(Geometer's Sketchpad Tessellations)
Have you ever made a tessellation out of fragmented parallelogram?
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Geometer’s Sketchpad can do the same thing but still has the power to manipulate the unit piece after you have
completed your tessellated plane.
o
Start by connecting several little line segments together to form the bottom
and left side of the parallelogram as shown at the right. (Please make certain
that each adjacent segments share the same endpoints) Label the 3 points that
will become the vertices of the parallelogram with the finger tool.
o
Select Point A and I in that order. Select Mark Vector “A –>I” under the
Transform menu. Watch your sketch as you select this from the menu
and you should see a little
animation of a dotted 'laser beam' going from A to I.
o
An easy way to select
many objects at once is
to "drag a box" around
the items you wish to be
selected as shown
below.
With the selection tool drag a box around all of the segments from A to E
as shown at the right
o
After the fragmented segment is highlighted, select Translate… from the
Transform menu. Make sure the radio button labeled “By Marked
Vector” is checked. Then, click Translate. This will create the top
segment of your fractured parallelogram.
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Next, we will need to mark the vector AE . Using the selection tool,
select Point A and E in that order. Select Mark Vector “A->E” under the
Transform menu. Again, watch your sketch as you select this from the menu and you should see a little animation of a dotted
'laser beam' going from A to E.
M. Winking
p.70
e-mail: [email protected]
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With the selection tool we will need to carefully need to drag a box around all of
the segments between A and I
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Once Again, after the fragmented segment is highlighted, select Translate… from
the Transform menu. Make sure the radio button labeled “By Marked Vector” is
checked. Then, click Translate. This will create the right hand side segment of
your fractured parallelogram.
o
Using the selection tool, deselect everything by clicking in any blank space of
the sketch. Next hold down shift (this will help incase you miss clicking on a
point you won’t deselect everything) and begin clicking on each point around
the perimeter of the fractured parallelogram in consecutive order ( We must click
on each point as we would if we were showing someone how to connect the dots to create
the parallelogram).
o
1
2
3
4
5
After all of the points are selected, under the Construct menu select Polygon
7
…..and so on all the way
around the parallelogram
interior. This will fill in the interior of the polygon.
o
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While the polygon interior is still highlighted under the Display menu we can
change the polygon’s interior color.
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The last vector we marked was AE . We can re-use this marking to translate our
polygon interior. Make sure the polygon interior is still highlighted. select
Translate… from the Transform menu and click Translate This will create a
new polygon interior directly to the right of the unit piece of our tessellation.
o
While the newly created polygon is still highlighted, change its color under the
Display menu and then, select Translate… from the Transform menu. Repeat
We can choose the
color of the interior
of the polygon under
the Display menu.
this process until there are about 5 – 8 parallelograms across the bottom. (The
tessellation may look best with alternating colors as shown at the right)
o
To better establish alternating colors we will need to mark a new vector EI . (It
may first be necessary to deselect everything by clicking in a blank area with the
selection tool) Using the selection tool, select Point E and I in that order. Select
Mark Vector “E->I” under the Transform menu. Again, watch your sketch as
you select this from the menu and you should see a little animation of a dotted
'laser beam' going from E to I.
o
Using the selection tool, select all of the polygon interiors by clicking on each of
them. After all of the interiors are selected, under the Transform menu select
Translate… and click Translate
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While the new row of polygon interiors are still highlighted, again select
Translate… from the Transform menu. Repeat this process until there are about
5 – 8 rows.
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How do you think we might continue to tessellate this space up and to the right?
Which vector should we mark? Which polygon interiors should we highlight and
then translate?
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DO NOT forget the most important aspect of Sketchpad, "Everything is dynamic." Play around with the points on the unit
piece by clicking and dragging the points. Experiment with setting tracks for original points and animating.
M. Winking
p.71
e-mail: [email protected]
Have you ever made a tessellation out of fragmented equilateral triangle?
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tape
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tape
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Geometer’s Sketchpad can do the same thing but still has the power to
manipulate the unit piece after you have completed your tessellated
plane.
o
Start by creating a several conjoined segments that would represent the fragmented
left side of the equilateral triangle.
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Double click on the top point (point B) of the fractured segment. This will mark
the point as the center of rotation.
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Highlight the entire fragmented side of the triangle by dragging a box around the
entire side.
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Select Rotate under the Transform menu. Change the angle to 60º and press the
Rotate button.
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Using the segment tool create a segment between the two bottom points which
should represent the two bottom vertices of the equilateral triangle. While the
segment is still highlighted, select Midpoint from the Construct menu.
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Click in a blank area to deselect everything. Highlight the bottom segment and
select Hide Segment under the Display menu. The midpoint should remain.
M. Winking
p.72
e-mail: [email protected]
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Make a set of conjoined segments between points A and D.
