Tag Archives: Continuous Line

Change a four sided continuous structure into a single surface Mobius torus, or Mobioid.

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Drawing of Doug Eglen continuous metal sculpture with sides in four colours. Mick Burton, continuous line artist.

In my last post, on 3 June 2025, I said that I would like to draw Doug Eglen’s 3/8″ square forged metal sculpture. 

Continuous Lines in forged metal, by Doug Eglen.

Here it is, but I have used separate colours for each of the four sides.  You can only see Red, Green and Blue because the metal remains flat overall and you cannot see the Black I allocated for the underside.

I have tried to reflect the 3D quality of the sculpture, with its Celtic over and under style, highlights and shadow.

When I first saw the structure, I wanted to understand what would happen if I applied twists to the sculpture.  I thought that there should be one or more types of twist which would turn the culture from being four sided with four surfaces into four sided with one surface.  I knew that the Mobius Strip has two sides before the half twist and one continuous surface afterwards and thought that the single surface should also be possible for Doug’s structure.

Another thing that I wanted do was to try and clarify my idea that his structure reflected the shapes of two Boomerangs.  I wondered what twists I would need to do this.

The poker which I made at the age of 12 at school appeared to have a half twist but it is difficult to visualise how this changed the position of the sides.

Brass handle with half twist on my poker, made at school when I was 12. Mick Burton, continuous line artist.

I had to use a pencil to draw along one surface to confirm that it changed the top side into the bottom side.  This meant that I could use two half twists near the centre of a new drawing of the sculpture to produce two boomerangs which had individual colours.

Using two twists to produce Black and Red Boomerangs on drawing of Doug Eglen’s metal sculpture. Mick Burton, continuous line artist.

Black has now appeared at the top for half the time, so we have four colours on view.  Another outcome is that the two sides, which are Green and Blue, swapped from one side to the other at the twists which has resulted in blue almost disappearing.  This is due to the close photographic angle by Doug showing most of the insides of each boomerang shape and I coloured them almost equally Green and Blue in my first drawing.  Green stayed the same within the Red boomerang here and Blue changed to Green in the new Black one.

I have mentioned the Mobius effect.  The Boomerang drawing has not produced a change in the number of surfaces but just altered their positions. Sides still equal surfaces.  This is because I have used two half twists.  If I had only used one half twist then the number of surfaces overall would become two surfaces, one colour for top and bottom and another for both sides.

It seemed to me that employing one quarter twist would produce a Mobius single surface throughout, as each time a surface comes round again it deflects a quarter and on the fourth approach it is back on top. Maybe a three quarter twist would also achieve this, but I found this more difficult to visualise.  First approach deflects to side 4, Second to side 3, Third to side 2 and fourth to side 1 again.

Anyway, I have drawn simply a single quarter twist as an addition to Doug’s sculpture.

Drawing of Doug Eglen continuous metal sculpture, but with added quarter twist. Mick Burton, continuous line artist.

I have started with black and the result is All Black.  This drawing depicts a 4 sided metal continuous sculpture which has only one surface.  It is a complete Mobius result.  Visually, you can see that it is a quarter twist and our knowledge of the general sculpture helps us to realise that there is in fact one surface.

Yes, a Boomerang drawing and an “All Black” drawing suggest that I have Australian and New Zealand interests.  My mother, maiden name Brenda Mace, was born in a pub in Bedale, North Yorkshire.  Nearly 100 years earlier six brothers were born in the same pub and four of them went to gold fields in both Australia and then New Zealand in the early 1860’s.  They were cricketers as well and Christopher Mace played for Victoria against the first team to visit from England and two years later John and Harry joined with him, in Otago New Zealand, against the first English team to play there.

Back to the drawings.  I looked on the Internet for images of general structures which included a quarter twist and failed to find any.  There were examples of the pure circular twist, including 4 sided, but these are difficult to visualise as there is no obvious start point. 

Mobius Strip Structure of Rectangle Geometric Shape. Issuu website.

