Chapter 2:

Transformations and Symmetry

The word "transformation" refers to a movement or change in a geometric shape. If we slide a geometric shape to the right or left, this is called a "translation". In the example below, the original triangle is triangle ABC, and the second triangle A'B'C' is a translation of the original triangle. In moving the first triangle two inches to the right, we say that we translated triangle ABC two inches horizontally to create triangle A'B'C'.

In the second example of translation below, triangle DEF has been translated one inch vertically to create triangle D'E'F':

The second kind of transformation is called a "reflection". When you look in a mirror, you see a reflection of yourself, but left and right are reversed. As you see in the diagram below, when we reflect triangle ABC, the reflected triangle A'B'C' is "flipped" or reversed. Line "m" is the mirror.

A third kind of transformation is called a "rotation". If you want to rotate an object, you must first choose the Center of Rotation. This point is the point that you will rotate the object around, just as the center of a wheel is the center about which the wheel rotates. In the diagram below, triangle DEF has been rotated 90 degrees counterclockwise to its new position, triangle D'E'F'. P is the center of rotation.

In the diagram below, a triangle has been rotated 6 times about point P, at 60 degrees each time. Since 6 times 60 is 360 degrees, we see that this completes a full circle:

The fourth transformation is called "dilation". When the eye doctor "dilates your eyes" during an eye examination, the pupils of your eyes become enlarged so that the doctor can see the veins of your eyes more clearly. Dilation, mathematically, means to take a shape and make it larger or smaller but keep it the same proportion. In the geometric example below, the triangle on the left is "dilated" by a factor of 2 (it is "doubled") to create the dilated triangle on the right. Each side of the new triangle is twice as long as the corresponding side of the original triangle. Point "P" is the center of dilation. We would therefore say that the "scale factor is 1 to 2" comparing the original to the new figure. Another way to compare the sizes of the figures is to say that each side of the smaller triangle is one-half the size of the corresponding side of the larger figure.

What do you think is true of the angles of the new triangle compared to the angles of the original triangle? If you answered "they are the same", or "the new angles are the same measure, or number of degrees, as the original triangle" then you are correct! Since the angles are the same, the two triangles are the same shape.

http://mathforum.com/sum95/suzanne/symsusan.html

Test Question #1: What is special about the word ATOYOTA, as explained on the web site linked above?

To create an interesting geometric, you can combine the two or three of the transformations (translation, reflection, rotation) and use a variety of shapes and colors as in the example below. Which of the transformations were used?

If you answered "reflection and rotation", then you are correct!

Translation and dilation were used to construct the "frieze pattern" below. A "frieze pattern" is a design made from horizontal or vertical repetitions of the same shape.

Frieze patterns are often used in fabric designs for clothing, draperies, and wallpaper, and even in architecture, such as the example below, a Tile Frieze from the Palacio de Velazquez in Madrid, Spain

(photo from http://www.fandm.edu/Departments/Mathematics/a.crannell/hm/Math/seven.html)

You will find a mathematical discussion about these and other transformations at the following web page, on the Math Forum website. The Math Forum is a fascinating web site, with thousands of pages about math.

http://forum.swarthmore.edu/sum95/suzanne/symsusan.html

Test Question #2: What type of transformation is discussed on the Math Forum web page that was NOT discussed on this MathArt page, what is the definition of that type of transformation, and how does it differ from the 3 types of transformation discussed in MathArt?

Symmetry

We say that an object has symmetry, or is symmetrical, if the two sides are "of similar measure". Many animals and insects are symmetrical; a beautiful example of this is the Butterfly below.

We say that a butterfly, and the human body, are symmetrical because the left and right sides appear to be very much alike. This type of symmetry is called "reflection symmetry" because the left side, if it were reflected across the middle line, would match the right side. You can see this general symmetry in the following famous drawing by Leonardo Da Vinci. In this drawing, Da Vinci illustrates the form with two positions of both the arms and legs, depicting the symmetry and proportion of the human figure.

