construction and iteration:
mathematical generalization
in Dynamic Geometry
PRISM II, Toronto
16 May 2006
Nicholas Jackiw
KCP Technologies
njackiw@keypress.com
http://www.keypress.com/sketchpad
Introduction. The Dynamic Geometry act of dragging a point, and varying a shape, offers a
powerful encounter with mathematical generalization, by visualizing the entire family of shapes
determined by a set of mathematical relationships. Students can experience this within minutes
of encountering the tool—it’s the very heart of basic Sketchpad activity—and quickly learn to
use dragging, and construction, to explore and analyze mathematical conjectures. At a
considerably more advanced level, however, Sketchpad offers even more powerful tools for
generalizing mathematical expressions, objects, and actions you’ve created in Sketchpad.
Geometric loci generalize a construction across an entire range of possible positions. Analytic
functions generalize an arithmetic relationship across an entire range of possible values. Iteration
generalizes an arbitrary mathematical relationship across an entire range of possible repetitions.
These forms of generalization are the historical building blocks of mathematical development.
By working first with a simple construction of a basic shape, and then iterating that shape to
produce a fractal of both visual and arithmetical beauty, in this workshop you’ll encounter
mathematical generalization at both ends of the Sketchpad spectrum.
Contents
simple constructions: how to build a square
advanced constructions: a pythagorean tree
about iteration
geometric iteration challenges
numeric iteration challenges
fractal iteration challenges
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simple constructions: how to build a square
why construct?
1. Choose the Segment Tool (in the toolbox); then click and
drag in your sketch (the white space to the right of the Sketchpad
toolbox) to draw a segment. With the tool still chosen, draw a
few more segments to get the hang of them.
2. Now try to draw a square, using only the Segment Tool. Once
you succeed, use the Arrow Tool (top of the toolbox) to drag
each of your square’s vertices. What happens to the square?
Probably, your square stretched and twisted—and no longer
stayed “square.” This is because when you drew it, you
constructed four segments connecting four points. That’s a good
mathematical definition of a quadrilateral, but not a good
mathematical definition of a square.
When you drag parts of drawings in Sketchpad, the program respects the mathematical
definitions you’ve provided in that drawing—and nothing more. (This is the Dynamic
Geometry principle.) Here since you’ve defined a quadrilateral, your diagram behaves as a
(general) quadrilateral.
Construction is the mathematical art of drawing by precise definition. To construct a square,
we’ll need to define elements that account for a square’s right angles (we’ll use perpendicular
lines) and for the fact that all of its edges have the same length (we’ll use circles).
to the square
1. Draw a segment AB with the Segment Tool. (Don’t worry
about getting the labels to show up yet.)
2. With the Arrow Tool, click A, B, and the segment between
them, so all three objects appear selected as at left. Choose
Construct | Perpendicular Lines.
3. Now use the Compass Tool (above the Segment Tool) to
draw a circle centered on A and passing through B. Be sure not
to let go of the mouse button until the mouse is directly over
point B (B will appear highlighted).
4. Construct another circle centered on B passing through A.
Then construct a segment between the intersections of each
circle with the perpendicular lines through its center.
5. Do you see a square? With the Arrow Tool, select construction elements like the circles that
you no longer want to see and choose Display | Hide Objects. Use segments to finish the
square’s edges. When done, drag the vertices with the Arrow Tool to see how your square
“behaves.”
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advanced constructions: a pythagorean tree
continuing from your square
1. Now let’s introduce some labels and color. Choose the Text
Tool, and click the four vertices of your square in the order you
created them. If your square’s still in its original position, it will
look like this one.
2. Back to the Arrow Tool to select the four points and construct
their Polygon Interior. This shades in the square.
3. Construct the Midpoint of the top-edge, and then select that
midpoint, D, and E, and construct the Arc on Circle centered at
the midpoint, starting at D, and arcing to E.
4. Use the Segment and Text Tools to construct a CE and ED,
where E is any point on your arc.
What angle is CED?
Let’s review what you’ve done. You began with a segment AB. On top of it you built a square,
and on top of that a right triangle. The right triangle has two “exposed” edges, CE and ED.
To create a Pythagorean tree, we’d like to repeat this process, but instead of starting it with AB,
we want to start it with CE. That will give us another smaller square “to the left.” Then we want
to repeat it again, this time starting with ED to give us a third smaller square “to the right.” Then
we want to do the same thing again—to each of these smaller squares!
Rather than repeat all these steps yourself, have Sketchpad do it, by iteratively transforming the
locations of starting points A and B.
5. With the Arrow Tool select only points A and B. Then
choose Transform | Iterate.
