mathematical illuminations - california institute of ...matilde/illuminationschapter.pdf ·...

MATHEMATICAL ILLUMINATIONS

MATILDE MARCOLLI

Contents

1. Illumination in history 22. Illuminated Mathematical Notebooks 17

1

2 MATILDE MARCOLLI

1. Illumination in history

Illumination of manuscripts is a form of visual decorations ac-companying and partly superimposed to a written text. It takesmany different forms including initials, marginalia, and minia-ture illustrations. The earliest examples date back to around400–600 CE, after codices replaced scrolls. The medium couldbe papyrus (a lass durable material of which there are rare sur-viving samples), or vellum and other parchment and, starting inlate Middle Ages, paper. The illumination techniques include theuse of graphite powder, ink, gold leaf, and paint. The use of illu-minated manuscripts was especially widespread in the EuropeanMiddle Age and in the Islamic Renaissance, but illumination asa form of art persists to modern times.

While people generally associate illuminations to religious texts,this is in general not necessarily the case. There are illuminationsof ancient Greek and Roman literature, and of many kinds of phi-losophy and science texts. In fact, it is the connection betweenillumination and science that we are going to be focusing on here.

What is the general purpose of Illumination? For the reader,the illustrations enrich and highlight specific aspects of text. Inthe case of scientific texts: scientific illustrations (in botany,medicine, anatomy) have often greater explanatory power thanthe accompanying written text. Sometime the purpose is purelythe enjoyment of the reader through aesthetic decorations (floralmotifs, decorated initials of paragraphs, fantastic animals). Forthe artist illumination is a form of meditation on a text using thecreation of visual images as amplification.

MATHEMATICAL ILLUMINATIONS 3

The first example we look at is taken from Ancient Greek lit-erature: an illustration of the Iliad, taken from the AmbrosianIliad a manuscript of the 5th century CE, on vellum.

Illustrations accompanying literary texts typically are figurativerenderings of some of the scenes described in the lines of the text.The most ancient examples we have of illuminated manuscriptsare typically of this kind.

Another class of ancient illuminated texts consists of philo-sophical treaties, including also various alternative religions, likeNeoplatonism and Gnosticism. An example of this kind is theHomiliae in numeri of Origen: the 7th century manuscript Bur-ney 340 of the British Library collection. Often the use of il-luminations in these early more philosophical works is limitedto decorated initials of paragraphs, as a way to create emphasisaround a part of the text.

4 MATILDE MARCOLLI

By the 8th century, one encounters illuminations in arithmetictexts, as in the case of Boethius, De institutione arithmetica.

It is interesting to notice how, in this example, the illuminationsof the arithmetic text are not, as one might expect with a moremodern sensitivity, mathematical diagrams, but realistic pictorial


scenes of the same kind that one finds accompanying literary textsin manuscripts of the same epoch.

Medical treaties are the first example of illuminations whoseprimary purpose is to directly illustrate a scientific concept. Forexample in this 10th century manuscript of Dioscurides, De Ma-teria Medica, the Greek text on medical plants is accompanied byaccurate reproductions of the form of the plants for the purposeof recognizing them. In this way, illumination leaves the purelydecorative function and acquires a functional explicatory role.

Along with texts on medicinal plants, a growing interest in thelisting of animals in bestiaries leads to the production of severalilluminated manuscripts, like the Aberdeen Bestiary of the 12thcentury or the Ashmole Bestiary, of the 13th century, respectivelyshown in the figures below.

6 MATILDE MARCOLLI


These Medieval bestiaries collected illustrations of animals bothreal and fantastic.

A further development of the expressive form of manuscriptillumination takes place at around the same time with astro-nomical treaties, like Ptolomy’s Almagest, in this 13th centurymanuscript of the Huntington Library collection.

This is now really an illustration of mathematical concepts bymeans of diagrams, exactly as we mean it in our modern under-standing.

Another “functional” use of illuminations developed at aroundthat historical time with the growing use of musical scores. TheOld Hall Manuscript, of the late 14th and early 15th century, insone such example.

8 MATILDE MARCOLLI

The connections between illumination of manuscripts and sci-entific texts also involved geographic atlases, which started to be-come more widespread and detailed as Europeans embarked intolonger and more substantial voyages, ushering the era of colonial-ism. Hand drawn maps already included Medieval reproductionsof Roman cartography, although European cartography startedto acquire greater prominence with the 14th century. The Cata-lan Atlas of the 14th century is a good example of this use of theillumination technique.


