Egyptian images were often designed using a grid that measured 18 units to the top of the head. Using this system they were able to achieve a visual harmony and consistency of image making over many years. The fact that they applied this grid was very influential in the development of canons for Greek sculpture. The Egyptians measured their figures in cubits and cubits were the length from the elbow to the tip of the thumb. Therefore there is a close correspondence with the human body and its natural proportions. The other units of measure were the palm, or hand width and the finger, again cementing ideas of body proportions into the way things were measured. This relationship of course still exists, our current measure of the 'foot' being an obvious example.
The 18:11 ratio as used between 2650BC and 2181BC
An Egyptian figure measured in cubits
The Greeks developed two competing canons of proportion: the original, developed by the sculptor Polykleitos in the 5th century BC and later in the 4th century by Lysippos. The Greek use of modes (ratios or measures) were designed to give the viewer a sense of harmony by making sure that ratios for parts of the body were both relative to each other and to the whole.
Comparison of the the Canons of Polykleitos and Lysippos
This Greek concept of ratios that give the viewer a sense of harmony by ensuring that all parts of the body were both relative to each other and to the whole, would be of vital importance to Renaissance art.
Leonardo: Vitruvian Man
Leonardo's drawing of the Vitruvian Man is based on his understanding of the writings of the Roman architect Vitruvius and Vitruvius took his ideas from the Greeks. In Leonardo's drawing the head measured from the forehead to the chin is exactly one tenth of the total height, and the outstretched arms are as wide as the body is tall. Vitruvius linked ideal human proportions to geometry in Book III of his treatise De Architecture, describing the human figure as being the principal source of proportion among Classical orders of architecture. Interestingly Vitruvius set out the ideal body as being be 8 heads high, agreeing with Lysippos rather than Polykleitos.
These are the actual measures as set out by Vitruvius.
A palm = four fingers
A foot = four palms
A cubit = six palms
Four cubits make a man
A pace is four cubits
A man is 24 palms
A palm = four fingers
A foot = four palms
A cubit = six palms
Four cubits make a man
A pace is four cubits
A man is 24 palms
As you can see the measures set out by Vitruvius are directly related to the Egyptian measurements that were laid down over 2,000 years before. He had to work hard to make some of these relationships work, in De Architecture he advocates a particularly difficult set of movements in order to make the theory fit, "If you open your legs enough that your head is lowered by one-fourteenth of your height and raise your hands enough that your extended fingers touch the line of the top of your head, know that the centre of the extended limbs will be the navel, and the space between the legs will be an equilateral triangle". These could be the directions for a contemporary dance piece or instructions given to an artist for a performative drawing.
These are the proportions that Vitruvius sets out as being ideal.
- the length of the outspread arms is equal to the height of a man
- from the hairline to the bottom of the chin is one-tenth of the height of a man
- from below the chin to the top of the head is one-eighth of the height of a man
- from above the chest to the top of the head is one-sixth of the height of a man
- from above the chest to the hairline is one-seventh of the height of a man.
- the maximum width of the shoulders is a quarter of the height of a man.
- from the breasts to the top of the head is a quarter of the height of a man.
- the distance from the elbow to the tip of the hand is a quarter of the height of a man.
- the distance from the elbow to the armpit is one-eighth of the height of a man.
- the length of the hand is one-tenth of the height of a man.
- the root of the penis is at half the height of a man.
- the foot is one-seventh of the height of a man.
- from below the foot to below the knee is a quarter of the height of a man.
- from below the knee to the root of the penis is a quarter of the height of a man.
- the distances from below the chin to the nose and the eyebrows and the hairline are equal to the ears and to one-third of the face.The concept of ratios has endured and the modern architect Le Corbusier came up with his own version of these types of measurements.
Corbusier's modular man
Francesco di Giorgio Martini
As an artist your work when it is shown will nearly always be shown in a gallery and that gallery will be part of an architectural complex. The architect may well have used a software program to help design the building and that program will have been developed around a grid, or series of grids based along the x, y and z axes. If as an artist you want to respond to the situation the work finds itself in, then you may want to measure the walls and think about the ratios set up by the architect. The doors and windows will have certain proportions and these will intrude on the space your work will be shown in. How might you reflect upon these issues? Would you go as far as contacting the architect and asking to see the original drawings? How would you develop a series of works that are both site specific and conceptually related to these ideas of measurement, ratio, harmony and proportion?
Robert Morris's 'Location Piece' is the first work I ever came across that raised this issue. You hang the work and then set the dials that indicate its position in relation to walls, ceiling and floor. As far as I know it is the only work of his in the Leeds City Art Gallery collection.
Robert Morris: Location Piece 1973
Occasionally I have tried to use computer modelling programs to develop ideas and of course if you do, you are immediately faced with having to think about how a grid can be used as a structural device. What interested me about this was that you could flatten out any 3D grids as nets, in doing this I was able to find a close association both with Egyptian images from thousands of years ago and Renaissance artists such as Uccello.
