MATHEMATICAL PROPERTIES OF THE GOLDEN RATIO - A FASCINATING NUMBER

Headnote

Abstract: The Golden Ratio appears in many situations: in geometry, in lists with special numbers drawn up by mathematicians, in art history lectures, in rose petals arrangement or spiral shells of mollusks. This paper presents definitions and algebraic properties of Golden Section and their applications in geometry, in the calculation pentagon and decagon sides and area, and volume of Platonic solids. Golden Ratio occurrences are analyzed in geometric shapes as Golden Triangle, Golden Rectangle and Golden Gnomon. Different expressions involving radicals that continue indefinitely and continued fractions with Golden Section value are given and discussed. It is also revealed the connection between the Fibonacci series and the Golden Ratio. Are presented two patterns that have symmetry of order five and that have incorporated in them the Golden Section.

Key words: proportion, pentagram, pentagon, Golden Triangle, Golden Gnomon, Golden Ratio.

1. INTRODUCTION

"Golden Number" represents a geometrical proportion known since antiquity. This number has received in the nineteenth century the honorary name "Golden Ratio" or "Golden Section".

In the mathematical literature, the common symbol for the Golden Ratio is the Greek letter tau (τ) which means "the cut" or "the section". At the beginning of the twentieth century, the mathematician Mark Barr gave the ratio the name of phi (?), the first Greek letter in the name of Phidias, the great Greek sculptor. Barr decided to honor the sculptor because a number of art historians maintained that Phidias had made frequent use of the Golden Ratio in his sculptures [1]. Golden Section is the core of an infinite number of phenomena, it can be found at the intersection of Euclidean geometry with fractal geometry or in a variety of human made objects and works of art.

The paper describes the fascinating properties assigned of number phi, or 1.6180339887..., widely known as "the Golden Ratio", "the Golden Section" or "the Golden Mean". This mathematical relationship was discovered by Euclid more than two thousand years ago because of its principal role in the construction of the pentagram. This number appears in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy.

This paper develops the relationship between the Golden Ratio and triangles (equilateral triangle, pentagon triangle, isosceles triangle, and whirling triangle), rectangles, pentagrams and Platonic solids.

The geometrical plane figure most directly related to the Golden Ratio is the regular pentagon, which has a fivefold symmetry. Also, the five regular polyhedra are intimately connected with the Golden Section.

2. ALGEBRAIC PROPERTIES OF THE GOLDEN RATIO

The first definition of the Golden Section was given around 300 B.C. by Euclid of Alexandria. The proportion derived from a simple division of a line with intermediate point C (fig. 1) such as:

... (1)

with

... (2)

was called by Euclid "extreme and mean ratio".

In other words, if the ratio of the length of AC to that of CB is the same as the ratio of AB to AC, then the line has been cut in a Golden Ratio.

In Euclid's Elements (a fundamental work on geometry and number theory) the Golden Ratio appears in several places. The first definition of the Golden Ratio ("extreme and mean ratio"), in relation to areas, is given in Book II. A second definition, in relation to proportion, appears in Book VI. Then, the Golden Ratio is used especially in the construction of the pentagon (in Book IV) and in the construction of the icosahedron and dodecahedron (in Book XIII) [1].

The precise value of the Golden Ratio (the ratio of AC to CB in fig. 1) is never-ending and never-repeating number:

? = 1.6180339887... (3)

Being an irrational number, the Golden Ratio cannot be expressed as a fraction (as a rational number). In other words, we cannot find some common measure that is contained in the two lengths AC and CB.

a) The Golden Ratio possesses unique algebraic properties. In fig. 1 we may take the length of the shorter segment, CB, to be 1 unit and the length of the longer one, AC, to be x units:

CB = 1; AC = x. (4)

The point C was chosen so that:

... (5)

By substitution we get:

... (6)

Multiplying both sides by x, we obtain x2 = x + 1 or the simple quadratic equation:

x2 - x - 1 = 0 (7)

The two solutions of the equation for the Golden Ratio are:

... (8)

... (9)

The positive solution ... ... gives the value of the Golden Ratio, that is irrational, being half the sum of 1 and the square root of 5.

