I

GOLDEN GEOMETRY

“Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.” 1

—Johannes Kepler

Although the proportion known as the golden ratio has always existed in mathematics, geometry, and nature, exactly when it was first discovered and applied by mankind is unknown. It is reasonable to assume that it has been discovered and rediscovered throughout history, which explains why it is known by several names. There’s some compelling evidence of awareness and application of the golden ratio by the ancient mathematicians of Babylon and India, but let’s first start with Greece.

This engraving by Jean Dambrun (1741–c. 1808) portrays Pythagoras as depicted on a Roman coin from the third century.

This painting by Russian artist Fyodor Bronnikov (1827–1902) shows the cult of Pythagoras celebrating sunrise.

ANCIENT GREECE

Most of the content in today’s geometry textbooks is derived from the discoveries of the ancient Greeks, and the earliest references to what we now know as the golden ratio may have come from the time of Pythagoras, a mathematician and philosopher who lived from about 570 BCE to 495 BCE. It is thought that the five-pointed star, or pentagram—in which the length of every line segment is in a golden ratio relationship to every other one, as shown below—was the symbol of his school, and that he and his followers were the first to discover some of the unique properties of the golden ratio.

The pentagon at the center of the pentagram makes an appearance in the work of the renowned Greek philosopher Plato (c. 427–347 BCE)—specifically his c. 360 BCE dialogue Timaeus, which describes a universe made up of four elements, represented by four fundamental geometric solids (now known as the Platonic solids). The fifth solid is revealed to be the dodecahedron—an assemblage of twelve pentagons intended to represent the shape of the universe. In his dialogue, Plato also wrote of a mean relationship between three numbers that might be a direct precursor to Euclid’s “extreme and mean ratio”:

The golden cut of the pentagram.

The ratios of the red segment to the green segment, the green segment to the blue segment, and the blue segment to the purple segment are all equal to phi (Φ).

This illustration of the five Platonic solids and their associated elements appears in Johannes Kepler’s Mysterium Cosmographicum (1596).

2“When the mean is to the first term as the last term is to the mean, … they will all by necessity come to be the same, and having become the same with one another will be all one.” 2

To this day, however, it is unclear whether this is a description of means in general, or whether this is a specific reference to the golden ratio.

Although little is known about his origins, Euclid lived in ancient Alexandria around the third century BCE, when Ptolemy I (c. 367–c. 283 BCE) ruled over the Hellenistic kingdom of Egypt. Comprised of thirteen books, Euclid’s Elements contains illustrated definitions, postulates, propositions, and proofs covering geometry, number theory, proportions, and incommensurable lines, which are those that cannot be expressed as a ratio of integers. It was a foundational work in the development of logic and modern science, and today it is regarded as one of the most influential textbooks ever written. First printed in 1482, it was one of the earliest books on mathematics to be produced after the invention of the printing press by German blacksmith Johannes Gutenberg, and it is likely second only to the Bible in the number of editions published. Abraham Lincoln studied it intensely to hone his logical thinking skills, and in 1922 the Pulitzer-winning American poet and playwright Edna St. Vincent Millay penned a poem entitled “Euclid Alone Has Looked on Beauty Bare.”

Plato’s Academy is portrayed in this first-century BCE Roman mosaic from Pompeii, Italy.

Flemish painter Justus of Ghent depicted Euclid in his c. 1474 series “Famous Men.”

This first printed edition of Euclid’s Elements from 1482 shows propositions 8–12 from Book III.

This Arabic translation of Eulid’s Elements was created by Persian polymath Nasir al-Din al-Tusi (1201–1294.)

In what Einstein referred to as the “holy little geometry book,” Euclid referred to “the extreme and mean ratio” a number of times, along with constructions (including the pentagram) showing how it is derived geometrically. Beginning a quick tour of Euclid’s fundamental work on the golden ratio, we find the following construction in Book VI: 3

Proposition 30.
To cut a given segment (AB) in extreme and mean ratio (E).

Here, Euclid asks us to construct square ABHC with sides equal to our initial segment AB, and then construct rectangle GCFD with area equal to that of ABHC, where GAED is also a square. When segment AC = 1, we find:

• The area of square ABHC = 1

• The area of rectangle CFEA = 1/Φ

• The area of both square GAED and rectangle EBHF = 1/Φ2

Euclid introduces this same construction in Book II before ratios have been introduced, creating the midpoint E of AC and then using EB as the arc to determine lengths of the segments EF and AF as follows:

Proposition 11.
To cut a given segment (AB) so that the rectangle (BDKH) contained by the whole (AB) and one of the segments (BH) equals the square (AFGH) on the remaining segment (AH).

