Tag Archives: Four Colour Map Theorem

Continuous Line Artist view of Haken’s Gordian Knot.

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.

 

Four Colour Theorem continuous line overdraw.

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

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.

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.

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.

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

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.

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.

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.

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

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.

 

 

 

 

Skydivers in Ten Petal Flower Formation, link to Four Colour Theory continuous line.

164 Skydivers head down record in Illinois, 31 July 2015.

164 Skydivers head down record in Illinois, 31 July 2015.

Two weeks ago 164 skydivers, flying at 20,000 feet and falling at 240 miles an hour, set the “head-down” world record in Illinois. The international jump team joined hands for a few seconds, in a pre-designed formation resembling a giant flower, before they broke away and deployed their parachutes.

I was intrigued by the design of the formation. I have found many qualities in ten petal (or star) designs and, of course, I look for continuous lines in all sorts of designs that I find and in particular the possibility of a Single Continuous Line.

Here is my sketch of the skydivers formation.  It is made up of many linked circles, starting with a central ring of ten circles which radiate out to ten “petals”.  The plan seemed to involve six skydivers forming each circle by holding hands.  Some extra skydivers started links between petals.  I checked to see if I had included all the skydivers, which made my sketch look like a prickly cactus.

Cactus Count, 164 Skydivers all Present and Correct. Mick Burton, continuous line artist, August 2015

Cactus Count, 164 Skydivers all Present and Correct. Mick Burton, continuous line artist, August 2015

This was such a tremendous achievement by very brave men and women.  My only experience of heights is abseiling 150 feet down a cliff in the Lake District.  I realise that the jump would have been very carefully planned using the latest science and involved a lot of training, etc. but I am particularly interested in the part that the formation design played.

The hexagon appears to be an essential element so that hands can be joined at 3 way junctions.  A core circle of hexagons would naturally be 6, but more would be required for this jump.  The next highest near fit would be 10, which is fine given the variations in human proportions.  This also naturally allows linking between middle circles in the petals to complete a second ring of circles, which was partly done in this jump.

Every participant would need to know exactly where their place would be in the design and yet it is so symmetrical that I struggle to get my sketch the right way up.  Also with the short time involved co-ordination of planned stages would be difficult.  This made me think of a flock (or murmuration) of starlings performing their remarkable patterns in the sky and how they manage to co-ordinate. Apparently each bird relates and reacts to the nearest birds around it.  Absolute simplicity and ruthless efficiency with no critical path.

If the skydivers have adopted a similar approach then the design is ideal. The design is basically 30 circles in sets of 3 in a row making up 10 petals. The process is fluid and adaptable, building outwards from the centre. Think 10 individuals linking hands to start off with, which then recognisably evolves into 10 petals, and think 6 individuals in each of the 30 circles.  Everyone is dropped (there were 7 aircraft I think) in an order which anticipates being able to take up a place a certain distance from the centre of the structure and within a specific circle. As they approach they can recognise the progress and assess whether they can link in as expected or whether a modified position may need to be taken up (and being guided by the people already in place). The last individuals to be dropped will not have  a planned position in any circle but will form the start of the links between the central of the 3 circles in the petals. They need to be prepared to become part of an outer circle which has not been completed.

I have done a sketch of how this may work, with the numbers indicating my thoughts on the expected order of arrival in the building of the formation.

Skydiver formation with possible roles and order of arrival of individuals. Mick Burton, continuous line artist.

Skydiver formation with possible roles and order of arrival of individuals. Mick Burton, continuous line artist.

I hope this was a useful exercise, in trying to work out how the formation worked, and not total tosh (if so my apologies to all concerned).

To help my attempt to apply my continuous lines to the design I have completed the links between middle circles, which was partially done this time and I suppose will be considered for the next larger attempt at the record (say 180 skydivers).  The Continuous Lines are intended to pass through every three handed junction once only (I normally would say three legged ! ).

The method I use to complete the overdraw was developed in the early 1970’s when I was working on trying to prove the Four Colour Theorem.   A single continuous overdraw throughout a map would split it into two chains of alternate colours which would demonstrate that only four colours were needed.  I will explain how this is done in a future post.

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

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

This overdraw has resulted in several continuous lines and no alternative would produce a Single Continuous Line. This is due to the lack of width going around the structure.

Consequently, I have extended the design further by adding linking lines between all outer petals and succeeded in drawing a Single Continuous Line on that. A future Skydiver jump completely assembling this design would require about 220 participants (I am not suggesting that this be attempted) !

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

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

Alternate Overdraw on Continuous Line Drawing

After my early attempts at continuous line drawing, and then alternate shading, I tried Alternate Overdraw on top of my continuous line drawings.  This produced some fascinating results which led to developments throughout my art.

Alternative Overdraw on Continuous Line of the Horse, start A to B.

Draw from A to B to start Alternate Overdraw on Single Continuous Line Drawing of the Horse.

Lets use the Horse as an example.  Here is a lightly drawn Continuous Line (sorry if you have to tilt your lap top to see it all).  Start at point A and use a thicker pen or marker and draw over the first section of line, in the direction of the arrow, between the two crossovers.  Then miss a section before overdrawing the next section of line.  Keep going overdrawing alternate sections to point B.  You will see that already some overdraw sections are forming closed lines.

 

Alternate Overdraw of continuous line of horse.

Complete Alternate Overdraw of Continuous Line Drawing of horse from point A.

 

 

The next illustration shows the complete Alternate Overdraw and all these new darker lines form closed lines.

 

 

 

Naturally, if we start the Alternate Overdraw in a section which was not overdrawn in the above example (eg at X below), then this produces a result where a completely different set of closed lines appear.

 

Other Alternate Overdraw on Continuous Line of horse.

Other Alternate Overdraw on Single Continuous Line Drawing of horse, starting at X.

 

 

 

 

 

 

 

 

 

 

From the above findings I first developed my Colour Sequence ideas, which I will expand upon in the next post.

Later I used the closed lines to help in the construction of large models of my continuous line drawings as well as to devise hidden drawings within continuous lines.  More later on those.

The Alternate Overdraw method also led to a way to find a path through Four Colour Theory maps as part of my attempt to prove that theory nearly 40 years ago.

So, watch this space !!

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 –    Skelldale. Continuous  line drawing.

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.

        Lizard. Continuous line with colour sequence. Mick Burton, 1971

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.

 

 

 

Flame on the Sun. Spherical continuous line. Mick Burton, 1972My “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. 

Africa. Four colour map theorem. Continuous line. Mick Burton, 1974.

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.