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NCB Deposit  # 140


Dr. Cye Waldman

cye@att.net


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Cornu-Voderberg Spiral Tilings . . . .

An Original Contribution
Voderberg

This paper is really about a new class of tiles for which two tiles can completely enclose one or two more copies of the same tile.  This remarkable property is no longer limited to the Voderberg tile.  There are no limitations as to the number of sides or the vertex angle, although only certain angles will admit radial and spiral tiling.  The Cornu tiles shown here are one case of the new tile family.
Dr. Cye Waldman
Click here to see a full "pdf" file.


Cornu Spiral
Cornu Spiral Animation  ( 3-degree vertex angle)


Cornu animation
Voderberg Application




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

Voderberg
References
Other Waldman contributions to the NCB:
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/waldman3/waldman3.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 >
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 >

Other Cardioid - Valentines -  in the NCB:
< http://curvebank.calstatela.edu/index/cardioid.gif

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