the art of spheres

Discovering Mathematical Ideals in Christian Abstract Art

By David Clayton
Second Spring

Most Christian art has been, so to speak, "representational". Whether stylised or naturalistic, it has attempted to represent people and material objects in a recognizable fashion. However, Christians are not bound to paint or sculpt representationally. The practice of non-representational art, usually called abstract art, is a perfectly legitimate pursuit for a Christian, provided that truth is not compromised in the process. But what is the relationship to truth implicit in abstract art?

To abstract means, literally, to draw out. Abstract art was given its name because it was felt to draw out a truth, the essence of something that is not immediately apparent when looking at it. In fact, to be authentically Christian any representational art must have a degree of abstraction. The artist is attempting to show the invisible through visible means.

By the same token, it would be wrong to represent anything that is material by nature and to ignore its materiality altogether. "Truth" in representational art corresponds to its degree of resemblance to a model. We can modify the appearance of things as they are seen in order to reveal metaphysical truths, but it must be a limited modification that does not distort the image beyond the bounds of truth.

Mathematics is an important link between representational and abstract art (including art with a mainly decorative function). The language of mathematics straddles the physical and the metaphysical realms. The number six, for example, can be used to denote a number of material objects – six apples. It also has a scriptural, and so metaphysical, significance – the creation of the world over a period of six days. And we can also conceive of the number six apart from anything else, without applying it to any material objects or investing it with any spiritual significance.

The fact that number can be conceived in the abstract and represented visually means that works of art based upon mathematical forms are would not contravene the principle of truth. This provides the key, then to one type of abstract art that is perfectly legitimate within a Christian milieu – or any milieu where truth is paramount.

The art of geometric pattern generally takes two forms (although most examples of abstract art contain elements of both): first, the representation of number as geometrical shape produces "hard" or "crystalline" works of art – examples of which would include the Islamic tiled patterns of the Alhambra and the Cosmati pavement in Westminster Abbey. Sometimes such designs are created as works of art in themselves, and sometimes as decorative borders around a piece of figurative art (and both forms of abstract art have been a basis for compositional design in figurative art).

The second is "soft" or "fluid" art. This is calligraphic art – the art of line. As the artist draws, he uses the circular arcs of flexing joints in fingers, elbows, and shoulders to produce parabolas and eccentric curves. These are in fact graphical representations of mathematical functions that describe the order of the natural world. However, the artists who produced these shapes are not necessarily aware of this correspondence. This type of art tends to be more intuitive and less prescriptive than the other.

We see this in Celtic illuminated manuscripts, in Islamic calligraphy and art based on "arabesques", in Chinese characters and in Baroque scrollwork. The beauty here is derived from an intuitive imitation of the natural order in the shapes of these curves. Baroque scrolls that appear in room decorations, picture borders, and picture frames are very abstracted, but nevertheless reminiscent of vegetation through the mimicry of growing vines with their twisting line and intertwinings, although this swirling decoration did develop into an overly ornate form in the later period. Unfortunately, it is this excess in the Rococo that in many peopleís minds typifies the Baroque and causes a negative reaction.

What is commonly described as "Baroque rhythm" is the imposition of eccentric curves and parabolas upon natural form. The sculptures of Bernini particularly are criss-crossed with these curves, at times forcefully and deeply applied, at other times as subtle as lacework across its surface. Bernini used them to introduce interest and vitality and to draw attention to particular aspects of form in his sculptures by using the line to guide the eye.

We talk about these lines as having a "natural grace". This is an appropriate phrase to use for it reinforces the fact that God is the ultimate author of all beauty (grace being an undeserved gift from God). It is interesting to note that the production of a graceful line requires a graceful movement in the hand that holds the pen. This movement becomes an art form in itself in, for example, dance. It is also, in part, what draws crowds to watch sport and causes a cricket commentator to purr in appreciation when a batsman plays a "graceful" classic off drive!

In painting, it seems to be a general rule that the more naturalistic the rendering, the less will the composition withstand obvious symmetry. So whereas Romanesque art would happily allow an obvious symmetry of placement, the later naturalistic artists sought to conform to the natural order but to incorporate it more subtly, so that the figures do not look as though they had been pushed into what would otherwise appear to be an ill-fitting box. Either they would arrange the figures according a geometric pattern and then break it up with the use of shadow and dynamic arcs moving in and out of the shapes, or they would arrange figures off the geometric pattern and allow the impinging line of shadow, drapery, limbs and so on to trace the geometry across the surface.

