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**Title:** Helward’s World

**Artist:** Madeleine Shepherd

**Designer:** Hugh Griﬃths

**Materials:** Debbie Bliss SoHo yarn

**Technique:** Knitting

**Description:** This piece is part of a larger portfolio of Alien
Surfaces, most of which are digital prints illustrating surfaces described
in quotations from some of Shepherd’s favourite works of science ﬁction. In
Christopher Priest’s **Inverted World**, the main character Helward
sees his home as a surface of revolution—speciﬁcally a rotation of *y* =
1/*x* around the *y*-axis. Using the
social networking site ravelry.com, Shepherd
recruited Griﬃths for pattern-writing collaboration. The calculations involved
were nontrivial as the knitted rows progress along the curve’s arclength rather
than the usual y-axis. For more details, please see the longer Artist’s Statement
accompanying this piece.

Early in 2007 I began work on Alien Surfaces – a portfolio of digital images illustrating surfaces described in quotations from some of my favourite works of science fiction. Mostly these involved looking at fairly flat surface texture. However one of the quotations I selected needed an entirely different approach:

“North of him the ground was level; flat as the top of a table. But at the
centre, due north of him, the ground rose from that flatness in a perfectly
symmetrical, rising and curving, concave spire. It narrowed and narrowed
reaching up, growing ever more slender, rising so high that it was impossible
to see where it ended. He saw it in a multitude of colours. There were broad
areas of brown and yellow, patched with green” Christopher Priest – *Inverted
World*

This is how the main character Helward sees his home and it is a surface
of revolution – specifically a rotation of *y*=1/*x* around the *y*-axis. The novel
was published when I first studied mathematics at university and the mathematical
basis always appealed to me. Now, as a member of staff at the International
Centre for Mathematical Sciences, it seemed an obvious choice for the exhibition.
I’ve been knitting and crocheting mathematical objects for the last few years
and so it was again obvious that I should knit it (for me that’s quicker).
All I needed was a pattern.

I could see that the level ground approximated to a circular plane and the
central spire tended towards a thin tube. Those bits would be straightforward.
I decided to start from the outside and work inwards on circular needles.
For the transition from flat to spire I reckoned I’d have the following
steps.

1. plot out the curve *y*=1/*x* using stitches as the units

2. at regular
intervals on the *y* axis, take the *x* value to be the radius of that circle
of knitting

3. work out the corresponding circumference and round to a whole
number of stitches and then

4. evenly space the decreases for each round.

Not being keen to reinvent the wheel, I asked around a few knitting groups for patterns in case something similar had already been done. To my astonishment, a reply came back, via ravelry.com, from Hugh Griffiths, a PhD student at the University of Edinburgh just a few miles from my studio. Hugh offered to run his pattern-writing algorithm on this structure. Hugh’s approach is more sophisticated than mine and this is what he had to say about it in an email:

“Being a surface of revolution simplifies the problem of writing down a pattern - you can work out the circumference at each row, convert this into a number of stitches, and work out how many stitches you need to decrease on each row.

The difficulty is that the rows in this case do not correspond to the coordinate
*x* but the arclength - the distance travelled along the curve from the first
row. In theory the change of coordinates is simple, but to do this explicitly
requires some numerical calculation. Using *Maple* it is possible to write
a short piece of code which will perform these calculations and return the
number of stitches on each row automatically.”

Hugh’s pattern was easy to follow and to incorporate into the larger structure. He’s written more about developing the pattern on his blog.

As I only ever intended making one of these, I did not keep notes that would constitute an exact pattern. The work was made up in assorted shades of Debbie Bliss SoHo on 8mm pins with the intention of felting it. However a test version was felted and the distortion due to different rates of shrinking in different directions was too great. So the knitted surface was given a circular wire frame around the base and needs to be hung under its own weight or supported inside by an appropriate length of dowel. We are both very pleased with the way it turned out.

Helward’s World was exhibited as part of Alien Surfaces during the Edinburgh Fringe Festival 2008. The entire exhibition can be found here.