Exercise Set 5.1
1. What is a plane tessellation?
2. Where does the word tessellation originate?
3. Who were the Moors? How did they contribute to the art of tessellation?
4. In general which polygons can tessellate the plane? Why?
5. Construct the triangle  ABC using Sketchpad. Construct point D outside of  ABC. We want to make
a copy of this triangle to be located in a new place where point D replaces point A. Mathematically
speaking, we want to translate ABC from A to D. For this, we select
points A and D but select A
first. Then we open the Transform menu and choose “Mark Vector”. Now, if we select  ABC and
choose “Translate” in the Transform menu and click on “OK” then the desired triangle will be
constructed.
Based on the above instructions, do the following (Hint: Use “Translate” function diagonally):
(a) Construct the scalene triangle  ABC. Use this triangle to tessellate the plane. For the first
step translate  ABC from point A to point B.
(b) Construct the parallelogram ABCD. Tessellate the plane using this parallelogram.
(c) Construct the right trapezoid ABCD (A trapezoid with a right angle). Tessellate the plane
using this trapezoid. We notice in this case that we should first translate some of its sides and not
the trapezoid itself.
(d) Construct the convex quadrilateral ABCD. Tessellate the plane using this quadrilateral.
(e) Construct the nonconvex quadrilateral ABCD. Tessellate the plane using this quadrilateral.
6. Construct a square and on one side of it construct an equilateral triangle and then hide this side. The
generated pentagon has all sides congruent but not all its angles are congruent. This pentagon tessellates
the plane. Construct this tessellation.
7. There is a special class of nonregular hexagons that tessellate the plane. Such a hexagon has the
property that its sides are pairwise parallel (and therefore opposite sides are congruent).
(a) Construct such a hexagon. For this, construct first a convex quadrilateral such as
quadrilateral ABCD. Then translate the three sides of BC, CD, and AD properly as is shown
in Figure 5.1.9 to construct the desired hexagon (Hide side AB).
(b) Tessellate the plane using this hexagon.
D
C
B
A
Figure 5.1.9
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