Thank you very much for your answer and your work.

I admire your work

and reading your papers has taught me a lot about the importance

and equations of minimal surfaces. The German school of Minimal surfaces

with Karcher, Webber and Gollek is truly a great one.

I believe I have developed a kind of generalized catenoid formula.

Since, the Tan(n*z) I have found several other catenoid types that are

periodic as well:

(f,g)=(1/Cos[n*z]^2,Cos[n*z]) : sets of even holes

(f,g)=(1/Sin[n*z]^2,Sin[n*z]) : sets of odd holes

and on that this polynomial derivative like:

(f,g)=( (n!)^2/x^(2*n),x^n/n!)

You may enjoy experimenting with these as I have.

Without Mathematica I would never have been able to see these surfaces.

I also developed a four dimensional catenoid formula

and used it to form a three dimensional projection.

If you liked the other result , this should also make you happy.

I started out trying to figure out how to make

minimal surfaces that were simple, that had n ( 0r 2*n) holes in them.

I got that as you have seen.

I'm ,now, working on seeing if I can get higher dimensional catenoid/

helicoids that can

be projected onto three dimensions as I did with the four space.

I used Peter Hennes' 6 space minimal surfaces to get a Hopf 6 map type

last night:

( I believe this is an SO(6) symmetry)

f1=-f2

g1=-I*g2

First the Catenoid:

x1=(1/g^2-1)/2

y1=*((1/g^2+1)/2

z1=1/g

second the Helicoid:

x2=-(1/g^2+1)/2

y2=-I(1/g^2-1)/2

z2=-I/g

These form the planes (plane waves)

x=x1+y1+z1

y=x2+y2+z2

r=Sqrt[x^2+y^2)

w=Atan(x,y)

(f3,g3)=(Log(r),w)

There are

6!/(3!*3!)=20

Combinations to give distinct planes.

The combinations of planes are

20!/(18!*2!)=190

The Hopf type of equation is:

(x1^2+y1^2+z1^2+x2^2+y2^2+z2^2)^3=x3^2+y3^2+z3^2=0

/It seems strange ( cheating?) , but the proof it that zero equals zero

in all cases.

All 190 of them. So if the helicoids are electromagnetic

and the catenoids are gravitational, there are 190

resulting three space fields for the interaction of photons with

gravitons.

Hubert Gollek wrote:

> I had a look at your formulas, turned it into a Mathematica notebook

Respectfully, Roger L. Bagula

> and studied

> its invariants, transformed the complex parametrization, trying to

> find out whether it is

> a some other known surface in a nonstandard parametrization.

> At first I had to impose some corections in the Weierstrass -

> representation

> (two signs and a factor 2 in the 3-rd component).

> Finally I got a nice picture that looks similar to other known ones.

> Probably some other parametrization of the catenoid?

>

> best regards

>

> Hubert Gollek

>

> Roger Bagula schrieb:

>

>> --

>> Respectfully, Roger L. Bagula

>> tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:

>>

>> 619-5610814 :

>> URL : http://home.earthlink.net/~tftn

>> URL : http://victorian.fortunecity.com/carmelita/435/

>>

>>

>

tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:

619-5610814 :

URL : http://home.earthlink.net/~tftn

URL : http://victorian.fortunecity.com/carmelita/435/