Brave New Whorled: Anamorphic Tilings

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NCB Deposit # 162

Dr. Cye Waldman
cye@att.net

"Brave New Whorled"

Anamorphic Tilings. . . .

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 gnomons generally increase 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

There are magnificent mosaics, or tilings, if you prefer, that continue to amaze us. In this paper we consider a particular whorled triangle and demonstrate that it can be generalized to arbitrary, indeed, even random growth, provided that the growth is monotonic. Thus we can create anamorphic tilings for which the ‘gnomons’ are all of different shapes, albeit within the same general family. Here, we are using the term anamorphic in the optical sense of having unequal magnifications along two axes perpendicular to each other. We show a generic anamorphic tiling and a single spiral. In fact, we demonstrate that these tilings support three different spirals and comment on the nature of those spirals.

The the equations and computer code for Anamorphic Triangles, Seed Triangles, Trapezoidal Gnomons, Primary, Secondary (complimentary) and Tertiary (full) Spirals are in the pdf file.

References

[1] M.J. Gazalé, Gnomon: From Pharaohs to Fractals. Princeton University Press (1999).

[2] Waldman, C.H., Gnomons Land (2016).
< ..//waldman16/waldman16.htm >

[3] "Fibonacci, Padovan, & Other Pseudospirals"
< ..//waldman6/waldman6.htm >

Please see other Waldman contributions to the NCB, e.g.,
Sinusoidal Spirals: < ..//waldman/waldman.htm >
Bessel Functions < ..//waldman2/waldman2.htm >

Other spiral Deposits in the NCB:
< ..//spiral/spiral.htm >
< ..//log/log.htm >

2017