Constructing Regular Hexagons, Equilateral Triangles, and Derivative Works

Last week, we explored the three regular polygons that tessellate: the square, the equilateral triangle, and the regular hexagon. Creating a regular hexagon or an equilateral triangle can be done with only a compass and straight edge. This was the only way the ancient Greeks explored geometry.

Constructing Regular Hexagon: Inscribing a regular hexagon in a circle is a lot of fun. A polygon is inscribed in a circle if each of its vertices (corners) touches the circle. The steps are simple:

1. Construct a circle, remember to mark the center.

2. Do not change the radius.

3. Pick any point on the circle and make a mark at that point.

4. Set the center of your compass on that point and use the compass pencil to mark a small arc that intersects with the circle. Mark a point on that intersection.
You can erase the arcs after you create the points.

5. Using the new point (created where the arc meets the circle) as the center and with the same radius, make another arc on the circle.

6. Continue making arcs around the circle until you have six points. These are the vertices of a regular hexagon. If your last arc is off by a few millimeters, do
not worry. If it is off by a lot, erase all of the points and start again (your compass may have become loose and the radius changed; check the circle radius
often during this process).

Constructing Derivative Artworks: This is the fun part. Now you are ready to create many derivative artworks just with your compass, straight edge, and pencil. There are four pages in the pdf which include 48 designs that the children can follow. It is very satisfying to be able to follow the construction of one of these designs. They can decorate them afterwards with color and cut them out as ornaments. It is also fun for the children to create their own original designs.

All of these designs are derived from the regular hexagon vertices. To find the lines and points
necessary to construct these works they should:

• Connect the adjacent vertices (points next to each other) to create the regular hexagon.
• Connect every other vertex to create an equilateral triangle, then use the vertices not used and repeat that process to create another equilateral triangle, the compound of which is a six pointed star. The popular name for this is The Star of David or Jewish Star, but its mathematical name in Greek, a Hexagram; and in Latin, a Sexagram.
• Notice that the intersection of these two triangles is another regular hexagon.
• Connect the smaller hexagon vertices to make another Hexagram; this process can be done
  infinitely just like the fractals the children explored earlier in the year.
• Alternatively, after the first Hexagram is made, they can connect opposite vertices to find the
  midpoints of the sides of the smaller regular hexagon. These midpoints are critical for many of the designs.
• At many of these intersections, they can use their compass to create more beautiful designs.
• A very simple construction is to make the original six vertices and then simply take the compass center (do not change the radius) to each of the hexagonal vertices and draw an arc inside the original circle (this will be a one-third arc); continue this from each of the six vertices and you will have a beautiful design that looks like the pedals of a flower (it actually creates six shapes called lunes).

Constructing Equilateral Triangle: The pdf also gives instructions for constructing an equilateral triangle; this is even easier than the regular hexagon.

The second pdf provides the students with circles marked with the vertices of a regular hexagon as if they had a compass. I recommend that the children buy a compass at the local CVS or Walgreens or on Amazon for a few dollars; it is an important tool for a mathematician. They do not need fancy compasses used by engineers and architects, but …..