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But Descartes also brought to being
the sort of analytical geometry that
breaks three-dimensional space into
the X-Y-and-Z coordinates and
allows us to manipulate that space
mathematically to make different
types of calculations. In my drawing
approach, I combine my
understanding of objects in
perspective space  with a kind of
Cartesian axonometry.  This method
allows me to view, understand, and
quantify any form and its
constituent  parts in a clear and
distinct manner.
Seventeenth century philosopher and
mathematician Rene Descartes is
famous for the maxim, "I think,
therefore I am," and he is often
blamed for introducing into Western
thought the mind-body split or
dichotomy that supposedly
introduced modernity.
I use axonometric thinking or visualization in order to make sense of complicated or even
simple forms that I wish to draw on paper -- for example, the facade and dome of
Saint Ivo alla Sapienza church in Rome center, illustrated above. The axonometric method
is particularly useful for drawing architectural subjects, especially in a city like classical or
baroque Rome, where spaces and monuments are built on a sort of CUBIC-like matrix.  
The trick is to imagine the object, then,  as
though mounted on or run through by the
X and Y and Z axes of space. On the left is
one such axonometric approach at
drawing a "lantern" on one of the bastions
of Castel Sant'Angelo in Rome. What I did
was, first, figure out the overall
proportions of the lantern (height by
width); next, I decided that I would view it
from the bottom up; finally,  I mentally
assigned each of the figure's various sides
to planes lying on the three Cartesian
axes.... and constructed the image in a sort
of transparent cube of spatial coordinates!
In the sketchbook page of my Rome
notebook, to the right here, I even drew
axonometrically a "cut" of the bridge
leading to the castle (center of the page).
To the left is another "moment" from my
sketchbook studies of the Church of Saint
Ivo : the drum-like decoration with the six
mounts and star of the Chigi family coat of
arms, high up on the facade of the church
from my ground viewpoint as I drew.  Using
the axonometric approach, I mentally
"flew" above the form. That is, I was able to
envision the figure and its curved ledge on
the X-Y-and-Z axes.
On the right is a section of the balcony in
the cloister surrounding the courtyard of  
Saint Ivo : one may see the faint axial lines
on which I "mounted" the plane of the
balcony above and that below. This method
makes the delineation of the angled arch
easier. The trick is to, first, envision the
balcony surface with its arch as one of the
sides of a cubeand then to lay that cube on
the tilted axis coordinates. From there, it
becomes easy and pleasurable to lay in the
arch as a circle (or half circle) inside the
square or rectangular surface of the space
mounted on the axis.
The applications of this axonometric
perspective approach are many, for
example, for situating the various parts
of a complex watercolor subject like that
of the Temple of Mars Ultor from the
Imperial Forum (above) or  the circular
interior of the 4th-century mausoleum of
Santa Constanza, in pencil.
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