Ratio & Proportion · 2D
Golden Triangle
And its gnomon, whirling inward
A tall triangle with a 36° apex, the point of a pentagram. Cut it from one base corner and it splits into a smaller golden triangle and a golden gnomon. Cut again and again, and the triangles whirl inward along a logarithmic spiral.
- Angles
- 36° · 72° · 72°
- Side ÷ base
- φ
- Gnomon
- 108° · 36° · 36°
Watch it drawn with compass and straightedge, hear the chord of its lines, and turn the Dimension dial to tilt it into space.
Facts
- Its long sides are φ times its base: every point of a pentagram is one of these.
- The cut bisects a 72° base angle, and falls exactly at the golden section of the opposite side.
- The piece cut off, the golden gnomon (108°, 36°, 36°), has its base φ times its sides: a pentagon splits into one golden triangle and two gnomons.
- Each smaller triangle is the last one turned 108° and shrunk by φ, so their apexes lie on one logarithmic spiral.
History and meaning
The Robinson triangles from which Penrose tilings are grown are these two: the golden triangle and the golden gnomon, each cut into smaller copies of both.
Learn it step by step
- Journey The Builders’ MeasuresStep 5: The golden triangle
