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Fibonacci Numbers, The Golden Ratio, and Platonic Solids Brief Lecture Notes Kieran O’Neill 1 1.1 Fibonacci Numbers A Reminder of the Deﬁnition of the Sequence Most of us are familiar with the Fibonacci Sequence. As a reminder, the sequence is deﬁned : F 1 = F 2 = 1; Fn = Fn−1 + Fn−2 , and was designed by Fibonacci during the Renaissance to model rabbit breeding. The Fibonacci Sequence can be illustrated geometrically, as follows: The start condition is the rectangle formed by placing two squares of side length 1 next to each other. Thereafter, each new number is formed by making a square of side length equal to the existing rectangle, then adding it to the side of the existing rectangle to form a larger one. 1.2 The Golden Ratio in the Fibonacci Sequence Observe that, in the above diagram, the ratio of the rectangle’s sides is the ratio between adjacent terms of the Fibonacci Sequence. Notice further, that, as the sequence progresses, it approaches a speciﬁc rectangle. In algebraic terms, as n → ∞, the ratio between adjacent terms of the Fibonacci Sequence approaches a value. √ This value is denoted τ , and is known to have the value 1+2 5 ≈ 1.618. It is also known as the golden ratio. 1 2 The Golden Ratio The Golden Ratio is that ratio such that, if a line is divided into two segments, of the Golden Ratio to each other, then the ratio of the line to the longer segment is also the Golden Ratio. This is illustrated below, where line AB has been divided into segments AC and CB, with AB:AC = AC:CB. 2.1 Mathematical Properties 1 τ • The Golden Ratio, τ , has the property = 1 − τ . This value, 1 − τ , is known as τ . τ n√ −τ 5 n • The value of the nth Fibonacci Number, is given by Fn = • τ satisﬁes the equation τ 2 = τ + 1. Solving this for its roots gives τ = √ 1± 5 2 , or τ = √ 1+ 5 2 , τ = √ 1− 5 2 2.2 Two Dimensional Geometric Properties The chord of a regular unit (sides equal to 1) pentagon is equal to τ . Where the chords intersect, they are divided into two lines, one of length 1, the other of length τ − 1. A regular pentagon can thus be constructed if the Golden Ratio can be constructed on a line. If a Golden Rectangle (one with sides of ratio τ : 1) is divided into a square and a rectangle, the rectangle is also a Golden Rectangle. This can be done recursively ad inﬁnatum. A logarithmic spiral with equation 2θ r = τ π (in radial coordinates) intersects with each subsequent division of a rectangle. 2 2.3 Interesting History and General Occurrence in Nature • The logarithmic spiral mentioned above is actually found in the cross section of the shell of the Nautilus, an aquatic snail. • The optimal dimensions for a box that will be used as a speaker cabinet are τ : 1 : τ . • The dimensions of the height, the base and the length of one side of Pyramid of Xheops at Giza are in the ratio τ : 1 : τ . • The Pythagoreans, a semi-religious cult of Ancient Greek mathematicians, founded by Pythagoras, used the pentagram as their sacred symbol. A pentagram is constructed from the ﬁve chords of a pentagon. The Pythagoreans used the knowledge of the correct construction of a pentagram (using the Golden Ratio) to identify themselves to other members. The pentagram contains many fascinating self similar ratios and proportions. 3 Platonic Solids 3.1 Introduction to the Platonic Solids • Recall that a polyhedron is a closed three dimensional shape, while a polygon is a closed two-dimensional (ie: planar) shape. • A platonic solid is a polyhedron, all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex. • There are only 5. These are the regular tetrahedron, octahedron and icosahedron, wherein 3, 4 and 5 triangles respectively meet at each vertex, the cube (or regular hexahedron), wherein 3 squares meet at each vertex, and the regular dodecahedron, wherein 3 pentagons meet at each vertex. These have 4, 8, 20, 6 and 12 sides respectively, and are named after the Greek words for these numbers (tetra- meaning four, octa- eight, etc). The regular means that all the polygons making up the polyhedron are congruent. From this point on in these notes, ”regular” will be dropped as a preﬁx, and all polyhedra referred to assumed to be regular. 3 Tetrahedron Octahedron Icosahedron Hexahedron Dodecahedron Polygon Sides 3 3 3 4 5 Polygons Meeting 3 4 5 3 3 Faces 4 8 20 6 12 Edges 6 12 30 12 30 Vertices 4 6 12 8 20 • For a polyhedron to be platonic, the internal angles of the polygons meeting at a vertex must add to less than 360◦ . This is because, if they add to 360◦ , they form a plane, and cannot be part of a polyhedron. Obviously, the internal angles around a vertex cannot add to more than 360 ◦ . • Consequently, there is no platonic solid formed from hexagons, or from any regular polygon of more than 5 sides. 3.2 Each Platonic Solid has a Dual Platonic Solid An interesting property of the platonic solids is that, if the centre of each face in a platonic solid is joined to the centre of each adjacent face, another platonic solid is created within the ﬁrst. These occur in pairs, in that a dodecahedron can be formed from an icosahedron, and vice versa, and the same for cubes and octahedrons. A tetrahedron can be formed within a tetrahedron. An example of an icosahedron formed within a dodecahedron is shown below. Notice also that the pairs of platonic solids that are duals of each other have mirrored face/edge/vertex numbers (see the table above). 3.3 The Golden Ratio in Icosahedrons and Dodecahedrons • An icosahedron may be thought of as having pentagonal ”caps” where ﬁve triangles intersect. If adjacent vertices on the cap are distance 1 from each other, then non-adjacent vertices are distance τ from each other, since they form chords of the pentagon. Thus, 4 vertices in an icosahedron may be connected by a rectangle of sides τ by 1, which is a Golden Rectangle. Since there are 12 vertices in an icosahedron, 3 such rectangles (which turn out to be orthogonal to each other) can describe all the points in an icosahedron. 4 This shows that the vertices of a unit icosahedron are (±1, ±τ, 0), (0, ±1, ±τ ), (±τ, 0, ±1). • A cube may be formed from the vertices of a dodecahedron, using chords of the pentagons making it up, of sides of length τ (when the dodecahedron has sides of length 1). This accounts for 6 of the 20 vertices in the dodecahedron. The remainder may be formed from 3 mutually orthogonal rectangles, as in the icosahedron, but of dimensions τ 2 by 1. This shows that the vertices of a unit dodecahedron are (±τ, ±τ, ±τ ), (±1, ±τ 2 , 0), (0, ±1, ±τ 2 ), (±τ 2 , 0, ±1). 5

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Fibonacci Numbers, The Golden Ratio, and Platonic Solids Brief

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