By Jan Krikke
With an isometrical perspective, the length and width of a cube are placed on the horizontal line of projection with
an angle of 30 degrees (see Fig. 1). Because of its standardized geometry, isometry is ideal for CAD applications.
The three dimensions of a cube are projected onto the picture plane without optical distortion. Height, width, and
length are true to scale. Isometry is especially valuable in architecture and technical working drawings.
The projection of three-dimensional space onto the two-dimensional picture planes is an age-old problem. In Europe,
the problem was tackled by the Renaissance artists Brunelleschi and Alberti. The most important development in this
period was the discovery of the so-called vanishing point, whereby the lines of projection meet at an imaginary point
at the horizon. This resulted in linear perspective, which came to be the basis of the pictorial language of European
art.
In the 17th and 18th century, linear perspective underwent greater development. The mixture of geometry and optics
gave linear perspective a mathematical foundation which could be easily followed by artists and architects. The
correctness of the system was confirmed in the 19th century with the advent of photography. The French artist
Delacroix, for instance, stated that painters could use photography as an aid to structure perspective in their paintings.
Axonometry in Chinese scrolls
Europe was not alone in developing a method to project space on the two-dimensional picture plane. The Chinese
developed axonometry, dengjiao toushi in Chinese, which translates as 'equal-angle see-through'. Unlike
linear perspective, axonometry is not based on optical principles. Axonometry has no vanishing point, and hence no
optical distortion (see Fig. 2). In a painting showing a building interior, structural members like pillars and beams
will remain strictly parallel if they are parallel in reality. Moreover, as can be seen in the Japanese print (Fig. 3),
beams and pillars do not taper off; their size and geometry remains constant.
Axonometry was used not only in wood block prints but also in the classic Chinese scrolls, the vertical hanging
scrolls and the horizontal hand scrolls. Classic Chinese hand scroll were up to ten metres in length, and are viewed
by unrolling them from right to left. Hand scrolls are based on a (pictorial) synthesis of space and time. Rather than
having a 'subject', the scroll is based on a 'scenario'. For instance, a scroll may depict 'life along a river.' Upon
unrolling the opening sequence of the scroll, we may see people boarding a boat on a river. As we unroll the scroll
further, we see the boat cross a lake, navigate rapids in the river, stop at a small harbour, and lastly arrive at its
destination at the sea shore. In other words, the scroll has taken the viewer through an experience in space and time.
(Importantly, scrolls were not a collection of separate pictures, but rather a continuous and seamless visual
image.)
The scroll as a textual format was, of course, also known in the Occidental world. But the Chinese also developed
the scroll as a pictorial medium. This partly explains the conceptual basis of axonometry. Unlike linear perspective,
axonometry has no vanishing point, and hence it does not assume a fixed position by the viewer. This makes
axonometry 'scrollable'. Art historians often speak of the 'moving' or 'shifting' perspective in Chinese paintings.
Axonometry was introduced to Europe in the 17th century by Jesuits returning from China. The Chinese projection
systems was initially used for technical and military purposes, such as diamond cutting and ballistic measurements.
(Axonometry eliminates blind angles and simplifies calculations, which explains its usefulness in CAD systems).
However, the wider acceptance of axonometry had to wait until it was given a mathematical foundation.
Western Architecture
It was Englishman William Farish who provided axonometry with its geometrical basis. In 1822, Farish published
a paper entitled 'On Isometrical Perspective'. Farish recognized the need for accurate technical working drawings
free of optical distortion. This would lead him to formulate isometry. Isometry means 'equal measures' because the
same scale is used for height, width, and depth.
From the middle of the 19th century, isometry became an invaluable tool for engineers, and soon thereafter
axonometry and isometry were incorporated in the curriculum of architectural training courses in Europe and the U.S.
(Please note that definitions of axonometry and isometry differ in the USA, Britain, and the continent of Europe.)
The popular acceptance of axonometry came in the 1920s, when modernist architects from the Bauhaus and De Stijl
embraced it. De Stijl architects Theo van Doesburg and Cornelis van Eesteren used axonometry for their architectural
designs, which caused a sensation when exhibited in Paris in 1923.
Despite its importance to modern architects, engineers, and graphic designers, the history of axonometry has been somewhat obscured. The reason may well be the mathematical treatment axonometry has received in Europe. Axonometry is often confused with orthographic projection (see Fig. 5). Orthographic projection is of Greek origin. It was originally a two-dimensional projection which, in the late Renaissance, developed into a three-dimensional system (see Fig. 4). However, it is important to distinguish between three-dimensional geometry and axonometry. The former is a mathematical, theoretical space, while the latter is a pictorial space. Axonometry as it was used by classic Chinese artists had its own (non-optical) pictorial grammar. The Chinese artist ignored the optical law of diminution, (whereby figures and objects in the background are smaller than those in the foreground), and the effects of light and shadow, (clair-obscure). Figures in the Chinese painting are not modelled in clair-obscure; they are rendered as flat, two-dimensional figures which are placed in 3-D axonometric space (see Fig. 5). This explains why computer graphics manuals often refer to axonometry as 2.5D.
Visual computing
With the advent of the digital media, and especially the latest techniques in visual computing, the age-old problem
of projecting space on the two-dimensional surface has gained a new topicality. Admittedly, both European and
Chinese perspective are based on a compromise between the two-dimensional picture plane and real three-
dimensional space. But axonometry illustrates that optical perspective is not the only, and not always the best, method
to create pictorial space. Moreover, axonometry may have a bright future in the artificial world of visual
computing.
After all, the digital media (computer graphics, Virtual Reality, and digital cinematography) no longer rely on the
input of a camera. Instead of optical input, digital artists can use either linear perspective or axonometry, and even
a combination of both systems. As William Mitchel wrote in his book The Reconfigured Eye: Visual Truth in
the Post-Photographic Era: 'The digital image blurs the customary distinction between painting and photography
and between mechanical and handmade pictures'.
Axonometry is already used in computer games, and computer simulations of industrial processes. It may also find
applications in Virtual Reality. Among those who anticipated the usefulness of the isometrical variation of
axonometry was American author and architect Claude Bragdon. In his book The Frozen Fountain,
published in the 1930s, Bragdon illustrated the spatial quality of isometry with an ingenious drawing (see Fig. 6).
Bragdon gave isometry a glowing description. He wrote:
'Isometric perspective, less faithful to appearance, is more faithful to fact; it shows things nearly as they are
known to the mind. Parallel lines are really parallel; there is no far and no near, the size of
everything remains constant because all things are represented as being the same distance away and the eye of the
spectator everywhere at once. When we imagine a thing, or strive to visualize it in the mind or memory, we do it
in this way, without the distortion of ordinary perspective. Isometric perspective is therefore more intellectual, more
archetypal, it more truly renders the mental image -- the thing seen by the mind's eye.'
Jan Krikke is a free-lance publicist. He can be contacted by email: jankrikke@hotmail.com