It is commonly supposed that abstract art - using the term here to mean non-figurative art - is an invention of the 20th century, starting, perhaps, with figures such as Kandinsky. It might surprise some to discover that this is not the case. The Christian tradition of abstract art which aims to represent number through geometric form has existed as long as there has been Christian art. And because, like Christian figurative art, it owes its inspiration to the ancient Greek and Roman art which preceded it, we can say that it has its roots in Western culture that go back well before the Christian era.
Examples of mathematical art in the Christian tradition are the decorative patterns that we might see in the architectural details, borders, and tiled floors of a Gothic or Romanesque church. If we do not know what we are looking at it, is easy to overlook these or to dismiss them as the frivolous self-indulgences of artisans. They are not. They are very likely carefully thought out and designed to connect the numbers or mathematical relationships they portray to the artifact they adorn, and in so doing, connect any symbolic association of the mathematics to that artifact too.
We might think of a simple six-fold design on a church floor. This one is a detail of one in Santa Maria in Cosmedin, in Italy.
In addition to this crystalline, geometric-based art, there is a softer, calligraphic decorative art that might also be considered mathematical in concept. The sweeping intricate lines of a Celtic illuminated manuscript, of an idealized acacia plant in a Romanesque mosaic, or in the baroque scroll of an ornate gilt frame are all inspired by the grace and flow of the patterns of nature particularly as observed in vegetation. The shapes are idealized by introducing symmetry to highlight order. This was done, one assumes, intuitively by the artists, but to the modern eye, they are evocative of the mathematics of nature as expressed by modern science; they bear the mark of the exponential curves, parabolas, and sine waves that the scientist uses to describe the natural order.
Here is a picture of idealized vegetation, a detail of a 12th century mosaic in the church of San Clemente in Rome. The crucifixion is a Tree of Life springing forth from idealized vegetation. There are additional limbs emanating from this base which morph into more abstracted decorative scroll designs.
In this illumination from the mid-15th century, we can see the border pattern extending the degree of idealization so that is closer to an abstract design. It is attributed to an anonymous artist called the Master of Girart de Roussillon.
In the border of this illumination of St Matthew in the 7th century Book of Durrow, we see similar smooth flowing lines which are pure form. They are not representations of vegetation but the flow of line is derived from it.
I am often struck by how much effort was put into such art, and how little it is noticed today. Look, for example, at this picture, one of the Raphael Rooms in the Vatican Museums. Some may recognize the fresco on the right as the School of Athens. Every square inch of this room has been decorated in some way.
The floor has ornate geometric art, and the patterned borders around the frescoes contain both geometric art and ornate patterns based upon idealized forms of the flowing pattern of nature. My guess is that for every commentary that discusses the basis of these patterns, there are a thousand that ignore them and focus exclusively on the figurative art.
Number Symbolism
Anyone who has read the Church Fathers’ commentaries on Scripture will be aware of the importance they attached to the symbolism of numbers. A number not only describes the quantity, but also can communicate a relationship with something else closely associated with that number, and so has a qualitative, that is symbolic, aspect to it. We could say that it not only tells us ‘how much’, but also ‘how’.
A word of warning here. This is not meant to be a secret code that only the cognoscenti can read. This is supposed to speak naturally of underlying truths. We should remember, for example, that when something is said to have happened three times, that doesn’t automatically mean that it is connected to the Trinity. The author might simply be giving us a historical detail that has no further symbolism. Symbolism works when a group of three seems naturally connected to the Trinity in other ways. So, for exmaple, the three angels who appeared to Abraham and Sarah in the book of Genesis are often seen as representing the Trinity, not merely because there were three of them, but because their words and conduct and the nature of their interaction with Abraham also indicated such a connection.
Nor is the numerical pattern a cause of such a connection. Again, we can’t make something trinitarian simply by doing it three times or having three of it. Rather, the numerical pattern is symbolic when it is a reflection, an outward sign, of what is already there.
In thinking about the interpretation of Scripture and how it might usefully translate into art, consider the genealogy of Jesus given at the beginning of St Matthew’s Gospel. He begins with Abraham and tells that there are 14 generations from Abraham to David, 14 from David until the Babylonian exile, and 14 from the Babylonian exile to Christ. The significance of the numbers here has been disputed, but the fact that they are important generates less controversy. Given that Matthew specifically mentions the numbers, and he seems to stretch and squeeze the lineage a bit so as to create these three even divisions in his presentation, one can assume at least that he thought it was important.
The interpretation that seems most convincing to me is that 14 is important because it is a number that symbolizes David. According to the Fathers of the Church, the Gospel according to Matthew was written in Palestine, almost certainly in Aramaic, and was addressed mainly to Jews living in that region. In this language, the characters of the alphabet are also used for numbers, and therefore, every name also has a numerical value. When the letters that comprise the name David are treated as numbers and added together, they create the number 14. This speaks additionally to the point that Christ is in the line of David, from whom, according to prophecy, the Messiah would emerge. But why three bundles of 14? The common explanation does connect this to the Trinity as the number of perfection.
