# Four Colour Theorem continuous line overdraw.

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.

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.

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.

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

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.

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.

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.

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

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.

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

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.

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

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.