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 geneate 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 Sub DrawCurve()
    ...

    ' Get the drawing parameters.
    GetParameters
    picCanvas.ForeColor = PointColor

    ' Compute the values.
    x = X0
    y = Y0
    z = Z0
    i = 0
    Do While Running
        ' Move to the next point.
        x2 = Sin(A * y) - z * Cos(B * x)
        y2 = z * Sin(C * x) - Cos(D * y)
        z = Sin(x)
        x = x2
        y = y2

        ' Plot the point.
        Select Case SelectedPlane
            Case plane_XY
                picCanvas.PSet (x * xscale + xoff, y * _
                    yscale + yoff)
            Case plane_XZ
                picCanvas.PSet (x * xscale + xoff, z * _
                    zscale + zoff)
            Case plane_YZ
                picCanvas.PSet (y * yscale + yoff, z * _
                    zscale + zoff)
        End Select

        ' To make things faster, only DoEvents
        ' every 100 times.
        i = i + 1
        If i > 100 Then
            i = 0
            DoEvents
        End If
    Loop
End Sub

For more information on fractals, see my book Visual Basic Graphics Programming.