Construction of Circles, Regular Hexagons and Derivative Works

"Construction" in geometry means to draw shapes, angles or lines accurately.

These constructions use only compass, straightedge (i.e. ruler) and a pencil.

This is the "pure" form of geometric construction - no numbers involved!

Constructing the Circle:        First, we worked on constructing circles with a compass. The key to this skill is to mark your intended center with a small dot on your paper. The compass is worthless if it is not tight; if it moves freely on the hinge, it will not hold your circle. Then, we set the center of the compass (this is the part without the pencil) on that dot. Planning your circle is important; make sure the radius (the distance between the center and the pencil) on your compass is not too big for the paper in all directions. It is important not to touch the side of the compass that contains the pencil. Keep slight pressure on the center, place the pencil on the paper, and turn the paper slowly to form a near perfect circle. 

Constructing a Regular Hexagon Inscribed in your Circle:       

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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 teh 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 milimeters, 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 cirlce 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. I have attached a pdf with 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:

1. Connect the adjacent vertices (points next to each other) to create the regular hexagon.
2. 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.
3. Notice that the intersection of these two triangles is another regular hexagon.
4. 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.
5. 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.
6. At many of these intersections, they can use their compass to create more beautiful designs.

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