# J.W. Power's Book on the Mathematics of Pictorial Construction

I recently became aware of the Australian artist J.W. Power's book on the mathematics of pictorial construction. Power's name was familiar to me, as his bequest let to the foundation of the Museum of Contemporary Art in Sydney, so as an ex-mathematician I was curious about the book.

J.W. Power was an Australian artist who, while at one time very significant, seems to have fallen out of the history books. Born in 1881, he studied medicine at the University of Sydney, and then moved to London in 1907 for further study. He was a military surgeon during World War I, and after the war decided to become a full-time artist, spending much time in Paris. He became a member of the avant-garde group Abstraction-Création in company with other leading artists in Paris, and was involved with both cubism and surrealism.

Power left a large sum of money to the University of Sydney for the purpose of making the latest ideas and theories in art available in Australia through lectures and through the purchase of artworks. The bequest eventually led to the establishment of the Museum of Contemporary Art in Sydney as well as the Power Institute within the University of Sydney. Power as an artist is currently attracting attention, with a major Power exhibition, in the form of a recreation of Power's solo exhibition in Paris in 1934, to be held at the University of Sydney's art gallery. [Edit: See http://sydney.edu.au/museums/exhibitions-events/abstraction-creation.shtml.]

My interest in Power comes about because in 1932 he published a geometrically-based book (in Paris; both French and English versions appeared). The English version has the title The Elements of Pictorial Construction: A Study of the Methods of Old and Modern Masters. I recently had a chance to examine copies of both versions at the University of Sydney, thanks to Anthony Green, Senior Librarian of the Schaeffer Library and Ann Stephen, Senior Curator of the University of Sydney Art Collection.

I also found that the whole work (in French and English) is available online through the National Library of Australia (go to http://catalogue.nla.gov.au/ and search for "elements of pictorial construction"). However, the physical book has a unique feature: six pockets at the back, each of which contains a photograph of an artwork and one or more transparent sheets with lines drawn on them, which are to be placed over the photograph according to instructions in Power's text. These reveal features of the construction of the work, according to Power's analysis.

Power considers that he is recovering a method of construction used by the Old Masters. He starts from the idea that all the significant points in a masterpiece of painting are carefully placed. He draws horizontal and vertical lines from each such point to the edges of the painting, and considers in what proportion the edges are divided, expecting this to be significant. Apart from the midpoint of the edge, the Golden Section point (approximately 0.618) is a natural choice, but Power introduces others, including a point on the long side which is √2 or about 1.414 times the length of the short side, being the diagonal of the square on the short side. Then Power has a method of "transfer": for example, take the distance between the √2 point and the end of the long side, and mark off this distance on the short side. He even makes a second transfer of the remaining distance on the short side back to the long side. The result is a large number of horizontal and vertical lines (27 horizontal lines and 16 vertical in the case of the "Mond" Crucifixion by Raphael, in Power's first detailed analysis). Power also identifies an equal-sided nonagon (nine-sided figure) connecting significant points within the painting, and a hexagon surrounding the painting.

Power also has an idea of "movable format", the same configuration appearing in different parts of the one painting. He applies this to a Last Judgement by Rubens, with the movable configuration consisting of a perspective drawing of a cone with an inscribed square pyramid, which he sees as being used in approximately eight different positions. Having a actual piece of cellophane to move about is a great help! According to Power, the positions are not arbitrary, but each is obtained by rotating the configuration about a specific point, or a similar such move.

There are some other ideas about construction in the book as well, but the division of the edges of the painting and the "movable format" are the most important.

Power sticks very closely to his subject of construction and does not consider colours in the book, let alone things like symbolic content; there is a complete absence of mystical waffle. The book is also clearly written and generally easy to follow. However, in my view Power has not made a convincing case for what he claims.

It is uncontroversial that large paintings were frequently first done at small scale and then transferred to a wall or panel by some process of drawing up a grid. However, as far as I know the grid was made up of equally spaced lines (extant examples indicate this). Also, although the golden section was known and used, again as far as I know there is no historical evidence for things like Power's √2 point, let alone his "transfers" of such points.

So if there is a lack of historical evidence, the evidence must be internal to the works; indeed Power says: "[The Old Masters'] studies and sketches handed down to us show very few traces of these methods, while the finished pictures show a great many."

