J. D. Sage sagama@charter.net 
Math Art 
Transformations  I have been exploring "Math Art" over the past 40 years, using mathematical methods to transform and represent images. My Math Art includes: (1) Transformations based on mathematical functions, (2) Progressions formed from numerical sequences, (3) Explorations of compression and reflection in time and space.  
Progressions  
Explorations 
Sage, Joseph D., 2003: MetaForms and MetaNudes etcetera Soft cover available from Sagama Publishing eBookIII is an extension available on iBooks 
My first Math Art was made in 1980. An outline of a photograph of
the author was placed onto graph paper to obtain the position of points
on the outline. The x and y coordinates of critical points on the image
were manually measured. A slide rule was then used to calculate the
logarithm to the base 10 of the y coordinates. The x coordinate and the
logarithm of the y coordinate were plotted to obtain a transformation of
the original image. The outline of the image was then filled in with
black India ink. The author was fascinated with the primal feel of the
transformed image or metaform (below). 
Additional metaforms were
stimulated by the following events. On any given day, I would develop on
a blackboard the mathematical description of a particular physical
process. When the blackboard surface was fully used, the board was
partially erased and the writing was continued over the erased board. At
the end of the lecture, the blackboards in the lecture hall would be
filled with mathematical symbols and drawings. The blackboard was then
partially or completely erased. A few days later the lecture would be
continued from where the previous lecture left off. The same set of
blackboards would again be filled with a different set of mathematical
equations, only to be erased at the end of the lecture. The full
mathematical development of the physical system under consideration
would often take three or four lectures. The record of these lectures
was only preserved in the
notes of the students. 

"Log Nude",
1980. 
An art piece was made using a
selected set of lecture notes which was consigned to paper or other
suitable surface prepared with a chalkboard surface. Drawings were made
to represent a series of one hour lectures on the development of a
particular process using mathematical symbols and occasionally graphs,
curves or drawings. The firsthour lecture was written in colored chalk
on three to six blackboard panels. The secondhour lecture was done in a
different colored chalk superimposed on the same three to six panels.
The third and occasionally fourthhour lecture was again done in a
different colored chalk and superimposed on the same three to six panels
until the development of the process was complete. Below is a
completed work consisting of three (48”x48”) panels based on the
theory of projection of images in space. In this work the panels were
constructed from fiberboard coated with a blackboard paint. 
"Projections and MetaNudes", 2002. Pastel on
chalkboardtreated fiberboard, three panels each 48"x48". 
Metaforms or transformations of images are not new.
A metaform or transform is an image, object or form which has
been in some way modified or transformed from the original. A circle
modified to appear as an ellipse; a woman transformed into a bird.
Shadows of images falling on oblique walls or curved walls are
transformed and represent the earliest transforms. Pliny indicated that
the early Greeks believed that the art of painting started with the
tracing of the outline of a shadow of a man [Pliny,
LCL 394 (1999) Natural History, IX. Book 35:15. Translated by
H. Rackman, ed. G. P. Goold. Cambridge, MA: Harvard University
Press]. Other early
transformations are images conceived and depicted on the walls of caves,
rock outcrops and other natural surfaces. They are collectively called
rock art. Some images are painted on rock and others called petroglyphs
are carved, pecked or scratched into the surface of rock. 
The use of number is innate. Early man might not have had the number system we are most familiar with, but even they had a way of signifying, one or none, more than one, and more. They probably used their toes and fingers or digits to count. The concept is carried over to contemporary man for which the term digital refers to the representation of information and data by discrete units or digits. Examples of the use of number as tokens on clay tablets are found in the Sumerian culture (5,000 BP) http://it.stlawu.edu/~dmelvill/mesomath/sumerian.html. Paleolithic man (20,000 BP) scratched lines and gouged holes in sticks and bones, possibly counting the days between phases of the moon http://www.physics.nist.gov/GenInt/Time/ancient.html. Without counting, we are aware of objects in groups of two, three, four and five. Beyond five, counting may be required or we assign the collection a name. 
One example of numerical art is the use of the stochastic progression.
This progression is a sequence in which each new term of the sequence is
obtained from a probability distribution and a relationship to
a number of earlier noncontiguous terms. This
procedure may be applied to generate a sequence of terms representing a
physical process. Here (left) is a drawing using an application of this
procedure. In this work,
both the speed and direction of the wind were treated as stochastic
variables. A sequence of wind direction and speed was generated using
the statistical values from the meteorological data for Akron, Ohio.
Colored lines having equal length were plotted and oriented in the
direction of the hourly wind. The
color of the line was based on the value of the generated wind speed. An
art work based on a new sequence would lead to a similar but different
form. The procedure may also be used to produce three and four
dimensional art works. 


Recently, I have been interested in iconographic ways to capture the
passage and representation of
time. The “Cubists” compressed space
into a single plane. “Timists” compress time into a single instant.
In the Kimberly's of
Australia, early artists used superimposition of
images spanning as much as 20,000 years [Walsh, Grahame (2000) Bradshaw Art of the
Kimberleys. North Carlton, Australia: Takarakka Rock Art Publications.] For composers,
time is compressed on a sheet of music. A visit to the Grand Canyon is a
“timist” event, particularly when we are made aware of the fact that
the canyon walls and gorge represent hundreds of millions of years of
geologic events.
The multidimensional work shown below contains two sets of glass.
Each set contains four panes of glass separated from each other, to
permit looking forward and backward through time. In the first set, an
image from 2 BP (years before present) is placed on the first pane of
glass, an image from 6,000 BP is placed on the second pane and an image
from 20,000 BP is placed on the third pane. The equations for time travel
are placed on the fourth pane. In the second set, the time sequence of
the images are placed on glass in reverse order and represent images
from 20,000, 6,000 and 2 years BP. Mirrors are incorporated in the work to
permit the viewer to be a part of the work. 
"Time Travel", 2002. Acrylic and gold leaf on
glass pieces, 9"x15". 