MATHS AS THE SECRET TO BEAUTY | GREEK ANTIQUITY
Updated: 3 days ago
In my opinion the exploration and development of mathematics applied to art and design truly flourished for the first time in history during Greek antiquity. One does not need to have ever researched this to know it. All you need to do is look at a Greek vase or any Greek depiction of human form. The immaculate sense of ‘perfected’ proportions that just feels ‘right’ and that overriding impression of perfect visual harmony can only be possible with the implementation of mathematics and its accomplishment has not been surpassed since.
It is impossible to go into any real depth exploring this topic without context. The context is the belief system the Greeks held. For the Greeks nature, mathematics, religion, spirituality, science and philosophy were all inextricably linked, they were one and that whole was called the Cosmos. Let us begin with nature, the Greeks viewed nature as being two-fold: the microcosm and the macrocosm: where the microcosm is the human (anatomy and soul) and the macrocosm is all remaining nature of the universe. They saw one as an identical mirror to the other, in that all components and workings of one, such as the elements of earth, fire, water and air, mathematics, geometry and the soul, exist and operate in a human as they do so everywhere else in the universe. They believed the universe had a world soul and humans had a fragment of that soul. The Greeks believed the divine realm had a language and that language was mathematics, since decoding nature's secrets, whether in music or astronomy, came down to the pureness of numbers. This process of decoding was seen as religious and spiritual practice as by understanding the workings of the planets or of anatomy one was uncovering the mathematical blueprint of the divine creator (pioneering Greek philosophers and scientists gravitated towards a singular cosmic creator). Yet even their polytheist religion directly correlated to their overall mind-set as their Gods and Goddesses were believed to take shape in human form, and that that form was idealised and mathematically perfect just like the rest of the cosmos. They believed there was a divine order to everything including beauty and maths was the key to that divine order and hence beauty. They believed the more they unlocked that truth and manifested it in art and architecture, the closer they were to the divine.
The main mathematical principles and themes discussed in this blog post are: integer ratio types (Pythagorean ratios, Plato’s Lambda, Symmetria), non-integer ratio types (root ratios, the golden ratio), the logarithmic curve, the Fibonacci Sequence, Contrapposto (Chiasmus, Antithesis and Eurythmia) and Perspectival Anamorphosis.
THE GREEK VASE
Let us begin with the Greek vase. From archaeologists and art historians to mathematicians, fascination over ‘the secret’ to the accomplishment of the perfected form of a Greek vase has persisted for generations.
There are two ways that the extent to which the application of mathematics by the Greeks in the design of their pots can be analysed. One is in relation to the actual making process and ascertaining if mathematical principles were employed during construction and the other is to analyse the pots’ forms after the fact, applying mathematical principles in retrospect to theorise if mathematical principles were used to design what would be the final result.
Proportions
Regarding the construction process, from what I have gathered from my research, the most basic question I have has not yet been investigated in any real depth. My question is whether or not the Greeks used a modular unit system to allocate their clay into parts when constructing a vessel. For example did they begin with 12 parts and allocate 8 or 9 to the body, 1 or 2 to the foot and neck and 1 or a ½ to each handle? This question is important to me because it will help to solve whether or not the immaculate proportions of a Greek vase, stem in part, to proportionate thinking when it came to the quantity of material used for each part. At present, I don’t think we have a definitive answer. Since we know the Greeks used Pythagorean ratios in design (explored further in architecture section), which were whole integer ratios such as 2:1, 3:2, 4:3, 4:9 and so on, it seems logical to me to pose this question.
However an area where there has been a fair amount of investigation in an attempt to uncover the secret to those idyllic proportions is extensive measuring of Greek pottery and analysis of those measurements to determine if there are ratios between things like width and height and so on. Two of the most well-known texts are Dynamic Symmetry: The Greek Vase by Jay Hambidge (1920) and Geometry of Greek Vases by Lacey Davis Caskey (1922). These texts conclude that based on their research, having measured and analysed multiple vessels, that there is a mathematical relationship between the measurements of the pots, based on both integer ratios, root ratios and the golden ratio.
