"... 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 rhombus' in two different ways as shown below.
Definition 1: A golden rhombus is a rhombus with its diagonals in the golden ratio φ.
Apart from directly constructing a golden rhombus by starting with the construction of perpendicular, bisecting diagonals in the golden ratio φ, a golden rhombus having these properties can also easily be constructed from a golden rectangle as follows:
Construction 1a: Construct tangents to the circumcircle of a given golden rectangle EFGH at its vertices as shown below.
Challenge 1a: Prove that the diagonals AC and BD of the formed rhombus ABCD in the construction above are in the golden ratio φ (as experimentally shown by the measurements in the sketch below).
Further Challenge 1a: Prove that the tan of the angle formed by the diagonals of the golden rectangle as shown in the diagram equals 2; i.e. tan(∠EKF) = 2.
Some Golden Rhombus Constructions
Construction 1b: Given a golden rectangle EFGH, then the rhombus ABCD formed by the midpoints of its sides is a golden rhombus. Click on the Link to Construction 1b button to view.
Challenge 1b (Very easy): Prove that ABCD is a rhombus with diagonals in the golden ratio φ.
Definition 2: A golden rhombus is composed of a 'golden triangle' - which is an isosceles triangle with angles 36°, 72°, 72° - reflected in its shorter side. Click on the Link to Construction 2 button to view.
Challenge 2: 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). So in this 'golden rhombus' we have that its sides are in the ratio φ in relation to its shortest diagonal.
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Critical Comparison
Critically compare the two possible definitions of a 'golden kite' 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 Rhombus (involves golden ratio φ)
Golden Quadrilaterals (generalizing a 'golden rectangle')
Visually Introducing & Classifying a Rhombus
Exploring Rhombus Properties
Free Construction of a Rhombus
Systematizing Rhombus Properties (Rethinking Proof activity; defining a rhombus)
Isosceles Trapezoid Midpoints (Rethinking Proof activity; generalizes to equidiagonal quadrilateral)
A diagonal property of a Rhombus constructed from a Rectangle
The Perpendicular Bisectors of an Apollonius Quadrilateral
Visually Introducing & Classifying Quadrilaterals
Introducing, Classifying, Exploring, Constructing & Defining Quadrilaterals
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Semi-regular Angle-gons and Side-gons
(Generalizations of rectangles and rhombi)
A Rhombus Angle Trisection Result
Rhombus Angle Quadrisection Result
The 120o Rhombus Theorem
External Links
Golden rectangle (Wikipedia)
Golden ratio (Wikipedia)
Golden triangle (Wikipedia)
Golden rhombus (Wikipedia)
Golden Rhombus (Wolfram MathWorld)
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, 16 February 2022 with WebSketchpad; updated 24 April 2026.