Tag Archives: abstract 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.

 

Barn Owl continuous line drawing at Harrogate and Nidderdale Art Exhibition

Continuous line drawing of Barn Owl onto Wet on Wet watercolour. Mick Burton.

Single continuous line drawing of Barn Owl onto Wet on Wet watercolour. Mick Burton continuous line artist, 2015.

This Barn Owl painting will be one of my eight pictures on display at  the Harrogate and Nidderdale Art Club Autumn Exhibition in Ripley Town Hall, near Harrogate, on 21 & 22 November 2015.  I then intend to submit it to the next Association of Animal Artists Exhibition.

Harrogate and Nidderdale Art Club exhibition at Ripley Town Hall, near Harrogate, 21 & 22 November 2015.

Harrogate and Nidderdale Art Club exhibition at Ripley Town Hall, near Harrogate, 21 & 22 November 2015.

Visiting demonstrators at art clubs are amazingly varied and it is usually useful to attempt whatever they ask the club members to do.  I have done some workshops myself and appreciate the efforts of club members who really have a go at continuous line drawings, and associated things I show, even though to is unlikely that any of them will take up my technique as a main style.  Hopefully people can pick up things which can apply to other styles, such as building abstract patterns, using colour sequences, drawing key identifying parts of a subject and trying to manage a picture which sometimes appears to be drawing itself !

Charles Kelly from Bradford, who I have seen doing demonstrations before, came to Stainbeck Arts Club a couple of months ago and said he was doing a workshop this time.  Watercolour tends to be the most popular topic at art clubs, but Charles has a spectacular approach to “wet on wet” and this time we were doing it too.  Here is an example of his work from a demonstration to Alwoodley Art Group in 2013.

A Pair of Geese, painted by Charles Kelly in a demonstration at Alwoodley Art Group in 2013.

A Pair of Geese, painted by Charles Kelly in a demonstration at Alwoodley Art Group in 2013.

My usual style of strong lines and flat colours (acrylic or poster colour) are poles apart from watercolour but I always learn something.  I have to say that using a big brush to coat large proportions of the paper with water in advance (up to selected boundaries of course) and then squeezing brush loads of watercolour in dollops all over is a bit “hairy”.  Then picking up the paper and waving it about so that the colour swishes around, like tides on a beach, reminds me of relatives of mine “panning” for gold in Victoria in the 1850’s.

Charles had brought many reference pictures which we could use and I chose one of a barn owl.  I thought that I could do washes up to the outline of the owl and also within the owl and later put a continuous line on top which more or less matched the washes.  Here is a copy of the wash I did initially, helped by some tips from Charles along the way.

Copy of Wet on Wet watercolour of Barn Owl, before I attempted the continuous line drawing. Mick Burton.

Copy of Wet on Wet watercolour of Barn Owl, before I attempted the continuous line drawing. Mick Burton.

Later, at home, I worked on the continuous line on top of the above copy.  I started by putting key lines along the outline of the owl, feathering and other features – to match the borders of colours as far as I could.  Then I added more connecting pattern and finally joined everything up and made sure I had a continuous line.

Once I was satisfied with this I traced the continuous line down onto my watercolour painting and drew over the lines in acrylic pen making final changes as I saw how the firm line was developing.

I think that the translucent effect of the feathering has worked well, although this view may not be appreciated by a victim mouse in its last moments.

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.

 

 

 

 

Spherical Continuous Line Abstract with Colour Sequence.

Spherical continuous line with colour sequence.  Flypast Over Rolling Hills. Mick Burton 2015.

Spherical single continuous line drawing with colour sequence. Flypast Over Rolling Hills. Mick Burton, continuous line artist 2015.

I have modified my Spherical approach to continuous line from the method I described in my Continuous Line Blog post of 9 July 2014, which did not quite reflect the reality I was seeking.

I have kept the idea that when you draw out of one SIDE of the paper you need to return at the opposite SIDE at the corresponding point, so that the pattern matches vertically and after colour sequence the colours also match if you pull the paper round into a tube shape.  This is similar to the equator on a globe of the world matching.

Previously I had said that when going out of the top of the drawing you also need to return at the corresponding place at the bottom.  I was correct to say that the colours would not match, which would be equivalent to the poles on the globe of the world not meeting, but the treatment of the lines needed to be modified.

I realised that the bunching effect of the top being pulled together totally separately to the bottom being pulled together was fine regarding separate sets of colours but matching the line patterns from top to bottom was the wrong approach.

So, when I go out at the TOP now I need to come BACK IN AT THE TOP at the corresponding distance from the other end of the top.   Similarly if I go out at the bottom I come back in at the bottom.  You could then imagine that folding the picture vertically down the middle would mean that both pattern and colour sequence would now match at the top and bottom respectively (don’t actual fold it and spoil the picture ! ).

