Knots, Klein Bottles, Hopf Links, Moebius Bands and Sketches.
My topologist father, John G. Hocking, looking very professorial watching over me from my pinboard.
The cool thing about topology is that a circle is a the same as a square, a ball is no different than a cube, a doughnut the same as a coffee mug. It is often referred to as rubberised geometry. And the really interesting thing is that it uses other dimensions where very odd things take place, such as knotting spheres and turning balls inside out without surgery. For the artist there is the enticing contradiction of freedom with a bevy of strange rules. For me there is the wonderful mystery my father instilled in me as a child. I was intrigued with the Moebius band and the Klein bottle when I was in primary school. His delighted explanations reserved a space deep in my imagination for the work I do now.
Seeking
These are Moebius bands seeking to find a partner of the opposite twist and to leave our world where, in 4-space, (the mathematical 4th dimension) they can join fully and become Klein bottles.
The Line-up
You can buy a glass facsimile of a Klein bottle, but you can’t make a real one. It is a two-dimensional surface that lives in 4-space. It looks like the neck intersects the bottle, but in 4-space it bypasses it.
Wannabe
This sculpture is a lone Moebius Band stretching and extending itself. It yearns to be a Klein Bottle.
Still Connected
This is one of several studies exploring the Moebius and the Klein
Encounter with a Fundamental Polygon
In this imaginary piece a swarm of Moebius bands are attempting to pass through into 4-space. They encounter the Klein bottle’s fundamental polygon and are cut into bits which then drift towards the unknown. Below is a quick study on a similar theme.
Knots 7_3, 8_4 and 9_24 as ribbons emerging from their matrix.
On the right is a finished plaque. On the left is its silicone mold, (mould in the UK) looking rather like an evil mask!
The wall plaque in clay, not yet finished.
Blue Trefoil
I suspended a paper strip with three crossings, a trefoil knot, against a blue background. I wanted the photos taken as it was spinning to seem as if the knot was becoming a knotted sphere. Knot theorist knot spheres. They say that all knots in 4-space are really knotted spheres.
Below are four pieces about the Hopf link, called Conversations in a Foreign Language. Taking advantage of topology’s licence to stretch and distort, all 15 images in each drawing are Hopf Links.
Three Solid Arguments
I made three clay models and drew each one from five different view points.
Twisting the Narrative
The three voices start out relaxed and become more tense as they are twisted. I formed them from paper strips, twisted them and drew each stage.
Scott’s Secret
These come straight from Scott Carter’s work. I just gave them some depth.
15 Statements, Some a Little Quirky
The top left is a simple Hopf link. The rest of the 15 are elaborations, all Hopf links, I promise.
Fantasy In this fantasy, delicate Hopf links drift towards 4-space. Our view is blocked by the four black screens. Some mystery is taking place behind the screens. Intense light casts strange shadows.
This display at a Dashwood exhibition in London 2018 is all about the knots with 6, 7 and 8 crossings.
These are all the knots with 6, 7 and 8 crossings as taken from the online knot index, KnotInfo. I gave them some style, of course. You can see that the knots have no loose ends and are closed.
On the left is the braid of three conjoined knots with 24 crossings. Basically a braid is a flattened out knot in the form of a diagram. The centre is the same braid but unravelling and becoming a ribbon knot with 24 crossings. On the right at the top are knots with 6, 7 and 8 crossings made from milliners mesh tubing, the only three-dimensional part of this piece. Below them them are my stylised 6, 7, and 8 crossing knots from the knot index
One of the millinery mesh knots
And these cheeky things are Naughty Knots! They are from the KnotInfo online index just slightly stylised.
Below are some ideas and sketches
My clay work station I made a relief sculpture wall plaque from the 8_4 knot drawing