Category Archives: Abstract

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.

Symmetry of Broken Plastic Table Tennis Balls – in Structure, Time and Space.

First broken Plastic table tennis ball.

First of two plastic table tennis balls broken during a match on 25 July 2017.  Manufactured by XuShaofa Sports in China, whose badge is fully shown on the separate piece .  Photo by Mick Burton, continuous line artist.

This match ball is made of plastic and my table tennis club in Leeds has used the XuShaofa ball for over two years since plastic balls were introduced internationally in place of the celluloid ball. This is my favourite type of new ball to play with and they are long lasting and I have rarely seen one break.

Last night I played a Leeds Summer League match at Moor Allerton Sports and Social Centre and the above ball was broken during a game between Kiran Babra of Leeds YMCA A and Liang Wong of Leeds Judean C team.  Kiran has an outstanding vicious forehand topspin stroke which he can play from anywhere, often from wide on his backhand side of the table.

In one point he hit the ball with the top edge of his bat and it flew up and hit the corner of the lighting casing on the ceiling.  When Liang picked up the ball from the floor it was in two pieces, as shown in the photo above.   Unlike the old celluloid balls, which had a seam around the ball, this plastic ball is seamless and so when it breaks a piece usually separates.

I noticed that the piece which came off nicely encompassed the maker’s badge, along with a dent from impact, and so I pocketed it as a natural artistic object for my collection and then produced my last new ball to continue the match.

Two points after the above incident, Kiran again went for his topspin and mishit the ball in exactly the same way and it flew up and hit the corner of the lighting casing again.  This time the ball was picked up and they played another point.  (see later note at end of this post).

Things did not seem to be quite right in this point and Michael Chang said “Can’t you see that the ball is dead”.  We looked at the ball and there was a big hole in it and a loose piece inside.  I announced that this time you could look through the hole and see the reverse of the XuShaofa badge at the opposite side.

Second broken plastic table tennis ball.

Second of two plastic table tennis balls broken during a match on 25 July 2017.   Manufactured by XuShaofa Sports in China, whose badge in reverse can be seen through the hole.  Photo by Mick Burton, continuous line artist.

Another specimen for my collection.   In fact in over two years I have only seen 2 or 3 balls XuShaofa balls break and each time commented that the broken ball was “A collector’s item”.  But these two balls are something else!  Think about the probabilities of the same mishit, followed by the same trajectory and point of impact and with the speed and distance involved resulting in the same type of break.  Think about the spin, all those revolutions per second, and the first point of impact being directly on the badge and then the impact on the second ball directly opposite that of the first.

I love symmetries, but have difficulty in working out what this is.  A piece of ball with the full badge on it and a similar hole in the other ball where you can look through and see the reverse of the badge on the other side.  Is that a reflection, a transformation, anti-symmetrical, or what?

Regarding probabilities, there is probably more chance that I will post pictures of symmetrical and anti-symmetrical prime numbers on this web site than the above happening again.

Just to put the old celluloid balls to bed, XuShaofa themselves welcomed the decision to ban them saying that “celluloid is flammable and has killed countless factory workers in China from fast-spreading fires”.  I have long known of this flammable nature and have often used broken celluloid balls as fire lighters at home.

If you would like to know what was in use before celluloid balls were introduced in 1901, one original choice was the ball shape cut from champagne corks which were hit around on dining tables with cigar box lids.   No surprise that Boris Johnson could refer to “Whiff  Whaff” when welcoming the 2012 Olympics to London.

Subsequently, Battledore bats were manufactured from around 1890.  These had parchment paper stretched around a frame and I am lucky to have one circa 1903, see below.  It has “PING PONG” stamped on it, which was in general use before  J. Jaques & Sons Ltd registered the name as a copyright in 1901.  

IMG_1905 - Battledore circa 1903

Table Tennis battledore, parchment paper stretched around a frame, stamped “PING PONG” circa 1903.  Photo Mick Burton, continuous line artist.

IMG_1905 - Ping Pong

“PING PONG” stamped on Table Tennis battledore, circa 1903.   Photo Mick Burton, continuous line artist.

Pimpled rubber was first used on bats in 1903.  Mr E. C. Goode from London was in his local chemist’s when he saw a pimpled rubber mat on the counter.  He purchased it and stuck it to his plain wood table tennis bat and found it produced fast spin on the ball.  He became Champion of England and others copied his idea.

I have never used my battledore for fear that a modern ball, such as XuShaofa, might burst through the parchment.   I don’t want a ball shaped hole in that !

