"... the algorithmically constructive and creative definition ... models new objects out of familiar ones." - Hans Freudenthal (1973:458) in Mathematics as an Educational Task.
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 as 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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Critical Comparison
Critically compare the possible definitions of a 'golden parallelogram' above, judging them not only by 'appearance' or 'shape', but also by their mathematical properties & the ease by which one can construct an example or deduce other additional properties (not contained in the definition).
a) Which one of these do you most prefer? Why?
Other Golden Quadrilaterals
To view, interact with & explore other golden quadrilaterals CLICK HERE.
Constructive Defining
Read more about constructive (and descriptive) defining and classifying in mathematics in De Villiers (2009).
References
De Villiers, M. (1994, 1996, 2009). Defining and Classifying (Chapter 2), pp. 13-28, from Some Adventures in Euclidean Geometry. (Free to download). Morrisville, NC: Lulu Publishers, Dynamic Mathematics Learning.
De Villiers, M. (2017). An Example of Constructive Defining: From a Golden Rectangle to Golden Quadrilaterals. At Right Angles, Vol. 6, No. 1, pp. 64-69; Vol. 6, No. 2, pp. 74-81.
Related Links
Golden Isosceles Trapezium (Trapezoid) (involves golden ratio φ)
Golden Quadrilaterals (generalizing a 'golden rectangle')
Similar Parallelograms: A Generalization of a Golden Rectangle property
Exploring Parallelogram Properties
Parallelogram Squares (Rethinking Proof activity)
Kite Midpoints (Rethinking Proof activity; generalizes to Varignon parallelogram)
Isosceles Trapezoid Midpoints (Rethinking Proof activity; generalizes to Varignon parallelogram)
Varignon Parallelogram Area (Rethinking Proof activity)
Point Mass (Vertex) Centroid of Quadrilateral
Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles
Area Parallelogram Partition Theorem
Feynman Parallelogram Generalization
A generalization of a Parallelogram Theorem by Apollonius
The surprising 3D parallelo-hexagon
Some Parallelo-hexagon Area Ratios
Some Trapezoid (Trapezium) Explorations
An Inclusive, Hierarchical Classification of Quadrilaterals
Some Possible Definitions of Quadrilaterals & some Properties (Relates to the above classification)
The Parallel-pentagon and the Golden Ratio
A diagonal property of a Rhombus constructed from a Rectangle
External Links
Golden rectangle (Wikipedia)
Golden ratio (Wikipedia)
Golden triangle (Wikipedia)
Parallelogram (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Created by Michael de Villiers, 17 February 2022 with WebSketchpad; updated 24 April 2026.