The Golden Ratio has some unique properties:

?2 = 2.6180339887 ... (10)

1 = 0.6180339887 ... (11)

both having precisely the same digits after the decimal point. The Golden Ratio has the unique properties that we produce its square by simply adding the number 1 and its reciprocal by subtracting the number 1. The negative solution of the Eq. (7), x2, is equal precisely to the negative of 1/?.

We defined ? to be:

?2 = ?+ 1 (12)

If we divide both sides of Eq. (12) by ?, we obtain:

?= 1 + 1 (13)

Here is another definition of ? - that number which is 1 more than its reciprocal.

b) The value of the following expression that involves square roots that go on forever is equal to ?:

... (14)

We denote the value of the expression by x. We therefore have:

... (15)

Let us square both sides of the last equation:

... (16)

We therefore obtain the quadratic equation:

... (17)

that is precisely the equation that defines the Golden Ratio.

c) The value of the following expression that involves continued fractions is equal to ?:

... (18)

Based on previous procedure, we denote the value by x. Thus:

... (19)

But the denominator of the second term on the right-hand side is in fact identical to x itself. We therefore have the equation:

... (20)

Multiplying both sides by x, we obtain x2 = x + 1 which is again the equation defining the Golden Ratio. We find that this continued fraction is also equal to ?. d) Also, we may use other definitions of the Golden Section:

... (21)

... (22)

... (23)

... (24)

e) The Fibonacci sequence is intimately connected with the Golden Section. The Fibonacci series:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

has the general property that each term of the sequence (from the third) is equal to the sum of the two preceding terms.

Fn+2 = Fn+1 + Fn (25)

where Fn is the n-th term of the sequence.

If we calculate the ratio of two successive numbers of the Fibonacci sequence, the ratio of the two numbers oscillate around Golden Section (being alternatively higher or lower than it), but tends to the Golden Section (fig. 2).

... (26)

So the ratio of successive Fibonacci numbers Fn+1/Fn converges on ?when n increases.

In the Fibonacci series, the numbers increase rapidly as a power of the Golden Section.

The value of any Fibonacci number, Fn, is entirely based on the Golden Section (Binet's formula):

... (27)

For relatively large values of n, the second term in brackets becomes very small, so that Fn can be determined as the integer number nearest to ....

3. THE APPEARANCE OF THE GOLDEN SECTION IN GEOMETRICAL SHAPES

In a regular pentagon, the ratio of the diagonal to the side is equal to ?, this fact providing a simple means of constructing the regular pentagon.

The triangle in the middle of figure 3, with a ratio of side to base of ?, is known as a Golden Triangle. This isosceles triangle has the two base angles of 72 degrees. The two triangles on the sides, with a ratio of side to base of 1/?, are called Golden Gnomons. The 36°-72°-72° triangle occurs in both the pentagram and the decagon.

Figure 4 demonstrates a unique property of Golden Triangles and Golden Gnomons - they can be dissected into smaller triangles that are also Golden Triangles and Golden Gnomons. If we continue in this fashion we get a set of Whirling Triangles.

Figure 4 shows how constructing a sequence of smaller Golden Triangle yields a spiral.

Figure 5 shows a way to draw a Golden Rectangle with the Golden Ratio:

* Draw a square (of size 1);

* Place a dot half way along one side;

* Draw a line from that point to an opposite corner (it will be 5 / 2 in length);

* Turn this line so that it runs along the square's side. Then you can extend the square to be a Golden Rectangle with the Golden Ratio.

The self-generating capability of the Golden Rectangle is frequently cited [1]. The lengths of the sides of the rectangle are in a Golden Ratio to each other. If we cut off a square from this rectangle we will obtain a smaller rectangle that is also a Golden Rectangle. The dimensions of this rectangle are smaller than those of the first rectangle by factor of ?. We can now cut a square from the second Golden Rectangle and we will obtain again a Golden Rectangle, the dimensions of which are smaller by another factor of ?. Continuing this process ad infinitum, we will produce smaller and smaller Golden Rectangle. Figure 6 shows how constructing a sequence of smaller Golden Rectangle yields a spiral.