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Double click on the midpoint (point D) until it flashes denoting that it has been
marked as a center.
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Highlight all of the conjoined segments between point A and D by dragging a box
around them. Then, select Rotate under the Transform menu. Change the angle to
180º and press the rotate button.
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Using the selection tool, deselect everything by clicking in any blank space of
the sketch. Next hold down shift (this will help incase you miss clicking on a
point you won’t deselect everything) and begin clicking on each point around
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the perimeter of the fractured triangle in consecutive order ( We must click on each
6
5
4
3
point as we would if we were showing someone how to connect the dots to create the
triagnle).
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2
After all of the points are selected, under the Construct menu select Polygon
interior. This will fill in the interior of the polygon.
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Click in a blank space to deselect everything. Then, double click on the top vertex
(point B). After marking point B as the center of a rotation, highlight the polygon
interior. Select Rotate under the Transform menu. Change the angle to 60º and
press the Rotate button. While the new image is highlighted, select Rotate again
from the Transform menu. Continue to use an angle of 60º and press the Rotate
button again. Repeat this process until the entire tessellated hexagon is completed
(3 more times). When complete, you may wish to alternate the colors. Experiment
with the colors under the Display menu.
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Next double click on a vertex of the hexagon (such as point A). Highlight all 6
polygon interiors and select Rotate from the Transform menu. Change the angle
to 120º and press the Rotate button. While the new image is still highlighted,
rotate the new image another 120º in a similar manner.
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Remember to ‘play’ with all of the original points. Try experimenting with
animating the unit piece.
M. Winking
p.73
e-mail: [email protected]
1
8
9
…..and so on all the way
around the triangle
Non Traditional Tessellations (Escher-esque)
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Start by creating a circle similar to the one shown at the right. Then, create two
radii of the circle similar to the diagram at the right.
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Using the point tool, place a point on radius AB somewhere towards the edge of
the circle. Label this point D.
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Click in a blank space to deselect everything. Using the selections tool, highlight
radius AB, radius AC and the Circle. Select Hide Path Objects from the Display
menu. Only the points should remain.
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Using the segment tool, create a set of conjoined segments between B and D.
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Click in a blank area to deselect everything. Using the selection tool, highlight
Point B, A, and C in that order. Select Mark Angle under the Transform menu.
You should see an animation on the sketch denoting the angle has been marked.
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Drag a box around the conjoined segment and all of the points between them
except for point B (shown at the right). Select Rotate… under the Transform
menu. Make certain the radio button is checked next to Marked Angle BAC
about Center A. Click on the button Rotate.
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Using the segment tool, create a set of conjoined segments between B and C.
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Click in a blank area to deselect everything. Using the selection tool, highlight
Point A, B, and D in that order. Select Mark Ratio under the Transform menu.
You should see an animation on the sketch denoting the angle has been marked.
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Drag a box around the conjoined segments between points B and C (shown at the
right). Select Dilate… under the Transform menu. Make certain the radio button
is checked next to Marked Ratio (AD/AB) about Center A. Click on the button
Dilate.
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Using the selection tool, deselect everything by clicking in any blank space of
the sketch. Next hold down shift (this will help incase you miss clicking on a
point you won’t deselect everything) and begin clicking on each point around
the perimeter of the fractured quadrilateral in consecutive order ( We must click
on each point as we would if we were showing someone how to connect the dots to create
the quadrilateral).
o
6
After all of the points are selected, under the Construct menu select Polygon
interior. This will fill in the interior of the polygon.
5
4
2
3
1
M. Winking
p.74
e-mail: [email protected]
8
7
…..and so on all the way
around the quadrilateral
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While the polygon interior is still highlighted select Dilate… from the Transform
menu. Make certain the radio button is still checked next to Marked Ratio
(AD/AB) about Center A. Click on the button Dilate. While the new image of
the polygon is highlighted select Dilate… again and continue dilating a few more
images until the images get relatively close to point A. After enough images have
been generated, try selecting every other polygon and changing the color under the
Display menu.
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Using the selection tool, highlight all of the polygon interiors and Point C. Select
Rotate… under the Transform menu. Make certain the radio button is checked
next to Marked Angle BAC about Center A. Click on the button Rotate. The
colors will of the new images will need to be changed so that they alternate
properly.
E
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After correcting the color sequence, select Points B, A, and E in that order. Select
Mark Angle under the Transform menu. Highlight all of the polygon interiors
and select Rotate… under the Transform menu. Make certain the radio button is
checked next to Marked Angle BAE about Center A. Click on the button
Rotate. Continue rotating each new image to complete the circle. Don’t forget to
alter the pre-image points.
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Try experimenting with copying and pasting some of M.C. Escher’s work into Geometer’s Sketchpad. Can you
find investigate how he made some of his more difficult hyperbolic tessellations?
M. Winking
p.75
e-mail: [email protected]