Regarding integer ratios, Caskey very thoroughly illustrates how there are such a great number of vases that either exhibit exact integer ratios between varying parts of the vase, or ratios that are very close to integers, that it cannot be a coincidence. I completely accept this. His evidence strongly supports that integer ratios were used by the Greeks when designing vessels. The fact that some vessels hold more exact ratios than others, in my mind simply means that some makers were using the guideline of integer ratios more meticulously whereas other makers were using it as a mere guide and did not mind veering off. Examples of this can be seen in all the vessel types. For example for many Amphorae the ratio between their width and foot diameter was 2:1, for other Amphorae and Oinochoai the ratio between the width or diameter and the diameter of shoulder was also 2:1. The same ratio of 2:1 was found in Pelikai between their width and smallest diameter of neck, and in Krateres between their height or diameter and the foot diameter. The same ratio of 2:1 was also found in Kantharos vases (whose proportions I particularly like) in the ratio between height aspects, between height and diameter and also between width and diameter. Another vessel whose proportions I love is the Kylix. Integer ratios often found in Kylikes were 1:1 between the height and diameter of foot. Some Stamnoi were found to have ratios of 2:2:1 between height, width and diameter of foot.
Diagrams depicting vase types and how parts of a vase are named
However the most extensive aspect of research in both books relates to root ratios and the golden ratio. One very simple example of a root ratio is through the use of root 2 as a proportion. For example Caskey's analysis of a classical period Attica Oinochoe with Trefoil Lip (see below), shows that the height vs. width ratio aligns perfectly with root 2 proportions. These mathematical applications of both Hambidge and Caskey were subsequently widely rejected by archaeologists, who, although admitting the Greeks were aware of certain relevant mathematical principles, did not believe there was enough historical evidence to support that the Greeks used such highly accomplished, deliberately mathematical means of design. They also argue that many complicated, mathematical, rectangular arrangements could be applied to the vase sections in order to find mathematical relationships, so Hambidge and Caskey’s research proved nothing. However root relationships were found on a number of vessels, so if for a moment we accept that those mathematical relationships are there it then raises a very interesting question: Did the Greeks derive these ratios through practical, making means which ‘accidentally’ produce mathematically perfect root proportions or were they mathematically calculated? Since archaeologists believe the second is highly unlikely I think it is worth considering that the first is plausible. So for example if we take the example discussed above all the potter would have needed at their disposal would have been a perfect square, let’s say a flat, square piece of wood. Where any side of the square was used as a guide for the vessel width. If that square were then cut diagonally, and then the diagonal length were used as a guide for the vessel height, you would have the exact root 2 proportions of the below illustrated example. I don’t find this a far-fetched possibility at all, in fact as a designer/maker myself I find it extremely credible based on how I work. I will often take a scrap of paper and use off cuts from the same piece as a guide when designing, for example.
Mathematical diagrams of root 2 including overlay on outline of Attic Oinochoe with Trefoil Lip {depicting seated elderly man and standing warrior}, 430-420 BC, Boston Museum, Page 138 of Caskey's Geometry of Greek Vases, analysing the vase & the vase illustrated
Geometry of Curvature
The other argument as to the ‘secret’ to the beauty of Greek vases is that of sound geometric forms. Curves that have been perfectly modelled by the potters on the wheel blending in and out of one another to form the foot, body, shoulder, neck, lip and so on creating one united complete form constituted of multiple curvatures.