If you want to look at a basic way of turning a 3D
object into a net you could try using 123D Catch, a free downloadable bit of
software. Simply make a model, take about 60 photographs of it from different
angles and load into the software. (Since writing this post 123D Catch has been discontinued)
Every screen is also a grid and every grid has an aspect ratio, which is the ratio of the width to the height. Thinking of working on computers reminds me of the one area of aspect ratio we are all faced with; the computer screen. However we tend to just accept it as given. I clearly remember having to learn programming on BBC computers in the early 1980s, which like so many computers before 2003 had a 4:3 aspect ratio screen. I remember also having an Apple Mac Classic in the late 1980s, the almost square screen meant that you had to think about images with that proportion, and this was difficult for someone used to paper proportions. Of course the dynamics of fast scrolling game play were unthinkable in those days. Even so I still managed to use Hypercard stacks to create some basic animations. Monitors with a 16:10 aspect then began to appear in the early 2000s as the first laptop computers came in, this meant if you were working on page layouts you could put two pages side by side. By 2010, virtually all computer monitor and laptop manufacturers had moved to the 16:9 aspect ratio because of the demand for screening movies on computers. Having recently made animations that are tall and thin and then attempting to show them on screens, I began to realise yet again how much the aspect ratio of computer screens was shaping ideas.
Climbing man animation
I needed a tall shape for the animation, but I
either have to turn a monitor on its side or place a black shape around the
animation to make the total image ratio fit a 16:9 screen ratio. What I really needed was a contemporary version of Magnascope technology. This was a very short lived projection technique that coincided with the release of the 1933 film 'King Kong', it allowed the scene where Kong breaks through the huge tall gates, to be projected vertically.
The grid is
though a pretty basic mathematical form and sometimes it's useful to get away
from it and explore other forms that are less static.
I at one point looked at some basic visual exercises related to the relationship between
squares and circles and the mathematics of the super ellipse extends these
ideas further.
Variations of the Super ellipse
Piet Hein used the shape of the super-elapse to design a road roundabout, this was a wonderful way of calming traffic and allowing for a smooth transition from cars moving in straight lines into tight curves. Piet Hein explained the need for this shape very eloquently and I see no need to add anything.
- "Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand — such as the patchwork traffic circle they tried in Stockholm — will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't esthetically satisfying. The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity". Piet Hein
One variation of the forms above is called the 'squircle'.
Squircles
The point about these shapes is that they lie
between one form and another, therefore they are visually unstable. For an
artist looking for 'life' within abstract imagery, these types of shapes are
fascinating. As well as operating between geometric 'mechanical' form and
biologically 'soft' forms, they can be arrived at by inductive means as well as
by mathematical formula.
There is however something more fundamental about an interest in forms that are ‘off grid’ so to speak. When your eyes receive information, that information is collected at the focal point of a lens, and each lens forms part of a spherical object. As you look you revolve your head to follow what is interesting, so looking tends to take place within a perceptually curved environment. Our life experience is not about flat grids, but about overlapping curvatures of perceptions. Therefore locking images into flat grids could be seen as trapping them into intellectual nets, or cages and more curved formats perhaps suggest that you are meant to have a more phenomenological experience of the image. I.e. you are to have a heightened physical confrontation with what you are looking at. For instance I sometime curve the space through 180 degrees in my drawings, so that an observer has to in their mind, envisage physically travelling through the spaces constructed.
In this detail from one of my own large drawings above, you can see the space being gradually curved, which causes the horizon line to eventually become re-positioned as the edge of a hole in space.
The maths of road designThere is however something more fundamental about an interest in forms that are ‘off grid’ so to speak. When your eyes receive information, that information is collected at the focal point of a lens, and each lens forms part of a spherical object. As you look you revolve your head to follow what is interesting, so looking tends to take place within a perceptually curved environment. Our life experience is not about flat grids, but about overlapping curvatures of perceptions. Therefore locking images into flat grids could be seen as trapping them into intellectual nets, or cages and more curved formats perhaps suggest that you are meant to have a more phenomenological experience of the image. I.e. you are to have a heightened physical confrontation with what you are looking at. For instance I sometime curve the space through 180 degrees in my drawings, so that an observer has to in their mind, envisage physically travelling through the spaces constructed.
As you pull back from the drawing you can see how the hole/sky works
This is another drawing, using a similar concept
Stepping back you can begin to see how the idea of up or down can be challenged
A flat space can
be described by Euclidean geometry, but as Einstein pointed out space, energy
and matter are intertwined and the descriptions of gravity that
he produced were effectively diagrams of curved space, or, more specifically,
the curvature or warpage of four dimensional space-time. On a curved surface, the shortest distance between two
points is actually a curve, technically known as a geodesic. When undertaking
measured drawing, because you are at the centre of the measuring process, effectively
you are measuring from the centre point of a huge sphere. The issues
surrounding this have been explored in earlier blog posts related to how
drawing used to be taught in the college; see how to draw a line and drawing a straight line .
What I’m suggesting is that opening out any research into underlying
mathematical relationships and forms, is a possibility that should be taken seriously. Maps and how they relate to the flat rectangular
nature of drawing papers, contradicts my experience that spatial understanding doesn’t come neatly contained within flat grids. It is in trying to reconcile the two ideas that for myself, an interesting drawing arises. The 'Squircle', may seem a long way from the sky holes in my drawings, but it is still hanging in there as an idea about how my eyes might move around a shape, even if that shape is now a curved horizon.
See also:
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