Olive & Oppong: Transforming Mathematics with GSP 4, page 55
Chapter 5: Exploring Transformations and Tesselations
The NCTM Principles and Standards (2000) recommend the following regarding the
study of transformations in the Geometry Standard for grades 9-12:
Apply transformations and use symmetry to analyze mathematical situations:

Understand and represent translations, reflections, rotations, and dilations of objects
in the plane by using sketches, coordinates, vectors, function notation, and matrices;

Use various representations to help understand the effects of simple transformations
and their compositions. (p. 308)
Body Motions
Stand up and face a wall of your room. Move forward still facing the wall. Move
sideways but still face the wall (do not turn). Move back, still facing the wall. All of these
motions are called translations. Now try moving back to your original position without
turning (still facing the wall). This final move was the inverse translation of the combination
of the first two translations.
Question: Could you move to any location in your room using only translations?
Work with a partner and, standing facing one another, try to mirror each other’s arm
motions. Now try moving and mirror each other’s motion in space. If your partner turns
his/her body make sure you turn your body as if you were the mirror image of your partner.
Your motions are the reflection of your partner’s motions. Where would the line of reflection
be drawn?
Question: Could you move to any location in your room using only reflections
(assuming you may draw the reflection line any where in the room)?
Working with a partner again, stand side by side with shoulders touching and both of
you face a specific stationary object in the room. Make an estimate as to how far away you
are from the object. One of you must now move away from your partner in such a way that
you remain facing the fixed object and also remain the same distance from the object as your
stationary partner. This motion is called a rotation. You should have moved as if you were on
Olive & Oppong: Transforming Mathematics with GSP 4, page 56
an imaginary circle whose center was your fixed object. Have your partner move to where
you are, keeping the same distance from the fixed object and always facing the fixed object.
Move away from each other again, still facing the fixed object and still maintaining the same
distance from the object.
Question: Can you move to any location in your room using only rotations (assuming
you may choose the center of rotation to be any where you wish)?
Have 3 or 4 students stand side by side, all facing the same way, and invent a simple
line dance (e.g 2 steps forward, step to the left, step to the right and one step backwards).
Have pairs of students dance together as partners facing one another. Try some swing
dancing where one partner swings around the other. Which of the 3 transformations were
used in each different kind of dancing?
Dancing with Sketchpad
The exercises with body motions should have given you some feeling for how the
different transformations affect the movement of a body in space. We’ll now use GSP to
create sketches in which we can explore the geometry of these motions in the plane. First
create a simple stick figure something like the one in Figure 5.1
Figure 5.1: Stick Figure Drawn in GSP
We are going to use the translation transformation in GSP to create a line of stick
figures from this one figure. There are several ways we could do this. The simplest way is to
translate the figure by a fixed distance horizontally. To do so, you simply select the whole
Olive & Oppong: Transforming Mathematics with GSP 4, page 57
figure and then choose Translate from the Transform menu. A dialogue box will appear with
a choice of Polar Vector or Rectangular Vector. Choose the Rectangular Vector button and
enter the horizontal distance you desire. Remember to enter zero for the vertical distance.
You should obtain a second figure the fixed distance from your original (see Figure 5.2).
Figure 5.2: A Translation by Fixed Horizontal Distance
Change your original stick figure. Shake a leg; wiggly its hips; move an arm up and
down. How did the translated image respond? Create a line of dancers by translating the
image figure again and again. Have some fun with your line dancers!
An alternative (and more flexible) way of creating your line of stick figures is to use
the end-points of a segment as a vector of translation. Start with a new sketch and a new
stick figure (you can copy your original stick figure from your first sketch and paste it into
this new sketch). Draw a segment underneath your stick figure and select its end-points (from
left to right). Choose Mark Vector from the Transform menu. The end-points of your
segment now define a vector for purposes of translation. A vector has both length and
direction. Select the whole of your stick figure and choose Translate from the Transform
menu. This time the dialog box should have By Marked Vector selected. Accept this
choice. A translated image of your stick figure should have appeared in your sketch. Move
one end of your vector segment. How does this affect your translated figure? Does it have
any affect on your original figure? Translate the image figure again and again to form a line
of stick figures as in Figure 5.3.
Olive & Oppong: Transforming Mathematics with GSP 4, page 58
R
S
Figure 5.3: A Line of Stick Figures Translated by a Vector RS
Use your vector segment to move your stick figures further apart or closer together.
