Friday, September 16, 2022

Creating Exotic Mathematical Surfaces Based On Generating Singularities & Cusps

 On perusing an issue of The Bulletin of the American Mathematical Society  from 2003, I came across a paper entitled:  THE HIRONAKA THEOREM ON RESOLUTION OF SINGULARITIES:

S0273-0979-03-00982-0.pdf (

 which featured examination of a number of singular surfaces characterized by exotic properties, especially singularities and cusps.  I decided to use my Mathcad program to attempt to create similar complex surfaces and polyhedra - with varied results- some of which I present in this post.  I also was able to generate contour maps of the 3D surfaces, which definitely manifested cusps and singularities.

The first complex polyhedron to be considered was:

z= f(x,y) =  tanh (x13 )    yielding:

But the contour map yielded so many folds and at high density it was impossible to parse.

A more tractable equation was then used:

 z = f(x,y) = sin(x3 +  y 3) tanh(x) tanh(y)

Which generates the surface:

And the corresponding contour map:  

For which we see the emergence of cusps at the corners and perhaps a singularity near the center, but nothing definite or specific yet.

Now we change the Mathcad equation to:

f(x,y) = sin(x+  y7) tanh(x3) tanh(y3)  

And obtain the surface:

And contour:

Note that cusps have definitely emerged at the corners, as well as potential singularities - which also appear at the left and right sides of the contour map.  

Next we generate a surface more in line with the cusped entity shown in the authors' Fig.7.  We use the equation:

f(x,y)  =   x-  y 3

And we find  for the surface:

And the contour:

For which the contour associated with the '0' line bears a remarkable similarity to the cusp resolved by dragging in the AMS paper Fig. 7

Another interesting find is when one goes to higher degrees of any of the polynomial equations the complexity appears to alternate, e.g. for the  shapes above and using instead:  f(x,y)  =   x11 -  y 13

 We find  for the surface:

And the contour:

But when we then use:  f(x,y)  =   x12 -  y 14

 We find  for the surface:

And the contour:

Examination shows each has changed radically merely by increasing the degree of x, y by one.   I also found it possible to alter the cusp extent and intensity merely by changing the factor for hyperbolic functions in some equations. E.g. for:

f(x,y) = tanh (x) exp( py) -  x4

We see the surface go from:

To the much more pronounced cusping:

Merely by changing to tanh (3x2 ) .   The cusp enhancement is also apparent on inspecting the two contour maps, with the altered factor tanh version to the right:

In a future post I will explore possible parameterizations of these (and other) examples, as well as testing the authors' speculation:

"Singular curves are the shadow of smooth curves in higher dimensional space."

Questions for readers:  Is the last contour a 'singular curve?  Is it the shadow of a smooth curve in a higher dimensional space?

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