Tag Archives: continuous line artist

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.    

Christmas Tree Frost Image on Car Roof

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Christmas Tree frost image on car roof, with photograph darkened.   Mick Burton, continuous line artist, 24 December 2018.

I have noticed hard frost images before on our car parked overnight in the drive. The car is on a slope facing upwards towards the south and the frost pattern seems to emanate from the high point of the roof, which is towards the higher front end.

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Car with frost image on roof, parked facing south up hill, showing the centre of the pattern at the high point of the roof.   Mick Burton, continuous line artist, photo taken 24 December 2018.

Here is a closer view of the roof.

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View of the frosted over roof of the car, 24 December 2018.   Photo Mick Burton, continuous line artist.

The pattern on the roof of the car reminds me of the view from a plane when flying over the Alps.

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A closer view can be likened to the view when flying over the Alps.   Photo Mick Burton, continuous line artist.

A very close view shows the granules of frost making up the effect.  This is the apex of the car roof and I presume that periodic melting had occurred followed by new freezing.

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Granules of frost making up the image on the car roof.   Photo Mick Burton, continuous line artist.

As the sun hits the roof it highlights the pattern and eventually starts to melt the frost around its edges.   A pattern similar to a fir tree starts to emerge.   The melting starts from the high end establishing the bottom of the trunk of the tree and begins to form the outline of  the rest of the tree.

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As the sun increasingly hits the car roof it highlights the effect and also begins to melt around the edges.  Photo Mick Burton, continuous line artist.  

Making the photograph darker produces the effect shown at the top of this post.

I prefer to leave a bit of the car in the photo myself, but here is a view without the top of the car windows.

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Tree effect without showing any obvious part of the car.   Photo Mick Burton, continuous line artist.

 

I am fascinated by intricate decorative patterns appearing in nature.   Sometimes my Single Continuous Lines include a hint of this naturalness.

The Secret Art Project in St Gemma’s 2018 Art Festival

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“Loch Knares Monster” submitted to St Gemma’s 2018 Secret Art Project by Mick Burton, Continuous Line Artist.

As the open session for bidding on e-bay for the A5 size card paintings and drawings submitted to the St Gemma’s Secret Art Project has now closed, I am free to say which two pictures were submitted by me.  Of course, they are not single continuous lines.

The first one, above, is called “Loch Knares Monster”.  I used coloured pencils to produce delicate shades in the water and sky in contrast to the acrylic pen outline on the Serpent.

I have always been interested in the Loch Ness Monster, so what is wrong with turning the famous railway bridge in Knaresborough into a giant serpent.  My first attempt at the monster, years ago, was to add one to a mural of Venice which had been painted all over the bathroom wall of a house I shared with several others in Leeds in the late 1960’s.

Also an artist friend of mine, Bryn Glover, constructed a sculpture of “Nessie” from a motor cycle chain in 1969.  He worked at Leeds General Infirmary and once used a huge pair of forceps in a sculpture of a pelican.

There had to be a train in it, particularly a steam train, but I did not feel that my favourite engine “Mallard” would be appropriate.  See photograph below of Mallard crossing the Knaresborough bridge.

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“Mallard on Knaresborough Viaduct” in 1987. Photograph by Jo Turner (https://www.geograph.org.uk/profile/14389).

One day in the mid 1950’s, when I was on holiday staying with relatives, my cousin John Simmister and myself wandered into the railway sidings in Peterborough.  We saw Mallard, all be itself and so dirty that you could hardly tell that it was green. We climbed into the cab and talked about what it would be like to travel at 126 miles an hour and break the world steam record.

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“Mange 2 – Rail Root North” submitted to St Gemma’s 2018 Secret Art Project by Mick Burton, Continuous Line Artist.

“Mange 2 – Rail Root North” was coloured in acrylic and in a style which I hoped would be very different to my Knaresborough picture.

The HS2 railway project from London to Leeds and Manchester has been dragging on for years and I thought that I would compare that train to a streamlined pea pod.  My wife Joan comes from Wakefield, which is Rhubarb Triangle country, and as she is a vegetarian I thought she may appreciate the “jokes”.