Studies of these curvatures have included sections of Greek vases being overlaid with mathematically perfect logarithmic curves, often demonstrating that their alignment is virtually exact. This then raises the exact same question as that just discussed in the previous section: Did the Greeks derive these curves through practical, making means which ‘accidentally’ produce mathematically perfect logarithmic curves or were they mathematically calculated? Jay Hambidge’s text argues that the curvatures were mathematically calculated. As well as applying root number proportions, he also looks at the golden ratio and implements that substantially into his theories. Again, there is no solid proof that the Greeks consciously worked with complicated ratios such as the golden ratio when designing. However a logarithmic curve is based on the golden ratio (golden rectangle). So even if the Greeks weren’t implementing these principle consciously, they could have been doing it ‘by chance’ by creating logarithmic curves through practical means. For example if one takes a plank or wood where the width of the plank is wider at one end and narrower at the other and one bends it, you get a logarithmic curve (provided the wood's thickness is uniform). Based on the same logic as in the previous section, I believe it is plausible that a plank of wood could have been used as a profile guide when modelling the vessel on the wheel. Additionally using a fixed wooden gauge or a set of callipers (both of which we know the Greeks used) set to a ratio of 2:3 (which would have been in accordance to Symmetria - discussed further in depiction of anatomy section) whilst spinning on the wheel to shape the vessel curve, would have also automatically have created a logarithmic curve.
The following is an explanation as to what the golden ratio is, a golden rectangle and a logarithmic curve. The rule of the golden ratio dictates that if you take a line and divide it at a point where the ratio between the shorter part of the line and the longer part of the line is the exact same ratio as the longer part of the line to the overall line length then that proportion must be 1.618. This point can easily be found by practical means by starting with a basic square measuring 1x1. If you take the midpoint on any side of that square and use that as a central compass point where the compass passes through the top right corner of the square, when you trace that circle, the point that crosses the bottom line is the lower right hand corner of the golden ratio line. Hence the creation of the square and adjoining rectangle is the creation of the golden rectangle. If you then sub-divide the rectangle into a perfect square, it’s ratio to the initial square is the golden ratio, and that process continues as you sub-divide every remaining rectangle into a square. The golden triangle follows the same principle. In both cases of the rectangle and the triangle, if you then draw a circular curve joining the same points of each square or triangle you have a logarithmic spiral. A logarithmic spiral dictates that every part of the curve bears the exact same proportions just to a different scale.
Diagrams illustrating golden ratio, golden rectangle, golden triangle and corresponding logarithmic spirals.
The exploration of these mathematical principles (golden ratio and root ratios) is not random, these mathematical models are derived from examining nature such as floral formations and shells. Hence being why the golden ratio came to be seen as a very holy mathematical principle in later eras such as the Italian Renaissance. Another mathematical principle which was understood later and closely aligns with the golden ratio, and is also very applicable as a mathematical model to nature, is the Fibonacci sequence. This is the principle that dictates in a numerical sequence that every number is derived as a sum of the two previous numbers, so: 0,1,1,2,3,5,8,13,21,34 and so on. As the sequence progresses, after the first few progressions, the ratio between the number and its predecessor always remains extremely close to the golden ratio, hovering just below or above it. In other words the numbers overall increase in line with the golden ratio. Examples in nature where formation growth can be roughly modelled by these principles are the nautilus shell and the sunflower.
Nautilus shell and the sunflower
It is also worth pointing out that there is an overlap between the golden ratio and root ratios in that a root 5 rectangle is identical to the golden rectangle where it's sub-rectangle is flanked on either side of the main square (not just on one side).
Diagrams illustrating golden rectangle with sub-rectangle flanked on either side & root 5 rectangle
THE GREEK DEPICTION OF ANATOMY
Proportions
Arguably, what the Greeks achieved in terms of their depiction of anatomical proportions, whether it be human or the depiction of a horse, to this day seems so perfect, particularly in their sculpture, but also in their painted depictions on vases, one wonders if those immaculate proportions have been surpassed in the art world since.
