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NCB Deposit  # 144


Dr. Cye Waldman

cye@att.net


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New Paradigms for Spiral Tiling . . . .

Rhombus Substitution

rhombus

This new paper brings together some spirals previously thought to be ad hoc tilings.  We have extended the concepts of triangle substitution and deformable tiles to the rhombus; this provides an infinite number of innovative possibilities.
Dr. Cye Waldman
Click here to see a full "pdf" file with other figures and Matlab code.


Rhombus Substitution Spiral
Cornu Spiral Animation with Rhombus Substitution


rhombicspirals





The Spiral of Cornu is named for the French scientist Marie Alfred Cornu (1841 - 1902).  He studied this curve, also known as a clothoid or Euler's Spiral, in connection with diffraction.  Euler applied a similar figure while measuring the elasticity of a spring.
The parametric equations for a generalized Cornu spiral are on the right.


Similar integrals are named for Augustin Jean Fresnel (1788-1827), one of the founders of the wave theory of light.
Cornu equations




The Spiral of Cornu, a.k.a. Clothoids   "are important curves used in freeway and railroad construction.  For example, a clothoid is needed to make the gradual transition from a highway; which has zero curvature, to the midpoint of a freeway exit, which has nonzero curvature.  A clothoid is clearly preferable to a path consisting of straight lines and circles, for which the curvature is discontinuous." (!!)
Alfred Gray  




Copyright Notice:  This animation and all images within are under copyright by Cye Waldman and may not be copied, electronically or otherwise, without his espress permission.
Dr. Cye Waldman
cye@att.net



A rhombus tile is two triangle tiles that are concatenated at the base. Symmetric and antisymmetric concatenations are both possible, but only the latter will tessellate radially. The parent triangles may be symmetric or antisymmetric tiles.

rhombus figure
References
Other Waldman contributions to the NCB:

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 >


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