Here is a clipping from this month’s Yorkshire Reporter, showing that HS2 is still big news.

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Yorkshire Reporter, November 2018, showing plans for Leeds Railway Station as part of the HS2 project.

Therasa May is one of the celebrities (including John Bishop and Kaiser Chiefs) who have agreed to submit a picture to the Secret Art Project, but there is probably no danger that people will think that she painted Mange 2.

 

St Gemma’s 40th Anniversary Arts Festival – 2018 Leeds Art Exhibition and Sale 2018

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Mick Burton, continuous line drawing artist, one of 40 artists selected to exhibit in the Leeds Art and Photography Exhibition 2018 as part of the St Gemma’s 40th Anniversary Arts Festival.

The 2018 Leeds Art and Photography Exhibition and Sale takes place on 26 – 29 October 2018 at The Grammar School at Leeds as part of the St Gemma’s 40th Anniversary Arts Festival.

Last year there were about 170 artists exhibiting about 900 pictures and photographs. However, as it is the 40th anniversary of the charity, they have decided to give the 40 artists who have sold the most pictures in recent years the option of submitting more pictures and to specify their own presentation across two stands.

I have been fortunate enough to be invited and look forward to the exhibition.

You can check out the details on the initial website

https://events.st-gemma.co.uk/events/leedsartfestival

This also shows pictures by over 30 artists from the 2017 exhibition, including my “Leeds Olympic Lion” above.

I have also entered the “Secret Art Project” where you draw or paint on a post card and all the entries are displayed anonymously.  During and for a week after the exhibition the display is on the internet so people can bid for the cards.  They publicise various celebrities who St Gemma’s have asked to enter a picture card to help inflate the bidding.

Of course, using my continuous line style would be a bit of a “give away” so I have used another approach.  All will be revealed in a post after the event.

Four Foxes Frolicking in Gledhow Valley Garden.

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Four foxes appear in Gledhow Valley garden on 6 June 2018.   Photo Mick Burton, continuous line artist.

I have mentioned the local Gledhow Valley foxes several times in my posts, but this is the first time that we have seen four at once in the garden, which backs on to the woods. There appear to be two young ones and two adults (or older siblings).

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Four Gledhow Valley foxes, with a lot of licking going on.   Photo Mick Burton, continuous line artist.

In the above photo the foxes are licking themselves and each other and this seemed to go on for ages.

Joan and I have been provided with all sorts of entertainment by the foxes over the last few years.

Our next door neighbour used to put out spare ribs in their back garden.  One time a magpie was enjoying these when a fox came along and picked one up and went away with it.  The fox returned several times for others and the magpie became more agitated each time, finally bouncing around the fox in circles and squawking loudly, as if to say “Those are my spare ribs, get off them, clear off ! “.

Twice when we have had relatives here for Xmas and eating Xmas lunch in the dining room we have seen a rare event.  Looking out of the French windows there would be a fox helping himself to all the bread scraps we had put out on the low bird table, as if taking part in the festivities.

Foxes are territorial of course and I we have seen one on the garage flat roof marking his territory up there.

Of course, they are also hunters and we have seen one trotting up the garden, spotting a squirrel to one side and shooting off sideways to grab it in its mouth and then continuing on its way.

As hunters, our foxes have an annual run in with the swans on the Gledhow Valley Lake.  Several years ago we were walking past the lake at about 7pm in mid May when we saw a fox standing nose to beak with a swan, on its eggs on the nest in a shallow part of the lake.  A photographer, who was watching, said that the fox would not take on a swan on its own, but suggested that when it was dusk, several foxes could work together to try to get the eggs.  The next day there were no eggs.  Something similar may have also happened this year.

Foxes are wary of cats.  Avril, who lives on Gledhow Valley Road was once at the end our garden asking if we had seen her cat, which she said was quite shy and usually stayed in the garden.  It was a large fluffy cat.  The next day I saw a fox at the top of the garden just to the right of the huge oak in the corner.  It then backed to the right and a big fluffy cat came slowly from behind the tree, nose to nose with the fox.  The fox eventually turned slowly away and left.  When I saw Avril again she said she was not surprised as the cat was in charge of all her dogs, including a large Alsatian.

