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 NCB Deposit  # 151

Cye H. Waldman
cye@att.net
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A New Twist on Möbius

moebiusstamp


In MATLAB . . . . . . .


animation

Möbius Animation


Copyright Notice:  The animation and all images within are under copyright by Cye H. Waldman and may not be copied, electronically or otherise, without his express permission.

Cye H. Waldman,  cye@att.net


toroid
toroid3

animation2

You are looking at an optical illusion.  The cross-section is an astroid.  The animation is a tromp l'oeil in that the object is rotating about the vertical axis.  However, there is a stroboscopic effect due to the fact that the rotation rate matches the radial ribs, which then appear to stand still.  The net effect is that the object seems to rotate about the circular ring, much like a ring vortex.

Three-dimensional Möbius forms, or "Mobioids" are created from toroids by a radial cut, one or more twists, and reattachment.   This is remarkably similar to a Penrose triangle.
mobiods

Interestingly, for a toroid with five surfaces, any number of twists where n is not disivible by 5 will produce a single surface.  In addition, the generating toroid need not be circular.
 
Look for a cross-section:
Two surfaces and two cusps

moebius2sided

Look for a cross-section:
An Astroid with  four cusps
moebius4sided


multiple surfaces

References
Other Waldman contributions to the NCB using MATLAB:
More Valentine/Hearts:   < http://curvebank.calstatela.edu/waldman8/waldman8.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 >
Voderberg Tiling   < http://curvebank.calstatela.edu/waldman9/waldman9.htm  >
Cornu-Voderberg Tilings  < http://curvebank.calstatela.edu/waldman10/waldman10.htm  >
Aleph Animation  < http://curvebank.calstatela.edu/waldman11/waldman11.htm  >

Other Möbius contributions in the NCB
< http://curvebank.calstatela.edu/moebius/moebius.htm >

Printed References      August Ferdinand Möbius
August Ferdinand Möbius  1790 - 1868
Alfred Gray, Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, CRC Press, 2nd ed., 1998.


Eric W. Weisstein, CRC Concise Encyclopedia of MATHEMATICS, 1999.

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