My first comment is that Power's method is on the whole not perceptually based. We know that a division of a line in a ratio somewhere in the range 3/5 to 2/3 is perceptually attractive, and this explains the photographer's "rule of thirds". But there is no reason to suppose that the exact value 0.61803... of the golden ratio is perceptually markedly better than say 0.625 (which is 5/8). There is no perceptual reason for things like the "transfer" of twice the short part of the golden section division of the long side (which occurs in Power's analysis of Raphael's Disputa).

So we are dealing with Power's geometrical ingenuity rather than perceptual givens, and Power has provided an array of lines and construction methods that I suspect can yield a good approximation to any ratio. The question is then whether the Old Masters used geometrical ingenuity in a similar way. It is not out of the question for a Renaissance master to be playing mathematical games in a painting, certainly with the golden section, but Power has not made a good case for the Old Masters to be playing *his* mathematical games. There is a range of geometrical ideas that Power could have discussed and didn't. Power doesn't provide the derivations for all of the lines in his analyses, but it is notable that he doesn't discuss equal divisions beyond the midpoint: there is no mention of thirds, quarters, fifths, etc. It is also notable that there is essentially no mention of the possibility of the same configuration occurring at different scales within the same painting. In geometrical terms, Power is concerned far more with congruence than with similarity. And there are always further possibilities: for example Power introduces ellipses but doesn't mention their focal points.

The "movable format" idea is intriguing; certainly for a swirling composition like the Last Judgement by Rubens, something like it appears more appropriate than fixed horizontal and vertical lines. However, I found the use of a pyramid drawn in perspective problematic. Power is not attempting to construct a three-dimensional model of the space implied by the painting; he is simply moving the perspective drawing of the pyramid as a flat object around the surface of the painting. This makes nonsense of any perspective within the drawing of the pyramid. On the whole, Power isn't much concerned with perspective. Raphael's Disputa has lines near the bottom whose spacing is determined by perspective, not by the sort of division that Power is interested in; Power doesn't consider these lines in his analysis.

I think that Power has driven his methods much too far, seeing things that are not there; I suspect that similar methods could provide quite different analyses of the same painting. The choice of significant points and lines is of course up to the judgement of the interpreter, and I am reluctant to challenge Power on this, but in Raphael's Crucifixion, for me the nail through Christ's feet is a prominent point in the painting; Power doesn't mention it, though he does mention other less prominent points. Then I could draw diagonal lines through the angels' feet and the nail to the faces of the kneeling figures, and start looking for similar triangles, and so on. If it is possible to give two quite different geometric analyses of the same work, both are likely to be illusory. I was somewhat more convinced by Power's discussion of cubist construction, since in cubist work it is reasonable to find both simple geometric shapes and the use of plan and elevation as described by Power.

As an aside, although the title The Elements of Pictorial Construction brings to mind Euclid's Elements of Geometry, Power is not using Euclid's methods (and does not claim to use them). The nine-sided figure Power finds in Raphael's Crucifixion cannot be constructed exactly by ruler and compass. Also Power has not read all of Euclid's Elements: he refers to the "dodecahedron and icosahedron, the proportions and structure of which had then [by the later Renaissance] been worked out". But the dodecahedron and icosahedron are discussed by Euclid.

It is interesting to consider what has happened in mathematics since Power's time. Power's geometric viewpoint comes across today as far too rigid. Power was not a professional mathematician, and it is not fair to the history of mathematics to take him as representative of his era, but certainly an emphasis on more qualitative approaches such as topology has become stronger. Topology was already well established by the 1930s, but may not have been accessible to people in Power's position, or may not have been seen as relevant. However, D'Arcy Thompson's On Growth and Form, published in 1917, might have caught Power's attention. Since Power's time a major impetus away from simple formulas has come from the computer. Chaos theory, fractals, and things like strange attractors and percolation theory have all given us new geometric objects that are quite different from the cubist cone and cylinder; although these new forms have been given solid mathematical underpinnings, the computer has helped in the discovery and investigation of the phenomena. And in general the absence of simple formulas in an area of investigation is much less of an obstacle than it was.