I recently drew the following for a demonstration/workshop at Stainbeck Arts Club in Leeds.  I started drawing the line a couple of inches in from the top left side and did a few rolling curves diagonally down from left to right, followed by several exits and returns to the picture – initially out at the lower right side and back in at the lower left side, then down and out at the bottom left and back in at the bottom right.

Spherical continuous line drawing with rolling and jagged lines.  Mick Burton 2015.

Spherical single continuous line drawing with rolling and jagged lines. Mick Burton, continuous line artist 2015.

I later tried some “shark fin” curves and a couple of large jagged sequences.

All the time I tried to draw the line cleanly through existing shapes (avoiding going near previous junctions) and being aware of areas I had not visited much.  Finally I needed to work out how to get back to my start point without spoiling the composition too much (here going out and back in can be handy).

I hope you can check the route of the line through the whole picture fairly easily.  I then applied my Colour Sequence to produce the picture at the top of this post.

The first stage is my usual alternate overdraw of the line (if you are overdrawing a section as you go out of the picture you need to continue to overdraw as you re-enter, or if not overdrawing going out it’s not overdrawing when you re-enter).  See my post of 10 September 2014 for the full ALTERNATE OVERDRAW process and my post of 27 September 2014 for the COLOUR SEQUENCE process.

I have used a series of 6 colours from Pale Yellow through greens to Prussian Blue which I have tried to work out in steps of tone.  This is partly to highlight the overlap effect of continuous lines and the natural depth of the abstract.  As always, there is choice of direction of colours – light to dark or dark to light.  Here it seemed best to have the single lightest area at the top and several darker areas across the lower part of the picture.  The picture also has an Optical Art look about it.

Printing the picture in Monotone is usually a good way of checking the steps of colour and light to dark.  So here it is.

Monotone of Spherical Continuous Line

Monotone of Spherical Single Continuous Line Drawing “Flypast Over Rolling Hills”. Mick Burton 2015.

I also produced another similar abstract for the Demonstration at Stainbeck Arts Club to show the Spherical approach with a different flow of lines and colours.  I had coloured the drawing with a sequence from Yellow through Reds to dark Brown.

Spherical Continuous Line with Colour Sequence.  Forest Fire.  Mick Burton 2015.

Spherical Single Continuous Line Drawing with Colour Sequence. Forest Fire. Mick Burton 2015.

Here is the Monotone of this picture.

Monotone of Spherical Continuous Line

Monotone of Spherical single continuous line drawing “Forest Fire”. Mick Burton 2015.

Haken’s Gordian Knot and the Twisting, Overlapping, Envelope Elephant.

I constantly look for Continuous Lines in many fields of art, history, mathematics – anywhere, as I just do not know where they are going to crop up.  Currently I am casting an eye on Islamic Art and Celtic art and am developing ideas on those.

Recently I glanced through a book called “Professor Stewart’s Cabinet of Mathematical Curiosities” and came across Haken’s Gordian Knot, a really complicated looking knot which is really an unknot in disguise – a simple circle of string (ends glued together) making a closed line. Here it is.

Haken's Gordian Knot, from Ian Agol.  A simple circle of string (an Unknot) formed into a complicated continuous line.

Haken’s Gordian Knot, from Ian Agol. A simple circle of string (an Unknot) formed into a complicated continuous line.

When I looked at the Knot, it reminded me of my “Twisting, Overlapping, Envelope Elephant” continuous line in that it has a lot of twists. I realised straight away that a narrow loop on the outside (left lower) seemed to lead into the structure with its two strands twisting as it went, each time in a clockwise direction.  I followed the two twisting lines throughout the drawing until they ended in a loop on the outside (left higher).

I wanted to draw and paint this knot. My first drawing was of the line on its own. The depth of some of the lines reminded me of one of my earliest paintings “Leeds Inner Ring Road Starts Here”, which was based upon a sign board which appeared near Miles Bookshop in 1967 informing us of the route the new road would carve through the City. This was several years before Spagetti Junction was built near Birmingham. My picture had lines swirling all over at various heights in one continuous line.

Leeds Inner Ring Road Starts Here. Use of varying thickness of continuous line, overs and unders.  Pre dates Spagetty Junction near Birmingham. Mick Burton, 1967.

Leeds Inner Ring Road Starts Here. Use of varying thickness of single continuous line drawing, overs and unders. Pre dates Spagetty Junction near Birmingham. Mick Burton, 1967.

