Back to . . . .
Curve Bank Home
Curve Bank Index
NCB
Deposit # 161
Dr. Cye Waldman
cye@att.net

"Gnomon is an
Island"
More members of
the Spiral Family of Plane Curves:
The
Spirals of Archimedes, of Fermat, of Euler, of Cornu,

Hyperbolic, Logarithmic, Spherical, Parabolic, Nielsen's, Seiffert .
. . .

Golden
Ratio

Click
here for the full article.
We are
familiar with the concept of gnomonic tiling in which a figure added to
another reproduces the shape of the original. The process can repeated
over and over to form a tessellated mosaic that covers the plane. The
initial tile is called the seed and the added piece is called the
gnomon. The gnomon generally increases in size geometrically, i.e., by
a power law. Figures thus created are called whorled figures, as the
gnomons are usually added in a circular fashion about the seed. See,
for example, Gazalé [1] and Waldman [2].

Our Figure 1 shows
a whorled plastic pentagon.

There are magnificent
mosaics, or tilings, if you prefer, that continue to amaze us. These
are the whorled plastic pentagon, with its equilateral triangle gnomon,
and the whorled golden rectangle, with its square gnomon, shown in
Figure 1 and Figure 2, respectively. The former has a growth rate of p,
the plastic number, and the later has a growth rate of , the
golden ratio.

Figure 2: Golden rectangle and its
square gnomon with a growth rate of the Golden
Ratio.
It’s fairly well established that
the whorled figures in Figure 1 and Figure 2 are the only ones whose
gnomons are regular polygons, by which we mean, of uniform sides and
internal angles.
Insofar
as we have developed a program for creating pseudospirals and their
attendant triangles (or squares), we wondered if we could find
additional mosaics that covered the plane. . . . . . . . . . . .
. . .

Figure 3: An
octagon and its golden triangle gnomon.
Voilà! We found by experimentation the
pseudospirals with a
sequence and rotation angle produce a whorled octagon
with an isosceles triangle gnomon. The growth rate q was determined
empirically. The result is shown in Figure 3. Interestingly, the gnomon
is the golden triangle.
This set the stage
for us to (a) determine the growth rate
analytically, and (b) see if the result could be generalized for other
growth rate and turn angles. The technical details and
nomenclature that followed are included in the attached pdf file.