NOTE ADDED on 15.6.2018.  A couple of days ago I bumped into Kiran, who had broken these two balls, and he had heard that he was on my website and I told him where to find the post.  I re-read it myself and realised that things had moved on since I wrote the post.

Kiran became well known after that for breaking balls in most matches that he played in. His topspin action is extreme in that the bat travels very fast past the ball, hardly touching it, and a slight error can mean the ball hits the top of the bat and smashes.  My original assumption that the ball hitting the light casing after Kiran hit the ball was incorrect.

Other players are also finding this happens to them.  In fact a player did it twice in a match I played in this week, and Kiran said that he had done it five times in practice the night before I spoke to him.  I would rather not say how much a ball costs!

 

“Vortex” by David Kilpatrick. Single Continuous Line and Alternate Overdraw colouring.

Vortex, David Kilpatrick. flat,1000x1000,075,f.u1

“Vortex” by David Kilpatrick, artist from Atherton, Australia.   Single Continuous Line using the Alternate Overdraw method to allocate colours.   March 2017.   Mick Burton blog.

I have been exchanging ideas with David Kilpatrick recently and he has agreed to let me put some of his pictures in my blog.  “Vortex” stands out to me, as I have been a fan of Vorticism for many years.  He has used Alternate Overdraw to allocate colours in sequence and it has worked well.

David’s design gives the impression of a sheet of plastic, coloured green on one side and red on the other, and each twist showing the other side.  With overlaps you get darker greens or darker reds.  Four internal areas let the background shine through.  The whole thing is very natural, including David’s own style of patchy colour radiating outwards.

Next is David’s “Knight’s Tour” which he is still working on.

David Kilpatrick knights tour. image011

“Knights Tour” by David Kilpatrick, artist from Atherton, Australia.   Single Continuous Line based upon moves of a knight and using Alternate Overdraw to allocate colour sequence.   April 2017.  Mick Burton blog.

I did a Single Continuous Line “Knight’s Moves on a Chessboard” in 1973 (see Gallery 1965-74) with the intention of colouring it, but never tackled it properly.  One of the problems was the number of fiddly small areas.  It led to my “Knight’s Tour Fragments” instead (see my previous post on 16.2.2017).

But now we have GIMP!  David said that he used this to move the lines about on his “Knight’s Tour”.  I googled GIMP and it means “GNU Image Manipulation Program”.  Some areas are still fairly small but he has produced a vibrant structure.

David says that these are trial colours (I presume from GIMP) and he intends to work out an improved scale of colours in his own style.

However, the colours shown already demonstrate the natural balance inherent in the Alternate Overdraw colour allocation.  The composition suggests to me an island with yellow “beaches” as well as reds within opposite “volcanic” zones.

There is a choice regarding background, which would naturally be the same colour as the light blue internal areas and result in a surrounding “sea”, or it could be left white as shown above.

I look forward to seeing the final version, which I am sure will be another splendid example of Vorticism.

Another picture that caught my eye was his “The Pram” which is based on a magic rectangle.

“The Pram” by artist David Kilpatrick, from Atherton, Australia.   Based on a Magic Rectangle. 2015.    Mick Burton blog.

This pram picture has lots of line ends in it and makes me want to attempt one myself using a Continuous Line animal.  Such a design would make you want to connect up so many loose ends.  My Spherical pictures already do this to an extent, as I take a line out of the picture at one side and bring it back in at the corresponding opposite side.

I think that David chose the positions of the displaced squares in a sort of random way.  Maybe I would want to be confident that I could move them around, in the way you could on the movable squares game of my childhood, and get back to the actual original picture.

You can see much more of the art of David Kilpatrick on

https://www.redbubble.com/people/fnqkid

Nessie the cockapoo visits Gledhow Valley

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Nessie the cockapoo arrives with a favourite toy. How did she know my favourite colour range. Mick Burton, continuous line artist.

Nessie the cockapoo has come to stay for a week whilst Helen and Janet are in California.  She arrived waving one of her favourite toys, which just happens to have a range of colours similar to those in a recent painting of mine.

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“Knight’s Tour Fragments”, acrylic on canvas. Exhibited at Harrogate and Nidderdale Art Club Exhibition in November 2016. Mick Burton, continuous line artist.

