuous Line gives the drawing an enclosed flexible structure, or environment, which in turn means that all parts are related to some degree.  If I modify sections of a drawing there can be ramifications elsewhere, which may be small or extensive.  “Skelldale” was one of my earliest continuous line drawings.

When I compose a drawing I have to bear in mind that the line must return to the start point.  This is a lot of fun when I do an abstract, but for a figurative drawing it is more difficult as I am aware that the drawing will constantly change.  However, I have learned that this difficulty can help to trigger the creative force that often lies within the drawing.

This discipline of having to make choices on the route of the line, or modifying the route, can produce an extra structural effect or dynamic of movement which I may not have foreseen.  This creation of a result beyond my intention is similar to what happens in nature, where the practical necessity of combining all the elements needed in an animal or a plant often evolve into a tremendous design.

A further result of the Continuous Line is its creative effect on colours applied.  I worked out a method of applying colour sequences which can further enhance the natural structure and dynamic effect.  A colour only occurring once can be in a key area, eg. the eye of the Iguana.  I used a special repeat pattern for the scale effect.

My “spherical” drawings, where the line goes out of one side of the page and in at the opposite side, also produce a special arrangement of colours which can apply to a sphere.  The “Flame on the Sun” has coloured areas which match if the picture has its sides pulled around to meet in a cylinder.  However, there is no such match top to bottom, where a “bunching” effect would form the “poles” to complete a sphere.

My knowledge of Continuous Lines also lead to my creating them within natural structures when I researched the Four Colour Map Theorem 40 years ago.  This single line along boundaries of the countries of Africa (from my 1950’s school atlas), goes through every boundary junction once only.  I could connect up the two loose ends to make it “Continuous” !  You can use two alternate colours for countries inside the line and another two alternate colours for countries (and the sea) outside the line and you have your Four colours.  Proving that you only need four colours for any map is a different matter !

So there we are, my fascination with my lines continues.  I will cover all these types of line further subsequently.