Suppose you perform a series of iterations of equations to generate points. Sometimes the points converge to one or more points. For example, the equations X(n) = X(n - 1) / 2, Y(n) = Y(n - 1) / 3 approach the point (0, 0) as n grows large.

The points to which the equations converge is called an attractor.

Some equations are drawn towards a collection of points that is not easily defined but that somehow has a coherent shape. These points are called a strange attractor.

Clifford Pickover discovered that the following equations generate points that are drawn to a strange attractor.

    X(n) = Sin(A * Y(n - 1)) - Z(n - 1) * Cos(B * X(n - 1))
    Y(n) = Z(n) * Sin(C * X(n - 1)) - Cos(D * Y(n - 1))
    Z(n) = Sin(X(n - 1))

Here A, B, C, and D are constants.

The following code plots these points.

// Draw the curve.
private void DrawCurve()
{
    // Get the parameters and otherwise get ready.
    Prepare();

    // Start drawing.
    double x = X0, y = Y0, z = Z0;
    while (Running)
    {
        // Plot a bunch of points.
        for (int i = 1; i<=1000; i++)
        {
            // Move to the next point.
            double x2 = Math.Sin(A * y) - z * Math.Cos(B * x);
            double y2 = z * Math.Sin(C * x) - Math.Cos(D * y);
            z = Math.Sin(x);
            x = x2;
            y = y2;

            // Plot the point.
            switch (SelectedPlane)
            {
                case Plane.XY:
                    bm.SetPixel((int)(x * xscale + xoff), (int)(y * yscale + yoff), FgColor);
                    break;
                case Plane.YZ:
                    bm.SetPixel((int)(y * yscale + yoff), (int)(z * zscale + zoff), FgColor);
                    break;
                case Plane.XZ:
                    bm.SetPixel((int)(x * xscale + xoff), (int)(z * zscale + zoff), FgColor);
                    break;
            }
        }

        // Refresh.
        picCanvas.Refresh();

        // Check events to see if the user clicked Stop.
        Application.DoEvents();
    }
}

Use the program's controls to plot X-Y, X-Z, or Y-Z projections of the points in different colors. You can also change the equations' constants and starting value to see what happens.