The Golden Rhombus

1. Introduction: the golden ratio

Divide a segment into two parts so that the whole is to the longer part as the longer part is to the shorter. Euclid called this a division in extreme and mean ratio [1]; today we call the ratio the golden ratio and write it φ (phi). Taking the shorter part as 1 and the longer as φ gives φ / 1 = (φ + 1) / φ, which is the quadratic equation

Two identities follow at once and are used throughout this note: 1/φ = φ − 1 ≈ 0.618 and φ² = φ + 1 ≈ 2.618. φ is also the limit of the ratios of consecutive Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, … (Fig. 1). Each new ratio overshoots or undershoots φ in turn, and the error shrinks by a factor of about φ² ≈ 2.6 at every step.

2. The golden rhombus

A rhombus is a quadrilateral with four equal sides; its diagonals bisect each other at right angles. A golden rhombus is a rhombus whose diagonals are in the golden ratio. Let the short diagonal be q = 1 and the long diagonal p = φ (Fig. 2). The diagonals cut the rhombus into four congruent right triangles with legs 1/2 and φ/2.

2.1 The angles

Half of the acute angle α satisfies tan(α/2) = (1/2) / (φ/2) = 1/φ. Using the double-angle formula and the identity φ² − 1 = φ,

so the acute angle is exactly α = arctan 2 ≈ 63.4349° and the obtuse angle is β = 180° − α ≈ 116.5651°. It is a small miracle: an irrational ratio of diagonals produces an angle whose tangent is the whole number 2. Equivalently, cos α = 1/√5 and sin α = 2/√5.

2.2 Side and area

By Pythagoras the side is a = ½ · √(1 + φ²) = ½ · √(φ + 2) ≈ 0.9511, which happens to equal sin 72°: a first hint of the five-fold symmetry still to come. The area is half the product of the diagonals,

Joining the midpoints of the sides of a golden rectangle always gives a golden rhombus (Fig. 3). The rhombus is the diamond hidden inside the most famous rectangle in art history, and it covers exactly half of its area.

2.3 Explore: why φ and not another ratio?

The interactive Fig. 4 lets you change the ratio of the diagonals. Watch two numbers: the sum of five acute angles and the sum of three obtuse angles. When five acute angles add up to exactly 360° (ratio ≈ 1.376, acute angle 72°), the pieces lie flat; this is the “fat” rhomb of the Penrose tiling. For the golden rhombus they add up to only 317.2°. The missing 42.8° forces the five pieces to rise into a shallow pyramid, and that is exactly what bends a flat star into a ball.

3. From flat to solid: curvature at the corners

Around every vertex of a convex polyhedron the face angles add up to less than 360°. The shortfall is the angle defect δ, and it measures how sharply the surface bends at that corner. Around 1630 René Descartes discovered that the defects of any convex polyhedron always add up to the same total: exactly 720°, the full curvature of a sphere. This statement is equivalent to Euler’s formula V − E + F = 2.

The rhombic triacontahedron has 32 vertices of two kinds. At 12 of them five acute angles meet, at the other 20 three obtuse angles meet. Their defects are

The α terms cancel, just as Descartes’ theorem guarantees. What the golden angle adds is harmony: for α = arctan 2 the 12 five-fold vertices sit exactly on a regular icosahedron, the 20 three-fold vertices on a regular dodecahedron, and all 30 faces are the very same golden rhombus. Euler checks out too: V − E + F = 32 − 60 + 30 = 2.

4. The family of golden polyhedra

Golden rhombi build a whole family of solids. Five convex polyhedra have only golden rhombi as faces, and all of them are zonohedra (their edges come in parallel bands): the acute and the obtuse golden rhombohedron, the Bilinski dodecahedron discovered by Stanko Bilinski in 1960 [3], the rhombic icosahedron, and the rhombic triacontahedron, already described by Johannes Kepler in Harmonices Mundi (1619) [2]. Stellating the triacontahedron gives the non-convex rhombic hexecontahedron, whose 60 golden rhombi form twelve five-pointed stars (Table 1, Fig. 6).

