SPUN AND A LITTLE BIT TWISTED

My father used to say that a if a mathematical proof was beautiful then it must be correct He was a topologist. Topologists LOVE knots. They even knot spheres for heaven’s sake! They do need 4-space, to do it however, but that is another thing altogether. One way topologists depict knotted spheres is by slicing them like a loaf of bread and laying out the slices to show the movement through the ‘loaf’ or knot. It is a wonderfully mind bending and engaging visual challenge. Click on an image and scroll through. Roll your mouse over the images and their descriptions will come up.

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Two-Twist Spun Trefoil Knot, Math for Art's Sake (Copy) — Two-Twist Spun Trefoil Knot, Math for Art's Sake

Take a trefoil knot, spin it twice while simultaneously turning it around an axis once, (I don’t know why they do it either!) As the knot turns, it’s leading edge sweeps out a surface. The knot would have to intersect itself to complete the turning and spinning. However intersecting is not allowed so it dips in and out of a spacial fourth dimension, 4-space, to avoid intersecting any other part of itself. From the outside the resulting object is a touch lumpen but slice it like a loaf of bread and you will see that the inside is complicated and lovely. I imagine the slices as sensual ribbons running through the core of the knot
This drawing is based on J. S. Carter, J H Rieger and M Saito’s two-twist spun trefoil knot.
The names across the bottom of the drawing are those of the mathematicians who, over several decades, have worked on the problem of the 2-twist spun trefoil knot.
HB, 4B, 9B Pencil on soft art paper

Rourke-Sanderson Slices (Copy) — Rourke-Sanderson Slices

Taken from Colin Rourke and Brian Sanderson’s diagram of their 2-twist spun trefoil knot, this is the heart of their 2-twist spun trefoil laid open in a series of wonderfully sensuous slices. The simple trefoil knots progress through their twists and turns and resolve themselves in the opposite orientation. It reminds me of the musical notation of an voluptuous dance.
HB and 4B pencil on soft etching paper.

Study after Ayumu Inoue's 2-twist spun trefoil knot (Copy) — Study after Ayumu Inoue's 2-twist spun trefoil knot

Study taken from Dr. Ayumu Inoue’s 2-twist spun trefoil knot. The swooping band is the path of the 4-space in the heart of the knot. The 4-space is expanding, stretching. You can see in it the twilight of our world including the early evening stars. I have discarded the knot itself and passed it through transparent panels to demonstrate that it is not affected by our 3 dimensional world.
Pastel 90 x 80 cm
Here is a link to follow showing a 2 minute movie of Dr. A Inoue’s 2-twist spun trefoil knot. https://www.youtube.com/user/ainoue2357 .

Kids Spun Doodle Six (Copy) — Kids Spun Doodle Six

Kids Doodle is a charming finger painting app I downloaded on my phone for my granddaughter. With Dr. Inoue’s 2-Twist spun trefoil knot in mind I did several three second bursts of finger ‘painting’ . I was in the spirit and having fun and I do hope Dr. Inoue will forgive me for the anatomically incorrect representation of his knot..
Available as a Giclée print 15 x 12 cm

Kid's Spun Doodle Seven (Copy) — Kid's Spun Doodle Seven

Here is another Kid’s Doodle finger drawing inspired by Dr. Ayumu Inoue’s 2-twist spun knot.
This image is available as a Giclée print 15 x 12 cm

Kids Spun Doodle Four (Copy) — Kids Spun Doodle Four

Again I was in the spirit and having fun and I do hope Dr Inoue will forgive me for the anatomically incorrect representation of his knot..
This image is available as a Giclée print 15 x 12 cm

Shadow One Carter-Hocking Slice (Copy) — Shadow One Carter-Hocking Slice

This series of photographs were taken when topologist Prof. Scott Carter and I were at my dining room table playing with red mesh tubing. We were twisting it in to contours inspired by his 2-twist spun trefoil knot. The shadows were so intriguing to me that I had to photograph them.
The photos are available as **Giclée** prints 25 x 25 cm

Shadow Two Carter-Hocking Slice (Copy) — Shadow Two Carter-Hocking Slice

Spun and a little bit twisted shadow of a Carter Hocking slice.
20 x 20 cm

Shadow Three Carter-Hocking Slice (Copy) — Shadow Three Carter-Hocking Slice

Spun and a little bit twisted from Prof Scott and my play.
25 x 25 cm

Shadow Four Twice-Twisted and Spun (Copy) — Shadow Four Twice-Twisted and Spun

Shadow of a 2-twist slice
The photos are available as **Giclée** prints on soft art paper 25 x 25 cm

Shadow Five Carter-Hocking Slice (Copy) — Shadow Five Carter-Hocking Slice

Shadow of a slice gone a bit nutty.
25 x 25 cm

Shadow Nine Twice-Twisted and Spun (Copy) — Shadow Nine Twice-Twisted and Spun

Smokey, spun and a little bit twisted.
For reasons I can not understand, Photoshop refuses to make the surround in this image white.
All of the shadow photos are available as **Giclée** prints 25 x 25 cm

Scott Carter's Two-Twist Spun Trefoil Knot (Copy) — Scott Carter's Two-Twist Spun Trefoil Knot

Professor Carter has reinterpreted his diagram of a 2-twist spun trefoil knot as a work of art. This image would make a fabulous stained glass window in a cathedral of mathematics!
This image is available through this website. I want to emphasize that this is not my work. Scott and I work together and he has authorised me to exhibit his image.
Available as a Giclée print 40 x 60 cm

Study, Two Curvy Sections (Copy) — Study, Two Curvy Sections

This a study for sculpture. Wire and shadow play in a loose version of C. Rourke and B. Sanderson’s work.

A Blue Disk Gone a Bit Nutty (Copy) — A Blue Disk Gone a Bit Nutty

You can stretch it any way you care to but it is still a disk. Not a 2-twist knotted creature just a fanciful drawing.
Pastel on heavy soft paper 90 x 75 cm