Leonardo Da Vinci is often called the "Renaissance Man", because he combined imagination, curiosity and creativity. He was a scientist, an inventor, and an artist. His most famous painting is The Mona Lisa. The web page below will give you an introduction to the artistic side of Leonardo:

http://www.mos.org/leonardo/artist.html

I would like you to read the following web page from the Leonardo web site, but you do not have to do any of the activities (although you will probably find them intriguing):

http://www.mos.org/sln/Leonardo/LeonardoRighttoLeft.html

You will find some interesting information about symmetry at the following web page, on the Da Vinci web site:

http://www.fi.edu/qa97/me3/

Now here's a challenge for you. Using the method explained in the web page above, "de-code" the following question and answer:

(photo from http://members.aol.com/alohaquilt/index.html)

Many pieces of art work are symmetrical, and so are many graphic designs. Some of the most beautiful examples of symmetry can be found in Hawaiian quilts, such as the ones above. You will find out how to create your own geometric quilt pattern at the web page below:

MACsw.html

You will find out more information about symmetry by reading the one page linked below, from "The Geometry Pages". You do not have to go to any other pages besides this one page; please come back to the MathArt pages when you have carefully read the page below. Click on this link now:

http://mathforum.org/~sanders/geometry/

Test Question #4: Name the two types of symmetry, and give an example (from the above web page) for each type.

From earliest recorded history, we find examples of artists and architects designing their paintings and buildings with a strong emphasis on symmetry. For example, most buildings built before the 20th century had very strong symmetry: the left side of the structure is exactly the same as the right side, as you can see in the images below:

(Photos from http://www.GreatBuildings.com/)

From left to right:

Torii of Itsukushima on Miyajima Island in Japan 10th century AD, Norwegian storehouse 13th century AD, Piazza of St. Peter's in Rome 17th century, S. Maria Novella Church in Florence Italy 15th century AD, The Taj Mahal in Agra India 17th century, United States Capitol in Washington D.C. 19th century.

The following two images are from http://homer.reed.edu/Parthenon.html and the text is from http://www.johnco.cc.ks.us/~jjackson/part.html

"The supreme achievement of the Greeks was their claim that everything--from the human body to the entire cosmos--was governed by an order accessible to human reason. That claim has been the basis for western civilization ever since. The Temple of Athena Parthenos, the Parthenon, now a ruin lying atop the Acropolis of Athens, is the testament in stone to that claim."

And this " order accessible to human reason" is mathematical symmetry! The Parthenon is one of the classical examples of ancient Greek civilization, built in 447 BC. The structure, with beautiful proportions and perfect symmetry, is also an example of the importance of symmetry and proportion, and therefore mathematics, in architecture.

The floor plan of the Parthenon below shows the very strong order and symmetry, and emphasis on geometric shapes that is found in most architecture:

You will find further information at the following web page, called "The Golden Section in Architecture".

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html#arch

This page is rich with information on both mathematics and architecture and you should spend quite a lot of time reading and looking through it. As you scroll down you will also find "The Golden Section in Art", which you should also read. I would like you to go to some of the links from this page, and read what you find there. You will find many good ideas for your project!

So what is the "Golden Section"? You will find the answer to this question, and some good project ideas, at the following web page:

http://mathforum.org/~sanders/geometry/GP21MoreCurves.html

As you have seen, many architects design their buildings so that they are symmetrical, and this has been true throughout history. You will also find many examples of artistic use of symmetry in much of the ornamental designs on the walls of many buildings. One very good example is the Alhambra Palace, in Spain. You will find some interesting information about this beautiful building, and examples of the symmetry in the ornamental tiles of the Alhambra by clicking on the link below:

http://en.wikipedia.org/wiki/Alhambra
The transformation called "dilation" produces figures that are the same shape, but not necessarily the same size, as explained earlier on this page. In the example above, the original figure was one-half the size of the new figure. This meant that each side of the original triangle was one-half as long as the corresponding ("matching") side of the "dilated" triangle.

This mathematical concept is used extensively in building model airplanes, dollhouses, and in architectural design as well. Certainly an architect cannot possible draw a building at its full size - it would take an enormous sheet of paper! So an architect (or an engineer, or an airplane designer, etc.) draws the building "to scale". The phrase "scale drawing" refers to a drawing that is smaller (or occasionally larger) than the real thing. The floor plan of the house shown below has been drawn at to scale.