In order to create an idealised depiction of human or animal form the Greeks believed in Symmetria which was about using mathematics in design to attain visual harmony, and for that they employed a system called Polykleitos’s Canon. This dictated that a single element was to be used as a proportional building block for the whole. Hence starting with a very small anatomical unit such as the little finger and use that as their basis for the entire form. A prime example of this are the Riace bronzes that date to 460-430 BC. The ratio between the tip of the little finger and its full length is 1:3. The ratio between the finger width and palm width is 1:4. The ratio between the widths of the palm to the forearm is 1:4. The ratio between the torso height and leg height is 1:2. The full height of each figure is 7 times the height of the head (or 6 times the length of the forearm). And very interestingly analysis of the facial features has concluded ratios between aspects of the face are exactly those of the golden ratio. Many scholars believe that ratios found are too precise to be coincidental, hence concluding, again, to the disagreement of many, that the Greeks were in fact consciously implementing the golden ratio in their designing.
The Riace bronzes, 460-430 BC, Museo Nazionale della Magna Grecia
Composition
Compositionally the Greeks also employed mathematics in the design of their sculptures. For example the Artemision Bronze that dates to 460-450BC, when viewed flat-on frontal or rear view, fits into a virtually exact square measuring 209cm x 210cm. Additionally many bronzes such as the Riace bronzes were compositionally designed and executed based on Chiasmus, Antithesis and Eurythmia. This was a mathematical design philosophy based on the rhythmic notion of counterbalance and weight redistribution between active and relaxed parts of the body, centred on the axis of an X.
Artemision bronze (460-450BC) with perfect square overlaid & Riace bronze (460-430BC) with X axis overlaid at intersection of tense and relaxed body parts
Spatial Context
Additionally, the Greeks were always very contextually sensitive when designing sculpture with regards to the human viewpoint, they would always factor in the principle of foreshortening when designing proportions if needed, so for example if the sculpture were to be seen at a height above the viewer, not at eye-level then they would apply elongation where they felt it necessary as is the case with the elongated body of the charioteer of Delphi (that was positioned at a height) or the increase in thickness of the Parthenon frieze as it increased in height and therefore distance from human eye-level on the ground, creating a slight angle to meet the viewpoint better.
This also applied to their painted depiction of anatomy on pottery, where they also applied Symmetria, and where they took the curvature of the vessels and the prioritised frontal viewpoint into account. In that figures to the left and right of the vase (where curvature was most prominent when being viewed from the front) were painted wider so as to compensate for the curvature and match the central figures. Also figures that were painting vertically down a curve were elongated to compensate for that curve's foreshortening as well. Below is an example of this kind of adjustment made in the depiction of horses where the legs on the lower mid-section part of the vase are elongated to compensate for foreshortening from the most prioritised viewpoint. The name for this kind of visual manipulation is Perspectival Anamorphosis.
The Greeks were the first to employ Perspectival Anamorphosis in their design and one could argue that this was the earliest precursor of Op Art (20th century optical illusion art).
Amphora attributed to the Kleophrades painter (c.525-500BC), The Met
GREEK ARCHITECTURE
Pythagorean ratios
The Greeks were the first to realise that nature could be decoded through mathematics with the discovery of Pythagorean ratios. This was when Pythagoras discovered that musical harmonies are governed by precise mathematical ratios of whole numbers. As someone who has a deep love for music I find this breath-taking. As one can therefore say that music was the first ever bridge between nature and what became science and later technology. Music together with the human mind birthed science.
These were the musical ratios Pythagoras deduced by studying the sound of instrumental strings:
1:1 – Unison (The same note)
2:1 – The Octave (The fundamental division)
3:2 – The Perfect Fifth (The most consonant interval after the octave)
4:3 – The Perfect Fourth (The complement to the fifth)
The Greeks therefore believed these ratios were an order of the divine and sought to take that order and translate it visually. The most well-known example of this is the Parthenon where architects Iktinos and Kallikrates sought to create a visual embodiment of music. They used the perfect fifth ratio of 3:2 squared as their guideline for design:
³/₂ x ³/₂ = ⁹/₄
Regarding the temple's ground plan, this therefore dictated the ratio between the length and width of the Parthenon's footprint. It also dictated the ratio between the overall area of the footprint and the inner area defined by the inner walls. The ratio between the small and larger area of the walled space is 9:4. These ratio relationships are endless and dictate every choice made for every line drawn in the design.