This pure circular twist is more understandable than most and the Issuu website shows how they built one which you can walk round in.  I am not sure if this includes walking upside down at one stage!

https://issuu.com/vsvu/docs/prof_is_1000/s/16586493

I feel that a twist anchored into an actual general structure, which has some straight and flat sections, is important.  It is easier to identify the type of twist and how the effect of the twist radiates through the whole structure in a more meaningful way.

Some other reference sites mention continuous surfaces, with more than the single edge and surface of a Mobius Strip, as Morbioids.  They compare their structure with the Torus (when it has parallel lines drawn around it which can be regarded as equivalent to edges which can produce the Mobius effect).

There are specific explanations of degree of twist, the number of sides, leading to the number of surfaces.  These confirm my assumptions about a quarter (90 degree) twist, or three quarter (270 degree) twist resulting in a single surface for the square structure.  A half (180 degree) twist has two surfaces.  Others explain 5 or 6 sides and there is a formula for n surfaces according to twist and sides.

A useful link that works is headed “Name for a 3 sided Mobius Strip?”  https://reddit.com/r/topology/comments/1bfdu7m/

SamwiseGanges    said that he was going to call them Mobius prisms.  When he referred to square Mobius prisms, he confirmed my assumptions about the effect of their twists.

AceThe Aro   said that Dr. Cye Waldman called them Mobioid’s in 2017 and you can click onto his moving images.  You can also click on Ace’s own slideshow which runs through all the different twist and number of sides possibilities.

I would like to refer to my drawing of Doug Eglen’s structure, with a quarter twist, as an “Anchored mobius torus”, or “Anchored Mobioid”.

Doug may consider loosening the join on his double boomerang ironwork and resetting it with a quarter (90 degree) twist.  That would produce a real single surface anchored Mobioid.  

 

 

Continuous Lines in forged metal, by Doug Eglen.

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Continuous Line knots in tempered metal forged by Doug Eglen. Photo by Doug sent to Mick Burton, April 2025.

Doug Eglen contacted me in March 2025 to say that he had recently started having an interest in knots and unknots and came across my painting of Haken’s Gordian knot, which you can see in my earlier posts of 9 May 2018 and 5 June 2015

Continuous Line Artist view of Haken’s Gordian Knot.

How do you construct Haken’s Gordian Knot?

He asked for my permission to paint his own version of the Gordian knot, based upon my painting.  He intended to exhibit his painting, along with the new metal knots which he was producing in his forge.  Doug has an exhibit case at Purdue University, Indiana, in the Math department library.  I agreed, and answered his questions about assumptions he had made about the construction of my painting. 

Doug later sent me a copy of his painting.  This is impressive, as are his metal knots, and you can see them on his website   

https://http://www.deglen.org/math-art

I particularly liked Doug’s photo of his 3/8″ square metal, in what I would call a sort of “double boomerang” shape without any twists.

Doug Eglen square 3/8″ metal without twists in a continuous line, copied to Mick Burton.

The alternate over and under style, like Celtic knotwork, produces great highlight graduations as well as suggestions of colour that Doug achieves with his firing treatments. I like the dark shade of the sides of the metal compared with the shadow of the piece.  I would like to draw this.

At school, when I was 12, I produced a metal poker with brass handle which has a half twist.  I have had no further experience of metal work.

Brass handle with half twist on my poker, made at school when I was 12. Mick Burton, continuous line artist.

I did do some wire bending to produce a single continuous line wire horse in 1967.  At first I placed it underneath a piece of clear glass covering the dining table, to hold it down.  Then I secured it to a wooden board through a centre page magazine photo of a horse grazing in a meadow.

Continuous line Horse in a length of garden wire done in 1967. Mick Burton.

Next, I went sculptural and used cardboard strips to take the Horse a stage further in 1970 when I lived in Nottingham.

Continuous line Horse using strips of cardboard 1970. Mick Burton.