In the triacontahedron every pair of neighbouring faces meets at a dihedral angle of exactly 144°, and the solid has full icosahedral symmetry: 6 five-fold axes, 10 three-fold axes and 15 two-fold axes, 120 symmetries in all when reflections are counted [4]. It can even be cut into 20 golden rhombohedra, 10 acute and 10 obtuse, a classic dissection that ties the whole family together.

5. Quasicrystals: the golden rhombus in science

In the 1970s Roger Penrose found tiles that cover the plane without ever repeating [5]. One version uses two rhombs with angles of 72° and 36°. They are not golden rhombi, but φ rules them as well: in a large patch there are φ times as many fat rhombs as thin ones. Robert Ammann, and independently Peter Kramer and Reinhardt Neri [6], carried the idea into three dimensions: space can be filled without repetition by the acute and the obtuse golden rhombohedra, whose faces are exactly our golden rhombus.

Nature got there too. In April 1982 Dan Shechtman saw sharp ten-fold diffraction patterns in a rapidly cooled aluminium–manganese alloy, a symmetry that was believed to be impossible for crystals. The discovery, published in 1984 [7], was met with disbelief, but it changed the very definition of a crystal and earned Shechtman the 2011 Nobel Prize in Chemistry. Three-dimensional tilings by golden rhombohedra became one of the standard models of these icosahedral quasicrystals.

6. Design analysis: why the golden rhombus makes a great toy

• One piece, many worlds. Children learn a single tile. Every rhombus tiles the plane, so flat patterns always work; thanks to the golden angle the same tile also closes into polyhedra with icosahedral symmetry.
• Corners that count. The two angles become two numbers children discover on their own: five sharp corners make a star, three wide corners make a round corner. Counting turns into geometry.
• Edges that hinge. Snowflake connectors clip onto the middle of each edge and let neighbouring pieces meet at any angle: 180° on the table, 144° in the triacontahedron. One joint serves both flat and 3D builds.
• Transparency shows the maths. Coloured, see-through faces reveal the hidden axes of symmetry and the overlapping stars inside the hexecontahedron.
• A proof you can hold. When the thirtieth piece clicks into place, Euler’s formula and Descartes’ theorem have been checked by hand: there is no other way for the ball to close.

Table 2 maps what children do with the pieces at each age to the mathematics underneath. The progression is natural because every stage uses the same piece: only the questions change.

7. Activities for home and classroom

Four short experiments that turn the results of this note into hands-on discoveries. Each needs only Golden Rhomboi pieces and, sometimes, a ruler or a protractor.

8. Conclusion

A single condition, diagonals in the ratio 1 : φ, fixes every property of the golden rhombus: an acute angle whose tangent is exactly 2, a side equal to sin 72°, and angle defects that make five pieces rise into a star and thirty pieces close into a ball. The same tile sits at the heart of Kepler’s triacontahedron, Bilinski’s dodecahedron and the rhombohedral models of quasicrystals. Golden Rhomboi puts this tile into children’s hands, so that the golden ratio is no longer a number to memorise but a shape to build with.

References

1. [1]Euclid. Elements, Book VI, Definition 3 (c. 300 BC).
2. [2]Kepler, J. (1619). Harmonices Mundi, Book II. Linz.
3. [3]Bilinski, S. (1960). Über die Rhombenisoeder. Glasnik Matematičko-Fizički i Astronomski 15, 251–263.
4. [4]Coxeter, H. S. M. (1973). Regular Polytopes, 3rd ed. Dover.
5. [5]Penrose, R. (1974). The role of aesthetics in pure and applied mathematical research. Bulletin of the Institute of Mathematics and its Applications 10, 266–271.
6. [6]Kramer, P. & Neri, R. (1984). On periodic and non-periodic space fillings of Eᵐ obtained by projection. Acta Crystallographica A40, 580–587.
7. [7]Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters 53, 1951–1953.
8. [8]Markowsky, G. (1992). Misconceptions about the golden ratio. The College Mathematics Journal 23(1), 2–19.
9. [9]Livio, M. (2002). The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. Broadway Books.
10. [10]Grünbaum, B. & Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman.