Tag Archives: elephant continuous line

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.

 

Twisting, Overlapping, Envelope Elephant. Continuous Line Drawing colouring.

“Fluorescephant”, the original version of “Elephant Grass” which is at the top of this continuous line blog, was my first successful Colour Sequence painting.  The sequence ran from yellow through greens to blues in steps of colour and tones which gave a natural three dimensional effect and dynamism.  Part of this was the overlapping nature of continuous lines which was reflected by the successive darker colouring.

The painting was accepted for the International Amateur Artist exhibition, in Warwick Square London, in February 1973 and then a month later in the National Society annual open exhibition in the Mall Galleries.

Fluorescephant.  Continuous line drawing with colour sequence.  National Society Open Exhibition, Mall Gallery, London, 1973.  Mick Burton.

Fluorescephant. Continuous line drawing with colour sequence. National Society Open Exhibition, Mall Gallery, London, 1973. Mick Burton.

I was never totally happy with the colouring.  I thought that there was an extra natural effect, on top of the overlapping, which I was missing.  When I started my art again in 2012, after a gap of nearly 40 years, I once more tried to sort this out.  I realised that I could enhance the twisting of the design and highlight gaps where the outside would show through.

Here is the result, “Twisting, Overlapping, Envelope Elephant”.  Imagine that the continuous lines are describing a sheet of plastic, which is coloured Blue on the front and Red on the back.  Each time a twist occurs, against the outside background, then I colour it Red.  When the overlaps build up, the shades of the blue front go darker blue, and the shades of the twisted areas become darker red.  Where the blue front and the red back occasionally overlap, then I use violet to reflect the mix.

This continuous line drawing is coloured to represent a

This 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.

You can see considerable areas of background colour within the animal showing through. This looks natural within the form of the elephant.

The blue areas, including darker blue overlaps, are the same as the blue areas in the “Fluorescephant”, so it is good to keep a large part of the original colour sequence in this change of style.

Colour Sequence on Continuous Line Drawing

Fig 1.  Completed Colour Sequence on Continuous Line Drawing of horse.  Mick Burton, Continuous Line Blog.

Fig 1. Completed Colour Sequence on Single Continuous Line Drawing of horse. Mick Burton, Continuous Line Blog.

How do I apply Colour Sequence to my Continuous Line Drawings, which I first developed in the late 1960,s ?  In my last blog post, about Alternate Overdraw of My Continuous Lines, I pointed out that Colour Sequence was the next stage and so here we go.  I will now show the stages involved in completing the colouring of this Horse.

Other Alternate Overdraw on Continuous Line of horse.

Fig 2. Alternate Overdraw on Single Continuous Line Drawing of the Horse, as the first stage of Colour Sequence. Mick Burton, Continuous Line Blog.

From the two Alternate Overdraw examples in the previous post, I have chosen Fig 2 commencing at point “X” for this example (either “A” or “X” would result in the same colour sequence).

We are going to number all areas of the drawing, commencing with the background which will be numbered “0”.  In this example the background will remain uncoloured but “0” will also occur within the drawing and have a colour. 

Fig 3.  Initial numbering (0 and 1) of channels between Alternate Overdraws on the Continuous Line Horse.

Fig 3. Initial numbering (0 and 1) of channels between Alternate Overdraws on the Continuous Line Horse.

You will notice that between all the closed lines, formed by the Alternate Overdraws, there are channels of areas.  These can be completely numbered alternately by only two numbers, which in this case are 0 and 1.  So, starting with 0 on the background, work through all these linked channels, see Fig 3.  This also sets the direction of the number sequence throughout the drawing.

 

Fig 4.  Colour Sequence numbers 2 and 3 on the Continuous Line Horse.

Fig 4. Colour Sequence numbers 2 and 3 on the Continuous Line Horse.

 

 

The numbering progresses both upwards through positive numbers and downwards through negative numbers.  We will start with the positive direction and allocate the next pair of numbers, 2 and 3.  By moving from an 0 area into a 1 area, and on through its Alternate Overdraw border, we will enter an un-allocated area we can mark 2.  Now deal with all the other areas in this new channel, marking alternate areas 3 and 2, to complete this allocation.  After this we need to check for any further Alternate Overdraw channel, or channels, at this level adjacent to 1 areas and then allocate 2 and 3 to them also, see Fig 4.

We then need to check for any further Alternate Overdraw channels enclosed within any of the 2 and 3 channels.  If we found one we would allocate 4 and 5 to the new channel or channels.  In this case there is no higher level channel. 

Fig 5.  Colour Sequence numbers (-)1 and (-)2 on the Continuous Line Horse.   Mick Burton, Continuous Line Blog.

