Medieval Rainbows at Cambridge Mathematics

The Cambridge University Mathematics Campus and one of its more famous denizens.
The Cambridge University Mathematics Campus and one of its more famous denizens.

Ordered Universe Co-investigator Tom McLeish was invited down to the Cambridge Department of Applied Mathematics and Theoretical Physics (DAMTP) to talk about the project in that famous institution’s regular ‘fluids’ seminar series.

Tom chose to have a go at seeing whether the audience would indeed recognise that ‘The greatest mathematician of the [13th] century’, as Roger Bacon hailed Robert Grosseteste, was thinking in ways that they would resonate with.  The story of colour as described in the very short treatise De colore and the longer and later De iride (on the rainbow) was Tom’s chosen theme.  The great advantage here is that the tight 400 word text of the De colore is manifestly mathematical.  We have earlier written about the three-dimensional combinatorial space of colour perception that Grosseteste adopts, tracing three independent ‘bipolar’ axes of colour co-ordinates:




He then counts colours descending from white by varying just one quantity, then two, then all

Grosseteste's three-dimensional colour space
Grosseteste’s three-dimensional colour space
Dimensions of Colour book_
Cover of ‘Dimensions of Colour’

three together, so calculating seven ‘colour directions’ descending from the white ‘pole’ of his system (and another seven ascending from the black pole).  All this and more is documented in the first of the project’s planned series of collaborative editions, translations and commentaries Dimensions of Colour.   It’s a good story to weave as well – to add narrative tension to the mathematical description we have at our disposal the story of the missing ‘obscura‘ in Bauer’s 1910 edition’s description of the qualities of blackness, identifying its absence from the combinatorial logic of the text as a whole, and Giles’ discovery of it in the earlier Madrid ms unknown to Baur.

The Friday afternoon gathering clearly had their interest piqued – at one point the questions appearing during the talk itself threatened to derail the story before it reached its greatest obstacle of all – the puzzle of how to map Grosseteste three qualities from De colore onto the three-dimensional colour space we know today.  As readers of the Journal of the Optical Society of America will know, the conundrum found its resolution in the closing sections of the later treatise on the rainbow, the De iride, where Grosseteste returns to his colour space, but this time referring explicitly to aspects of the space of all possible rainbows.

Oxford Co-investigator Hannah Smithson had plotted the spectra from different scattering angles and droplet sizes in rainbows into the perceptual colour space generated by human colour

Grosseteste's rainbow co-ordinates mapped onto perceptual colour plane by H. Smithson
Grosseteste’s rainbow co-ordinates mapped onto perceptual colour plane by H. Smithson

vision with its short, medium and long wavelength retinal cone-cells – and found the stunningly beautiful interlocked spirals spanning the colour plane. Changing scattering angle from within a single rainbow  gave spirals that wound one way, and from different droplet sizes the other.  Even the idea that reading medieval science can stimulate new research such as this was intriguing to the seminar – even more so that it gave rise to an object of such conceptual and aesthetic beauty.

Especially for the mathematical audience, it was worth pointing out the route by which these abstract rainbow-spirals suggested a new co-ordinate system for colour space.  For it turns out that the familiar polar system is only the extreme member of a family of systems generated from the familiar Cartesian grid, in its case by the conformal transformation:

z → exp(z)

But if the Cartesian grid is first rotated by an angle φ, so that

z → exp(exp(iφ)z)

then a double, interlocking orthogonal co-ordinate systems of spirals emerges – an ordered version of the system generated from the rainbows. These systems have a name – they are the log-polar co-ordinates.  They also satisfy the requirements of Grosseteste’s coordinates in the De colore, that all axes converge on whiteness (both sets begin at the origin, unlike polar coordinates such as hue and saturation for which hues just orbit the origin at constant radius).

Question time was predictably challenging but delightfully interdisciplinary.  We discussed Grosseteste’s theological motivations for doing science, and their later re-articulation by Francis Bacon.  One careful listener had not forgotten the ‘seven colours descending from whiteness’ of the De colore, and asked in the light of the analysis of De iride, if we now understood what they were.  Surprisingly, they are not the seven ‘colours of the rainbow’ as we know them.  For these are all contained, as Grosseteste is careful to point out, within the single multa-pauca co-ordinate.  Instead, all these occur along just one of the ‘descending colours’.  Other directions take us along other combinations of hue and saturation – unsuggestive of single colours to the modern eye, but to a medieval mind whose colour reference was the rainbow, emergent from single, and combined actions.

We are left with the fascinating combination of the strange and the familiar within medieval science.

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