Tag Archives: Mick Burton

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.

 

Kaleidoscopic Wild Horses, continuous line drawing with colour sequence.

Wild Horses, June 2017

Kaleidoscopic Wild Horses. Single Continuous Line Drawing with colour sequence in acrylic on canvas.  I happened to have a canvas 36″ x 10″ previously intended for an upright picture idea.  Mick Burton continuous line artist 2017.

This painting originated from a continuous line drawing which I produced for a demonstration at Stainbeck Arts Club, Chapel Allerton in Leeds in May 2017.  

IMG_0883 -Horses line.

Wild Horses,  single continuous line drawing. Demonstration at Stainbeck Arts Club. Mick Burton, continuous line artist 2017.

When I was thinking about a subject for the demonstration I saw an advert on the TV for the Cheltenham Festival which just showed loads of horses running – why there were no riders or jumps I do not know.  This also reminded me of one of my favourite paintings – “Scotland Forever” by Lady Butler in Leeds Art Gallery, painted in 1881.  A bit like “Charge of the Light Brigade” but straight at you, with the horses wild eyed and seeming to leap out of the painting.

See it at   http://www.leedsartgallery.co.uk/gallery/listings/l0081.php

Lady Butler painted a lot of war scenes and of course she had no military experience.  She was, however, married to a General and she persuaded him to let her watch manoeuvres.  In preparation for this picture she asked that the cavalry ride straight towards her so that she could get the feel for facing a charge.

When I had finished the demonstration, which was a result considerably rougher than the above, the members asked about colours.  I had not intended to talk much about colours, as I thought that my approach to drawing the lines would be enough at this session, but we had a solid half hour talking about my method and ideas about colour.  They said that they looked forward to seeing the image in full colour, so here it is.

My original intention was to do a black and white alternate shading version only, and this is shown below.  The tweaking which I did on the horses heads to achieve a better result in black and white was essential both to improve the continuous line and later to enhance the colouring.

IMG_0888 - Horses black & white

Wild Horses, single continuous line drawing with black and white alternate shading.  Mick Burton, continuous line artist 2017

Initially I did my normal approach to colour sequence, where I devised a 6 colour range (white, lemon, golden yellow, orange, vermilion red and crimson alizarin) to fit my alternate overdraw template for this image.  

This resulted in gold and vermilion appearing on all outer areas and I thought that I needed a darker effect in the lower half of the image.  So I substituted cobalt blue for gold along the bottom legs of the horses and finished up also substituting, on an ad hoc basis, some dark blue, violet and green to try and naturally leach colour balance upwards to meet existing vermilion and gold.

A fellow artist who likes my alternate overdraw and colour sequence method has told me that I should always apply it fully to get the natural result.  Generally I would agree, but thought that I needed to break some rules on this occasion.  I try and mirror nature in my art and of course nature evolves by breaking a few rules. 

Joan and I visited my Aunty Ann a couple of weeks ago.  She is 99 years old and still as bright as a bobbin.  She is a good artist and only gave up painting relatively recently, and always wants to see my latest stuff.  i took the Wild Horses along.  It took up the length of the settee and she was delighted with the colours.  I then realised that the painting’s reflection in the shiny metal fire surround made the composition even more abstract.

Aunty Ann’s shiny metal fire surround reflecting Wild Horses. Mick Burton, continuous line artist, 2017.

Two different Reflections of Wild Horses on metal fireplace surround, detail strips. Mick Burton, continuous line artist.

Leeds Olympic Lion, a new single continuous line painting by Mick Burton.

Leeds Olympic Lion.   Mick Burton

Leeds Olympic Lion, coloured in many shades of red, white and blue to commemorate all the Leeds based athletes and swimmers who brought back medals from the Rio Olympics, 2016.  Mick Burton single continuous line drawing with colour sequence.

I did a demonstration at Farsley Art Group on 12 July 2016 and the continuous line drawing I used as an example was the basis for the above painting. The Group showed a lot of interest and produced many fine attempts at continuous line during my workshop. The club kindly featured me on their website, showing some of my drawings as well as work by members.  I gave them a free hand to put their own stamp on their continuous lines so we had some great variations.

Joan, my partner, watched many swimming and diving events on the TV during the Olympics broadcasts.  She worked at the Leeds International Pool and the new John Charles Centre, in various swimming organising roles, before she retired in 2012 and was delighted with the results of the Leeds members of the Great Britain team and their coaches.