Other examples involving the extreme and mean ratio appear in Book XIII, illustrated below:

Proposition 1.
If a straight line (AB) is cut in extreme and mean ratio (C), then the square (DLFC) on the greater segment added to the half of the whole (CD) is five times the square (DPHA) on the half (AD).

Proposition 2.
If the square (ALFB) on a straight line (AB) is five times the square (APHC) on a segment of it (AC), then, when the double of the said segment (CD) is cut in extreme and mean ratio (B), the greater segment (BC) is the remaining part of the original straight line (AB).

Proposition 3.
If a straight line (AB) is cut in extreme and mean ratio (C), then the square (ABNK) on the sum (BD) of the lesser segment (BC) and the half of the greater segment (AC) is five times the square (GUFK) on the half of the greater segment (AC).

Proposition 4.
If a straight line (AB) is cut in extreme and mean ratio (C), then the sum of the squares on the whole (AB) and on the lesser segment (BC) is triple the square (HFSD) on the greater segment (AC).

Proposition 5.
If a straight line (AB) is cut in extreme and mean ratio (C), and a straight line equal to the greater segment (AD) is added to it, then the whole straight line has been cut in extreme and mean ratio (A), and the original straight line (AB) is the greater segment.

In Proposition 6, Euclid introduces the concept of the apotome, which he defines as each “irrational” segment that makes up a “rational” line that has been cut in extreme and mean ratio. Jumping ahead to Propositions 8 and 9, we discover the golden properties of the pentagon, followed by the golden relationship between the sides of the six-sided hexagon and ten-sided decagon.

Proposition 8.
If the straight lines of an equilateral and equiangular pentagon (AC, BE) subtend two angles, then they cut one another in extreme and mean ratio (H), and their greater segments (HE, HC) equal the sides of the pentagon.

Proposition 9.
If the side of the hexagon (CD) and that of the decagon (BC) inscribed in the same circle are added together, then the whole straight line (BD) has been cut in extreme and mean ratio (C), and its greater segment is the side of the hexagon (CD).

Are you ready for the jump into three-dimensional space? This last proposition describes the golden ratio relationship between a cube and a dodecahedron:

Proposition 17.
To construct a dodecahedron and inscribe it in a sphere … and to prove that the side of the dodecahedron (UV) is the irrational straight line called apotome. Corollary: Therefore, when the side of the cube (NO) is cut in extreme and mean ratio, the greater segment (RS) is the side of the dodecahedron.

In the last example, Euclid shows that the side of the dodecahedron (e.g., segment UV) is an apotome—that is, the greater of two irrational segments that make up a rational line equivalent in length to the side (e.g., segment NO) of the inscribed cube. In order to illustrate this relationship, the sides of the cube are bisected at G, H, K, L, M, N, and O, and then GK, HL, HM, and NO are connected to form segments representing the width of the cube. Then the segments NP, PO, and HQ—which represent half the width of the cube—are cut in extreme and mean ratio at points R, S, and T. Since segments RU and SV are at right angles to the cube, the length of segment RS, which is the greater apotome to the rational line NO, is equal in length to the segment UV, which represents a side of the equiangular and equilateral dodecahedron UBWCV.

CONSTRUCTING THE GOLDEN RATIO

Euclid gave us a wonderful foundation for understanding the many appearances of the golden ratio in geometry. But we can make this even simpler. Let’s look at some of the other simple geometric constructions that can be used to create a golden ratio, starting with the line, and then proceeding to the three-sided triangle, four-sided square, and five-sided pentagon. Unlike David Letterman’s “Top 10” Lists, I’m going to start with one that is perhaps the most amazing, by virtue of its sheer simplicity. (I like to describe this approach as “incredibly simple, yet simply incredible.”)

THREE LINES

If Euclid had seen this elegant little construction, history probably would have recorded him rather than Archimedes as the one running naked through the streets, shouting, “Eureka!”

1. Gather three sticks (dowels, chopsticks, straws, or what have you) of equal length.

2. Place the first one in a vertical position.

3. Lay the second one against the midpoint of the first.

4. Lay the third one against the midpoint of the second, so that one end of each stick is lined up, as shown.

Figure 1. The golden ratio cut of line AC is point B.

THREE SIDES: TRIANGLE

Here’s another geometric construction that is simpler than any of those provided by Euclid.

