Supplementary Investigation 11

Tessellations Based on Rotations & Equilateral Triangles

Tool(s) Geometers Sketchpad

Data File(s) si11 & si11a

Focus A curve is defined from the endpoint to the midpoint of one side of an equilateral triangle. This curve is extended over the second half of the side using a rotation. When the curve is replicated on the other sides of the triangle, the resulting object tessellates the plane.

Tasks

1. Open the file si11 and vary the segment labeled "side of triangle" and the positions of one or more of the marked vertices along the perimeter of the propeller-like object. Measure and compare the areas and perimeters of the objects and the "negative spaces" between them. State a conjecture based on your observations. To prove your conjecture false, what sort of counter example would be necessary?

2. Repeat these activities using file si11a.

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