Regarding the façade, again the ratio between the width and height of the rectangular segment (ignoring the steps and pediment) is 9:4. This ratio also dictates the relationship between the column spacing (distance from the centre of one column to another) and the column width itself (widest point).
Plato’s Lambda
78 years after the Parthenon was completed the Pythagoreans developed another mathematical framework know as Plato’s Lambda (as it was popularised by Plato).
The principle dictates the following numerical progressions, which can create a geometric Lambda shaped design: Λ, hence its name.
2⁰ = 1
2¹ = 2
2² = 2 x 2 = 4
2³ = 2 x 2 x 2 = 8
3⁰ = 1
3¹ = 3
3² = 3 x 3 = 9
3³ = 3 x3 x 3 = 27
Just like Pythagorean ratios, Plato's Lamda holds an entire philosophy in its thinking which has a deep interplay with music. The numeric progression based on 2, is an even number progression and to the Greeks represented the material world. The numeric progression based on 3 is an odd number progression and represented intellect and spirituality. The relationships between the numbers in these progressions, within the same progression or across both progressions align with the Greek musical scale:
The doubling interval (1 → 2) represents a musical octave.
The tripling interval (1 → 3) represents an octave plus a perfect fifth.
The ratio 2 → 3 forms the perfect fifth, while 3 → 4 forms the perfect fourth.
The Greeks' study of the planets led them to believe their movements could be directly mapped by these numbers, making them conclude that the universe in motion played a song known as The Music of the Spheres.
When I visited the Polycentric Museum in Vergina, I saw that according to archaeologist Dr. Angeliki Kottaridi this concept too was applied in the design of architecture. A diagram displayed theorised how the concept was used to guide the ground plan design of the Palace of Agai of King Philip II of Macedon (father of Alexander the Great), built in the 4th century BC.
Geometric design guide correlating to Plato's Lamda overlaid onto the ground plan of King Philip II of Macedon’s Palace of Agai, 4th C. BC
It therefore makes sense that squares were used for the progression of even numbers as squares to the Greeks represented stability and the material world: earth, water, fire and air. Correspondingly dodecagons were used for the progression of odd numbers as the number 12 represented harmony and cosmic wholeness as it referenced: the number of times the moon orbits the earth annually and the number of zodiac signs, hence also being their choice for the number of Olympian Gods. The circle represented infinite movement and heaven, hence the dodecagons were a bridge between the square (earth) and a circle (heaven).
SUMMARY
The Greeks' application of mathematics to art and design is vast and endless as it was an intrinsic part of their philosophy, religion and spirituality. Numbers and geometry were not a cold and meaningless means to an end, they held philosophical meaning and symbolism.
This blog has proved that within a single work of design whether a vase or the Parthenon, the Greeks' application of mathematics was extremely layered ranging from: initial building blocks in design being guided by a mathematical principle and geometry, to the development of that design, be it the aspect of a vase painting or the Parthenon frieze, where other mathematical principles were at play, be it the tweaking of numbers or optical illusion. That even within this incredibly orderly and idealised philosophy of design, there was always the acknowledged in the designing process for adaptation so as to balance the rigidity of maths with the actual perceived human experience. Hence the Greeks held constant awareness that what materialised in the visual arts had to be a balance between mathematical order and contextual sensitivity relating to spatial human perception.
Maths was a seamless part of an integral mind-set where science, art, music and religion were all one and whatever mathematical principles were uncovered in one area were automatically considered innate in all the others and hence adapted and applied accordingly across the board.
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