We have also seen the effects of mange on an adult fox, which we saw for some time a few years ago, and it seemed to survive.

The fox below appeared on 4 July 2018, and may be one of the four we saw in early June.

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Healthy fox in Gledhow Valley garden 4th July 2018. Photo Mick Burton, continuous line artist.

This fox was up and down the garden a few times.

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Side view of healthy Gledhow Valley fox. Photo Mick Burton, continuous line artist.

Shortly afterwards, Joan was at the back of the garage with her watering cans and I walked down to the front and between the garage and the house only to come face to face with this fox.  It veered off and went between our garage and next door’s (a gap of about 8″).  I shouted to Joan “Fox coming through”, which of course meant nothing to her until it burst out at the top end of the garage.

And now for a different view of a young fox.

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Young fox in Gledhow Valley appears 5th July 2018.   Photo Mick Burton, continuous line artist.

The next day, a young fox appeared, which I think may be a younger sibling of yesterday’s fox, and headed for the pond, which we have in a half wooden cask, for a drink.  There were also  five magpies going in and out of the bird bath.

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Poor young fox with scanty fur in Gledhow Valley. Photo Mick Burton, continuous line artist.

It was soon obvious that this was not a well fox, certainly compared to the one we saw the previous day.  Lacking in fur and with a red sore patch on its back.

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Unhealthy young fox wants a drink from the bird bath in a Gledhow Valley garden. Photo Mick Burton, continuous line artist.

The fox now wants to have a drink from the bird bath, which the magpies have temporally vacated.

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The magpies have not left much water for Gledhow Valley fox.   Photo Mick Burton, continuous line artist.

The fox can’t quite reach for a drink as the magpies have splashed away most of it.

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Young fox in Gledhow Valley showing all the signs of mange, with open wound made worse by scratching.    Photo Mick Burton, continuous line artist.

This close up of the fox shows all the caking and cracking of the skin and the red open wound which are advanced signs of mange.  Of course it is made worse by constant scratching, and the longest period of hot dry weather since 1976 would have contributed.

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Young fox with mange, really struggling, with attentive magpies (there were five of them altogether).   Photo Mick Burton, continuous line artist.

All five magpies were nearby.  They were quiet, seemingly knowing that one of their rivals was really struggling.  Eventually the fox wandered off up the garden and into the woods.

Sadly, the following morning we found the body of this young fox on the lawn near to the pond.  Possibly the adult foxes would not let it into the den if they thought that it may die and would need to be removed, and so it returned to our garden.

I buried the fox in the garden, near to Tufty my cat.

But what about the other young fox?  

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The other young fox later appeared, also with signs of mange in Gledhow Valley garden. Photo Mick Burton, continuous line artist.

On 15 July 2018 this young fox appeared and it showed similar signs of having mange just like its sibling.  I is seen here eating some scraps of bread left by the magpies.  There was no sign yet of an open wound and so hopefully the mange was not as bad as with the other fox.

We all hope that this one survives.  

Gledhow Foxes Sunbathing, continuous line. Mick Burton

“Gledhow Foxes Sunbathing”. Association of Animal Artists “British Wildlife” exhibition, February & March 2015.   Mick Burton, Continuous line drawing.

I drew this picture over three years ago and I sold it at St Gemmas Leeds Art & Photography Exhibition & Sale last year.  I was also asked to produce a similar one as a commission. 

This year it is the St Gemma’s 40th Anniversary Arts Festival and I am one of the 40 artists invited to take part.  As I have intended to do a colour version of my foxes single continuous line for some time I will now complete one to include in my exhibits.

Hot Cross Bunny and the psychology of colour

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“Hot Cross Bunny”, single continuous line drawing painted in psychological colours. Mick Burton, continuous line artist.

In my posts I have said a lot about colour sequence and, along the way, talked about selecting appropriate ranges of colours for my drawings.  Here are some more colour comments, leading to the one about the bunny above.

I might consider that a yellow, red and brown range would be good for my horse. These have a similarity to its actual colours and give a warm and friendly feel which reflect the horse’s nature and temperament.

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Colour Sequence on Single Continuous Line Drawing of horse. Mick Burton, Continuous Line Blog.

A strong harsh colour seemed to be best for my roaring lion and simple black and white achieved this.  In the mid 1960’s when I drew the lion, Bridget Riley had been doing many black and white hard edge pictures, and I did several of my animals in this colouring.  I feel that this worked best for the lion amongst my drawings.

016. 1967-9. Lion. Alternate shading, black.

Lion, single continuous line drawing with alternate shading in black and white. Mick Burton, continuous line Artist.

With my “Flame on the Sun” painting, the sort of anti magnetism represented by complementary red and green hopefully reflect the explosive violence required.

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

Flame on the Sun. Spherical single continuous line drawing, with complementary reds and greens expressing explosive violence.  Mick Burton, continuous line artist.

For a more subtle result – my still life of a radish, apple, mushroom and flower heads – I used water colours to help to show the floppy translucent nature of the radish leaves.

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Radish, apple, mushroom and flower heads still life. Water colour used to show floppy, translucent nature of radish leaves. Mick Burton, continuous line artist.

Sometimes I find that I can use almost actual colours.  Here is a commission drawing, with the continuous line running through both robins and the branch.  I  was asked to do only a hint of pink on the Robins’ chests.  This is fine.  However, I had to have a go at a full colour result for myself.  The perky nature of robins is reflected pretty well, I think, by these “near” natural colours.

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Pair of Robins, single continuous line drawing. Full near natural colour. Mick Burton, continuous line artist.

My yellow, green and blue sequence of colours fits well for “Nibbles”, a friendly rabbit who likes nothing more than eating her greens.

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“Nibbles”, single continuous line drawing.  The rabbit has a suitable range of colours to reflect contentment just eating her greens. Mick Burton, continuous line artist.

However, for a rabbit drawn with exactly the same single continuous line as for Nibbles, but who has a completely different temperament  –  RED, BLACK and WHITE fits the bill.

This is, of course, “Hot Cross Bunny” who lurks at the top of this post.  A real, full on, “Psycho”.

The two Rabbit paintings and the Pair of Robins accompanied several other of my pictures at the Harrogate and Nidderdale Art Club exhibition a week ago at Ripley Town Hall.

At the Preview Evening various prizes are awarded.  One was the annual prize presented at the Spring Exhibition by Sir Thomas Ingleby, the club’s patron, for his own personal choice for the best picture on show.  This was won by Julie Buckley for her “Black Labrador”.  

Sir Thomas also mentioned other pictures which caught his eye.  He said that he liked all the paintings by Mick Burton, but never thought that he would ever consider buying one called “Hot Cross Bunny”.

Here is a bit of background to the Rabbit paintings.  Nibbles and Hot Cross Bunny are based upon my daughter Kate’s rabbits, Harriet and Clover.

Harriet was friendly and cuddly and Clover might have been better named “Cleaver”.  We kept them both in the garage – in separate cages.

When we bought Clover, a lop eared rabbit, the breeder was saying how friendly and harmless the baby rabbit was.  I asked if it was related to an adult lop eared which had just tried to bite my finger off and the answer was “Yes, it’s the granny”.  We still bought Clover!

She was alright at first but later became very aggressive.  Every time we opened her cage for any reason, she would bite viciously.  We also realised that some one else would have to take care of the rabbits when we were on holiday.

Strangely, I found that if I put a hand on Clover’s head as soon as I opened the door she would stay still and relaxed as long as I kept the hand there.  With the other hand I could top up food and water or clean out the cage.  This worked for all of us.  Fortunately, our neighbour was delighted to be able to do this too and things were fine when we were away.

After Clover died and I had buried her in the garden, Kate prepared a wooden plaque and nailed it to the fence “Here lies Clover Burton the rabbit”.

An interesting consequence of keeping the rabbits was that straw from the bale became piled on the floor of the garage.  One day the straw was seen to be moving and we feared that we had rats and so I was deputed to check it out.  I found a nest of baby hedgehogs.

 

 

 

 

 

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.