1. With the aid of a compass, draw a circle. Then inscribe an equilateral triangle inside it.

2. Draw a line through the midpoint of two sides of the triangle, extending the line to the edge of the circle, as shown.

Figure 2. The golden ratio cut of line AC is point B.

FOUR SIDES: SQUARE

This construction is closely related to Euclid’s propositions that apply an arc to the midpoint of a square, but we’re doing the construction in reverse.

1. With the aid of a compass, draw a circle. Then divide it into two semicircles.

2. Insert a square inside one semicircle, as shown.

Figure 3. In this construction, the golden ratio cut of line AC, again, is point B.

FIVE SIDES: PENTAGON

This construction is the first contained in Elements, appearing as Proposition 8 of Book XIII.

1. With the aid of a compass, draw a circle. Then create a pentagon by inside it by connecting five equally spaced points on the circle.

2. Connect two of the vertices with a line, and then connect another two vertices with another line, as shown.

Figure 4. The golden ratio cut of line AC is point B, where the two lines intersect.

See how easy this is? Golden ratios just seem to appear without much planning or effort. See Appendix B to explore other geometric constructions of the golden ratio.

PYTHAGORAS AND KEPLER WALK INTO A … TRIANGLE?

Have you heard the joke that starts, “Pythagoras and Kepler walk into a bar”? Probably not, but as you will discover, the findings of these two historical mathematicians helps to illustrate one of the golden ratio’s unique properties. Pentagrams aside, Pythagoras is best known for his eponymous theorem, which states that a right triangle with sides of length a, b, and c (where c is the hypotenuse), has the following relationship:

a2 + b2 = c2

As stated in the introduction, we also know that phi is the only number whose square is one more than itself:

Φ + 1 = Φ2

Two thousand years after Pythagoras devised his famous theorem, German mathematician Johannes Kepler (1571–1630) noticed the similarity between these two equations. This led to his discovery of a unique triangle, now appropriately known as the Kepler triangle, with sides equal to 1, √Φ, and Φ.

This 1610 portrait of Johannes Kepler by an unidentified painter comes from a Benedictine monastery in Kremsmünster, Austria.

Kepler observed another characteristic of this triangle and wrote to his former professor Michael Mästlin:

“If on a line which is divided in extreme and mean ratio one constructs a right-angled triangle, such that the right angle is on the perpendicular put at the section point, then the smaller leg will equal the larger segment of the divided line.” 4

Here, he is referring to the two legs of the triangles below with a dimension of 1.

As shown, when you draw a line perpendicular to the hypotenuse of the Kepler triangle through its right angle, the segments on either side of the line have a golden relationship, and the resulting two triangles have identical proportions to that of the original Kepler triangle.

The Pythagorean 3-4-5 triangle is the only right triangle whose sides are in an arithmetic progression, in which each successive term is created by the addition of a common difference:

3 + 1 = 4
4 + 1 = 5

Curiously, the √Φ-1-Φ Kepler triangle is the only right triangle whose sides are in a geometric progression, in which each successive term is created by the multiplication of a common ratio. In this unique case, that ratio is the square root of the golden ratio:

1 × √Φ = √Φ
√Φ × √Φ = Φ

Circling back to Pythagoras, in the pentagram we find two other triangles with golden ratio proportions—that is, two triangles with a Φ to 1 relationship between the base and sides.

The pentagram (above) can be divided into several golden triangles (below) and gnomons (below), each of which has at least one 36-degree angle.

The obtuse triangle above, center, is known as a golden gnomon. The acute isosceles triangle on the right is known as a golden triangle. These, in turn, form the basis of an important mathematical discovery, Penrose tiling (see here).

THE GOLDEN RATIO, ORIGAMI-STYLE

If you know someone who gets tied up in knots by math or geometry, try sharing this last golden ratio construction with him or her, because it requires neither. All you need is a strip of paper. Fold a paper into a simple knot and press to flatten. (Don’t overthink it!)

This “knot” forms a pentagon, with both variations of the golden triangle, whose base and side proportions are defined by the golden ratio.

HARMONY OF THE SPHERES

Both Pythagoras and Kepler saw mathematics everywhere, from the vibrations of a stringed instrument to the motion of the planets. Though no one knows for sure, it is believed that Pythagoras was the first to identify the inverse relationship between the pitch of a musical note and the length of the string producing it, and he may have gone further in linking the orbital frequencies of different planets to inaudible hums—a theory that has persisted through the ages under such names as musica universalis and “Harmony of the Spheres.”