The Golden Rectangle is the only rectangle with the property that cutting a square from it produces a similar rectangle. If we draw two diagonals of any pair of rectangles in the series, as in figure 6, they will all intersect at the same point. The series of continuously diminishing rectangles converges to that never-reachable point.

On a five-pointed star pattern (fig. 7), each of the five isosceles triangles that make the corners of a pentagram has the property that the ratio of the length of its longer side to the shorter one (the base) is equal to the Golden Ratio, ?.

The pentagram is related to the regular pentagon. If it connects all the vertices of the pentagon by diagonals, we obtain a pentagram (fig. 8). The diagonals also form a smaller pentagon at the center, and the diagonals of this pentagon form a pentagram and a yet smaller pentagon. This progression can be continued ad infinitum, creating smaller and smaller pentagons and pentagrams. All of these figures have the property that every segment is smaller than its predecessor by a factor that is precisely equal to the Golden Ratio, ?.

The ratio of the length of diagonal AC to the length of side CD of first pentagon is ?, the ratio of the lengths of CD to DR is ?, the ratio of the lengths of DR to RQ is ?, and so on.

If we inscribe a regular decagon in a circle, the ratio of the radius of the circle to the side of the decagon forms the Golden Ratio (fig. 9):

... (28)

Figure 10 shows how we can use any circle to construct on it a hexagon and an equilateral triangle. Joining a pair of three points, it reveals a line and its Golden Section point as follows:

* On any circle, construct a regular hexagon ABCDEF;

* Choose every other point to make an equilateral triangle ACE;

* On two of the sides of that triangle (AE and AC), mark their mid-points M and N by joining the centre of the circle to two of the unused points of the hexagon (F and B);

* The line MN is then extended to meet the circle at point P. N is the Golden Section point of the line MP.

The diagram shown in figure 11, containing only equilateral triangles, has many Golden Sections.

Figure 12 shows the sketch of a quadrilateral pyramid (tetrahedron) in which a is half the side of the base BC, s is the height of the triangular face VBC and h is the

If the square of the pyramid's height is equal to sxa (the area of the triangular face), then the ratio s/a is precisely equal to the Golden Ratio. We have that:

h2 = sx a (29)

Using the Pythagorean theorem in the right angle triangle VOM, we have:

s2 = h2 + a2 (30)

We can substitute for h2 from the Eq. (29) to obtain:

s2 = sx a + a2 (31)

Dividing both sides by a2, we get:

(s/a)2 = (s/a) + 1 (32)

In other words, if we denote s/a by x, we have the quadratic equation:

x2 = x +1 (33)

which is precisely the equation defining the Golden Ratio. The Great Pyramid of Cheops, built before 2000 B.C., has been measured and the ratio s/a = 1.62 which is very close to the Golden Section (differing from it by less than 0.1 percent). However cannot say that the ancient Egyptians knew the significance of the Golden Ratio, ?.

4. THE GOLDEN SECTION AND THE PLATONIC SOLIDS

Platon and the Golden Section are linked through the Platonic solids. A regular polyhedron is a three-dimensional shape whose edges are all of equal length, whose faces are all identical and equilateral, and whose vertices all touch the surface of a circumscribing sphere. There are only five regular polyhedra: the tetrahedron (4 triangular faces, 4 vertices, 6 edges); the cube/hexahedron (6 square faces, 8 vertices, 12 edges); the octahedron (8 triangular faces, 6 vertices, 12 edges); the dodecahedron (12 pentagonal faces, 20 vertices, 30 edges); and the icosahedron (20 triangular faces, 12 vertices, 30 edges). In each case, the number of faces plus corners equals the number of edges plus 2. These five polyhedra are also known as the platonic or Pythagorean solids, and are intimately connected with the Golden Section, ?.

The ancients considered the tetrahedron to represent the element of fire; the octahedron, air; the icosahedron, water; and the cube, earth. The dodecahedron symbolized the harmony of the entire cosmos [3].

In particular, the surface area and volume of a dodecahedron and an icosahedron with an edge length of one unit are defined in Table 1.

The symmetry of the Platonic solids leads to other interesting properties. For example, the cube and the octahedron have the same number of edges (twelve), but their number of faces and vertices are interchanged (the cube has six faces and eight vertices and the octahedron eight faces and six vertices).