Fig 5. Colour Sequence numbers (-)1 and (-)2 on the Single Continuous Line Horse. Mick Burton, Continuous Line Blog.

Having  completed the numbering of areas in the positive direction, we now go into the negative in Fig 5.  By looking at an area 1 and moving through a 0 area with an Alternate Overdraw border we can cross through that into a (-)1 and (-)2 channel.  Mark the initial one (-)1 and then allocate alternately through the channel with (-)2 and (-)1.  After completing that channel, look for other un-allocated channels adjacent to 0 areas and allocate (-)1 and (-)2 to them.  Now look for further channels in the negative direction enclosed within a (-)1 and (-)2 channel.  There is one such, a single area (enclosed by its own Overdraw) in the front leg of the horse, which I have left blank in Fig 5 , which will be (-)3.

Fig 6.  Colour Sequence colour chart for Continuous Line Horse.  Mick Burton, Continuous Line Blog.
Fig 6. Colour Sequence colour chart for Continuous Line Horse. Mick Burton, Continuous Line Blog.

I was inspired by Rainbows in deciding on the sort of Colour Sequences I wanted to use for my Continuous Lines.  For shorter sequences, I settled for “partial rainbows” involving two prime colours only with a progression of colour mix and tones from light to dark.  For the Elephant I used yellows, greens and blues and for the Horse it was yellows, orange, red and browns in Fig 6.

I have carefully selected colours which have a stepped progression, both in colour and tone, and where possible I apply them from the tube (poster colour in the late 1960’s or acrylic now) to achieve an even and solid result.  I avoid mixing if I can, to retain the pure consistency of colour application across the painting, but sometimes it is necessary.

 

Fig 7.  Black and white photocopy of Colour Chart for Continuous Line Horse.
Fig 7. Black and white photocopy of Colour Chart for Continuous Line Horse.

 

To assess the accuracy of the progression steps of my Colour Sequence chart, I do a black and white (or grayscale) photocopy of my chart to check that the steps still work in monochrome, see Fig 7.

Having produced the Colour Sequence chart, we need to decide the direction of the colours matched to the numbers, ie. Light to dark in an upward or a downward direction.  Generally I see whether a scale would mostly coincide with where a natural highlight would be, or have more darks towards the lower parts in a drawing to infer shadow.  Usually it is fairly obvious, but you can always start again with the other direction of colours.  Note that my style may take advantage of natural hints of highlight or shadow on a subject, but generally these aspects (along with perspective) are absent.

I remember that when doing equations at school, which produced two answers (+ or (-) ), was a puzzle to me which no one could explain.  I understand the concept of a practical outcome from having two answers a bit better now.

Fig 8.  Initial Colour Sequence pair of colours on Continuous Line Horse.   Mick Burton, Continuous Line Blog.

Fig 8. Initial Colour Sequence pair of colours on Continuous Line Horse. Mick Burton, Continuous Line Blog.

Once we have decided on the colour match with the numbers, the initial two colours can be painted in, ie.  Vermillion = 0 and Orange = 1, see Fig 8.

 

 

 

 

Fig 9.  Second Colour Sequence pair of colours on Continuous Line Horse.
Fig 9. Second Colour Sequence pair of colours on Continuous Line Horse.

 

We can then match numbers 2 and 3 in areas to the colours required in the next channels up, or simply apply Golden Yellow to areas across the overdraw from Orange and then its alternate colour Permanent Yellow, in Fig 9.

 

 

 

 

Fig 10.  Third Colour Sequence pair of colours, in the negative direction, on the Continuous Line Horse.   Mick Burton, Continuous Line Blog.

Fig 10. Third Colour Sequence pair of colours, in the negative direction, on the Single Continuous Line  Drawing of Horse. Mick Burton, Continuous Line Blog.

Now we can match numbers (-)1 and (-)2 in the negative direction, or simply apply Light Brown to areas across the Overdraws from Vermillion.  When these Light Brown and Burnt Sienna channels have been completed the last channel colour is (-)3 which is Burnt Umber.  In Fig 10 I have left this final area blank  (on the front left leg of the Horse).

So you have seen my Colour Sequence method, using Alternate Overdraw, for Continuous Line Drawings.  Sorry if it has been a long explanation (particularly if you grasped it quickly or had already come across parts of it), but I have tried to pitch it as helpfully as I can, based on my demonstration sessions.    

A couple a years after I started Colour Sequence I came across the Winding Number Theory.  There is a connection and I did pick up one or two ideas from it.  I will talk about this in a later post, but as always I am not a trained mathematician and so I will keep talking in pictures. 

 I hope that you will give it a try and I am sure you will enjoy the ride, as I have for so long.

If you display or publish your results, it would be great if you could specifically acknowledge me and my ideas.