**Spiral** 1. *Geom.* one of several plane curves formed by a point winding about a fixed center at an ever increasing distance from it. 2. A curve that lies on a cylinder or cone, at a constant angle to the line segements making up the surface. (Collins Concise English Dictionary).

The above definition is too general to be of use when modelling organic shells and horns that grow by accretion of material. The spirals of nature are of a very specific kind, and all follow one mathematical rule of growth: the size increases but the shape remains unaltered.

The only curve to obey this rule is the *logarithmic spiral* which has inspired artists and mathematicians for millenia.

#### Polar Equation of The Logarithmic Spiral

The *logarithmic spiral* is known by different names. Descartes called it the *equiangular spiral* because the radius always cuts the curve at a constant angle; it is sometimes called the *geometrical spiral* because the radius increases in geometric progression. Bernoulli was so enamoured with its mathematical perfection that he described the curve as *spira mirabilis*. *(Ref: The Logarithmic Spiral, from 'The Divine Proportion' by H. E. Huntley, Dover Inc, New York, 1970).*

I have been interested in the logarithmic spiral since art school days and my early ventures with turtle geometry (LOGO) were done principally to generate horns and shells - but all of these were in 2D. Whelk is my attempt to take the code into 3D using the *intrinsic* geometry of a 3D turtle - *id est*, no reference to an *external* cartesian coordinate system is required because all geometric operations are achieved through the standard Yaw, Pitch and Roll navigational routines. This seems ideally suited to the accretion of material and adornments found in Nature.

#### Visualising The Spiral in Whelk

Before building the mesh, it is useful to familiarize yourself with the Spiral Controls which change the polar equation parameters of the logarithmic spiral. This is best done by hiding the mesh and viewing the spiral alone. Uncheck ‘Mesh’ in the View menu, and check ‘Spiral Path’. This draws the spiral path at the profile center.

#### Golden Spiral `ON`

The yellow curve is the golden spiral generated by Whelk. The animation overlay frame in white shows construction of the golden spiral using ruler and compasses. `AB:BC = φ:1`

. Through `E`

, which is the golden cut of `AB`

, draw line `EF`

perpendicular to `AB`

, thus cutting off from the rectangle `ABCD`

a square `AEFD`

. The remaining rectangle `EBCF`

is a golden rectangle. When `EBGH`

is cut off, the remaining rectangle `HGCF`

is also a golden rectangle. The process may be repeated *ad infinitum* until the desired limiting point is reached. The sides of the rectangle are almost but not entirely tangential to the curve. Note that any two radii separated by `90°`

are in the ratio of `φ:1`

. *(Ref: The Logarithmic Spiral, from 'The Divine Proportion' by H. E. Huntley, Dover Inc, New York, 1970).*

#### Chromatic Spiral `ON`

The Chromatic spiral results from plotting the wavelengths of the Evenly Tempered* scale on polar graph paper, where the radii are proportional to the note wavelenths. The constant angle `α`

for the equiangular spiral of music is `77°50'`

or `77.833333r`

decimal degrees.

*k*=

`1.0595`

times that of the semitone above it:
*k*

------------------------ c' 1.0000 0° b 1.0595 15° a# 1.1225 30° a 1.1892 45° g# 1.2099 60° g 1.3348 75° f# 1.4141 90° f 1.4983 105° e 1.5870 120° d# 1.6818 135° d 1.7819 150° c# 1.8878 165° c 2.0000 180°

*The Evenly Tempered or Chromatic scale is a mathematically accurate exponential scale where the musical interval of the semitones remains constant.

*(Ref: Musical Scale and The Spiral, from 'The Divine Proportion' by H. E. Huntley, Dover Inc, New York, 1970).*

#### Custom Spiral `ON`

To form your own spiral, use the Custom option. Higher `α`

values make the spiral tighter; as the spiral tightens, more stacks must be built before it becomes visible; alternatively, you may reduce the 'Stack Divisor' number, which lowers the curve resolution by increasing the length of each segment in the curve. The spiral may also be enlarged by increasing the radial scaling factor *a*.

You may also specify a spiral by its rate of radial expansion ** R** on the ‘Generator’ panel. This rate is given as the ratio between two spiral radii 360° apart. See Generator Controls for details.

#### Spiral Tips

In general, spirals with low `α`

values are used to create cockles, bivalves and uncoiled gastropods and scaphopods, whereas higher `α`

values are used to create coiled cephalopods and gastropods. The following cockle form has an `α`

value of `33.2684248`

. It is composed of just `154 stacks`

, with a Whorl scale factor of `0.971`

.

When making bivalves, care must be taken with the parameters to ensure that the protuding umbos do not overlap the base of the shell. Higher ε values will raise the umbos. See Generator Controls for details.