Kepler’s own interests ranged into the mystical, and he explored the idea of the universe as a harmonious arrangement of geometrical forms in his 1596 book Mysterium Cosmographicum (Cosmographic Mystery), as well as his 1619 book Harmonices Mundi (Harmony of the World). In the former, Kepler proposed that the relative distances between the six planets known at that time could be understood through a nesting of the five Platonic solids (see here), each enclosed within a sphere that represented their orbits, with the final sphere representing the orbit of Saturn. This model turned out to be inaccurate, but he continued in his pursuit to explain the cosmos, and in 1617 he published the first volume of Epitome Astronomiae Copernicanae, in which he unveiled his most important discoveries: the true elliptical nature of planetary orbits and the first of his three laws of planetary motion.

This reproduction of Kepler’s model of the solar system shows the five Platonic solids in a nested formation.

Even though the hypothesis of nesting Platonic solids in Mysterium Cosmographicum did not hold up to scrutiny in the end, Kepler’s early model of the universe was mathematically brilliant in its own right. A unique property of these solids, which include (below, from left to right) the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, is that each can be constructed with identical faces meeting at each vertex.

Two of these five beautiful Platonic solids, the dodecahedron and icosahedron, are geometrically based on the golden ratio. Each of their vertex points can be determined by a simple construction using three golden rectangles (i.e., rectangles whose length-to-width ratio is equal to phi).

The three golden rectangles on the left can be assembled into the interlocking shape on the right. This interlocking shape creates the basis for the twelve-sided dodecahedron and the twenty-sided icosahedron.

In the case of the dodecahedron,
the 12 corners become
the 12 centers of each of
the 12 pentagons that form
the 12 pentagonal faces.

Dodecahedron.

In the case of the icosahedron,
the 12 corners become
the 12 points of each of
the 20 triangles that form
the 20 triangular faces.

Icosahedron.

If we map the interlocking golden rectangle construction in three-dimensional Cartesian space, the coordinates of the 12 (X, Y, Z) vertices of the icosahedron with an edge of length 2, centered at the origin, are represented as follows:

x-z plane (green, y = 0): (±1, 0, ±Φ)
y-z plane (blue, x = 0): (0, ±Φ, ±1)
x-y plane (red, z = 0): (±Φ, ±1, 0)

Next, mapping the dodecahedron in three-dimensional Cartesian space provides the following coordinates for the 20 (X, Y, Z) vertices of a dodecahedron enclosing a cube with an edge of length 2, centered at the origin: 6

orange cube: (±1, ±1, ±1)
y-z plane: (green x = 0): (0, ±Φ, ±1/Φ)
y-z plane: (blue, y = 0): (±1/Φ, 0, ±Φ)
y-z plane: (red, z = 0): (±Φ, ±1/Φ, 0)

Given what we know about the proportions of a pentagon, a dodecahedron that encloses a cube with edges of length 2 should have edges of length 2/Φ.

GOLDEN TILES

Mapping the surfaces of each Platonic solid in two-dimensional space, as shown here, area can be filled completely and symmetrically with tiles of three, and four, sides, but what about tiles in the shape of a five-sided pentagon? The lines of a pentagon star, or pentagram, have beautiful, golden ratio proportions, but it long appeared that they could not be tiled like triangles, square, and hexagons. Enter English mathematical physicist Sir Roger Penrose (b. 1931). In the early 1970s, Penrose noticed that the two triangles within the pentagon that have golden proportions (see here and below, top left) can be assembled in pairs, forming all-new symmetrical tiles that can be combined into different patterns. For example, two acute golden triangles can be combined to form a “kite” (the gold part in figure b), while two obtuse triangles with golden proportions can form a “dart” (the red part in figure b). The kite and dart can be combined to form a rhombus with sides of length Φ, as shown (figure c). The two triangles can also be combined to form diamond-shaped tiles, as shown (figure c). Although pentagons alone will not completely fill a two-dimensional space, these “Penrose tiles,” which have golden proportions, will (figure d).

As you expand the tiling to cover greater areas, the ratio of the quantity of the one type of tile to the other always approaches 1.618, the golden ratio. Depending on how they are arranged, the tiling may exhibit five-fold rotational symmetry. Small pockets of five-fold symmetry, such as stars and decagons, may also occur. As we will see in chapter 5, this same kind of five-fold symmetrical arrangement also appears in nature.

Various formations of Penrose tiles. Notice the proliferation of five-sided figures like the pentagram and pentagon.