I showed it to someone at work who’s Dad owned a company which produced steel castings. He said he would ask his Dad about the possibilities of doing a metal horse.  The answer was a “Yes”, but it would cost me £3,000 !  I now know someone who’s cousin does 3D printing, so maybe I can get a more reasonable quote now.

Of course, this is all “small beer” compared to Doug Eglen’s metal workings.

Another Artist in the Burton Family

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Fathers’ Day card, from my daughter Kate, painted by my grandson Lucas.      Mick Burton, continuous line artist.

This action painting, by my grandson Lucas, reminded me of a fast rushing stream.

My daughter Kate is a professional artist and film maker (see my post of 2 February 2016 – “BB” by Kate Burton, Glasgow film maker, at London Short Film Festival),

“BB” by Kate Burton, Glasgow filmmaker, at London Short Film Festival.

Three year old Lucas receives plenty of encouragement to make pictures.

Lucas also squeezes paint tubes directly onto a piece of card to create mixes of paint, before Kate provides another piece of card to squash the paint.  There is a sort of ink blot “butterfly” copy effect.

Lucas squeezes paint directly from the tube.        Mick Burton, continuous line artist.

Here are some results, along with “Grandad” type observations.

Lucas watery landscape.         Mick Burton, continuous line artist.

This result made me think of a sort of Chinese or Japanese landscape painting.  Here is a modern version for comparison.

Autumn Mountains at Dawn, by Zhang Daqian.      En.wahooart.com.

The next picture has the appearance of a flying elephant, and I have done a few continuous line elephants in my time.  

Lucas’s flying elephant.    Mick Burton, continuous line artist.

Lucas’s flying elephant.  Naturally, he has a squash copy flying the other way.  See below Dumbo, the most famous flying elephant.

Dumbo the Elephant, from Tim Burton’s remake of Dumbo Takes Flight.

Another Lucas watery image reminded me of a Monet.

Lucas – garden pond squash painting.    Mick Burton, continuous line artist.

Here is one of Monet’s many paintings of water lilies.

Water Lilies Giverny – by Claude Monet.

One day I may tell you about a picture Kate painted when she was four.

Pateley Bridge Art Club Continuous Line Drawings

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Pateley Bridge Art Club member’s single continuous line Rhino with internal pattern.

I was at Pateley Bridge Art Club earlier this month with a Demonstration and Workshop for the members.  They have covered the evening in their News Blog with a report by Charles Mellor, which included 16 examples of members’ pictures.  You can find the report at >      https://www.pateleybridgeartclub.org      .

I include here an additional three pictures by members which illustrate how you can devise an internal pattern which represents aspects of the subject.  Members could base their initial attempt at a continuous line, or lines, on a subject of their choosing or one or more of several faint outlines provided by myself.

I demonstrated how the basic ongoing line can involve curves, loops, angles, etc. taking in parts of the outline itself and involving where possible a novel pattern for that subject.  At the same time there is a need to leave spaces for the line to return to the start.  I said that members were free to use rubbers to correct their drawings or have more than one continuous line in their picture to get used to the process.

One member drew a Giraffe which incorporated some of the distinctive inner pattern of that animal.

Giraffe by Pateley Bridge Art Club member, showing inner pattern.

My third example is a Zebra where the member has included the famous stripes.

Head of a Zebra, by Pateley Bridge Art Club member.

The Rhinoceros above has a good inner pattern reflecting the armour plating appearance of the animal.  It reminds me of the woodcut by Albrecht Durer, who was amongst the first artists to portray this animal.

Rhinoceros woodcut by Albrecht Durer in 1515.

And that’s not all.  The Pateley Bridge member’s Rhino at the top of this post has a clear single continuous line and includes two specific features I have used in the past which appear in pictures which I displayed on the evening but may, or may not, have have influenced the drawing of this Rhino –

a.  The Rhino’s left eye is depicted by a gap in the outline, similar to my Panther’s right eye below, where I show only the pupil of the right eye.   