Can you find a way to have all of the stick figures piled on top of one another so that you
only see one figure? Change the direction of your vector segment. How does this affect the
line of stick figures? Adjust your stick figures and vector segment so that they are all holding
hands!
Partner Dancing with GSP. Start with a new sketch and a new stick figure. This time
create a reflected image of your stick figure. You will need to draw a line or segment for
your mirror. You can mark your mirror line or segment as a mirror for reflection in GSP by
simply double clicking on it (you can also select it and use the Transform menu to mark it as
a mirror). Having marked your mirror, select the whole of your stick figure and choose
Reflect from the Transform menu. You should have a reflected image something like Figure
5.4.
R
mirror
S
Figure 5.4: Reflected Stick Figure in GSP
Olive & Oppong: Transforming Mathematics with GSP 4, page 59
Move parts of your original stick figure and observe how the reflected image moves.
Move your mirror segment to move your dancers further apart or closer together. Change the
orientation of your mirror segment. What happens to your image figure?
Create a second mirror segment (approximately) parallel to the first mirror and to the
right of your reflected figure. Reflect your reflected image about this second mirror (see
Figure 5.5). What do you notice about the relation of the last image to your original stick
figure? Try drawing a translation vector that would translate your original figure onto this
third figure.
R
mi rror 1
T
mi rror 2
S
U
Figure 5.5: Double Reflection in GSP
You can also rotate your stick figures about a designated center of rotation. Any point
in a sketch can be selected as the center of rotation simply by double clicking on it. In a new
sketch create a stick figure and a free point, marked as a center of rotation. Select your stick
figure and choose Rotate from the Transform menu. The default angle of rotation is 45º. You
can change this to any angle you wish. Enter your own angle of rotation. Which way did your
stick figure rotate (clockwise or counter-clockwise)? Return to your sketch with the double
reflection as in Figure 5.5. Move the second mirror segment so that it is no longer parallel to
the first mirror segment. How does the third stick figure relate to the first figure? Try and
find a center of rotation and a rotation angle that will rotate your original stick figure onto the
third figure.
Assignment 5.1: The Burning Tent Problem
Imagine you are camped about 50 yards from a river. You decide to go looking for
more dry wood to keep your campfire burning. You are carrying an empty 5-gallon bucket in
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which to place the dry wood. You are about 100 yards from your tent and about 40 yards
from the riverbank when you notice that your tent has caught fire. You decide to run to the
river, fill your empty bucket with water and dash to your tent to try and dowse the fire. Draw
a sketch in GSP to represent this situation and find the shortest path to the river and tent.
Assume that the stretch of river between you and the tent is fairly straight and that you and
your tent are on the same side of the river. Vary the situation (distance of tent from river,
your distance from tent and river). Can you find a general construction that will guarantee the
shortest path to the river and tent from any starting position?
Dilations by Fixed Ratio
Dilations are similarity transformations: they preserve shape but do not preserve size.
Shrinking and enlarging are dilations. In order to construct a dilation, a center of dilation
must be designated. An object can then be dilated with respect to this center point by a
designated ratio of dilation. A ratio greater than one results in an enlargement, whereas a
ratio less than one results in a shrinking of the object. Negative ratios have some special
effects that we shall be exploring. We shall also explore the relationship of the center of
dilation to the pre-image and dilated image of an object.
1.
2.
3.
4.
5.
On a new sketch create a polygon with an interior.
Create a free point and label it Center.
Select the point and under the Transform menu choose Mark Center.
Select your whole polygon and under the Transform menu choose Dilate…
A dialog window appears as in Figure 5.6. The default choice is by fixed ratio and the
default ratio is 2:1. You can change the ratio by changing the numbers in the New and
Old boxes. For this first exploration accept the ratio 2:1 by clicking on the OK button.
Figure 5.6: Dilate Dialog from GSP
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6. The dilated image of your polygon should appear. Move the point Center until the two
polygons are separated as in Figure 5.7.
Dilation of Polygon 1 about the
point Center by a ratio of 2:1
P olygon 1
Center
Figure 5.7. Dilation by Fixed Ratio 2:1
Explore what happens as you move the point Center around the screen. Move your
original polygon towards the center point and away from it. What happens to the relative
positions of the two polygons? Make a conjecture as to what you think will happen as you
move your polygon slowly through the center point to the other side of it.
Measure corresponding segments on the two polygons. Make a conjecture about the
ratio of the lengths of corresponding segments. Write down your conjecture for the ratio of
the areas of the interiors of the two polygons. Measure these areas. Check your conjecture by
using the GSP Calculator to calculate the ratio.