As the production of this flow of line is a rather intuitive process I will concentrate here on art and design based on geometric shape. After considering why number is seen as significant, I will discuss some numbers and how they relate to the physical, the metaphysical, and the mathematical world that exists apart from these. Finally I will discuss how number has manifested itself in works of art of the past.

The Cosmic Liturgy

The natural order can be described mathematically. Even before the advent of modern science, the ancients were aware of this, as they observed changes and movements of the constellations in the night sky. Most ancient peoples (Babylonian, Egyptian, Chinese, Indian, Greek, Roman, Mayan, Inca, and Aztec) observed these in great detail. They believed that the celestial bodies and the seasonal changes were controlled by mysterious powers or gods. For Christians, a single God controlled all, but the stars and the planets were signs of the rhythms of heaven, to which the material world conformed.

For Christians the focal point for the meeting of the material and the spiritual is in the liturgy. All creation participates in a liturgy of praise to God. The book of Revelation describes the timeless heavenly liturgy; the Mass and Divine Office are a participation in this same liturgy. The physical and the spiritual come together in a single point in the body and blood of Christ, in the Eucharist. Everything else unfolds from this. Liturgy is not something that is confined to the services taking place in a church. Creation, through its being, is seen as giving liturgical praise to God. As Erik Peterson writes:

"The worship of the Church is not the liturgy of a human religious society, connected with a particular temple, but worship which pervades the whole universe and in which sun, moon, and all the stars take part ... The Church is no purely human religious society. The angels and saints in heaven belong to her as well. Seen in this light, the Church's worship is no merely human occasion. The angels and the entire universe take part in it." (1)

The Canticle of Daniel,(2)which is chanted in the Divine Office on Sundays and feast days, calls upon all of creation to bless the Lord, including the sun and moon, stars of the heavens, clouds of the sky, showers and rain. Man and the angels give praise through their existence too. But they have free will and so have the additional capacity to praise God through choice. In discerning how to harmonize his liturgical activity to that of heaven, man takes his cue, as it were, from the cosmos.

Christian cosmology is the study of the patterns and rhythms of the planets and the stars with the intention of ordering our work and praise to the work and praise of heaven, that is, the heavenly liturgy. The liturgical year of the Church is based upon these natural cycles. The date of Easter, for example, is calculated according to the phases of the moon. Pope Benedict XVI is sensitive to this dimension of Christian life. As Cardinal Ratzinger he discussed the importance or orienting church buildings and the Mass to the East, to face the rising sun, the symbol of the Risen One:

"The cosmic symbol of the rising sun expresses the universality of God above all particular places ... But ... this turning toward the east also signifies that cosmos and saving history belong together. The cosmos is praying with us. It, too, is waiting for redemption. It is precisely this cosmic dimension that is essential to Christian liturgy. It is never performed solely in the self-made world of man. It is always a cosmic liturgy. The theme of creation is embedded in Christian prayer. It loses its grandeur when it forgets this connection."(3)

But why would we want to have a liturgical life at all? One reason is the desire of believers to praise God well, as an end in itself simply because we love God. Another reason is that if we participate in the liturgy fully, it becomes an ordering principle for the whole of our lives; that is, by participating in an earthly liturgy that is in harmony with heaven, we receive grace that flows through our lives and overflows into the world. The liturgy is a portal that ushers the presence of God into our lives and (through our participation) the lives of others around us.

If we want to increase our collective ability to conform to grace, we should strive to make our liturgy conform to that in heaven. Canon law is the way that we do this. The rubrics of the Mass are gifts from God that can guide us so that we can love him more, and open us and so the world up to the grace of God. And number is as essential part of this, through the rhythmical repetitions of prayer and words, through posture, and in the production of beautiful music, art, and architecture that is "liturgical" even when it has a secular use.

The beauty of number is that not only can it be conceived in the abstract, but once conceived it can then be applied to every aspect of our lives – time, space, art, music. This is its special mystery. When we apply the liturgical numbers of the cosmos to the rhythms and actions of our lives extending beyond that part lived in the church building, the whole of life becomes infused with a liturgical rhythm.

