Equations for the Lawson Klein Figure

[lxln]

Hi,

Does anybody know the equations for the Lawson Klein bottle ? I am trying to construct it in a maths graphic application.

Just to be precise its the version of the Klein bottle that is composed of two Sudanese Mobius bands, also it may be that the application will only cope with ordinary 3-d space. The simpler the better !

The two best references I have found are here:
http://plus.maths.org/content/os/issue26/features/mathart/index
-towards the bottom of the page also has a link to a great 3D model.

and here:
http://en.wikipedia.org/wiki/Möbius_strip
-not so pretty but has parametric equations, unfortunately while I have some 'in principle' understanding of what is going on the math is a little beyond me and maybe even the graphic application too.


Thanks in advance.

 

[lxln]

Hi,

Does anybody know the equations for the Lawson Klein bottle ? I am trying to construct it in a maths graphic application.

I should add that the programme i am trying to construct it in is spacetime.us: the wikipedia equations appear to refer to a 4 dimensional embedding which can be projected into 3d space. Is there a simple 3D formulation I can simply plug into spacetime.us ?

 

[bcourter]

Here's the code I used to generate a recent sculpture. It might not be the world's most efficient, but it got the job done. It actually came from Daniel Piker's explorations, which you can find here: http://vimeo.com/2495945

This is C# written for the SpaceClaim API, but I reckon it's pretty readable:

public static Point Evaluate(PointUV uv, double p, double q, double circleAngle, Vector inverseOffset, bool isInverted) {
double u = uv.U;
double v = uv.V;

Circle circle = Circle.Create(Frame.World, 1);
circle = circle.CreateTransformedCopy(Matrix.CreateRotation(Line.Create(Point.Origin, Direction.DirY), circleAngle));

return LawsonTransform(circle.Evaluate(u).Point, v, p, q, inverseOffset, isInverted);
}

public static Point LawsonTransform(Point point, double t, double p, double q, Vector inverseOffset, bool isInverted) {
double xa = point.X;
double you = point.Y;
double za = point.Z;

//reverse stereographically project to Riemann hypersphere
double xb = 2 * xa / (1 + xa * xa + you * you + za * za);
double yb = 2 * you / (1 + xa * xa + you * you + za * za);
double zb = 2 * za / (1 + xa * xa + you * you + za * za);
double wb = (-1 + xa * xa + you * you + za * za) / (1 + xa * xa + you * you + za * za);

//now rotate the hypersphere
double xc = xb * Math.Cos(p * t) + yb * Math.Sin(p * t);
double yc = -xb * Math.Sin(p * t) + yb * Math.Cos(p * t);
double zc = zb * Math.Cos(q * t) - wb * Math.Sin(q * t);
double wc = zb * Math.Sin(q * t) + wb * Math.Cos(q * t);

//then project stereographically back to flat 3D
double xd = xc / (1 - wc);
double yd = yc / (1 - wc);
double zd = zc / (1 - wc);

point = Point.Create(xd, yd, zd);

//spherical inversion
if (isInverted) {
point += inverseOffset;
point = Point.Origin + point.Vector.Direction * 1 / point.Vector.Magnitude;

if (
Double.IsNaN(point.X) || Double.IsNaN(point.Y) || Double.IsNaN(point.Z) ||
Double.IsInfinity(point.X) || Double.IsInfinity(point.Y) || Double.IsInfinity(point.Z)
)
return Point.Origin;
}

return point;
}

And I call Evaluate() with:

double p = 0.5;
double q = 1;
double circleAngle = Math.PI / 2;
Vector inverseOffset = Vector.Create(0.5, 0, 0);
bool isInverted = true;

If you're curious, here's the sculpture:
http://www.coroflot.com/public/individual_details.asp?individual_id=260043

And here's a video of the tabs growing on it:

 

[lxln]

Thanks for that !

I had a look at your sculpture and models and they are pretty effing cool ! Out of interest how did you make the physical model ?

My question is a bit disingenuous - I find the figure fascinating and am studying my way to understanding it -I think there is in the end nothing for it but to build up the maths to understand at least the Wikipedia entry. Nevertheless I was wondering if there was any interesting scientific or mathematical insight that could be picked up on this forum over and above Daniel Pikers work - no such luck so far...

I am not familiar with script (yet!) but from reading through yours there are some mathematical components that are familiar. It seems conventional to create the figure in 4 dimensions and project back into 3 - but if it can be manifested in 3 should there not be a 3D description/equation (albeit intersecting) ? Would tide me over until I work out how to create and use 4D data sets and project them back into 3D !