Construct a ray from the point Center passing through a vertex of your original
polygon. Where does the ray intersect with your dilated image? Move your original polygon
around; change its shape. Do the corresponding vertices still lie on the ray? Move the center
point around. What appears to be happening to the dilated image?
Measure the distance from the center point to each of the vertices on the ray. What
do you notice? Make a conjecture about the ratio of the distances of corresponding vertices to
the center of dilation.
Explore dilations using different ratios. Choose some ratios where the “New” number
is less than the “Old” (this gives a ratio less than one). Try entering a negative number in the
“New” box. Take measurements, construct rays and test all of your conjectures in these new
situations.
Create a dilation that is equivalent to a rotation of 180 degrees about the center point.
Olive & Oppong: Transforming Mathematics with GSP 4, page 62
Dynamic Transformations
The Translation by Marked Vector and the Reflection by Marked Mirror are both
dynamic transformations in the sense that you can dynamically change the translation vector
or mirror line and observe the effect this creates on the transformed image. Such dynamic
manipulation of the actual transformation is not possible with the rotation by fixed angle and
dilation by fixed ratio (other than changing the position of the marked center point for these
transformations). It is possible, however, to define both rotations and dilations using dynamic
angles and ratios (respectively).
Dynamic Rotations
Any three points in a sketch can be used to mark an angle of rotation. The order in
which the three points are selected determines the angle and its direction. The second point is
the vertex of the angle; the direction of rotation is from the first to the third about the second.
It is usually helpful to see the angle using two segments (or rays) with the vertex of the angle
as a common end-point. With the three points selected you choose Mark Angle from the
Transform menu. You will still need to mark a point as the center of rotation. This does not
have to be the vertex point of your marked angle (although for some investigations or
demonstrations it might make sense for the vertex of the angle of rotation to be the center of
rotation). You can measure the angle that you have marked simply by selecting the three
points (in the same order) and choosing Angle from the Measure menu. Figure 5.8 illustrates
a rotation of a concave quadrilateral by the marked angle EFG about the marked Center of
Rotation.
E
G
F
C enter of R otation
Figure 5.8: Dynamic Rotation in GSP
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Dynamic Dilations
In order to define a dilation using a dynamic ratio we need some way of constructing
a dynamic ratio. The simplest way is to use two segments to define the ratio. Simply select
any two segments in your sketch and choose Mark Ratio from the Transform menu. The
first segment selected will determine the numerator of your ratio. With a point in your sketch
marked as a center you can now dilate any figure about this center by the ratio determined by
your two segments. In figure 5.9, the rotated quadrilateral (p1’) has been dilated (about the
center of rotation) by the ratio of the two segments I used to define my angle of rotation to
give quadrilateral p1”!
p1
p1 ’
E
p1 ’’
j
G
k
F
Cente r o f Ro ta ti o n
Figure 5.9: Dilation of the Rotated Quadrilateral in GSP
Explore the result of this combination of a dynamic rotation with a dynamic dilation.
Create your own sketch and manipulate the segments that define both the angle and the ratio
of dilation. Change the position of the marked center point. In the next section you will learn
how to define the result of these two transformations as a custom transformation.
Custom Dynamic Transformations
A custom transformation is simply a transformation that you define yourself. It can be
a single transformation or a combination of many transformations. We shall define the result
of the rotation followed by dilation illustrated in Figure 5.9 as a custom transformation and
then apply this two-step transformation over and over to create dynamic spirals.
All you need to define a custom transformation is the original object and the final
object resulting from your sequence of transformations. In our case this will be our original
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quadrilateral (p1) and the dilated quadrilateral p1”. Because our intention is to create a spiral
from our custom transformation it will be better to hide the rotated quadrilateral (p1’) before
defining the custom transformation. It will also help to hide the label of the final quadrilateral
(just click on the quadrilateral with the Label tool).
Step 1: Select the original and final objects for your transformation (the original and
final quadrilaterals in figure 5.9).
Step 2: Choose Define Transform from the Transform menu.
Step 3: Name your custom transformation by typing in the dialog box.
When you go to the Transform menu you should now see the name of your custom
transformation at the bottom of the list of options. You should also notice the command-1
symbol (Ctrl-1 on a Windows machine) to the right of the name of your customized
transformation. This means that you can use the Command-1 keystroke to invoke your
transformation instead of having to go to the Transform menu. As you define more custom
transformations these will be assigned command-key numbers in order of definition.