My first picture of the Gordian Knot, in black and white, concentrated on the heights of the lines following the overs and unders shown by Haken.

Depth of lines in black and white, in Haken's Gordian Knot.  Mick Burton, continuous line drawing.

Depth of lines in black and white, in Haken’s Gordian Knot. Mick Burton, single continuous line drawing.

But my main aim now was to use blue and red to show the twisting nature of the pair of lines running between the starting loop and the end loop.  This was intended to allow the viewer to more easily follow the loop and the twists throughout the structure.

Twisting, overlapping colouring of Haken's Gordian Knot.  Mick Burton continuous line.

Twisting, overlapping colouring of Haken’s Gordian Knot. Mick Burton single continuous line drawing.

Just like viewing my “Twisting, Overlapping, Envelope Elephant”, from my previous post, imagine that you have a strip of plastic which is blue on the front and when you twist it over it is painted red on the back.  Where blues cross each other you have darker blues, and correspondingly with reds.  Where blue crosses red you have violet.  I show the strips feeding through each other, like ghosts through a wall.  There are some darks and lights in there as well.  Most usefully, the background shines through to help make the strips stand out.

You can now get more of a feel for what is going on.  I counted 36 clockwise twists and one anti-clockwise (number 26).  Continued twists in the same direction tie in the ongoing loop, when it feeds through the two strands of its earlier route at least 12 times.  Twist  number 26 probably cancels out the effect of number 25.

This is a preparatory painting, in acrylic but on two sheets of copy paper sellotaped together.  When I exhibit these pictures they will be hung as portrait, rather than the landscape shown here for comparison with Haken (as you will note from where my signature is).  I think they look a bit like the head of the Queen in portrait mode !

Having got this far, I realised that I should find out more about the Haken knot (or unknot), beyond Professor Stewart’s brief introduction.  How did Haken construct the knot and why?

Please see my next post, on this continuous line blog, to see how I got on.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Continuous Line Drawing Alternate Overdraw embedded image.

 

Abstract before Alternate Overdraw  embedded Dog appears.  Mick Burton, Continuous Line.
Abstract before Alternate Overdraw embedded Dog appears. Mick Burton, Single Continuous Line Drawing.

 

 

Here is an abstract Continuous Line Drawing which conceals an image of a dog.  Possibly you can see some clues as to where the outline of the dog is.  The idea is to carry out an Alternate Overdraw along the line throughout the picture which will produce the image of the dog.  Start the overdraw with the arc marked with chevrons.

After I developed Alternate Overdraw in 1970, which enabled me to allocate my colour sequence to continuous line drawings, other possibilities started to occur to me.  The first was that you could hide an image of an animal within what looked like a totally abstract continuous line drawing.

See below how the dog finally appears.  In fact it looks a bit like Ben, who we have been looking after this week.

Alternate Overdraw embedded dog appears.  Mick Burton, Continuous Line Drawing.

Alternate Overdraw embedded dog appears. Mick Burton, Single Continuous Line Drawing.

The start of the process was to sketch a simple dog outline and then , knowing that the outline had to include many crossing lines, I broke the outline down into short lines or arcs which were at an angle to each other.

Next, I needed to run lines through the dog, from various directions, which used these arcs.  For it to work, there had to be an odd number of arcs in each line which went across the dog, between each two outer arcs.  This was so that, after completion, when you drew an alternate overdraw across the dog, both arcs on the outline were overdrawn.  This would result in all the arcs around the outline being overdrawn, thus forming one loop of overdraw. 

Within the body of the dog, several overdraw loops were also formed.  Similarly in the background a number of overdrawn loops and inner loops resulted. 

Of course, trial and error is involved in connecting up all the loose ends (of the lines running through the dog) to achieve a single continuous line through the whole drawing.

Finally, having completed a continuous line, I needed to check that there were no obvious sections which would indicate that an animal was in there. 

I did one more embedded image in 1970 before moving on to other things.  This time, instead of making the abstract continuous line from flowing curves with no straight lines, I decided to use mostly straight lines and right angles.

Abstract before Alternate Overdraw embedded steam engine appears.  Mick Burton, Continuous Line Drawing.

Abstract before Alternate Overdraw embedded steam engine appears. Mick Burton, Single Continuous Line Drawing.

 

 

 

 

 

 

 

 

 

 

 

This time I  was concealing a steam engine and the Alternate Overdraw result is shown below.

Alternate Overdraw embedded steam engine appears.  Mick Burton, Continuous Line Drawing.

Alternate Overdraw embedded steam engine appears. Mick Burton, Single Continuous Line Drawing.

 

 

 

 

As always with my various styles, I wonder who else may be using them  and whether they were in use long ago.