Building Dynamic Tessellations Using Dynamic Custom Translations

A Sketchpad Exploration by John Olive, Ph.D.

University of Georgia

Dynamic Custom Translations

1. Create a simple figure on a new sketch (e.g. create four points with the shift key down and construct the polygon interior.)

2. Create a segment. Select the two endpoints of your segment (note the order of selection).

3. Under the Transform menu select Mark Vector. (Note the name of the vector.)

4. Select your simple figure. Under the Transform menu select Translate.

5. When the Translate window appears choose By Marked Vector. What happens?

6. Move one end of your segment which defines the translation vector. What happens to the translated image of your figure?

7. Select one point or the interior of your figure and then select the corresponding point or the interior of its translated image (using the shift key to maintain selection of the original). [Note: If the objects selected are colored then the custom transformation will maintain the color. Points have no color, so selecting corresponding points will not preserve color.]

8. Under the Transform menu choose Define Transform. Name your translation by the labels of the endpoints of the translation vector (e.g. Translate by EF).

9. Select the translated image of your figure and choose your custom translation from the Transform menu (e.g. choose Translate by EF). What happens?

10. Use the command key number for your custom translation (probably Command-1) and repeat the translation of your figure several times (see Figure 1). What happens when you change the size or direction of your translation vector?

Figure 1: Custom Translation by Vector EF



 
 
 
 
 

Translation tessellation of a parallelogram

1. On a New Sketch, create two translation vectors which share a common point (see Figure 2).

Figure 2: Two Translation Vectors

2. Select points B and A (in that order) and choose Mark Vector under the Transform menu.

3. Select segment BC and translate by marked vector BA.

4. Select segment BC and its translated image in that order and define your own custom translation. Call it Translate by BA.

5. Select points B and C (in that order) and choose Mark Vector under the Transform menu.

6. Select segment BA and translate by marked vector BC.

7. Select segment BA and its translated image in that order and define your own custom translation. Call it Translate by BC.

You should now have a parallelogram as in Figure 3.

Figure 3: Parallelogram formed from two vectors

8. Use repetitions of your two custom translations to tessellate the plane with your parallelogram as in Figure 4. One efficient way to do this is to select all sides except BC and repeat the Translate by BA translation until you reach the top of your screen, then select all segments except the left side of your figure (do not select BA and its translated images) and Translate by BC until you have filled the screen to the right.

9. Change the position of your original points. What happens?

Figure 4: Tessellation of the Plane with a Parallelogram

Escher tessellation of a fractured parallelogram

1. On a new sketch, start with three points A, B and C as in the last section.

2. Create a "fractured" path from A to B and another from B to C (see Figure 5).

Figure 5.

3. Select points B and A (in that order) and choose Mark Vector under the Transform menu.

4. Select point C and translate by marked vector BA.

5. Select point C and its translated image in that order and define your own custom translation. Call it Translate by BA.

6. Select points B and C (in that order) and choose Mark Vector under the Transform menu.

7. Select point A and translate by marked vector BC.

8. Select point A and its translated image in that order and define your own custom translation. Call it Translate by BC.

9. Apply Translate by BA to the segments in the fractured side BC, and Translate by BC to the fractured side BA. You should now have a "fractured" parallelogram which will be the base figure for your tessellation.

10. Select all the segments along the left side and top of your figure and apply your custom translation Translate by BA repeatedly until you reach the top of your screen.

11. Select all your current segments by clicking on the segment tool and then choosing select all segments from the Edit menu Use repetitions of your Translate by BC until you have filled the screen. You should now have a tessellation similar to Figure 6.

NOTE: To create the interior of your shape, translate the points as well as the segments to complete your first base shape and then select each point in cyclic order going around your shape with the shift key depressed. With all your points (and no segments) selected choose Polygon Interior from the Construct menu. Apply each of your custom transformations to the interior of your shape to fill in your design.

12. Move any of the points on your fractured sides to create a desired design. Have fun!

Figure 6: Tessellation Based on Translation of a "Fractured" Parallelogram

Click here to explore a dynamic tesselation based on translation.

Explore the Rotation-translation tessellation in the file Rotation_trans_tess.gsp. What geometric shape is the tessellation based on? What has been rotated and what has been translated?

Explore the file Glide_reflect_tiling.gsp file, created by the designer of GSP, Nick Jackiw.