Nessie is two and a half and lives in a village in Worcestershire in a house almost surrounded by common land.  Strangely, there are no cats in the village and no squirrels (although several years ago one appeared in the garden the day we arrived for a visit, and it was suggested that it had been a stowaway in our car).  Hens roam free in the garden – so where are the foxes?  No greater spotted woodpeckers, they are all green.

Nessie’s favourite spot in our house is by the French Windows at the back.  She watches the birds and squirrels endlessly, and it is good to lie down on the job.

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Nessie, the cockapoo, watching birds and squirrels. Why not take it easy?   Mick Burton, continuous line artist.

But Nessie is not used to seeing cats, and we have plenty of those.  Suddenly we hear barking and scraping at the window.

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Nessie spots a cat and all hell breaks loose.  Mick Burton, continuous line artist.

Hopefully one of the foxes will turn up whilst Nessie is here.  We often see one or more during the day, and we even had one on the garage roof marking its territory.  Here is a photo of one in the garden in late January 2017.

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A Gledhow Valley fox in the garden in January 2017.  Mick Burton, continuous line artist.

Nessie eats sensibly and feels that there my be more nurishment in the cardboard box than in the breakfast cereals themselves.

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Nessie tucking in to a cardboard box which had contained breakfast cereals.  Mick Burton, continuous line artist.

Of course the highlight of each day for Nessie is the walk through Gledhow Valley Woods to the lake.

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Nessie, the cockapoo, can’t wait to go to Gledhow Valley Woods, and the lake, with Joan and me.  Mick Burton, continuous line artist.

We are used to seeing the odd rat scamper across the path by the lake, as well as seeing how well they swim.  One rat dashing across suddenly realised that Nessie was passing and took off, missing Nessie’s nose by a whisker.  I am not good at taking photos of flying rats, so here is one nearby wondering what is going on.

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A rat peeping from behind a tree on the banks of Gledhow Valley Lake.  Mick Burton, continuous line artist.

Twenty ducks who were sitting on the bank and the path fly off when they see Nessie, and Joan has brought some oats to feed to the Swan.  There is only one swan left at the lake just now and it is still in its first year.

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Young swan, now living alone on Gledhow Valley Lake.  Mick Burton, continuous line artist.

We have been concerned for some months about the swans, particularly since the water level dropped after a digger cleared rubbish from the dam end. Large areas of silt have been on view where the swans nest.  Here are the adults and one youngster in late January 2017.

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Family of swans on Gledhow Valley Lake. Photo taken in late January 2017 before the adults abandoned the lake.  I hope the swans did not have to pull bread slices from this wrapper themselves.   Mick Burton, continuous line artist.

At the time of this photo, showing the two adults and the above young swan in late January 2017, the second youngster had been ostracized and was sitting in a corner of the lake.  When we were litter picking this Sunday on the monthly action day with Friends of Gledhow Valley Woods, they told us that soon after the photo foxes killed this young bird and then the adults left the lake.  One adult was found wandering in the Harehills area and the RSPCA took it to Roundhay Park lake.  A lady told us that the other adult was walking past her house in Oakwood, presumably heading for Roundhay Park lake too.  So we hope that things work out well for the adults at Roundhay and our young  survivor here in Gledhow.

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Nessie spots a cat she has not seen before. Henry adopts defensive mode.  Mick Burton, continuous line artist, Leeds.

On the way home from the lake, Nessie confronted a cat.  This is Henry and he stood sideways and seemed to double in size.  Nessie was on her lead, which was probable just as well for Nessie.

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Henry the marmalade cat from Gledhow Valley.  Dogs beware.   Mick Burton, continuous line artist.

Anna and Emma, the children next door, went to the woods with Nessie today and had been looking forward to it for days.  Nessie gets on well with everyone.

She has enjoyed her holiday in Gledhow Valley and we are taking her back to the land of green woodpeckers.

Four Colour Theorem continuous line overdraw.

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

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

My recent post about the formation design used by the record breaking skydivers included a continuous line overdraw of their design (modified slightly be me to complete links which would have been present with more skydivers).  I said that I would explain how the overdraw (above) was completed.

The structure is made up of circles which have 3 way junctions throughout (3 handed in the case of skydivers ! ).  This can be regarded a map and so I will apply my Four Colour Theorem continuous line overdraw which I devised in the early 1970’s.

I was trying to prove the Four Colour Theorem, which states that no more than four colours are required to colour all the regions of a map.  My basic idea was that drawing a single continuous overdraw throughout a map would split it into two chains of alternate regions, which would demonstrate that only 4 colours were required.  If more than one continuous overdraw resulted then there were still only two types of chains of alternate regions.

