**2.5 Fivefold and Spiral Symmetry Associated with Fibonacci Sequence**

The ratio of adjacent elements of the *Fibonacci* sequence approaches the irrational number *t* =*(1+Ö 5)/2* in the limit. The number, *t* is the solution to the algebraic equation

1 + x = x^{2}

As a result* t* has the property

1+ t = t ^{2}

Therefore, the double geometric sequence

is the *Fibonacci *sequence since it has the property that each term is equal to the sum of the earlier two terms and also the ratio of each term to the earlier term is equal to the *golden mean t *. It is the only geometric series which is also a *Fibonacci* sequence (Kappraff, 1992 References). The *Fibonacci* numbers can be represented geometrically in polar coordinates in two dimensions by the equiangular spiral **R _{o}R_{1}R_{2}R_{3}R_{4}R_{5}...** drawn with origin

**R _{o}R_{1} **may be considered to be the tangent at

The initial radius **OR _{o} **equal to

FIGURE 5

Therefore

tan (a )=**R _{1}X/AR_{o}**

**XR _{o} = Rdq **for the arc

tan (a ) = a in the limit for small values of a .

Therefore

or

Integrating for growth of radius from **r to R **associated with angular turning from **0 **to** q **

or

Geometrical consideration for generation of the *Fibonacci* spiral in three dimensions specify a constant angular turning *dq *equal to *1/t *between successive radii and therefore a constant crossing angle, also equal to *1/t* The *Fibonacci* equiangular spiral is then given by the relation

The angle subtended at the center between two successive radii is therefore equal to the *golden angle*

The *Fibonacci *equiangular spiral as shown in *Figure 6* has intrinsic internal structure of the quasiperiodic Penrose tiling pattern (see Section 2.6) and associated long-range spatial and temporal correlations.

FIGURE 6

The* Fibonacci *spiral is traced with mathematical precision in nature in the dynamical growth processes of plants as seen in the geometrical placement on the shoot, of primordia, which later develop into the various plant parts. In a majority (92%) of plants studied world wide, successive primordia always subtend angle equal to the golden angle at the apical center (Jean, 1994 References). Primordia placement in space and time may therefore be resolved into the precise geometrical pattern of the quasiperiodic *Penrose* tiling pattern.