In the sixth century AD, St Benedict, the founder of the Benedictine Order, underlined an aspect of "liturgical number" in chapter 16 of his Rule by looking to the Old Testament. "the prophet says: 'Seven times daily I have sung your praises' [Psalm 119:164]. We will cleave to this sacred number if we perform our monastic duties at Lauds, Prime, Tierce, Sext, None, Vespers, and Compline." Man cannot address his attention to prayer constantly, but must attend to the needs of life. These seven occasions of prayer during the day are seven portals through which grace pours into the daily life, sanctifying the times between prayer by integrating them with the cosmic rhythm of the liturgy.

Reviving the Tradition

Revelation through scripture and observation of the natural world are two ways to discern the significance in number. The other is by the consideration of numbers significant in the abstracted world of mathematics, in the tradition that stems from Pythagoras.

Pope Benedict XVI discusses the mathematical ordering of time, space and matter in his book the Spirit of the Liturgy. An extended quotation from this work is justified, in order to sum up and extend all that has been said so far.

"Among the Fathers, it was especially St Augustine who tried to connect this characteristic view of the Christian liturgy with the world view of Greco-Roman antiquity. In his early work 'On Music' he is still completely dependent on the Pythagorean theory of music. According to Pythagoras, the cosmos was constructed mathematically, a great edifice of numbers. Modern physics, beginning with Kepler, Galileo and Newton, has gone back to this vision and, through the mathematical interpretation of the universe, has made possible the technological use of its powers. For the Pythagoreans, this mathematical order of the universe ('cosmos' means 'order'!) was identical with the essence of beauty itself. Beauty comes from meaningful inner order. And for them this beauty was not only optical but also musical. Goethe alludes to this idea when he speaks of the singing contest of the fraternity of the spheres: the mathematical order of the planets and their revolutions contains a secret timbre, which is the primal form of music. The courses of the revolving planets are like melodies, the numerical order is the rhythm, and the concurrence of the individual courses is the harmony. The music made by man must, according to this view, be taken form the inner music and order of the universe, be inserted into the 'fraternal song' of the 'fraternity of the spheres'. The beauty of music depends on its conformity to the rhythmic and harmonic laws of the universe. The more that human music adapts itself to the musical laws of the universe, the more beautiful it will be.

"St Augustine first took up this theory and then deepened it. In the course of history, transplanting it into the worldview of faith was bound to bring with it a twofold personalization. Even the Pythagoreans did not interpret the mathematics of the universe in an entirely abstract way. In the view of the ancients, intelligent actions presupposed an intelligence that caused them. The intelligent, mathematical movements of the heavenly bodies were not explained, therefore, in a purely mechanical way; they could only be understood on the assumption that the heavenly bodies were animated, were themselves 'intelligent'. For Christians, there was a spontaneous turn at this point from the stellar deities to the choirs of angels that surround God and illumine the universe. Perceiving the 'music of the cosmos' thus becomes listening to the song of angels, and the reference to Isaiah chapter 6 ['Holy, holy, holy is the LORD of hosts; the whole earth is full of his glory.' Isaiah 6:1-3] naturally suggests itself.

"But a further step was taken with the help of the Trinitarian faith, faith in the Father, the Logos, and the Pneuma. The mathematics of the universe does not exist by itself, nor, as people now came to see, can it be explained by stellar deities. It has a deeper foundation: the mind of the Creator. It comes from the Logos, in whom, so to speak, the archetypes of the world's order are contained. The Logos, through the Spirit, fashions the material world according to these archetypes. In virtue of his work in creation, the Logos is, therefore, called the 'art of God' (ars = techne!). The Logos himself is the great artist, in whom all works of the art – the beauty of the universe – have their origin. To sing with the universe means, then, to follow the track of the Logos and to come close to him. All true human art is an assimilation to the artist, to Christ, to the mind of the Creator. The idea of the music of the cosmos, of singing with angels, leads back again to the relation of art to logos, but now it is broadened and deepened in the context of the cosmos. Yes, it is the cosmic context that gives art in the liturgy both its measure and its scope. A merely subjective 'creativity' is no match for the vast compass of the cosmos and for the message of beauty. When a man conforms to the measure of the universe, his freedom is not diminished but expanded to a new horizon."(4)

In the West, art and music has, for the most part, become detached from the great ordering and unifying principle of beauty. But potentially any work of art or artefact can be infused with the natural order by the presence of harmony within it: an icon, a landscape painting, a cheap poster in a card shop – even a toothbrush! St Augustine famously said that those who sing their prayers pray twice. The mathematical order of the universe adds grace to the visual imagery contemplated during prayer, to the architecture of the building in which we pray, and to the rhythm of our voices. Perhaps it is not fanciful to imagine prayer being "multiplied" not just twice but many times through the effect of cosmic resonance.

