From the beginning of time, those among us we now call artists have tried to capture in two dimensions what they saw of the real, three-dimensional world in which they lived. Almost from the very beginning, space, the immaterial stuff that envelops us, became the object of their fascination; how could they represent it?

The creativity of the early Homo sapiens , allowed them, for the first time, to paint animals on irregular walls, using, for example, a depression to represent the prominent belly of a bison. The overlapping images of animals on a cavern wall allowed them to reproduce the top down vision, which a hunter might have seen from the top of a promontory, as they watched the plain below them.

Sahara, Tassilin’Ajjer, Jabarun. Shepherds and hunters.(-10 000 ?) Photo by Dominique Lajoux

Homo sapiens was not only a sculptor but was also able to sketch on a flat surface. For this they created special tools, anticipating the efforts needed to execute such designs.

Sahara. Iheren. (Between -3000 and -300). Photo by Gabriel Camps.

So our ancestors manipulated abstract concepts such as space with great skill. Our view of the way they might have lived is in part shaped by the fact that they carried this baggage of such abstract and relational concepts.

It seems that this natural skill was several times lost to humanity. If, for example, we move forward on the timeline to the first “mother” cities of Mesopotamia around 5000 years ago, one finds they made no effort at all to represent physical space— at least, not that we have found thus far. For this we have to wait until the Hellenistic period when documented works begin to appear in the Mediterranean basin and also further east in China. It must be noted that social organization played a fundamental role in the aspirations of artists. The complexity of life developed together with appropriate techniques for the representation of space. In short, we can distinguish three main types of perspective: the Curvilinear, the Rectilinear and the Real. These three techniques coexisted over time.

The mosaic of Alexandria was probably a reproduction of Greek fresco (-400). ( Public Domain )

Astronomy and Trigonomitry: The Curvilinear Perspective

This technique consists in using curves emanating from a single vanishing point. We first notice the Greeks used curves to construct temples whose columns curve outward towards their midpoint or zenith and are narrower or concave near ground level. We see this principle at work in some existing monuments such as Trajan’s Column, built in Rome in AD 107, that followed hidden rules of trigonometry.

The Column of Trajan, in Trajan's Forum, Rome (Alvesgaspar/ CC BY-SA 4.0 )

These principles of trigonometry were collected in AD 150 by Ptolemy in Alexandria, in the form of a "table of circular arcs" originally as used by astronomers.

The Column of Trajan; an observer sees the bottom and the top of the frieze under the same angle. (Diagram by Xavier Bolot)

During the Renaissance, artists did not concern themselves with the principles of trigonometry, it was not really part of their training. Thus they were hampered by the fact that they could not define the fugitive curves which they observed when they viewed a landscape through a window. Artistic masters such as Brunelleschi and Alberti opted instead for an approximation known as rectilinear perspective, limiting their works to a narrow observational window.

15th century illustration from the Old French translation of William of Tyre's Histoire d'Outremer. ( Public Domain )

However, many famous painters, from Fouquet (1420 - 14781) to Matisse (1869-1954), intuitively use a magnifying glass effect or a curvilinear perspective, without dwelling too much on any underlying theory.

The Young Sailor I, Oil on canvas. 99 x 77.5 cm. Private collection. Henri Matisse, 1906. ( Public Domain )

By Royal Edict: The Rectilinear Perspective

Alberti, by representing here, in the same drawing, a view facing a grid on the left and profile view on the right, gives a method of constructing of the enlargement of squares when they are closer to the observer. (Diagram by Xavier Bolot)

The principle consists in saying that in a landscape everything converges to infinity towards a vanishing point from which start receding straight lines. The lines of equal magnification of the objects are also straight . This technique had the advantage of simplicity. It also avoided the magnifying glass effect that was too close to the raw reality.

The Genoese architect Alberti (1404-1472) elaborated a method, showing the steps in the mapping of space. The French architect Desargues (1593-1693) developed the theorem further showing it by a construction of the shadows cast by the sun on the floor.