Association of Animal Artists Annual Competition Winner !

b.  The Rhino’s armour pattern involves the outside running into the interior of the animal, similar to my Zebra below.

 

 

Continuous line Zebra without crossover lines.

Continuous Line Zebra

I have requested that the Pateley Bridge member, who did the Rhino, copies the line and then applies alternate shading to highlight the “inside outside” effect.  If not I may have to do it myself.

Picasso was certainly aware of this “inside outside” situation.  At my Demonstrations I pass around Picasso’s book “One-liners” which has 50 pictures.  Most have the line starting at one end and then stopping on completion elsewhere, but there are several which are single continuous lines.  The last two pen drawings in the book are as follows.

Picasso “Studies for Mercury”. Picasso’s One Liners. Artisan, 1997.

On the left, the head, arms, body and legs are all inside the single continuous line.  At first glance it is the same on the right, but in fact there is no “inside”.

Red Alert, Continuous Line Detected on Train Tracks Puzzle.

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I started doing Train Tracks puzzles in the Daily Mail a few months ago and then moved onto Medium puzzles (dimensions up to 10 x10) on puzzlemadness.co.uk and a month ago tried Large difficulty (dimensions up to 12 x 12).

You start off with a grid which states the number of cells which occur vertically or horizontally and they give you some bits of track initially, including start and end track at the edge.

Train tracks from puzzlemadness.co.uk Large difficulty 12.12.2020. Mick Burton, continuous line artist.

There are many attractive elements to this sort of puzzle, including the possibility of solving them totally without trial and error.  The first thing to do is to add initial offshoots for all these start tracks.  Next look for any rows which already meet the number of cells containing track, such as the right hand vertical which has the required two including the offshoot.  This allows you to allocate spaces to the remaining 10 cells. 

Being an artist, I know the value of space in a picture and it is particularly important here.  Then you have to consider the various types of track and on you go.  Constantly checking and rechecking is the key as you add pieces.  Bear in mind that the aim is to end up with one route from start to finish, avoiding dead ends, and use that to your advantage.  Finding dead ends is also useful as you can allocate spaces.

It is best to start off with smaller easier Train Tracks puzzles to get used to the process.

I attempt my puzzles on paper where I draw the grid and enter the numbers and given track pieces.  My fingers are too wide and clumsy to do much prodding on my mobile phone and if I complete the puzzle I then tap in the answer.  Here is my initial drawing of the above puzzle.

Initial attempt at the rail track puzzle (large difficulty) of 12.12.2020. Mick Burton, continuous line artist.

When I loaded this on my phone, I expected that as I tapped in the last piece the completed puzzle would disappear to be replaced by congratulations across the screen, for completing a route from start to finish.  Instead I saw the Red Alert.  It is not normally an offence to produce a continuous line in this blog.

I am good at mending this sort of thing of course and here is the final result – there is a X (space indicator) so that you see the complete shape before the last piece goes in causing the whole thing to disappear.

Correct completion of Train Tracks puzzle, with just the last bit to go in. Mick Burton, continuous line artist.

I am interested in various stand alone structures which have an environmental feel to them, where all the different elements can produce a surprising result.  

As it has been Lockdown etc,  I have completed 94 in about 10 weeks scoring 17,925 points, which put me at position 272 out of 863 listed.  Top is Stirlingkincaid with 2,766,965 !

On the monthly list I am 91st with 7,650 points.  Stirlingkincaid has 228,640 – does this person ever sleep?

Personally, I will probably move on now, looking for more structures which I can unlock with my continuous line knowledge.  Also, I need to finish my current work about Drawing Prime Numbers.

Tawny Owl window impression in Gledhow Valley.

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Window impression of Tawny Owl.  Front garden in Gledhow Valley.  Mick Burton, Leeds continuous line artist.

Yesterday morning Joan said that there was a mucky mark on the lounge window and that the Window Cleaner had only just been the other day for the first time since the start of Lockdown.  Maybe a bird had done it. 