As the athletes all had the red, white and blue lion on their track suits I felt I had to colour my Lion in a range of similar colours and call it the Leeds Olympic Lion.  The painting will be exhibited in the Stainbeck Arts Club Annual Exhibition on Saturday 3 September 2016.  The exhibition is part of the Chapel Allerton Arts Festival taking place in north Leeds this week.Stainbeck Arts Club Poster

Joan’s daughter, Helen Frank, represented Great Britain in the 100 metres breast stroke in the Seoul Olympics in 1988 and was one of five swimmers from Leeds.  Adrian Moorhouse won a swimming gold medal, in the 100 metres breastroke, in 1988.  A gold by a British swimmer was not achieved again by a  British swimmer until 2016.

Leeds Olympic Swimmers at Seoul 1988

Helen brought back a commemorative plate from Seoul, which is part of Joan’s collection of Olympic Plates.

Seoul 1988 Olympics plate

 

Mallard steams by towing eight ducklings in a continuous line.

Mallard steams by towing eight ducklings in a continuous line. Photo Mick Burton, 7.8.2016, Well House Drive, Gledhow Valley, Leeds.

Mallard steams by towing eight ducklings in a continuous line. Photo Mick Burton, 7.5.2016, Well House Drive, Gledhow Valley, Leeds.

After a couple of hours digging rocks and roots in the back garden, I emptied the roots into the brown bin and glanced towards the road at the front. There was a gap between the car and the gate post and I glimpsed a line of ducklings going by on the other side of the road.  What were the chances of that happening?

View across the road where a line of ducklings went by. Photo Mick Burton, continuous line artist,7.5.16.

View across the road where a line of ducklings went by. Photo Mick Burton, continuous line artist,7.5.16.

I rushed inside the house for my camera and ran to catch up with the female Mallard mother marching down the road with her brood in tow.  There were some kids in gardens not yet aware and cars going up and down the road.  I took the photo which appears at the top of this post.

But what was this Mallard mother doing coming down our road with her ducklings, when I knew of no streams or ponds in the streets higher up?

Anyway, I knew where she was likely to be going.  If she turned right at the bottom of the road she was on Gledhow Valley Road and the houses along there all had the Gledhow Beck flowing through their gardens.  She started to move out towards the curb to assess when to cross the road.

Mallard mum starting to move out towards the curb. Photo Mick Burton, continuous line artist.

Mallard mum starting to move out towards the curb. Photo Mick Burton, continuous line artist.

I took another closer view of the bird as she headed along parallel to the curb.  There is an eighth duckling just out of shot at the back.

Closer view of female Mallard with seven ducklings in the picture. Photo Mick Burton, continuous line artist, 7.5.16, Gledhow Valley.

Closer view of female Mallard with seven ducklings in the picture. Photo Mick Burton, continuous line artist, 7.5.16, Gledhow Valley.

Looking at these photos reminds me of my  train spotting days as a youth when I saw the  A4 pacific steam engine “Mallard” at speed pulling many carriages.  No wonder they decided to name the fastest ever steam locomotive after such a sleek bird.  Here is a photo I took of the “Mallard” steam engine at the National Railway Museum in York three years ago, along with my son Matthew.

Mallard, the fastest ever steam engine, at York National Railway Museum along with my son Matthew. Photo Mick Burton, continuous line artist.

Mallard, the fastest ever steam engine, at York National Railway Museum along with Matthew Burton.  Photo Mick Burton, continuous line artist.

Now back to the ducks and the dangerous task of crossing the road.  The road seemed clear so Mum decided to cross.  A car was signalling to turn left into our road and I flagged it down just as it was turning, which gave me little time to get a photo of the ducks in the middle of the road.  The result was a bit shaky.

Mallard Mum and eight ducklings crossing the road in front of a waiting car in Gledhow Valley. Photo Mick Burton, continuous line artist.

Mallard Mum and eight ducklings crossing the road in front of a waiting car in Gledhow Valley. Photo Mick Burton, continuous line artist.

When you are a very small duckling it’s not easy to mount a pavement you can’t really see over, and there’s not much sympathy from Mum.

Ducklings struggle to mount a pavement higher than they are. Photo Mick Burton, continuous line artist.

Ducklings struggle to mount a pavement higher than they are. Photo Mick Burton, continuous line artist.

Pause to regroup and a crowd of kids is starting to gather.

