Professor of Pure & Applied Mathematics, Trinity College Dublin
Mathematics and Perfection
Ideas and ideals of perfection are found throughout mathematics – in the conception of new theories and in the quest for “elegant” and “beautiful” proofs. A mathematical theorem holds with no obligation or need for validation in the real world. Nevertheless these “perfect” results very often have deep and unexpected applications in our imperfect world. The theoretical physicist, Eugene Wigner captured this in his article The Unreasonable Effectiveness of Mathematics in the Natural Sciences where he wrote “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a gift we neither understand nor deserve. We should be grateful for it”.
Perfection is often expressed in a desire not just for a true result but also a beautiful one. But what does this mean in mathematics? Essentially a combination of creativity and depth with simplicity and clarity – the same characteristics of beauty in art or music. The remarkable result, mathematician Euler’s identity: eiπ + 1 = 0 is often given as an example of deep mathematical beauty, exposing a profound connection between fundamental numbers: e, i and π, yet expressed in a simple formula.
Associate Professor, History Of Art, Trinity College Dublin and Director of Irish Art Research Centre
The circle has a long provenance in art, both as a framing device (a tondo) and as an element within an image. Giorgio Vasari (1511–74) recounted the fable that the artist Giotto (c.1266–1337) attracted the patronage of a Pope when his drawing of a circle, without mechanical aids, was accepted as demonstration of his skill as an artist. The salient element of this legend was recounted in various versions in relation also to other artists, such as Apelles, but the question it raises is why this particular proficiency – the capacity to draw a perfect circle freehand – came to symbolise artistic skill in the first place.
A circle may be regarded as the most perfect of geometric forms, comprising a single boundary line or circumference whose infinity is indicated by its apparent seamlessness without evident start or finish, and therefore emblematic of eternity with all of its connotations. A perfect geometric circle exists where the distance from the centre to the circumference – the radius – is consistent and identical in every direction and is perfectly symmetrical. While human anatomy, intellect and creativity renders a perfect circle challenging to draw free-hand, its achievement infers an exceptional level both of control and commitment to repeated practice. While such characteristics may be considered anathema to contemporary ideals of visual interpretation and representation, nonetheless numerous artists have alluded to the geometric circle to infer skill, harmony, spirituality, and infinity, and to serve exploratory, scientific purposes both abstract and material.
The many artists who conscripted the circle in their art include Leonardo da Vinci, whose Vitruvian Man (c.1490) is understood to correlate human proportions with geometry – humanity proposed as a microcosm of the universe. Vasari’s tale was recounted by Carel van Mander (1548–1606) ensuring its familiarity to artists in seventeenth century Netherlands. Rembrandt’s Self-Portrait with Two Circles (c.1665-69), depicts an aging artist, palette in hand, defiantly confronting the viewer. The muted circles inscribed on the surface behind have prompted art historical enquiry as to their meaning, and research reveals that the capacity to draw a perfect circle was associated both with calligraphic ability and artistic skill. Rembrandt’s self-portrait has been associated with the Giotto legend, to geo-metric (in the literal sense of measuring the world) capacities, in forms that referenced the cartographic globe, as well as to cabbalistic symbolism, though their defiant incompleteness as infinite circles has raised unanswered questions. At a more mundane level, they have been identified as studio props, and we are reminded of the circular mirror in Francis Bacon’s studio with all of its connotations of illusion, reflection and vision. More recently, circular forms abound in visual art – the concentric circles of Sonia Delaunay’s Electric Prisms (1914) or Jasper John’s provocative Target series of the 1950s and 60s. Ben Nicholson, Patrick Scott, Richard Long – there are numerous examples of artists conscripting the circle as the epitome of completeness at one level, but embodying also the impetus to challenge its perfection at the other.