The impression was pretty detailed and we thought about pigeon size. We looked outside, as a blackbird had smashed into the window a few years ago and did not survive.  No sign this time so we hope that this bird is ok.

Here is a closer version of the impression, which I have darkened a bit so the grey impression, lit by the sun, shows up more against our evergreen hedge.

Full window impression of Tawny Owl, with good body, head and wing detail.  Mick Burton, Leeds continuous line artist.

I know that we have Tawny Owls in Gledhow Valley, Leeds, but I have only seen one sitting in a tree at dusk and of course heard them.  I looked up Tawny Owl in our RSPB Handbook of British Birds, by Peter Holden and Tim Cleeves.  “37-39cm.  As large as a Woodpigeon.  Has a tubby body, large round head and rounded wings.  Its face is surrounded by a ring of dark feathers….”  I took a closer photo of the body shape to look at this sort of detail.

Close up of the body of the Tawny Owl window impression, showing feathers around the head, beak and tubby body.  Mick Burton, Leeds continuous line artist.

You can see the faint ring of feathers around the head and where the beak has hit the window.  Also the tubby body and chest.  I suppose the impression is made by grease and dust off the feathers.

The impression of the body did not seem to be full size, so this may have been a youngster which first flew at the end of April.

I had been sorting out some of my framed paintings and had left one out on the guest room bed.  It was my single continuous line Barn Owl, which has a virtually identical composition to the window impression, with the sun shining through its wings.  See also my Post of November 2015 about this owl.

Continuous line drawing of Barn Owl onto Wet on Wet watercolour, 2015. Mick Burton.

The Sunbathing Foxes are back in Gledhow Valley.

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IMG_5567

Two foxes sunbathing in my garden in Gledhow Valley on 15 February 2019.  Photo by Mick Burton, continuous line artist, Leeds, Yorkshire.

This burst of warm spring weather is bringing out the foxes just about every day. These two spent about 30 minutes wandering around, sitting and just relaxing.

Not too different to the painting which I did of a couple doing similar things a couple of years ago.

IMG_4534 Best - Copy

My single continuous line painting of two foxes sunbathing a couple of years ago.   Mick Burton, continuous line artist, Leeds, Yorkshire.

The first time I saw foxes in Gledhow Valley was in the late 1970’s, when I lived near the middle of the valley and brought the dog to this end of the woods, above the Well House, and spotted a pair of cubs playing.

To see more animals go to the Friends of Gledhow Valley Woods website    http://www.fgvw.co.uk      .

Updated WordPress Galleries for continuous line artist Mick Burton

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071. 1971-10. Iguana. Colour sequence. Blue, green, yellow, red.

Iguana single continuous line drawing with colour sequence. Mick Burton, Leeds, Yorkshire, 1971.

At last I have got round to updating my galleries of both “New Work since 2012” and “Gallery 1965-74”.   Here one of the additions to my older ones – Iguana, which I did show in an early post in July 2014 but did not include in the Gallery.

I now have a fantastic gallery with a tiled format where every picture has been sized to fit and those in short related sequences seem to be logically placed as well.  Exactly what I needed and way beyond what I expected.

Do have a look.  You can click on any picture to see it in detail and then click on arrows to see the whole gallery in turn.

 

Hot Cross Bunny and the psychology of colour

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IMG_3425 Hot Cross Bunny

“Hot Cross Bunny”, single continuous line drawing painted in psychological colours. Mick Burton, continuous line artist.

In my posts I have said a lot about colour sequence and, along the way, talked about selecting appropriate ranges of colours for my drawings.  Here are some more colour comments, leading to the one about the bunny above.

I might consider that a yellow, red and brown range would be good for my horse. These have a similarity to its actual colours and give a warm and friendly feel which reflect the horse’s nature and temperament.

Fig 1. Copy of IMG_5869 Horse complete, furst sequ

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

A strong harsh colour seemed to be best for my roaring lion and simple black and white achieved this.  In the mid 1960’s when I drew the lion, Bridget Riley had been doing many black and white hard edge pictures, and I did several of my animals in this colouring.  I feel that this worked best for the lion amongst my drawings.

