GraphingCalculator 3.0;
Window 53 15 832 681;
PaneDivider 146;
SignificantDigits 14;
HideAxes 1;
FontSizes 12 9 9;
BackgroundType 1;
Slider 0 1;
SliderSteps 50;
SliderControlValue 9;
SliderThrottle 10;
SliderVariable b;
3D.X -1.5 1.5;
3D.Y -1.5 1.5;
3D.Z -1.5 1.5;
3D.U 0 6.283;
3D.V 0 6.283;
3D.Depth 2.33309097429715;
3D.View -0.638449963702045 -0.733757517689039 0.232339297376356 0.683095395986021 -0.679295885721698 -0.268212191457366 0.354629840611436 -0.0125301595554171 0.934922815671791;
3D.Rotation 0.999972937129389 0.00735616471861366 0.00010885521131104 -0.00735643982851647 0.999969048895568 0.00278998995398077 -8.8328216457267e-05 -0.00279071523565567 0.999996102045703;
Parameter a = 3;
Text "** ""Möbius Twist"" family of curves";
Color 14;
Grain 1;
Expr vector(r,theta,z)=vector((n-50)/20+sgn([cos(b*v)])*[cos(b*v)]^4*[cos(a*u)]+sgn([sin(b*v)])*[sin(b*v)]^4*sin(a*u),u,[[-sgn([cos(b*v)])]*[cos(b*v)]^4]*sin(a*u)+sgn([sin(b*v)])*[sin(b*v)]^4*cos(a*u));
Text "This is a variation on my ""Möbius Triangle"" curve. The curve |x|^0.5+ |z|^0.5=1
(in parametric form) is shifted horizontally, then revolved about the z-axis
with a twist of ""a"" rotations about the figure's center for each revolution.
Play with the parameter values for interesting effects. ""a"" determines the number of
rotations of the cross section per revolution. I like a = 0, 0.25, 0.5, 0.75, 1, 2, 3, 4 etc.
(a = 0 gives a revolution with no rotations.) The parameter b, taken from 0 to 1,
determines the fraction of |x|^0.5+ |z|^0.5=1 that will be drawn. Try a = 0 and b << = 0.25.
I usually use b = 1.
Values of ""a"" greater than 6 have edges that are line segments rather than curves. This,
I presume, is due to the size of the ""step"" used by the software when graphing parametric
equations. This effect causes some striking patterns for certain values of ""a"". Try a = 9.75,
19.5, 29.25, and 39, as well as values of ""a"" that are 0.25, .5, or 1 away from these values.
Spinning the figure with a = 0.25, 0.5, or 0.75 in conjunction with monochrome coloring
(especially violet) gives an interesting optical illusion.
Finally, the value of ""n"" seen on the slider governs the translation of |x|^0.5+ |z|^0.5=1
from the origin. Click the ""play"" button on the slider to see a ""pulsing"" version of
your 3-D graph.
See my website for QuickTime movies of several variations on the ""Möbius Twist""
family, as well as dozens of other Graphing Calculator movie clips.
Bruce Simmons
Austin, TX USA
http://home.earthlink.net/~besimmons/movies/moviehome.html
bsimmons@sss.austin.tx.us
December 11, 2000";
**