This is the classical system we know and which prevailed in the European academies from the eighteenth to the middle of the nineteenth centuries.

But it is remarkable that this system was not one willingly chosen by the artists themselves. It had to be imposed by the royal edict of the French king Louis XIV. He did this in order to put an end to the tiresome polemics between the partisans of the curvilinear perspective, in particular Canon Bosse, and those of rectilinear perspective, such as Lebrun, director of the Royal Academy.

Louis XIV, King of France, in 1661 ( Public Domain )

On their side, Chinese artists represented space with parallel lines, which does not really correspond to our physiological perception. In the picture below, the tables appear as if they have shifted out of line; the overall effect is still quite pleasing if not exactly what the artist intended.

Eiri, 1790-1800, Japan. Tea by the seaside. Far East Rectilinear Parallel Perspective as we see for the tables. (Private collection via author)

Lessons from our Eyes : The Real Perspective

The first systematic attempt to define the vanishing lines and the steps of magnification was that of Barre and Flocon who in 1930, took note of the fact that our human perception is constructed in the spherical space of the physical eye-ball.

Curvilinear Spherical Perspective by Barre and Flocon (1930).The bottle is behind the sketcher.

At the same time, in 1947, M.C.Escher, in his lithograph, Above and below, represented a patio as a cylindrical perception space. According to his friend, Bruno Ernst, he had used sinusoidal arcs (repeating mathmatical curves, such as a sine wave), but without being aware of it.

Sine waveform. ( GNU Licence )

M.C. Escher ‘Up and Down’ or ‘Above and Below’. ( Source)

In 1992, David Hockney photographed the Grand Canyon of Colorado with 200 shots. He did this because he maintained that we are surrounded by a cylindrical space because of the presence of our two eyes.

David Hockney, photographs of the Grand Canyon ( Source)

The present author, writing in 2002 Drawing in Real Perspective , demonstrates that, in cylindrical space, the receding lines are trigonometric arcs, and the curves of constant magnification are ellipsoids.

We then find again:

- the principles of construction of the Greek temples and of the column of Trajan,

- what appears when Leonardo da Vinci transferred a landscape through a window pane,

- and the magnifying effect used intuitively by Matisse.

Perspective at Home

The application of this technique is actually surprisingly easy to do. All that is required is for the artist to look again with renewed attention at all the angles and curves. The neurosciences have shows us that our body is made for this task. The eye sees curves; the brain captures them, even the movement of the joints in our arms works on these principles. When an angle meets a wall we draw a trigonometric, ie curved line, almost without thinking. With a little retraining, these natural curves are used to make images on paper in Real Perspective . After a while it becomes considerably faster to work this way; the risk of making an error, virtually non-existent.

Constain magnification curves. (Drawing by Xavier Bolot)

Three minute drawing (with practice) (Drawing by Xavier Bolot)

The newly rediscovered techniques of Real Perspective, based as they are on the natural techniques determined by our physiology and perception, in part, returns us to the mind set and wondrous skill of our prehistoric ancestors.

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Chris ‘Mogg’ Morgan is a respected independent scholar, former Wellcome student, and holder of an advanced degree in Oriental Studies from University of Oxford. He’s author of Phi-Neter: The Power of the Egyptian Gods .

Xavier Bolot leads his research on the various problems of perception with the support of the National Fine Art School of Bourges in France. Xavier is a painter, Engineer of the INPG National Polytechnic Institute of Grenoble, France, Professor of Electronics at the University of Montreal, Canada, President of the French Association of Industrial Advertising in Paris, and Consultant in communication and behavior, Orleans, France. He is a researcher at the CEAQ (Center of Studies on the Present and Everyday Life) University of Paris V Descartes Sorbonne.

‘Drawing in Real Perspective: A new approach to space with natural and immediate application’ by Xavier Bolot is available now at Mandrake of Oxford .

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Top Image: Pietro Perugino's use of perspective in this fresco at the Sistine Chapel (1481–82) helped bring the Renaissance to Rome. ( Public Domain )

By Chris Mogg Morgan & Xavier Bolot