016. 1967-9. Lion. Alternate shading, black.

Lion, single continuous line drawing with alternate shading in black and white. Mick Burton, continuous line Artist.

With my “Flame on the Sun” painting, the sort of anti magnetism represented by complementary red and green hopefully reflect the explosive violence required.

Flame on the Sun. Spherical continuous line. Mick Burton, 1972

Flame on the Sun. Spherical single continuous line drawing, with complementary reds and greens expressing explosive violence.  Mick Burton, continuous line artist.

For a more subtle result – my still life of a radish, apple, mushroom and flower heads – I used water colours to help to show the floppy translucent nature of the radish leaves.

IMG_20180510_Raddish

Radish, apple, mushroom and flower heads still life. Water colour used to show floppy, translucent nature of radish leaves. Mick Burton, continuous line artist.

Sometimes I find that I can use almost actual colours.  Here is a commission drawing, with the continuous line running through both robins and the branch.  I  was asked to do only a hint of pink on the Robins’ chests.  This is fine.  However, I had to have a go at a full colour result for myself.  The perky nature of robins is reflected pretty well, I think, by these “near” natural colours.

IMG_3417 (1) Best. Pair of Robins.

Pair of Robins, single continuous line drawing. Full near natural colour. Mick Burton, continuous line artist.

My yellow, green and blue sequence of colours fits well for “Nibbles”, a friendly rabbit who likes nothing more than eating her greens.

IMG_3498 Nibbles

“Nibbles”, single continuous line drawing.  The rabbit has a suitable range of colours to reflect contentment just eating her greens. Mick Burton, continuous line artist.

However, for a rabbit drawn with exactly the same single continuous line as for Nibbles, but who has a completely different temperament  –  RED, BLACK and WHITE fits the bill.

This is, of course, “Hot Cross Bunny” who lurks at the top of this post.  A real, full on, “Psycho”.

The two Rabbit paintings and the Pair of Robins accompanied several other of my pictures at the Harrogate and Nidderdale Art Club exhibition a week ago at Ripley Town Hall.

At the Preview Evening various prizes are awarded.  One was the annual prize presented at the Spring Exhibition by Sir Thomas Ingleby, the club’s patron, for his own personal choice for the best picture on show.  This was won by Julie Buckley for her “Black Labrador”.  

Sir Thomas also mentioned other pictures which caught his eye.  He said that he liked all the paintings by Mick Burton, but never thought that he would ever consider buying one called “Hot Cross Bunny”.

Here is a bit of background to the Rabbit paintings.  Nibbles and Hot Cross Bunny are based upon my daughter Kate’s rabbits, Harriet and Clover.

Harriet was friendly and cuddly and Clover might have been better named “Cleaver”.  We kept them both in the garage – in separate cages.

When we bought Clover, a lop eared rabbit, the breeder was saying how friendly and harmless the baby rabbit was.  I asked if it was related to an adult lop eared which had just tried to bite my finger off and the answer was “Yes, it’s the granny”.  We still bought Clover!

She was alright at first but later became very aggressive.  Every time we opened her cage for any reason, she would bite viciously.  We also realised that some one else would have to take care of the rabbits when we were on holiday.

Strangely, I found that if I put a hand on Clover’s head as soon as I opened the door she would stay still and relaxed as long as I kept the hand there.  With the other hand I could top up food and water or clean out the cage.  This worked for all of us.  Fortunately, our neighbour was delighted to be able to do this too and things were fine when we were away.

After Clover died and I had buried her in the garden, Kate prepared a wooden plaque and nailed it to the fence “Here lies Clover Burton the rabbit”.

An interesting consequence of keeping the rabbits was that straw from the bale became piled on the floor of the garage.  One day the straw was seen to be moving and we feared that we had rats and so I was deputed to check it out.  I found a nest of baby hedgehogs.