As you will probably know, this theorem has many complexities which I will not attempt to cover here.  In the mid 1970’s I corresponded with two mathematicians at the Open University about my approach, Robin Wilson and Fred Holroyd, who were both very helpful and encouraging.  The theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken, running one of the biggest computers for over 1000 hours.  I soon decided that it was time to go onto other things!  However, my journey had been fascinating with numerous amazing findings which have been so useful in my art.

I can keep to relatively simple methods for my pictures.

 Here is the design, used above, with my initial overdraws shown in red.

Assumed formation design used by Skydivers, with initial overdraws. Mick Burton four colour overdraw.

Assumed formation design used by Skydivers, with initial overdraws. Mick Burton four colour overdraw.

On final completion of the overdraws, every junction should have two of its three legs overdrawn and so the start decision (1) above overdraws two legs and this means that the third leg, which I call a “spar”, links to another junction where the other two legs must be overdrawn.

We then carry on making decisions which trigger other overdrawn lines across spars.  Usually there is a “knock on” effect where new overdraws connect with already overdrawn lines which then trigger more overdraws.

If we go wrong and a junction is triggered which has all three legs overdrawn, or none, we have to go back and change earlier decisions in a controlled process.  I usually photocopy the overdraws completed, every two or three stages, so that going back is not too time consuming.

Here is the situation after decision (3).  Decision (2) in blue had only triggered two overdraw sections but decision (3), in green, has triggered ten sections to be overdrawn in green.

Four Colour Overdraw decision 3 triggers 10 further overdraws, in green. Mick Burton, continuous line artist.

Four Colour Overdraw decision 3 triggers 10 further overdraws, in green. Mick Burton, continuous line artist.

Here is the completed overdraw.  It can be seen that some decisions still only trigger one or two overdraws, but decisions 5 and 7 triggered 13 and 12 overdraws respectively.  There are 80 junctions in the design and it took 11 decisions to complete the overdraws.

completed Four Colour Theorem overdraw, on design based upon Skydivers formation design. Mick Burton, continuous line artist.

completed Four Colour Theorem overdraw, on design based upon Skydivers formation design. Mick Burton, continuous line artist.

The completed overdraw has several continuous overdraws.  I tried other variations but had to accept that this design cannot be overdrawn with a Single Continuous overdraw.  This is due to the design having basically only two full rings of circles, which means that some tips of petals cannot be included in a continuous overdraw.

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

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

This situation can be overcome by adding links between the tips of the petals to produce that extra ring of areas.  Here is the expanded design and the stages of overdraw.  I managed to complete the Single Continuous overdraw in one sequence without having to go back to change any decisions.

Increased size design with successful Single Line Overdraw using Four Colour Theorem method. Overdraw decisions shown. Mick Burton.

Increased size design with successful Single Line Overdraw using Four Colour Theorem method. Overdraw decisions shown. Mick Burton.

Of course it looks better with one solid colour overdraw and no decision numbers.

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

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

I have said that the method of overdraw was developed with Four Colours in mind, and so you could use one pair of colours alternately within the above overdraw and another pair of colours on the outside of the overdraw (which can include the background).

I have found another interesting result in that if you use strong colours inside the overdraw, as it is the main image, and neutral colours outside (or even leave the outside blank) then the gaps between the “petals” show good use of space.  Here is the design simply coloured in strong red inside the overdraw, which creates a good contrast as the background seeps in. 

Solid colour within single continuous overdraw, with Four Colour method, showing good use of space. Mick Burton.

Solid colour within single continuous overdraw, with Four Colour method, showing good use of space. Mick Burton.

The chains of areas produced by the continuous overdraws can be coloured, not just in two pairs of colours to demonstrate Four Colours, but with a colour sequence or a mixture of sequence, alternate colours or even one colour.  In the last picture I have used colour sequence on main chains of areas related to the central space and, as a contrast,  light grey on the chains connected to the outside of the design.

Star Burst. Four Colour Theorem applied to a map of shell shapes wound round from the centre. Rainbow sequence of colours. Mick Burton, 1971

Star Burst. Four Colour Theorem applied to a map of shell shapes wound round from the centre. Rainbow sequence of colours. Mick Burton, 1971

This is one of the first paintings that I produced after discovering my Four Colour Theorem overdraw in 1971. I called the picture “Star Burst”, one of my first planetary pictures.

 

 

 

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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