Multiple Transformations

Objective: When done with this lesson, you will be able to apply a combination of translations, rotations, and reflections to an object and give the resultant coordinates. Also you will be able to create a design with or without technology using a combination of two or more transformations

Approximate completion time: 6-10 hours

(The tile design above is a type of "Penrose Tiling". If you'd like to try your hand at Penrose Tilings, then try to cover the floor in this activity.)

Floor tilings and patterns often intrigue the mind. We will look into some tiling work in this lesson as we study Multiple Transformations.

To start, check out some of the work people have done with tilings and patterns. (Yes, we have looked at some similar work in Unit 1.)

• Saxe-Patterson tile work
• The Alhambra
• M.C. Escher's work in "symmetry"

So how do people come up with those amazing designs? Let's try our hand at it!

To keep it simple, let us work with only one shape at a time. Can you think of a shape that would fit together perfectly with copies of itself so that an entire floor could be covered without gaps? Such a shape is said to "tessellate".

Shapes that tessellate by themselves	Shapes that don't tessellate by themselves
Take any one of the shapes above and you could make copies of it to completely cover a floor as large as you want. No gaps or overlapping of the shape need happen. (Depending on the shape of the floor, you may have to trim some of the pieces on the outside edges of the room.)	You cannot take a single shape above to use in tiling a floor without gaps or overlapping. You can try, yet you will find that you will need another shape to fill in the gaps. This is not to say that you couldn't make a fabuluous design, but that you would need more than one shape to do it.

Example 1.

Here is a sample of how a triangle could be used as the basic starting place for a tile. Since the equilateral triangle would tessellate, some careful "morphing" of the triangle will still tessellate.
	Click on the word "morph" to see the triangle transformed into another tessellating shape. Do you see the rotations? What do you notice about the areas of the triangle and the finished tile?

Do you think that shape will really tessellate? Click the tile below to watch!

 

Example 2.

Here is another sample of how a simple shape we know tessellates can be "morphed" into an interesting tile that tessellates.
	Do you see the translations? What do you notice about the areas of the rectangle and the finished tile?   You could do the same type of work with a piece of cardstock, some scissors, and some tape!

 

Hopefully that gives you some insight into how a simple tile could be made by using transformations. In your assignment you will be practicing using multiple transformations and will create your own design.

Before you start into your assignment, let's look at a tile that is made by using multiple transformations.

Example 3.

In this example notice the two different types of transformations: reflection and translation (Click on the parallelogram to watch!)

 

Example 4.

Here is one last example for you to think about. Would this idea work? (Click the square to start.)

Since the original square tessellates and the area of the tile is the same as the original square, the tile should tessellate through rotations.
(Here's one example of what a square with two such rotational morphings looks like.)

 

Now it's your turn to practice multiple transformations.

Assignment 4.05a - Multiple Transformations.

First, before you make your design, you should practice multiple transformations. With the help of a coordinate system you will be rotating, reflecting, and translating shapes. Download Assignment 4.05a here.

 

Assignment 4.05b - Tessellation Design.

Create a design of your own using at least 2 transformations.
To be successful, follow these steps:

• Using cardboard, cardstock, notecards, or whatever material you choose to make your tile.
• Create your tile by drawing a basic tessellating shape and then morphing it as shown in the examples above.
  • Sketching a design on one side of the shape to start is a good idea.
  • Cut along your design so that you can rotate, reflect, or translate the small piece.
  • Use tape to hold your "morphed" tile together.
  • Apply a second transformation by morphing another side.
• After you have created one tile, simply use it as a template. By tracing around your template, you can create amazing designs!

If you think you might need them, here are some other helpful online resources:

• http://www.lwcd.com/paper-folding/tessellations.html
• http://www.iproject.com/escher/teaching/maketessel.html

 

Keep in mind that you will be turning in two parts for this design:

1. A digital image of your final design. (A digital camera or scanner would work well for this. If you're in need of one, ask a friend or a local library to assist you.)
2. A three to five paragraph written explanation of your design project. Open up a new word document, and in it you should include the following:
   • A description of what basic shape with which you started.
   • A detailed description of how you "morphed" your shape to arrive at your tile.
   • An explanation of how you made the design. (You should be using words like translate, rotate, and reflect.)
   • If you want, you can insert pictures of your project along the way to help explain what you did.

Feel free to add color and other artistic designs to your project. Have fun with it.

When your design and writeup are completed, please submit both your digital image of your final design and your three to five paragraph written explanation.