How do we represent this artistically? There would be many reasons why one might wish to represent this geometrically and associate it with Christ, but if we try to do so, we can see quickly that the number 14 does not lend itself to geometric representation. If you were to see a design with a 14-sided geometric shape (a tetradecagon) most of us would have to stop and carefully count the sides in order to know which number it represented. It isn’t obvious in the way that we can instantly see that a triangle speaks of three. Also, they are difficult to construct geometrically, and difficult to incorporate into repeat designs that would enable tiles to be created and pieced together. Nevertheless, I did find this pattern from in a Gothic church in the Italian town of Civita Castellana, which has a 14-pointed start at its center.
The number 8 is another number associated with Christ. His life ushers in a new covenant, and His life, death and resurrection are described as the Eighth Day of creation. This connection is more widely accepted and more broadly rooted in the tradition than that of the number 14, and so its symbolic power is greater. Furthermore, octagons are easily created geometrically, and more easily recognized than tetradecagons. This is why baptistries and fonts are traditionally octagonal and not 14-sided. Also, octagonal shapes are relatively easily incorporated into repeated patterns that can be used on tiled floors. So we very often see patterns built around the multiple octagons in the central aisle of a church. It is a geometric representation of Christ who described himself as ‘the Way’. Here is an example from the cathedral of Monreale in Sicily.
Both of these last two designs are created in an art form called ‘cosmatesque’, which is characterized by colored marble cut to shape before being placed into the design. The name comes from the Roman Cosmati family, which over four generations in the 13th century provided seven expert artisans capable of doing such work. Cosmati work is different from a mosaic in which the design is made of different colored pieces of glass or stone called tesserae, which are all the same size.
This point about the ease of design and of reproducing it in the material suitable for a floor is an important one. It doesn’t matter how expansive or imaginative the design is; if it is too difficult to turn it into a cut stone floor it is never going to be made. Similarly, this might also mean that the design is more decorative than symbolic. We should not assume that every geometric shape is imbued with deep spiritual significance. One could imagine that at times the designers of church floors, as much as the hallway in a suburban house, were happy to make use of those shapes that were most easily made into tiles and fitted together without gaps - triangles, squares and hexagons, or a combination of octagons and squares - and content merely to not run contrary to any spiritual principle. The beauty they lend to a place is profound nevertheless, and will effect on us for the good.
The number six is important in geometric art because of the ease of creation of patterns with sixfold symmetry. It has a symbolic importance in that six is a ‘perfect number’ in traditional mathematics. A perfect number is one in which the sum of all numbers that are perfect divisions of the number equals the number itself. That sounds complicated in words, but more easily explained with examples. The numbers that are perfect divisions of 6 ie can be divided into it perfectly, are 1, 2, and 3. The sum of 1, 2 and 3 is 6. So six is a perfect number. Similarly, the numbers that are perfect divisions of 28 are 14, 7, 4, 2 and 1. The sum of 14, 7, 4, 3, 2 and 1 is 28. So 28 is a perfect number.
The Church Fathers commented on how the world was made in six days. This period is called the hexaemeron. St Augustine tells us of the hexaemeron that ‘These works are recorded to have been completed in six days (the same day being six times repeated), because six is a perfect number — not because God required a protracted time, as if He could not at once create all things, which then should mark the course of time by the movements proper to them, but because the perfection of the works was signified by the number six. For the number six is the first which is made up of its own parts, i.e., of its sixth, third, and half, which are respectively one, two, and three, and which make a total of six.’
Examples of mathematical art in the Christian tradition are the decorative patterns that we might see in the architectural details, borders, and tiled floors of a Gothic or Romanesque church. If we do not know what we are looking at it, is easy to overlook these or to dismiss them as the frivolous self-indulgences of artisans. They are not. They are very likely carefully thought out and designed to connect the numbers or mathematical relationships they portray to the artifact they adorn, and in so doing, connect any symbolic association of the mathematics to that artifact too.
We might think of a simple six-fold design on a church floor. This one is a detail of one in Santa Maria in Cosmedin, in Italy.
Here is a picture of idealized vegetation, a detail of a 12th century mosaic in the church of San Clemente in Rome. The crucifixion is a Tree of Life springing forth from idealized vegetation. There are additional limbs emanating from this base which morph into more abstracted decorative scroll designs.
Number Symbolism
Anyone who has read the Church Fathers’ commentaries on Scripture will be aware of the importance they attached to the symbolism of numbers. A number not only describes the quantity, but also can communicate a relationship with something else closely associated with that number, and so has a qualitative, that is symbolic, aspect to it. We could say that it not only tells us ‘how much’, but also ‘how’.
