Mary Martin's "Inversions" - a Mathematical Sculpture

[image from http://www.tate.org.uk/art/images/work/T/T01/T01198_9.jpg]

This beautiful  sculpture is currently on display at Tate Liverpool as part of the free exhibition DLA Piper Series - Constellations. It's huge.  It's over 24ft long.

I wrote about it three years ago in Thoughts on Mary Martin (artist), when it was on display in London.

Tate has an extensive catalogue entry for it, where the artist's use of permutations is fully explained. I will paraphrase that description in what follows.

The sculpture is made up of 24 columns. Each column comprises 4 aluminium plates mounted so as to face in different directions. In fact, each plate can face right, left, up or down.

For example, in the first column of the sculpture, reading down from the top, the aluminium plates are oriented to face right, left, up and down respectively.

This next image, taken from the side,shows the first column a little more clearly.

[photograph taken from http://kieranleahy.blogspot.co.uk/2011_02_01_archive.html]

Mary Martin has carefully arranged the 24 columns of the sculpture in such a way that the following two properties are implemented

1. All 24 columns are different
2. No two adjacent plates have the same orientation.

To develop this design the artist used the mathematical concept of permutations. These are the first few permutations that Mary Martin constructed

    1 2 3 4
    4 3 1 2
    3 2 4 1
    4 1 2 3
    2 4 3 1
    .....

She uses the numbers 1, 2, 3 and 4 to represent the orientations of the plates in the sculpture. The first row above represents the first column in the sculpture, reading from the bottom upwards, with the following interpretation

1. plate faces down
2. plate faces up
3. plate faces left
4. plate faces right

It may be easier to compare if I turn the permutations on their side and show you the first 5 columns of the sculpture alongside

          4 2 1 3 1 ...       R U D L D ...
          3 1 4 2 3 ...       L D R U L ...
          2 3 2 1 4 ...       U L U D R ...
          1 4 3 4 2 ...       D R L R U ...

Perhaps to make things a little clearer, take a look at the following image. I have taken the full set of Mary Martin's permutations and constructed a coloured chart of the orientations

The squares in this image are coloured so as to represent the orientations of the plates in Mary Martin's sculpture, with the following coding

1. red - plate faces down
2. green - plate faces up
3. blue - plate faces left
4. black - plate faces right

Here is the full set of permutations

    4 2 1 3 1 4 2 3 2 1 4 3 1 2 4 3 4 1 2 3 2 4 1 3
    3 1 4 2 3 2 1 4 3 4 2 1 3 4 1 2 3 2 4 1 3 1 2 4
    2 3 2 1 4 3 4 2 1 3 1 4 2 3 2 4 1 3 1 2 4 3 4 1
    1 4 3 4 2 1 3 1 4 2 3 2 4 1 3 1 2 4 3 4 1 2 3 2

There are some interesting properties of these permutations that the artist has used.

Firstly, there are only 24 different permutations of 4 distinct things (here, the orientation of the plates). Secondly, while there are many arrangements in which these permutations can be listed, only a few of them have the property that adjacent permutations do not have the same entry in the same place. Mary Martin constructed one of those arrangements, presumably the one that suited here best.

Apparently the sculpture can be exhibited in two halves. I am not entirely sure how it is constructed physically, although there are descriptions on the Tate website, but another interesting property of the arrangement that the artist chose is that if the sculpture is divided into four as follows, then these four sections can be reassembled in any order without destroying the property that no adjacent plates face in the same direction

 4 2 1 3 1 4   2 3 2 1 4 3   1 2 4 3 4 1   2 3 2 4 1 3
 3 1 4 2 3 2   1 4 3 4 2 1   3 4 1 2 3 2   4 1 3 1 2 4
 2 3 2 1 4 3   4 2 1 3 1 4   2 3 2 4 1 3   1 2 4 3 4 1
 1 4 3 4 2 1   3 1 4 2 3 2   4 1 3 1 2 4   3 4 1 2 3 2

I expect the artist knew that.



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Notes:

For the purposes of checking, here are the full set of permutations as described on the Tate website referred to above

1 2 3 4
4 3 1 2
3 2 4 1
4 1 2 3
2 4 3 1
1 3 2 4
3 4 1 2
1 2 4 3
4 1 3 2
2 3 4 1
3 1 2 4
2 4 1 3
4 2 3 1
1 3 4 2
3 2 1 4
1 4 2 3
2 1 3 4
4 3 2 1
3 1 4 2
4 2 1 3
1 4 3 2
2 3 1 4
3 4 2 1
2 1 4 3

I have been helped in this analysis by using the descriptions on the Tate website and also the artist's description of her use of permutations in other sculptures in the catalogue for the 1984 exhibition of Mary Martin's work at the Tate.