The concept of a 'golden rectangle' - a rectangle with its sides in the the golden ratio φ = (√5 + 1)/2 - can be extended to that of a 'golden parallelogram' in different ways as shown below.
Definition 1: A golden parallelogram is a parallelogram with both its sides & diagonals in the golden ratio φ.
Dynamic Investigation
1) Use the dynamic sketch below to drag the vertices of the parallelogram until the ratio of the sides as well the ratio of the diagonals are both approximately in the golden ratio φ. What do you notice about ∠CBA?
2) Navigate to an accurate sketch by clicking on the 'Link to Definition 1' button. Check your conjecture in 1) above by clicking on the 'Show Angle CBA' button.
Some Golden Parallelogram Constructions
Challenge 1
Prove that if a parallelogram ABCD has both its sides BC/AB & its diagonals BD/AC in the golden ratio φ, then ∠CBA = 60°.
Challenge 2
Prove that if a parallelogram ABCD has its sides BC/AB in the golden ratio φ and ∠CBA = 60°, then its diagonals BD/AC are also in the golden ratio.
Challenge 3
Prove that if a parallelogram ABCD has its diagonals BD/AC in the golden ratio φ and ∠CBA = 60°, then its sides BC/AB are also in the golden ratio.
Challenge 4
Can you subdivide (partition) the golden parallelogram ABCD into smaller golden parallelograms? By continuing this subdivision (partitioning) can you generate an approximate 'golden spiral' similar to the one obtained from a golden rectangle?
Alternative Definitions
Either of the two results in Challenges 2 & 3 could also be used as alternative definitions for a golden parallelogram. Which of the three possible alternatives would you PREFER to choose as a definition for a golden parallelogram? Why?
Definition 2: A golden parallelogram is composed of a 'golden triangle' - which is an isosceles triangle with angles 36°, 72°, 72° - rotated through 180° around the midpoint of one of its two equal sides. Click on the 'Link to Construction 2' button to view.
Note that in this case the diagonals are not in the golden ratio, but the sides are, and one of the diagonals, AC, is in the golden ratio to the shorter side, AB, of the parallelogram.
Challenge 5: Prove that the sides of the golden triangle are in the golden ratio; i.e. that BC/AC = CD/AC = φ (as experimentally shown by the measurements in the sketch above).
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Other Golden Quadrilaterals: To view & interact with other golden quadrilaterals click here.
Paper: Read my 2017 paper An Example of Constructive Defining: From a Golden Rectangle to Golden Quadrilaterals in the journal At Right Angles.
Created by Michael de Villiers, 17 February 2022 with WebSketchpad.