6. With the Iteration dialog box showing, click C and E in
your sketch, to complete the “mapping” of A and B’s
location to the locations of C and D (see picture at left).
7. In the Structure menu, choose Add New Map to create
a second “map” for the locations of A and B. This time,
map them to E and D (as shown).
8. Increase the number of iterations with the Display menu
a few times.
9. Then close the dialog box and drag point E.
If ABCD has 1 square unit of area, what is the area of the
full tree shown at left?
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about iteration
In mathematics, iteration refers to the act of performing of some mathematical process—a
computation, algorithm, or construction—on some initial value, and then repeating that process
on the result. Each repetition of the process is a single iteration of the process. For example,
counting whole numbers—0, 1, 2, 3, 4—can be thought of as starting with the initial value of
zero, and then iterating the operation of “adding one.”
Iteration in Mathematics
Iteration appears across mathematics, from the simplest arithmetic operations (such as counting)
to very recent ideas about fractals and chaos. In the school curriculum, iterative processes may
show up in:
•
•
•
•
•
•
•
•
Arithmetic Procedures (e. g. counting, long division, etc.)
Sequences and Series
Calculus — Limits and Approximations
Fractals & Chaos
Optimization
Geometric Art & Design
Recursion
Dynamical Systems, Statistical Mechanics, etc.
In addition, iterative phenomena and processes frequently appear in physics, biology, and
chemistry.
Iteration and Mathematical Thinking / Problem Solving
The repetitive nature of iterative processes makes them appropriate to a variety of problem
contexts and problem solving strategies:
•
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Extrapolation:
If you can’t find an answer, can you find a process that leads to an answer?
(If so, iterate that process until you reach the solution.)
Incremental Approximation:
If you can’t find an answer, can you find a process that allows you to improve
an arbitrary guess? (If so, iterate that process until your approximation is
“close enough” for the purposes of the problem.)
Divide & Conquer:
If a problem is too hard to solve, can it be split into two smaller problems
which are easier to solve? If so, iteratively “divide” the problem into small
enough parts that the parts can be solved, and then assemble your result
from the “conquered” problem parts.
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geometric iteration challenges
For geometric iteration, use the Iterate dialog box to map pre-image points to image points.
Sketchpad repeats your construction as these iterated points move to their images. After iterating,
experiment with the level of iteration by pressing + or – when your iterated construction is
selected.
Midpoint Polygon
Hint: Map AE, BF, CD.
A
A
F
C
F
D
E
B
D
C
E
B
Rotation & Dilation:
F
F
E
E
D
D
B
B
C
C
A
A
Golden Spiral:
A
B
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numeric iteration challenges
For numeric iterations, use the Iterate dialog box to map one “parameter” pre-image to a
calculation based on that parameter. As you increase the level of iteration (n), Sketchpad displays
a table containing the values of all iterated measurements and calculations. Numeric iteration can
be used to explore numeric sequences and series.
Basic Function Iteration
Hint: First define pre-image seed (using Graph | New Parameter); then define f(x) (New
Function), then calculate f(seed) (Calculate). Iterate seedf(seed). Then try changing your
function f (by double-clicking it) to explore other iterated functions.
seed = 1.00
f x  = 2 x
f seed  = 2.00

n
f see d
0
2.0 0
1
4.0 0
2
8.0 0
3
16.00
4
32.00
5
64.00
Approximating the Square Root of p (Hero’s Method)
p
gu ess0 +
n
p = 5.00000
2
guess0 = 1.00000
p
guess0+
guess0
2
gu ess0
0
3.0000 0
1
2.3333 3
2
2.2381 0
3
2.2360 7
4
2.2360 7
5
2.2360 7
= 3.00000
p = 2.23607
Fibonnaci Numbers ( fn = fn-2 + fn-1 )
f 0 = 1.00
f 1 = 1.00
f 0 +f 1 = 2.00

n
f 0 +f 1
0
2.0 0
1
3.0 0
2
5.0 0
3
8.0 0
4
13.00
Numeric iteration can be combined with geometric visualization: Plot your individual calculated
values before iterating, and Sketchpad will plot the entire series after iterating.
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fractal iteration challenges
Fractal iterations frequently involve more than one map—that is, the same pre-image or set of
pre-images maps to two or more sets of post-images simultaneously. To create multiple
mappings in the Iterate dialog box, use Structure: Add New Map.
Sierpinski Gasket:
Hint: Make 3 Maps of ABC: ABCADE, ABCDBF, ABCECF.
A
A
E
D
C
F
B
E
D
F
B
C
Koch Curve:
A
B
A
B
Pythagorean Tree:
E
C
D
E
A
B
C
A
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