GEOMETER'S SKETCHPAD
ASSIGNMENT #5 (Youngberg)
Note the following instructions as you complete this assignment and all other GSP assignments:
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Although you will be emailing your assignment to me, save a copy of all of your sketches to your disk.
Answer all of the questions posed to you in a text box at the bottom of the sketch you are currently working on.
Make sure that all of your sketches are constructions, not drawings. That is, create all sketches so that the required
geometric relationships are maintained when any object is “dragged.”
1.
We learned in section 12.3 that any triangular tile will tessellate the plane. Construct an arbitrary triangle and it’s interior.
Transform this interior several times to create a tessellation. See the top of page 794 of your text if you need help getting
started. (Note: Just construct a few rows of the tessellation. Your tessellation should have at least 4 rows with at least 4
shapes per row.) Save your sketch as GSP 5A (triangle tiling).
2.
Construct a regular tessellation. There are three possibilities. See p. 791 of your text if you don’t remember what they are.
(Again, just construct a few rows of the tessellation.) Save your sketch as GSP 5B (regular tiling).
3.
We learned in section 12.3 that it is possible to tile the plane with any triangle or any quadrilateral. We also learned that
regular hexagons will tile the plane and certain non-regular pentagons and non-regular hexagons will tile the plane.
The Conway Criterion, named for the English mathematician, John Horton Conway, describes rules for hexagonal tiles that
will tile the plane. This criterion states simply that any hexagon with a pair of opposite sides that are parallel and congruent
will tessellate the plane.
Construct a non-regular hexagon ABCDEF (something like the one below) so that sides AB and ED are parallel and
congruent. (Note that AB and ED need to stay parallel and congruent as you drag points of your figure, but point A should
NOT be stuck directly above point E and point B should NOT be stuck directly above point D.) Construct the interior of
your hexagon in any color you choose. Because of the Conway Criterion, this hexagon will tessellate the plane no matter
how much you distort it by dragging points.
Construct the midpoint M of segment AF and the midpoint N of segment BC. Mark the point N as a center of rotation.
Select the polygon interior and the point M and rotate them 180° about the point N. Give the rotated image a different color
or shade.
Make sure that the image of M is labeled M’. Mark the vector MM’. Select the two polygon interiors and translate them by
this marked vector.
You now have a row of polygons. Can you see how they fit together? Mark the vector AE. Select all four polygon interiors
that you have constructed so far and translate them by the marked vector. Translate again two more times so that you have
four rows total. Change the colors of the polygon interiors as necessary so that you can tell them apart. Your sketch should
look something like this:
Drag any of the vertices of your original hexagon. The hexagon should still tile the plane—i.e. there should be no gaps
between polygons and there should be no polygons overlapping each other.
QUESTION 1: If you drag point B so that it lands on top of point A, what kind of tessellating shape do you have?
QUESTION 2: Leaving B on top of point A, if you then drag point F over top of points B and A, what kind of tessellating
shape do you have?
Move points A, B and E so that you have a hexagon again.
QUESTION 3: How could you turn the hexagon into a pentagon that will tile the plane by moving a point or points? (Try it.)
Move your points back so that you have a hexagon again.
Save your sketch as GSP 5C (Conway Criterion).
4.
In a new sketch, construct a “Conway-criterion hexagon” (a hexagon with one set of parallel sides) as you did in question 3.
This time use dashed lines for the hexagon.
Construct a jagged edge from A to B consisting of 3 line segments and translate this edge by the directed line segment
(vector) AE to obtain something like this:
Construct midpoints of the other four sides. From point B to the midpoint of BC construct a jagged edge consisting of two
segments:
Rotate this jagged edge along with the point on it 180° about the midpoint of BC:
Similarly, construct a jagged edge through the midpoints of other three sides of your original hexagon:
Construct the polygon interior and tessellate as you did with the hexagon in problem 3:
Note: Don’t try to make your figure look just like mine—be creative and see what sort of interesting figure you can come up
with by dragging the points around.
Save your sketch as GSP 5D (fun tessellation).
5.
For this problem, do either part (a) or part (b) below. You do not need to do both parts.
(a) Create a tessellation that looks just like the diagram below so that when points A, B, C, D, E or F are dragged, the scale
and/or orientation of the tessellation may change, but otherwise the tessellation remains the same. (Hint: The hexagon
ABCDEF is a regular hexagon.)
(b) Do problem 23 on page 803 of your text.
Save your sketch as GSP 5E (Escher-like design).