Tag Archives: single continuous line

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.

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 Lines in Escher Islamic Mosaic painting, STAGE 1.

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.

I look for continuous lines in all forms of art.  I first saw this design in my daughter Kate’s book “Escher, The Complete Graphic Work”, by J.L. Locher.   We are both long term admirers of this artist.  Escher did this detailed painting  in 1922 when in Granada at the Alhambra, and its quality really hit me.  It was of an Islamic mural Mosaic tile,  which was made up of those geometric lines which are often seen in Islamic art, and I assessed it for continuous lines.  

I could see that the overall symmetrical  pattern and I saw that Escher had painted the design BORDER, which seemed to indicate what happened to the lines after they hit the sides of the square.  I then worked out, from the Border Pattern, that the lines were fed back in the same routes on all four sides of the square.  From the point of view of finding a single continuous line, in my experience, such overall symmetry of the structure meant that it was very unlikely that there was only one line. 

Here is the basic structure which I arrived at, which shows the “wiring” connections indicated by the border.  Let’s see how many continuous lines there are.

Escher Islamic Tile.  Basic line structure, with border connections. Mick Burton continuous line study.

Escher Islamic Tile. Basic line structure, with border connections. Mick Burton continuous line study.

When I traced over the lines I found that there were in fact two continuous lines making up the whole design.  Here are the two results, a Main continuous line (in red) and a Minor one (blue).

Main continuous line, one of two.  Escher Islamic tile design.  Mick Burton continuous line study.

Main continuous line, one of two. Escher Islamic tile design. Mick Burton continuous line study.

Minor continuous line, 2nd of two.  Escher Islamic tile design.  Mick Burton continuous line study.
Minor continuous line, 2nd of two. Escher Islamic tile design. Mick Burton continuous line study.

 

By experimenting with border changes, a bit like swapping wiring connections, I did come up with a single continuous line, but the borders were no longer symmetrical.  It seems likely that the artist realised that two continuous lines was the best he could hope for whilst retaining overall symmetry.   In a LATER POST I will show how a border can be “tweaked” by a slight alteration to make one continuous line in the mural mosaic, and how this answer is achieved.  I will also show how the artist is likely to have worked out how to achieve two continuous lines by connecting up the correct loose ends.

I now needed to know  “How important continuous lines were, within this design, to the artist?”   It could be that Continuous Lines were incidental to other aims, or they may have been of prime importance.

In my NEXT POST I will apply my Alternate Overdraw technique to produce a Template of closed lines, which I use to decide upon the colours to allocate.   I will also suggest what the artist’s ideas were for the design and his colour selection.  In a FURTHER POST you will see how my colour allocation compares with the original colours and to what extent I feel that my ideas were the same or similar to those used by the artist.

All this has been done without any reference to the construction of the original line structure.  I have taken the completed structure as a starting point to apply my ideas.  I did not research in any detail on Islamic line construction, until after my whole study was completed.

I have recently found YouTube demonstrations by Eric Broug entitled “How to Draw a Mamluk Quran Page” and “How Grids and Patterns Work Together”, which gave me a good insight into pattern construction and include an explanation of a larger tile containing this Escher Mosaic design as a section.  This is a fascinating process used by the Islamic artists over 500 years ago.  Otherwise, I have not found any reference to borders, colouring, or specific meaning of this design.

Possibly my ideas will generate a new view on aspects of the creation of this and other Islamic designs. 

Mick Burton, Continuous Line Blog. Continue reading