Of course, it is hard to see how many artist can truly reunite his art with the principle of liturgical number if he is not himself living a life infused with liturgical rhythm. For Catholics, this starts with the Mass and the Divine Office. From that foundation in Christ, we may begin to integrate all the other aspects of life.

Discovering Number Symbolism

The following is a brief sketch of some aspects of number as understood in the West around the 13th century.

One
This number traditionally signifies the unity of being, transcending all that exists. It is often represented by a circle, or else by a point. One is the number that when "squared", i.e. multiplied by itself, produces itself. Symbolically, the One is not merely the first in a series of numbers, but the number-beyond-number that includes all others, equivalent in that sense to the modern conception of Infinity. There are circular windows in Gothic cathedrals that symbolise this unity, for example at Chartres, Notre Dame and Rhiems.

Two
This is the number of polarity and division. In a Christian context it often represents the separation of matter and spirit. It can also symbolize the beginning of the process of creation, which in the Book of Genesis is described as taking place through a series of polarizations (heavens and earth, light and dark, etc.). Geometrically it appears as a line between two points, or as the central point and circumference of a circle. Interestingly, the length of the diagonal across a square made from sides one unit in length is √2, and the diagonal can be used to form the side of a second square exactly twice the area of the first (to put it another way, twice the square of the side equals the square of the diagonal). The medieval cathedral builders were familiar with this ratio, among others, which they constructed geometrically and employed in the design of spaces and arches to evoke a sense of harmony in the building. They saw as √2 representing the temperate mean between two extremes, 1 and 2, because the ratios 1:√2 and √2:2 are numerically equal.

Three
The Triad, of course, is manifested as a triangle. This presents a graphic image of how the number three can be seen as returning polarity to unity. The triangle and the circle together are used to indicate one God in three persons. Plato thought all four natural elements were made up of particles of different regular shapes constructed from triangles. An equilateral triangle has a particular significance in the natural order, as it is the simplest shape that can be repeated on a two-dimensional plane without leaving any space (just as the tetrahedron can be stacked in three dimensions without leaving space between). As such the triangle is one of the fundamental building blocks of patterned art.

Four
Quaternity represents the earth, or the entire material order, and is manifested as a square. The four traditional "elements" of earth, water, air and fire seem to correspond not to the types of atom described by modern science (hydrogen, helium, etc.) but to the four basic states of matter which we call solid, liquid, gas and plasma. Four is also the number of the Evangelists whose foundational writings are spread throughout the four corners of the world. The fourfold nature of the Gospels was established as a canonical principle by St Irenaeus in the second century.

Five
Five is traditionally seen as the number of the flesh, of life and of love. It is manifested as a regular pentagon or a five-pointed star, perhaps more often used in Islamic art than Christian. Inanimate nature conforms to the order of four- and six-fold symmetry, as in crystal structures. Living nature, on the other hand, appears to be ordered more often by five. The traditional association is with the five senses and the five fingers on each hand. In Christianity, there are also the five wounds of Christ.

Six
Six is both the sum and the product of the first three numbers: one, two and three. It is represented geometrically as a regular hexagon. The hexagon, comprised of six equilateral triangles, is the third shape that can be repeated in a plane without leaving space. If equal sized spheres are allowed to stack along any plane they will form repeated patterns of either six-fold or four-fold symmetry.

Seven
The seven days of creation (six days of work and one of rest) became the model for the seven days of the week that we still use today. This number appears many times in Scripture, being a particular favourite of St John. There are seven sacraments, seven deadly sins and seven virtues. Seven represents totality, since it is the sum of four, the number of the material order, and three, the number of the Trinity, and is related to twelve, as the product of these same numbers. It appears in iconographic and Western naturalistic art as the combination of a square and a triangle.

Eight
The so-called "eighth day of creation" is the day God became man. It is represented geometrically as a regular octagon. Thus the baptistry of the Duomo in Florence is an octagonal building.