A word of warning here. This is not meant to be a secret code that only the cognoscenti can read. This is supposed to speak naturally of underlying truths. We should remember, for example, that when something is said to have happened three times, that doesn’t automatically mean that it is connected to the Trinity. The author might simply be giving us a historical detail that has no further symbolism. Symbolism works when a group of three seems naturally connected to the Trinity in other ways. So, for exmaple, the three angels who appeared to Abraham and Sarah in the book of Genesis are often seen as representing the Trinity, not merely because there were three of them, but because their words and conduct and the nature of their interaction with Abraham also indicated such a connection.
Nor is the numerical pattern a cause of such a connection. Again, we can’t make something trinitarian simply by doing it three times or having three of it. Rather, the numerical pattern is symbolic when it is a reflection, an outward sign, of what is already there.
In thinking about the interpretation of Scripture and how it might usefully translate into art, consider the genealogy of Jesus given at the beginning of St Matthew’s Gospel. He begins with Abraham and tells that there are 14 generations from Abraham to David, 14 from David until the Babylonian exile, and 14 from the Babylonian exile to Christ. The significance of the numbers here has been disputed, but the fact that they are important generates less controversy. Given that Matthew specifically mentions the numbers, and he seems to stretch and squeeze the lineage a bit so as to create these three even divisions in his presentation, one can assume at least that he thought it was important.
The interpretation that seems most convincing to me is that 14 is important because it is a number that symbolizes David. According to the Fathers of the Church, the Gospel according to Matthew was written in Palestine, almost certainly in Aramaic, and was addressed mainly to Jews living in that region. In this language, the characters of the alphabet are also used for numbers, and therefore, every name also has a numerical value. When the letters that comprise the name David are treated as numbers and added together, they create the number 14. This speaks additionally to the point that Christ is in the line of David, from whom, according to prophecy, the Messiah would emerge. But why three bundles of 14? The common explanation does connect this to the Trinity as the number of perfection.
How do we represent this artistically? There would be many reasons why one might wish to represent this geometrically and associate it with Christ, but if we try to do so, we can see quickly that the number 14 does not lend itself to geometric representation. If you were to see a design with a 14-sided geometric shape (a tetradecagon) most of us would have to stop and carefully count the sides in order to know which number it represented. It isn’t obvious in the way that we can instantly see that a triangle speaks of three. Also, they are difficult to construct geometrically, and difficult to incorporate into repeat designs that would enable tiles to be created and pieced together. Nevertheless, I did find this pattern from in a Gothic church in the Italian town of Civita Castellana, which has a 14-pointed start at its center.
This point about the ease of design and of reproducing it in the material suitable for a floor is an important one. It doesn’t matter how expansive or imaginative the design is; if it is too difficult to turn it into a cut stone floor it is never going to be made. Similarly, this might also mean that the design is more decorative than symbolic. We should not assume that every geometric shape is imbued with deep spiritual significance. One could imagine that at times the designers of church floors, as much as the hallway in a suburban house, were happy to make use of those shapes that were most easily made into tiles and fitted together without gaps - triangles, squares and hexagons, or a combination of octagons and squares - and content merely to not run contrary to any spiritual principle. The beauty they lend to a place is profound nevertheless, and will effect on us for the good.
The number six is important in geometric art because of the ease of creation of patterns with sixfold symmetry. It has a symbolic importance in that six is a ‘perfect number’ in traditional mathematics. A perfect number is one in which the sum of all numbers that are perfect divisions of the number equals the number itself. That sounds complicated in words, but more easily explained with examples. The numbers that are perfect divisions of 6 ie can be divided into it perfectly, are 1, 2, and 3. The sum of 1, 2 and 3 is 6. So six is a perfect number. Similarly, the numbers that are perfect divisions of 28 are 14, 7, 4, 2 and 1. The sum of 14, 7, 4, 3, 2 and 1 is 28. So 28 is a perfect number.
The Church Fathers commented on how the world was made in six days. This period is called the hexaemeron. St Augustine tells us of the hexaemeron that ‘These works are recorded to have been completed in six days (the same day being six times repeated), because six is a perfect number — not because God required a protracted time, as if He could not at once create all things, which then should mark the course of time by the movements proper to them, but because the perfection of the works was signified by the number six. For the number six is the first which is made up of its own parts, i.e., of its sixth, third, and half, which are respectively one, two, and three, and which make a total of six.’
As a final example in this brief run through the subject of number symbolism and its representation in art, another common shape that we see in geometric art is the quincunx. This originated in Roman times and began as a pattern on a 5 denarii coin. It seen today in the arrangement of five dots on a domino, which has a central circle surrounded by four with their centers tracing a square. It came into the Christian tradition and is common in sanctuary floors and churches. This might be considered to be a geometric representation of a common figurative image of Christ in Majesty, in which the central figure Christ, the Word, sends the four Evangelists out to the four corners of the world. This one is from a 13th-century German manuscript called the Codex Bruschal. The four evangelists are depicted as the four faces of the angels described by Ezekiel in his vision. The Church Fathers differ on which Evangelist each represents, but today we generally go with the symbolism that was adopted by St Jerome: St Matthew is the man, St Mark is the lion, St Luke is the ox, and St John is the eagle.