The original scale at which this floor plan was created was 1/4 inch represents 1 foot, which means that every 1/4" inch on the drawing represents 1 foot in the real house. (Note: to fit the drawing onto a web page, it has been reproduced, so is not measurable as you see it now.)

Now, some mathematics: if 1/4 inch represents 1 foot, then it would also be true that 1/4 inch represents 12 inches. We can simplify this equation by multiplying both sides by 4 as shown below:

Therefore, the ratio is 1 to 48, and the actual drawing was one forty-eighth the size of the real house.

Here's another example: The length of a stadium is 200 yards and its width is 120 yards. If 1 inch represents 10 yards, what would be the dimensions of the stadium drawn on a sheet of paper?

Answer: 20 inches by 12 inches.

http://regentsprep.org/Regents/math/scale/Lscale.htm

In searching on in the internet for information about architects and architecture, I found the following website. It is fascinating! If you are interested in possibly becoming an architect, please visit this website.

http://www.abc.net.au/architecture/

As you have seen, architects, artists, and designers use the mathematical concepts of transformations - translation, reflection, rotation and dilation - in creating their work. You, too, can create beautiful designs using the mathematical concepts of transformations and symmetry. In your project for this chapter, you will design a beautiful geometric graphic. You will find out more about how to create these geometric graphics at the following web page:

http://mathforum.org/~sanders/geometry/GP05Constructions.html

Chapter 2 Project

The project for this chapter is to create an artistic symmetrical design, using geometric shapes and transformations. First do some research on the Internet, and look at geometric artwork (net search: geometry art OR geometric art). Read some information about the geometric artwork that you find, and explore any aspects that interest you. Then design and create a symmetrical geometric design yourself. If you would like to see some examples of graphics created by students like yourselves, please go to the Student Work section from the Table of Contents.

• You must use only geometric shapes in your design (triangles, pentagons, hexagons, circles, half-circles etc.)
• Create the graphic using compass, ruler, and tracing paper, in pencil on unlined white paper. If you are using pencil and compass and ruler, draw a geometric figure (an equilateral triangle, for example) then trace it onto a small piece of tracing paper. Draw a point on the original paper, and place the tracing on top so that it lines up with the original triangle. Put the point of your compass at the center, and rotate the top sheet about this point 30 degrees, 45 degrees, or 60 degrees (or any number that divides evenly into 360 degrees).For angle measures, you will need to use a protractor - or you will need to construct the angles you need as follows: construct a hexagon with compass and that construction will give you a circle divided into 6 equal parts, each of which is 60 degrees. Constructing a square and drawing both diagonals will give you 45 degree angles at each corner, etc.). If you are using The Geometer's Sketchpad, follow the instructions on the GSP pages - you will find a link to those pages in the Introduction to MathArt Connections.
• Make your design more interesting by constructing a second geometric shape (a square or diamond, for example) and then use tracing paper to rotate and/or reflect copies of the shape, around that same central point, OR use The Geometer's Sketchpad.
• When you are satisfied with your design, trace it in ink (or go over the pencil lines in ink and then erase those that still show). Then color it using colored pens or pencils OR if you are using The Geometer's Sketchpad, hide all the points, print it out and then color it with colored pens or pencils.
• If you use pencil, paper and compass, lay out your design with light pencil lines, then complete the graphic using compass and ruler, in pen, then color it using colored pens
• If you use Sketchpad, hide all the points before you print it. (Click on one point. Then use the Edit menu to Select All Points. Then use the Display menu to Hide the points.)
• For both of these options, complete your geometric graphic using a black pen for the outlines, and colored pens or colored pencils for shading.
• Write at least 30 words about the transformations that you used, the kind of symmetry that your design has, and the names of the geometric figures in your design (for example: "I drew three connected lines on the left side of the rectangle and translated the lines to the right. Then I drew a half-circle and a triangle on the top of the rectangle ..." etc.)

I certainly hope you have fun doing this project!

Go To Chapter 3: Tessellations