Nine
As the product of three multiplied by itself, and at the same time the sum of three threes, nine echoes the Triad, indicating the impress of the Trinity on creation. It features most famously in the nine choirs of angels in the celestial hierarchy of Dionysius the Areopagite.

Ten
The Decad or Tetractys is the sum of the first four numbers, and represented by a arrangement of ten points arranged in an equilateral triangle: one over two over three over four. In so far as it is composed of a one and a zero, it also represents the return to unity in another way. As a doubling of five, it echoes the form of the human being, or the symmetry of hands and feet.

Twelve
Outside the Decad, twelve is probably one of the most richly symbolic of numbers – especially in a Christian context. There are twelve lunar months in the year, and the heavens have been divided into twelve signs of the zodiac. It is represented geometrically by a regular solid with twelve sides, the dodecagon, or by a twelve-pointed star composed of equilateral triangles.

A Way Forward

The use of proportion and number is a very useful tool for artists and designers, but it is possible to use it badly. Its employment does not guarantee beauty (just as knowing the rules of harmony will not necessarily make you a composer).

In many art schools today, students are encouraged by their teachers to produce shapes that do not look like and are not even reminiscent of the world we see around us. Their aim is to encourage the artist to create his own order. This is an impossible task. There is no "order" that is distinct from the natural order, only disorder. Works of art that manifest disorder can only be ugly and cannot be good. Even purely abstract art succeeds in being decorative only to the degree that it corresponds to the natural and metaphysical order. It is therefore important to study the art of number for two reasons. Not only will it enable us to re-establish a neglected Christian tradition, but it will help our figurative artists to infuse their work with beauty through a correspondence to the natural order.

There has, of course, been much speculation (especially in New Age circles) about the sophisticated mathematics employed by the ancient and medieval masons. My impression is that much of this goes too far. Sacred number was a tool to use in the pursuit of beauty rather than an end in itself. The schemata employed in the construction of most of the great buildings tended towards simple arrangements of shapes based upon combinations of largely whole numbers. This was, after all, the Pythagorean tradition. In the Renaissance with the development of the new mathematics and a more elaborate musical theory we get a tendency to complicate things much more (possibly sometimes to the detriment of beauty)

There are three areas of study that would help us fully to establish the principles that govern the creation of the art of "liturgical number" as a living tradition. These are developed in the theoretical course for artists and patrons offered by the Maryvale Institute (Art, Inspiration and Beauty) and will help to shape some of the work that we hope to do in connection with the new school of art we are creating in Oxford.

•   Study of the great Christian art of the past: in order to discover the principles by which this tradition operated.

•   Study of the art of traditional non-Christian cultures. Islamic art would be a prime example. Islam, for the most part, forbids any figurative art, so the manifestation of number in abstract art has been of much greater interest to Islamic artists than to Christian. As such, Islam not only preserved the art of geometric pattern, but developed it into its own highly complex and distinctive art form that far more comprehensively manifests number than its Christian forebears in Byzantium. Christians would be helped to rediscover the art of number through the study of Islamic art – even though they should also be aware of underlying theological differences that will have shaped the tradition in subtle ways.

•   Study of the history of science and mathematics, and of these subjects in their own right. It may be possible not only to re-establish a tradition, but develop it creatively in the light of new knowledge.

Perhaps through this process we will come to a renewed appreciation of the importance of beauty and harmony in what we do and make, and of the cosmic significance of this harmony – as expressed by Lorenzo, in Shakespeare's The Merchant of Venice, gazing at the starry sky from the garden of Belmont, a villa near Venice:

"Look how the floor of Heaven
Is thick inlaid with patines of bright gold
There's not the smallest orb that thou beholdest
But in his motion like an angel sings
Still quiring to the young-eyed cherubims
Such harmony is in immortal souls."

A version of this article is scheduled to appear in Second Spring issue 8 towards the end of 2006. David Clayton is Managing Director of ResSource Ltd.

NOTES

1 Erik Peterson, The Angels and the Liturgy (Herder & Herder, 1964), pp. 22, 50. back

2 Dan 3:57-88, 56. back

3 Joseph Cardinal Ratzinger, The Spirit of the Liturgy (Ignatius Press, 2000), pp. 70, 76. back

4 Joseph Cardinal Ratzinger, The Spirit of the Liturgy (Ignatius Press, 2000), pp. 152-4. back

Back to Articles