A very interesting mixture of scientific facts, philosophical re-flections, and magical and religious speculations occurred withthe development of Alchemy. With it came a production of sev-eral texts, alchemical treaties, typically illustrated by a wealthof symbolic images, referring to chemical processes, as well asto spiritual stages of a form of mystical enlightenment, which ishowever entirely based in the observation of nature and in exper-imenting with natural material (the precursor of modern chem-istry). The Lumen Naturae, the light emanating not from a tran-scendental deity but from matter and nature, is released throughthe alchemical process, as a form of entirely natural enlighten-ment, a non-supernatural transcendence, capable of profoundly

10 MATILDE MARCOLLI

transforming the self. One of the most famous alchemical textsis “Aurora Consurgens”, dating back to the 15th century, fromwhich this image is taken.

Carl Gustav Jung famously reinterpreted the alchemical processand the images of alchemical texts as symbolic representationsof psychological processes and the “individuation process” of theSelf. Many of the symbols used in Alchemy, such as the greenlion that eats the sun in the following image taken from anotherfamous alchemical texts, the “Rosarium Philosophorum” of the16th century, live on many different levels of interpretation. Atsome levels they are a language of pictorial symbols for specificprocesses of transformation of matter. At this level, one can arguethat the modern language of chemical formulae and chemical re-actions is much simpler, more transparent, and has the advantageof being quantitative not qualitative: not only it gives us signi-fiers denoting which process of transformation of matter we arereferring to, but also a way to quantitatively assess the process,


and a precise description of what is happening based on atomicand molecular models of matter. At another level, however, asJung pointed out, the alchemical symbols stand for symbols of aninternal process, more akin to stages of enlightenment in easternphilosophical traditions. This significance was entirely lost whenthe language of alchemical symbols was replaced by the languageof chemical reactions.

The image of the green lion thus refers to the distillation of greenvitriol (Iron(II) sulfate, FeSO4·7H2O) and the eating of the sunstands for the precipitation of gold by green vitriol. In Jungianpsychology these become symbols of the Ego and the Self.

By the end of the 18th century, illumination of manuscriptsreappears in the visionary poetry of William Blake, who com-bined words and images to create a special resonance betweenhandwritten text and painted images that blend one into theother.

12 MATILDE MARCOLLI

We see an example in William Blake’s “The Tyger”, from“Songs of Experience” of 1794, and in “The Argument” from“Visions of the Daughters of Albion” of 1793, shown here.

Blake’s mystical visions and evocative use of illumination op-posed the scientific use of manuscript illustration. Blake himselfopposed the new materialistic and non-religions visions of theEnlightenment, and portrayed Newton as part of an “infernaltrinity” with Francis Bacon and John Locke. This portrait ofNewton, realized by Blake in 1795 and shown here below, showsthe scientist as a naked figure with a white drape on his side,suggesting the ancient Greek roots of the modern science of theEnlightenment era. He is bent forward, in the act of drawinggeometric figures, and illustrating what appears to be the scrollof a scientific manuscript, a reference to the use of illumination


for scientific purposes. The author’s ideological position notwith-standing, we can interpret this image as witnessing the transitionto modern science and the double use of illustration as both a po-etic and a scientific technique.

In even more recent times, illuminated manuscripts reappear inthe context of psychoanalysis. We have already mentioned Jung’sinterest in alchemical manuscripts and illustrations. In the years1915–1930, Jung himself engaged in a very sophisticated use ofillumination techniques, in his long manuscript, “The Red Book”,which was only recently published in its entirety. Based on hisown dreams and visions, active imaginations, and reflections, thecollection of illuminations of the Red Book constitute probablythe most elaborate use of the technique in modern times. Theimages are at the same time a meditation technique and a wayof amplifying unconscious contents. Their wealth of symbolism,

14 MATILDE MARCOLLI

ranging from Gnosticism to alchemical references, is extremelyelaborate and interesting. One of Jung’s illuminations of his RedBook is reproduced here.

Jung viewed the illumination technique primarily as a methodof exploration of the unconscious. In his writings, he explicitlyrejected the artistic quality of his illustrations of the Red Book,even though they clearly are of considerable interest from a purelyaesthetic point of view, and he preferred to attribute to them apurely “scientific” connotation, in relation to understanding theworkings of depth psychology.

This continuous dialog between the artistic and the scientificaspects of manuscript illuminations, which has unfolded throughhistory, was taken over again, in the 20th century by Max Ernst,in his “Natural History” series, of which we reproduce below“L’evade” (1926).


In this portfolio of 34 collotypes realized with his “frottage” tech-nique, Ernst refers to the tradition of scientific illustration in en-cyclopedias and to the ancient tradition “Natural History” com-pendiums (Pliny the Elder) of botany and animal life. However,while the folio is designed so as to evoke in style these traditionalforms of illustration, the resemblance is immediately dispelledby the surrealistic style of the illustrations and the aleatory ele-ments introduced by the use of frottage. This technique consistsof placing various textured objects like wood or leaves under asheet of paper and rubbing (frotter) over the paper with a pencilof crayon so that the underlying structure becomes visible, in anew and often not immediately recognizable form, on the paper.