In order to use your new transformation, simply select the object you want to apply it
to (in our case the final quadrilateral, p1”) and press the Command key and the number 1 (at
the same time) or Ctrl-1 if on a Windows machine. A rotated and dilated image of the object
should appear. Continue pressing the Command-1 key combination. A sequence of rotated
and dilated quadrilaterals should appear as in Figure 5.10.
p1
E
G
j
k
Center of Rota tion
F
Figure 5.10: A Spiraling Quadrilateral from GSP
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Glide-Reflections
A very special combination of a translation followed by a reflection across the vector
of translation is called a Glide-reflection. Start with a segment and mark its endpoints as your
vector of translation. Then mark the same segment as a mirror. Create a non-symmetric
figure (such as the concave quadrilateral in Figure 5.8), translate it by marked vector and
then reflect the translated image across the marked mirror. Hide the translated image. Select
your original and final images to define your custom transformation as a glide reflection.
Apply your glide reflection to your final image and to subsequent images several times.
Write a brief description of the pattern that emerges from repeating a glide reflection several
times.
Transformation Isometries with GSP
A transformation (or combination of transformations) is isometric to another
combination of transformations if both sets of transformations end up with the same final
image for any given pre-image. In the following assignment you will test the mathematical
assertion that “All shape-preserving transformations can be achieved by at most three
reflections.”
Assignment 5.2: Construct the reflection lines for the following transformations:
- a simple translation (defined by a given vector)
- a simple rotation (defined by a given angle)
- a translation followed by a rotation (defined by a given vector and a given angle)
- a glide reflection (defined by a given vector)
For each of the above constructions, start with the GSP object(s) that define the
original transformation. Using these objects, construct the reflection lines that will produce
the same effect as the original transformation. Test your construction by creating a nonsymmetric polygon in your sketch and applying both the original transformation to this
polygon and the sequence of reflections to this polygon. The final images should overlap
exactly and stay overlapped when the original image is moved or when the defining objects
for the original transformation are changed.
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One way to begin this activity is to define your original transformation (say an angle
of rotation using three points) and apply this to a polygon. Play around with the angle and
observe the relative motion of the rotated image. Try and imagine where two reflection lines
would need to be constructed to produce the same image. Construct a first reflection line that
relates to your angle in some way. Reflect the polygon about this line. Now play with
another reflection line until you get the second reflection to overlap with the rotated image.
What do you notice about the relation of the two reflection lines to the angle of rotation? Use
what you find out from this play phase to construct the two reflection lines based on the
original angle of rotation.
Challenge Assignment 5.3: Given three arbitrary mirror lines, construct the glide-reflection
vector that would produce the same image as the three reflections. Note: Does the order in
which you carry out the reflections change the position of the final image?
From Transformations to Tessellations
Translation tessellation of a parallelogram
1. On a New Sketch, create two translation vectors which share a common point (see Figure
5.11).
A
B
C
Figure 5.11: Two Translation Vectors
2. Select points B and A (in that order) and choose Mark Vector under the Transform
3.
4.
5.
6.
menu.
Select segment BC and translate by marked vector BA.
Select segment BC and its translated image in that order and define your own custom
translation. Call it Translate by BA.
Select points B and C (in that order) and choose Mark Vector under the Transform
menu.
Select segment BA and translate by marked vector BC.
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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 5.12.
A
B
C
Figure 5.12: Parallelogram formed from two vectors
8. Use repetitions of your two custom translations to tessellate the plane with your
parallelogram as in Figure 5.13. 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?
A
B
C
Figure 5.13: Tessellation of the Plane with a Parallelogram
Olive & Oppong: Transforming Mathematics with GSP 4, page 68
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 5.14).
A
B
C
Figure 5.14.
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
5.15.
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.
Olive & Oppong: Transforming Mathematics with GSP 4, page 69
12. Move any of the points on your fractured sides to create a desired design. Have fun!
A
B
C
Figure 5.15: Tessellation Based on Translation of a “Fractured” Parallelogram
Explore the Rotation-translation tessellation in the file Rotation/trans.Tess. What geometric
shape is the tessellation based on? What has been rotated and what has been translated?
To be added:
Rotation tessellation based on any triangle (two sides have point-symmetry)
Rotation Tessellations based on the techniques developed by M. C. Escher
Reflecting on connections between art and mathematics