**3.1.2 Historical Overview of Transformational
Geometry
****
***Printout*

*I begin to understand that while logic is a most
excellent guide to governing our reason, it does not, as regards stimulation to
discovery, compare with the power of sharp distinction which belongs to
geometry.
—*

During the 17^{th}
century, *
*René Descartes (1596–1650), a French mathematician and philosopher, laid the stepping
stones for modern mathematics, specifically in analytical geometry, through the
use of the Cartesian coordinate system. That is, every point of a curve is
given
two numbers that represent its location in a plane. Descartes' system was more
general than the usage of horizontal and vertical axes as we know it today.
This new coordinate system helped to show that there was a link between geometry
and algebra, starting with a geometric shape or curve and assigning an ordered
pair to each point so algebraic techniques could be assigned to the figure.

*
*Pierre
de Fermat
(1601–1665), a French jurist, proposed a system of analytical geometry similar
to the one noted by Descartes. Fermat’s system used a more direct approach and
is more similar to the system currently used. Fermat and Descartes are both
credited with independently developing the ideas of analytical geometry. Fermat
is credited with contributions to every field of mathematics known in the 17^{th}
century.

This idea of assigning
algebraic ideas to geometric figures led to the study of group theory in
geometry. *
*Felix Klein (1849–1925), a German geometer, showed the importance of groups in geometry.
This new idea allowed Klein to unify geometry. Klein’s 1872 address in Erlanger,
Germany proposed that the study of geometry be defined as the study of
transformations that leave objects invariant (unchanged). This viewpoint is now
known as the *
Erlanger Program.* Klein’s address was able to rearrange what
seemed to be unrelated geometries known at this time into a cohesive system.
His lecture classified geometry to include both Euclidean and non-Euclidean
geometries.

Transformational and analytical geometry are not new branches of geometry; they are considered to be part of Euclidean geometry. These approaches include new methods of working out geometric questions. The geometric designs involved used many ideas that have long been a part of art and culture.

Different transformations have been used in art, architecture, crafts, and quilts throughout history. Historians have found numerous transformation designs in pottery, architecture, rugs, quilts, and art pieces from almost every known culture. The designs used can help to determine where and to whom an artifact belonged.

There are many reasons
why historians believe that cultures started to use transformations in their art
work. For example, *
*arabesque
developed in the 7th, 8th, and 9th centuries from beliefs
that creating living objects in art was blasphemous or that only
God should create animals and other living objects. Due to this belief,
many
did not use living creatures in art work, instead they used different transformations
and geometric designs to increase the appeal of their art and architecture.

Another example why
historians believe transformations were used was a lack of materials, such as in
the history of quilting. When there was a shortage of quilting materials, women
would often turn to a technique called appliqué to visually increase the appeal
of their work. *
*Appliqué, a process when one piece of fabric is sewn onto
another and then stitched together with an intricate design, traditionally had
elaborate geometric transformations that were typically symmetric; however,
modern appliqué has been expanded to include pictures of everyday items, such as
flowers and houses that are not necessarily symmetric.

Another reason
historians believe cultures would use transformations was the aesthetic appeal
that geometric designs create. Different cultures loved the visual interest and
equality that geometric transformations made in the art and architecture of
their communities (Islamic
Art). These techniques were typically inexpensive to add to
crafts, such as pottery. They also liked the aesthetic appeal of symmetric
buildings. This idea can be seen in buildings throughout history, from
the *
*Parthenon of ancient Greece to *
*St.
Peter’s Basilica in Rome from the 17^{th}
century to *
*St.
Peter's Cathedral in Adelaide in the 19^{th}
century, as well as over many centuries in *
*Islamic
architecture. Even modern buildings from the 19^{th} and 20^{th}
Centuries are built on the basis of symmetry such as the *
*United States Capitol
building in Washington, D.C.

*
*Transformational
geometry is quite important in many fields, such as the study of architecture,
anthropology, and art, to name a few. The study of which forms of
transformations were used helps to distinguish time frames for artifacts and
helps to illustrate which cultures may have made the item being studied.

For example, architects
are able to study the history of very old buildings, taking note of which
transformations were used. A classical example that involves this study is the
illustration of the study of the history of the *
*Parthenon in Athens, Greece.
This was a shrine constructed in honor of the goddess *
*Athena, built between 447
and 436 B.C. Historians look at the different *
*frieze patterns throughout the
temple to help understand the culture and architecture. They are able to
compare the designs from the Parthenon to other buildings throughout Greece and
Europe to help see which cultures may have had contact with the Athenians. They
can also compare the frieze patterns used in other buildings to see how similar
they are to the designs in the Parthenon and are able to help narrow in on a
time frame for when the other building was constructed.

Historians can also
use different transformations to help learn for what a building may have been
used. Historically, buildings which were very well constructed and decorated
typically
served religious purposes, honored a god or king, or served a wealthy or
important person in the community. If a building has lavish frieze patterns or
other transformations, historians are more thoroughly able to recognize how the
building was used and understand the importance of the building in the
community studied. A familiar instance of this throughout history is the importance
of churches and palaces to a particular culture. For example the *
*Alhambra
Palace, built in Granada, Spain in the 13^{th} or 14^{th}
century, has many extravagant symmetric and tessellation designs throughout the
tile work, windows, and ceilings in the building. This lavish detail throughout
historic architecture is usually because of the importance of the building for
the community. The Alhambra Palace, which includes many intricate designs
throughout its construction, helps to show how important the building was to the
community and dynasty as well as the wealth and importance of the population.

Transformations are also studied heavily by anthropologists. Many cultures throughout time have used transformations in their craft and pottery work. Anthropologists are able to study the artifacts of cultures through pottery; pottery is a common remain due to the durability of materials used. Anthropologists are able to study these design features to help understand which cultures had contact with each other and present evidence of their lives. Pottery artifacts are studied a great deal to help understand the culture. One way they are able to do that is by the transformations that were used. Cultures typically used the same designs in most of their work, so anthropologists are able to match a design with a particular culture. If they find the same design in an artifact from the site of a different culture, they are able to establish that the cultures had contact either through trade or education. This is important for anthropologists to understand how a culture lived and worked and to track the interactions and movements between several cultures.

Another area that
transformational geometry is commonly used is in art. There have been many
artists throughout history that have used different techniques of
transformations, such as symmetry and tessellations, in their pieces. A well
known artist using transformations was *
*M. C. Escher (1898–1972), a Dutch graphic
artist. He would typically use symmetry and tessellations in his art. The
Alhambra Palace, previously mentioned for its lavish transformational detail,
was a common place that Escher would go to work on his pieces. Escher read a
few mathematics papers regarding symmetry, specifically *
*George
Pólya’s
(1887–1985) 1924 paper on 17 plane symmetry groups, and although he did not
understand many of the ideas and the mathematical theory of why it worked, he
did understand the concepts of the paper and was able to apply the ideas in his
work. These concepts helped him to use mathematics more extensively throughout
many of his later pieces.

Transformational geometry has been extensively used throughout history by cultures from all regions to create works of art and decorate their architectural buildings. This use of transformations, no matter the reason, is very helpful for historians and anthropologists in aiding with the research of different cultures and their traditions.

3.1.1 Introduction to Transformational Geometry3.2.1 Preliminary Definitions of a Transformation |

© Copyright 2005, 2006 - Timothy Peil and Nancy Martin |