16 MATILDE MARCOLLI

Max Ernst further developed this interplay of reference to sci-entific illustration, manuscript illumination, and visionary sur-realistic imagery in his later work “Maximiliana ou l’exerciseillegale de l’astronomie” (1964). Already the title hints to animproper use of the scientific reference. In the work, an exampleof which is reproduced below, fragments of poetry lines alternatewith cosmic images, allusions to scientific diagrams, and myste-rious scripts in an incomprehensible alphabet.


2. Illuminated Mathematical Notebooks

A mathematician’s notebook is a sketchpad for thoughts, cal-culations, examples that are worked and reworked, diagrams,drafts of arguments, all merging and superimposed to one an-other, coasting on the verge of the unintelligible: a laboratoryon paper, a chaotic evolution of ideas, a cauldron, like the al-chemist’s vessel, out of which a clean final result will hopefullyeventually emerge.

There are several intermediate stages in between what getswritten in the notebooks and what ends up in a publishable pa-per. These many processes of reworking, refining, and writing,happen elsewhere. The notebook provides only the prima mate-ria. However, throughout the process one keeps returning to thisraw material, revisiting it anew. It is during these stages that theillumination of notebooks takes place. At other times it happensafter the whole process is completed, again while revisiting theoriginal material.

Abstract thought requires abstract art. The underlying writ-ings in the notebooks provide the shape, illumination providescolor, a kind of synesthesia: is mathematics naturally colored?The technique is simple and suitable for a written backgroundon pages of thin paper: wax pastels and watercolors. All theexamples discussed in the rest of this chapter are taken from myown mathematical notebooks of the years 2014–2016.

We are going to discuss a first example of such illuminations,which has to do with the concept of DG-algebras which combinean algebra structure (a multiplication operation) with a compat-ible differential and homology structure.

18 MATILDE MARCOLLI

The concepts of complexes and homology/cohomology play a fun-damental role in modern mathematics. The origin of the idea isgeometric: associating to a manifold or to a triangulation itsboundary (or zero if it has no boundary) and the crucial factthat the boundary of a boundary is zero (a boundary has nofurther boundary itself). A cycle is a formal combination of geo-metric objects such that its overall boundary is zero, but it needsnot be itself a boundary. This makes it possible to measure thediscrepancy between cycles and boundaries, which is called ho-mology. It measures the number of “holes” of a given dimen-sion in a given geometric space (topological space, triangulation,


smooth manifold). The resulting form of a complex consists of asequence of vector spaces (corresponding to the different possiblevalues of the dimension of the geometric objects considered) witha boundary maps that lowers the dimension by one (a surfacehas a one-dimensional boundary, so an N -dimensional manifoldhas an (N − 1)-dimensional boundary). As shown in the figurebelow, the boundary maps send the whole vector space of for-mal combinations of N -dimensional objects (called N -chains) to(N − 1)-dimensional boundaries and sends N -dimensional cyclesand N -dimensional boundaries to zero. Homology at each levelis determined by those cycles that are not boundaries.

On the other hand, in mathematics it is customary to describealgebraic operations through diagrams that illustrate the proper-ties of those operations: for example a multiplication operationthat has an identity element is illustrated by a diagram

20 MATILDE MARCOLLI

The painted image above hints at the mathematical structure ofa DG-algebra that combines the homology structure describedabove and the multiplication structure mentioned here.

A second example is the following notebook illustration, whichsurrounds with multicolored smooth shapes the computation ofa zeta function of a Dirac operator.

Dirac operators are very natural operators in the theory of smoothand Riemannian manifolds. We already encountered them in pre-vious chapters, in discussing the spectral action, where we men-tioned that they can also be defined on other kinds of geometric


objects, like noncommutative spaces or fractals. However, theirprimary existence is in the world of smooth geometry. They arein fact related to the geometry of spinors. These entities, roughlyillustrated in the image as pointing arrows, are somewhat mys-terious objects.

As a famous mathematician describes them:

“their significance is mysterious. In some sensethey describe the ‘square root’ of geometry: justas understading the square root of −1 took cen-turies, the same might be true for spinors”(Michael Atiyah)

We have already seen in previous chapters how a smooth man-ifold can be endowed with a metric structure (Riemannian man-ifold) and how the metric is used crucially in General Relativityto describe the gravitational field. We have also seen how onecan consider the spectrum of an operator defined by the met-ric (Laplace operator) as the basic frequencies of vibration ofour space, thought of as a drum. The same can be done with theDirac operator, which is a square root of the Laplacian, acting on

22 MATILDE MARCOLLI

the spinors (which is the reason why one should regard spinors as“a square root of geometry”). These frequencies, the spectrum ofthe Dirac operator, can be assembled together to form a function.We have seen this in the case of the spectral action. Another,closely related, function formed from these frequencies is the zetafunction

ζD(s) = Tr(|D|−s) =∑

λ∈Spec(D)

λ−s.