Ducklings regroup on the pavement after mounting to kerb. Photo Mick Burton, continuous line artist.

Ducklings regroup on the pavement after mounting to kerb. Photo Mick Burton, continuous line artist.

The worst is over and the female Mallard turns right into Gledhow Valley Road.

Mallard and ducklings turn right into Gledhow Valley. Photo Mick Burton, continuous line artist.

Mallard and ducklings turn right into Gledhow Valley. Photo Mick Burton, continuous line artist.

The Mallard Mum and her train of ducklings can speed up as they enter the home straight.  I am sure now that they will turn right into a drive soon.

The train of Mallard ducklings speed up as they enter the home straight. Photo Mick Burton, continuous line artist.

The train of Mallard ducklings speed up as they enter the home straight. Photo Mick Burton, continuous line artist.

The crowd of kids is becoming a bit intrusive and I ask them to keep their distance, but without much effect.

Kids arriving from all directions are beginning to crowd the ducks. Photo Mick Burton, continuous line artist.

Kids arriving from all directions are beginning to crowd the ducks. Photo Mick Burton, continuous line artist.

The Mallard Mum’s preferred garden is a bit further along I think, but she decides that she will turn into this drive anyway to escape the kids and go down to the stream.

The Mallard and her brood turn down a drive to escape the kids and to head for their stream. Photo Mick Burton, continuous line artist.

The Mallard and her brood turn down a drive to escape the kids and to head for their stream. Photo Mick Burton, continuous line artist.

Just another incident with wildlife in the valley.  Not long ago Joan and I helped a family of swans to cross Gledhow Valley Road, whilst policemen held up traffic from both directions, but that’s a story for another day.

“BB” by Kate Burton, Glasgow filmmaker, at London Short Film Festival.

BB-poster-online

Poster For ‘BB’ Written and Directed by Kate Burton. Poster Design by Jen Davies

My daughter Kate has recently attended the London Short Film Festival where her latest film “BB” is showing.  The film was also selected by the Glasgow Short Film Festival and the Inverness Film Festival, and was nominated for Channel 4 innovation in storytelling award 2015.

The action starts with a bee buzzing around, the synopsis  reads “When Anna discovers an unwelcome intruder in her home she enlists the help of her mild mannered neighbour.  Frank enters into Anna’s strange and chaotic world and finds himself well removed from his comfort zone.  Amusingly awkward social challenges follow and an unlikely relationship is formed.”

The film was shot in Kate’s previous flat in Glasgow, and in the local area.  She was determined to make the film before she left.

Joanne Thomson in the Kate's flat, featured in "BB", short film by Kate Burton, Glasgow, 2015..

Joanne Thomson in Kate’s flat, featured in “BB”, short film by Kate Burton, Glasgow filmmaker, 2015. Photograph by Helena Ohman

I asked Kate to give me the background to the film. 

“At the time I wrote the script I was living in a small one bedroom flat in a Victorian townhouse conversion in the west end of Glasgow.  The ceilings were high and there was a split level mezzanine where I worked and completed the script.

There was a huge rose garden opposite the flat and in the summer confused bumble bees would fly inside my flat and roll up into fuzzy balls on my window ledge.  This and many other elements of my surrounding environment fed into the content of the script.”

Kate Burton watches as David Liddell and Helen Ohman McCardle take a close up for "BB", a film by Kate Burton, Glasgow.

Kate Burton director, David Liddell cinematographer and Hannah Kelso assistant camera on the set of  “BB”, a film by Kate Burton, Glasgow. Photograph by Helena Ohman

“I knew the characters very well after a lengthy development time, but the story for the short came relatively fast and after two months writing, in between filmmaking jobs, I was finally ready to let other people read it.

‘BB’ is a character driven story capturing the awkwardness of first attractions.  It takes place mainly in one location.  I wanted the cinematography and style to reflect the simplicity of the story and I decided that I wanted to shoot in black and white to give a heightened feel, emphasising the unique lines of the interior space.  I was interested in long duration takes and capturing physical and theatrical comedy within the frame.  Some of my research references were Woody Allen’s ‘Annie Hall’ and ‘Manhattan’, ‘Breakfast at Tiffany’s’ by Blake Edwards and ‘Francis Ha’ by Noah Baumbach.”

Kate04

Kate Burton, writer/director, on set of ‘BB’ 2015.

