Equilateral Triangles and Beyond

Topics in this lesson

1. Introduction to constructions with straightedge and compass
2. Constructing an equilateral triangle
3. Definition of tessellations
4. Constructing a regular tessellation of the plane by equilateral triangles
5. Related tessellations

Introduction to constructions with straightedge and compass

In constructing geometric figures with straightedge and compass, the allowed tools are a straightedge (you can use a ruler, but you can't use its markings) and a compass for drawing circles. If you have two points marked, you can use the straightedge to draw the line containing them. You can also use the compass to draw the circle with one of the points as center, and the other point on the circle. If two lines or two circles or a line and a circle intersect, you can use the intersection points to construct more things.

Using these basic tools and operations, you can draw a surprising number of things. But there are some things that can't be drawn with these tools.

It is useful to think of a compass as a tool for copying lengths; after all, (Definition:) a circle is the set of all points that are the same fixed distance (radius) from a given point (the center).

Construct an equilateral triangle

Construction steps

Draw a circle.
Without changing the compass opening, pick a point on the circle as a new center and draw another circle with the same radius.
Pick one of the two points where the circles intersect. Connect the two centers and your chosen intersection point with line segments.

You have just constructed an equilateral triangle.

Definition: An equilateral polygon is one whose sides all have the same length.
An equiangular polygon is one whose angles all have the same measure.
A regular polygon is a polygon that is both equilateral and equiangular.
We will deal with the definition of a polygon later. For the time being, a polygon is a plane figure with flat sides, like a triangle, square, or trapezoid.

Definition: A tessellation of the plane is a collection of plane figures that cover the plane without gaps or overlaps. A tessellation is also called a tiling. A tessellation need not have any repeating pattern, but most of those we study will.

Group problems, to be handed in at the end of class

1. How do you know that the triangle you constructed is an equilateral triangle? (That is, prove it.)
2. What would you call a regular triangle? A regular quadrilateral? An equilateral quadrilateral? An equiangular quadrilateral? Draw two examples of equilateral hexagons that have different shapes.

Homework problems

1. Construct a tessellation of the plane by equilateral triangles. Start by constructing an equilateral triangle, then construct new circles with the same raduis at all new intersection points.
2. How are the dots in isometric dot paper related to your triangle tessellation? Be specific.
3. Find a regular hexagon in your triangle tessellation. Find a tessellation of the plane by regular hexagons in your triangle tessellation. (Outline the hexagons in a different color.)
4. Invent a method to construct a single regular hexagon with straightedge and compass, using as few steps as possible. (A step is drawing a line or a circle, or part of one.) Do the construction and write clear instructions.
5. The pattern below is an old Moorish pattern from Spain. Find it in your construction of the tessellation by equilateral triangles. Can this pattern be continued to tessellate the entire plane? Why or why not?

   

6. Choose one of the patterns below and reproduce it either from a copy of your equilateral triangle tessellation or from isometric dot paper. Are desgins a and b the same or different? Why?

   a.

   b.

   c.

   d.