The same is true for the dodecahedron and icosahedron. Both have thirty edges, and the dodecahedron has twelve faces and twenty vertices, while it is inversely for the icosahedron. Because these two solids are dual to each other they have the same symmetry group. The order of the group of direct symmetries (all rotations) is S = 60. The elements are: 4 rotations (by multiples of 2π/5) about centers of 6 pairs of opposite faces; 1 rotation (by π) about centers of 15 pairs of opposite edges; 2 rotations (by ± 2π/3) about 10 pairs of opposite vertices. Together with the identity, these rotations totalize 60 elements [2]. These similarities in the symmetries of the Platonic solids allow for mappings of one solid into its dual (reciprocal) solid.

We define the dual of a regular polyhedron to be another regular polyhedron, which is formed by connecting the centers of the faces of the original polyhedron [1]. For example, if we connect the centers of all the faces of a cube, we obtain an octahedron, while if we connect the centers of the faces of an octahedron, we obtain a cube (fig.13).

The same procedure can be applied to map an icosahedron into a dodecahedron and vice versa, and the ratio of the edge lengths of the two solids (one embedded in the other) can be expressed of the Golden Ratio, as ?2 / 5

The te t rahedron is self-reciprocating (joining the four centers of the tetrahedron's faces makes another tetrahedron). A pyramid (a tetrahedron), an octahedron, an icosahedron and a dodecahedron are all beautiful because of the symmetries and equalities in their relations.

The twelve vertices of any icosahedron can be divided into three groups of four, with the vertices of each group lying at the corners of a Golden Rectangle. The rectangles are perpendicular to each other, and their one common point is the center of the icosahedron (fig. 14). Similarly, the centers of the twelve pentagonal faces of the dodecahedron can be divided into three groups of four and each of those groups forms a Golden Rectangle.

Finally, two isosceles triangles (the 36°-72°-72° triangle and 36°-36°-108° triangle) are the basic building shapes of Penrose tilings [5]. Penrose tiles can be pairs of several different shapes, though the two most interesting are those presented in figure 15a, known as "darts" and "kites" [1], [4]. The two shapes were derived from a pentagon as is shown on the fig. 15c. The dart is produced by adding two of the central triangles together and the kite by the addition of the two side triangles. The dart and kite shapes, when added together, form the rhombi that are known as Penrose rhombi.

The darts and kites can be obtained from a rhombus with degree measures of 72° and 108° by dividing the long diagonal into two segments in the Golden Ratio ?= 1.618... then joining the dividing point to the obtuse corners as shown in fig. 15c. Both rhombuses are composed of two Golden Triangles. The ratio of the areas of the kite and dart is the Golden Ratio as well.

5. CONCLUSION

The Golden Ratio is a very special number. The Golden Ratio can be found not only in natural phenomena but also in a variety of human-made objects and works of art.

This number appears in numerous situations: in geometrical constructions, in lists of "favourite numbers" processed by mathematicians, in the works of many artists, architects and designers, in the animal kingdom, and even in famous musical compositions [1].

The Golden Ratio combines two definitions of a proportion: to express the comparative relation between parts of objects with respect to size or quantity or to describe a harmonious relationship between different parts (to one another and to the whole).

References

REFERENCES

[1] LIVIO, M.: Sec?iunea de aur - Povestea lui phi, cel mai uimitor num?r, Ed. Humanitas, Bucure?ti, 2005.

[2] DOBRE, D., SIMION, I.: Symmetry - Mathematical concepts and applications in technology and engineering. Journal of Industrial Design and Engineering Graphics, no. 5, 2009, pp. 21-26, ISSN 1843-3766.

[3] WEYL, H.: Simetria, Ed. ?tiin?ific?, Bucure?ti, 1966.

[4] LOCKERBIE, J.: Notes for a Study of the Design and Planning of Housing for Qataris, London, 2010.

[5] PENROSE, R.: The role of aesthetics in pure and applied mathematical research, Institute of Mathematics and its Applications Bulletin 10, no. 7/8, pp. 266-271, 1974.