Wolfram Language

Perfect Triangle Dissections

RandomInstance can create visualizations of geometric relationships that would be complicated to write algebraically. For example, a perfect dissection of a polygon is a decomposition of into internally disjoint polygons () that are mutually similar but pairwise incongruent.

The triangle is divided into six triangles similar to itself.

This construction is based on the plastic constant ρ, one of the Pisot numbers. The plastic constant ρ is the real root of α3-α-1, roughly equal to 1.32472.

Extract the coordinates of the points in the dissection.

Use them to compute the areas of all seven similar triangles (including the one being dissected).

Take the ratios of successive areas to see that each ratio is either ρ or ρ3 (since ρ3ρ+1).

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