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

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.

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

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.