Ben Clifford and Joanne Thomson in "BB", as short film by Kate Burton, Glasgow, 2015.

Ben Clifford and Joanne Thomson in “BB”, by Kate Burton, Glasgow filmmaker, 2015. Still from film. 

One of Kate’s earlier films was “The Ice Plant”.  I thought that this was a strange title until she told me that it was about ice cubes and that her research led her to  Highland Ice Ltd, a factory in Aberdeen which makes ice cubes, which kindly agreed to let her film there.  She has a good photo of the film crew standing around in woolly hats and scarves.

ice plant still 04

ICE PLANT.022

The Ice Plant 2007.mov.01_02_35_04.Still010

Festivals showing “The Ice Plant” were – Edinburgh International, Glasgow International, Clemont Ferrand International, Seattle International Film Festival, preceding a feature presentation at the GFT Glasgow.

Current television period dramas “War and Peace” and “Dickensian”, featuring Tuppence Middleton, have reminded me of a film Kate made in 2010, “Ever Here I Be”, a 16 minute ‘Digicult’ Film, UK Film Council & Scottish Screen.  Shown at Edinburgh International Film Festival, Nashville International Film Festival, Palm Springs Short Film Festival , Portabello Film Festival London and Inverness Film Festival Scotland. 

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Tuppence Middleton, Ever Here I Be, photograph by Janet Johnstone

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Ever Here I Be Master - cow.mov.10_01_07_15.Still031

Ever Here I Be starred Tuppence Middleton and Christopher Simpson and was nominated for the Underwire ‘Best Female Character’ award 2012.

Kate was a talented artist from a very young age and it was a delight to watch her progress.

She went to Allerton Grange High School in Leeds, where Damien Hirst was a former pupil, and then completed a Foundation course in Fine Art at the Leeds College of Art and Design, where she increasingly studied film.  Kate then moved to Glasgow in 1999 to study Environmental Art at Glasgow School of Art, during which she went on exchange to Massachusetts College of Art and Design, Boston to study filmmaking.

Kate graduated in 2002 and has written and directed a number of shorts, documentary and promotional films.  These include several commissions from Glasgow School of Art to make documentaries and promotional films for them.

Kate has a passion for film education and has taught filmmaking and screenwriting to children and young people for the past 8 years through Into Film, Project Ability Glasgow, Glasgow Film, Edinburgh Filmhouse, Glasgow Media Access Centre and Visible Fictions theatre Company.

Her website is  www.kateburtonfilm.com  which highlights all her main films.  You can watch many, such as “Mrs Pickering’s Music Cabinet”, 2015, where Helen McCook, Artisan Embroidery & Artist, has been requested to reproduce the lost textile screen to go into an original Rennie Mackintosh cabinet.

Kate is currently working on an outline for her first feature film script.

 

Stainbeck Arts Club Exhibition at Chapel Allerton Arts Festival

Stainbeck Arts Club Exhibition at Chapel Allerton Arts Festival, Leeds.  on Saturday 5 September 2015.  Paintings by Mick Burton, continuous line artist, are included.

Stainbeck Arts Club Exhibition at Chapel Allerton Arts Festival, Leeds. on Saturday 5 September 2015. Paintings by Mick Burton, continuous line artist, are included.

Stainbeck Arts Club, my local art club which is based in Chapel Allerton in north Leeds next to Gledhow Valley, is holding its Annual Art Exhibition as part of the Chapel Allerton Arts Festival.  The Exhibition held in Chapel Allerton Methodist Church, which is situated at the lower end of the main Festival area, will be open from 10.00 to 16.00 on Saturday 5 September 2015.

There will be many quality pictures on display, so if you are attending the Arts Festival do pop in and have a look.  Admission is free.

Chapel Allerton Arts Festival 2015, front cover of brochure.

Chapel Allerton Arts Festival 2015, front cover of brochure.

The Chapel Allerton Arts Festival was started in 1998 by members of the local community in a small way, with a few stalls and two bands playing from the back of a lorry in the evening.

It now runs from Monday to Sunday and involves many parts of the community at lots of venues during the week, including a Short Film Festival and an Art Trail. At the weekend two streets are closed off and the central parking areas taken over for all the stalls and events, including a main stage for the many quality bands. The festival is still run entirely by volunteers.

This year the Festival starts on Monday 31 August and the main day will be Saturday 5 September when people will arrive from all over the place to attend one of the star attractions in the Leeds calendar.

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.