When the underlying geometry is a circle, this zeta function isthe Riemann zeta function we discussed in a previous chapter.

The specific computation of zeta function that forms the back-ground of the painted image above comes from a cosmologicalmodel we already discussed briefly in previous chapters, the frac-tal Packed Swiss Cheese Cosmology obtained from an Apollonianpacking of spheres


The painted forms in the image suggest the spinors on whichthe Dirac operator acts, the underlying smooth manifolds, andthe beginning steps of a packing construction. A related imagefrom the same notebook is also accompanying the same compu-tations of zeta function and spectral action for the Packed SwissCheese Cosmology, where we see packed configurations of sphereswith varying radii.

The length spectrum is the list of the radii of all the spheres inthe packing PD,

L = L(PD) = {an,k : n ∈ N, 1 ≤ k ≤ (D + 2)(D + 1)n−1}

24 MATILDE MARCOLLI

where the numbers an,k are the radii of spheres SD−1an,kin the n-

iterative step in the construction of the Apollonian packing andin the k-th place in the list of all the spheres that contribute tothe same level n of the packing. The number D is the dimensionof the spheres. In the cosmological models this is either D = 3(when considering spatial sections) or D = 4 when consideringthe full spacetime. There is a zeta function formed out of theseradii, called the zeta function of the “fractal string” (Lapidus),

ζL(s) =∞∑n=1

(D+2)(D+1)n−1∑k=1

asn,k.

For s a sufficiently large real number, this series converges. Theboundary of the set of values of s that give convergence (theMelzak’s packing constant) is related to the fractal dimension ofthe resulting Apollonian packing.

As we discussed in a previous chapter, the Packed Swiss CheeseCosmology models are an elaborate example of what happensif one replaces the stadard cosmological assumptions of homo-geneity and isotropy of the universe with a situation where theisotropy assumption is retained but the requirement of homo-geneity is removed. We also mentioned that, if one proceeds inthe opposite way and retains homogeneity but drops isotropy,then another interesting class of cosmological model arises, theBianchi IX cosmologies (and the related mixmaster universe mod-els mentioned before). The next example from the notebooks isan illumination that accompanies computations performed in oneof these homogeneous but non-isotropic Bianchi IX cosmologicalmodels.


In particular, the kind of spacetimes considered here are gravi-tational instantons: these are solutions to the Einstein equationsin vacuum, which look like a flat 4-spacetime far away (asymp-totically). There are several classes of examples : Taub-NUT,Eguchi-Hanson, and more general Bianchi IX that include theprevious ones as special cases, Multi Taub-NUT. For example,the Taub-NUT spacetimes have a shape that exhibit a numberof singularities and a global symmetry as shown in the figure.

26 MATILDE MARCOLLI

The general form of the metric in Bianchi IX spacetimes isgiven by

g = F (dµ2 +σ21

W 21

+σ22

W 22

+σ23

W 23

),

with an overall conformal factor F and with three different scal-ing factors W1,W2,W3 in the three independent spatial direc-tions (anisotropy). The asymptotical behavior of these BianchiIX spacetimes that satisfy the Einstein equations and a “self-duality” equation (gravitational instantons) recovers the Eguchi–Hanson and Taub-NUT solutions, with two of the scaling factorscoming together

W1 ∼π

2, W2 ∼ W3 ∼

1

µ+ q0

This is the calculation that lies in the background and motivatesthe notebook image shown above.

Another example of illumination from the mathematical note-books series is related to the mathematical theory of Bruhat–Titsbuildings, which generalize graphs in higher dimensions.


In this case, the computation refers to a specific class of buildingsof rank 2.

Bruhat–Tits Buildings generalize the one-dimensional case ofBruhat–Tits trees and their quotient graphs. Instead of patternsof edges in the tree and in the graphs, in the higher rank build-ings one has patterns of other geometric objects, for exampleplains tessellated by triangles as painted in the image above, as-sociated to certain group actions. These arrangements are called“apartments”. Examples of these building blocks (apartments)for buildings arising from the groups PGL3 and Sp4 are shown inthe figure.

28 MATILDE MARCOLLI

As in graphs it is important to specify how edges are linked to-gether at vertices, in the theory of Bruhat–Tits Buildings anotherimportant ingredient is the way apartments are assembled to-gether. This is specified by linking properties, a topological andcombinatorial structure that provides the gluing instructions forthe apartments of the building.


For example, the Bruhat–Tits Building of the group PGL3(Qp)with the appropriate linking structure taken into account has theform shown in the figure.

The following related image in the same notebook, referringto the same computation, highlights this linking structures ofbuildings.

30 MATILDE MARCOLLI

mathematical illuminations - california institute of ...matilde/illuminationschapter.pdf ·...

Documents