Back to . . . .

Curve Bank Home
Curve Bank Index


 
NCB Deposit  # 146


Dr. Cye Waldman

cye@att.net


NCB logo

 
New Paradigms for Spiral Tiling . . . .

Waldman Parquet Tilings

icon2

This paper introduces spiral tilings possibly never seen before.  In the animation notice the spiral arms can be discontinuous in spite of the fact that they were created in the same way as many other spirals.  They are unique.
Dr. Cye Waldman
Click here to see a full "pdf" file with other figures and Matlab code.


WaldmanParquetSpiral
A Waldman Parquet Demo


parquettrio



Abstract

Parquet, or parallelogram substitution, tiling is a new paradigm for radial and spiral tiling.  It is inspired by, but decidedly different than rhombus substitution tiling. We describe the tiles that are suitable for this tiling and how to create them. We demonstrate how to produce radial and spiral tilings with the Goldberg shift.


Copyright Notice:  This animation and all images within are under copyright by Cye Waldman and may not be copied, electronically or otherwise, without his espressed permission.

Dr. Cye Waldman
cye@att.net


Historical Sketch

From the legendary Delian problem in antiquity to modern freeway construction, spirals have attracted great mathematical talent.  Among the more famous are Archimedes, Descartes, Bernoulli, Euler, and Fermat, but there are many more whose work has enormously influenced pure mathematics, science and engineering.

The name spiral, where a curve winds outward from a fixed point,  has been extended to curves where the tracing point moves alternately toward and away from the pole, the so-called sinusoidal type.    We find Cayley's Sextic, Tschirnhausen's Cubic, and Lituus' shepherd's (or a bishop's) crook.  Maclaurin, best known for his work on series, discusses spirals in Harmonia Mensurarum (1722).  We find parabolic spirals.  In architecture there is the Ionic capital on a column.  In nature, the spiraled chambered nautilus is associated with the Golden Ratio, which again is associated with the Fibonacci Sequence.
        
References
Other Waldman contributions to the NCB:

Rhomus Tilings:  < http://curvebank.calstatela.edu/waldman14/waldman14.htm >
Infinity Paradox Tilings:  < http://curvebank.calstatela.edu/waldman11/waldman11.htm >
Cornu-Voderberg Tilings:  < http://curvebank.calstatela.edu/waldman10/waldman10.htm >
Voderberg Tilings:  < http://curvebank.calstatela.edu/waldman9/waldman9.htm >
Sinusoidal Spirals:  < http://curvebank.calstatela.edu/waldman/waldman.htm >
Bessel Functions    < http://curvebank.calstatela.edu/waldman2/waldman2.htm >
Gamma Funcions   < http://curvebank.calstatela.edu/waldman5/waldman5.htm >
Polynomial Spirals and Beyond   < http://curvebank.calstatela.edu/waldman4/waldman4.htm >
Fibonacci and Binet Spirals with a touch of Mondrian  < http://curvebank.calstatela.edu/waldman6/waldman6.htm >
"Other" Fibonacci Spirals and Binet Spirals  < http://curvebank.calstatela.edu/waldman7/waldman7.htm >
A Mathematician's Valentine < http://curvebank.calstatela.edu/waldman8/waldman8.htm >
                                              < http://curvebank.calstatela.edu/waldman12/waldman12.htm >
The NCB thanks Dr. Waldman for his strong contibutions.

Other spiral Deposits in the NCB:
< http://curvebank.calstatela.edu/spiral/spiral.htm >
< http://curvebank.calstatela.edu/log/log.htm >

index icon  
NCB Home logo
    signature      2015