The document discusses transformation geometry, a mathematical branch focused on the movement and alteration of spatial figures through various transformations such as reflection, rotation, translation, and enlargement. It highlights the contributions of notable mathematicians like René Descartes and Felix Klein in developing analytic geometry and the concept of symmetries. Additionally, the importance of transformational geometry in fields like art, architecture, and anthropology is emphasized, as it aids in identifying cultural and historical contexts of artifacts.

1. Reported By:
Junila A. Tejada

2. Measurement of the Earth.
In today’s usage, it is a branch of
mathematics dealing with spatial
figures.

3.  a process which changes the position
(and possibly the size and orientation)
of a shape. There are four types of
transformations: reflection, rotation,
translation and enlargement.

4. Historical Overview of Transformation
Geometry
 17th century
Mathematician.
 Made a great contribution
in analytic geometry.
 First used the Cartesian
coordinate system.
Every point of a curve is
given two numbers that
represents its location in a
plane.
Rene Descartes

5. Historical Overview of Transformation
Geometry
 Proposed a system of analytic
geometry similar to Descartes.
 Credited because of his
independent developing ideas in
analytical geometry.
invented modern number
theory virtually single-handedly.
Formulated several theorems
on number theory, as well as
contributing some early work on
infinitesimal calculus.

6. Historical Overview of Transformation
Geometry
 assign algebraic ideas to
geometric figures led to the
study of group theory in
geometry.
 Enlanger Program
Study in geometry defined
as the study of transformations
that leave objects invariant.
 Rearranged the unrelated
geometry know at his time
into a cohesive system.
Felix Klein ( 1849-1925)

7. Klein’s Idea
• A geometry is a set of objects with the
rules determined by its symmetries, i.e., its
transformations. Two geometries may have
the same objects but different
transformations.
• The properties of the geometry are
properties that are not changed by the
transformations.

8. 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 culture. The
design used can help to
determine where and to
whom an artifact belonged.

9.  Developed in the 7th, 8th,
9th centuries from beliefs
that creating a living
objects in art was
blasphemous or that God
should create animals and
other creatures in art work.
 Due to this belief, many
did not use living creatures
in their art work, instead
they used different
transformations and
geometric designs to
increase the appeal of their
art and architecture.
Arabesque

10. 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; such
as flowers and houses
that are not necessarily
symmetric.

11. 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.

12. M. C. Escher (1898–1972)
 Dutch graphic artist.
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.

13. Types of
Transformation
 Reflection
 Translations
 Rotation
 Dilation

14. Reflection
You can reflect a figure using a line
or a point. All measures (lines and
angles) are preserved but in a mirror
image.
Example: The figure is reflected
across line l .
You could fold the picture along
line l and the left figure would
coincide with the corresponding
parts of right figure.
l

15. moves a shape by
sliding it up, down,
sideways or
diagonally, without
turning it or
making it bigger or
smaller.
Translation

16. Rotation
Rotation (also known as
Turn) turns a shape through a
clockwise or anti-clockwise angle
about a fixed point known as the
Centre of Rotation. All lines in the
shape rotate through the same
angle. Rotation, (just like
reflection) changes the orientation
and position of the shape, but
everything else stays the same.

17. Dilation
A dilation is a
transformation which
changes the size of a
figure but not its shape.
This is called a
similarity
transformation.

18. “Do not just pay attention
to the words;
Instead pay attention to
meaning behind the words.
But, do not just pay
attention to meanings
behind the words;
Instead pay attention to
your deep experience of
those meanings.”
Tenzin Gyatso, The 14th
Dalai Lama
END