Minimal Surfaces and Geodesic Forms

Minimal Surfaces are the most economical connections between loops or lines in 3-d space.

"The definition of a minimal surface is any surface that has a mean curvature of zero.
Physically this means that for a given boundary a minimal surface cannot be changed without increasing the area of the surface."
The most obvious minimal-surface is the Sphere of a soap bubble - it takes this shape naturally as the most economical and space/energy conserving form as the gasses inside the bubble equalize against the air pressure outside.

Surface of Least Area is always formed by a bubble. As a result, the soap film joining two parallel circles has the shape of a catenoid. A tetrahedron and a cube give rise to complicated arrangements of nearly flat surfaces that meet at characteristic angles.
Soap film has been used for years in experiments to illustrate & devise generalized theories about minimal surfaces.



They are included for their elegant beauty and fundamental significance to geometry and physics, but I know way less about Minimal Surfaces than the other visual math forms in this section, therefore, for accuracy most of the commentary will be quoted from books & websites on this topic:

Most of the WONDERFUL images below are from the comprehensive source for Minimal Surface data available at
http://www.indiana.edu/~minimal/toc.html the Images on black backgrounds lower on the page are (c) by Paul Bourke, Swinburne University AU. Check his site for authoritative information. There are also some great animations at the very bottom of the page

All the surfaces listed below were found before 1900.

# The Catenoid (at left)
# The Helicoid (at right)
# The singly periodic Scherk surface
# The doubly periodic Scherk surface
# Riemann's minimal surface
# Enneper's surface

Enneper's Surface - single and multiple crossings

"The surface below has one end of Enneper type and one planar end."


Below
"Riemann found a family of singly periodic minimal surface
whose intersections with horizontal planes are circles."

"The Costa surface has two catenoidal ends and one planar end. From far away, it looks like the intersection of a catenoid with a horizontal plane. The surface has two straight lines on it and the vertical coordinate planes as symmetry planes."


Minimal Surfaces give me the same feeling of 'dejas vous' that I get when looking at the mandelbrot set
Although the images are new I have a feeling of familiarity when viewing them, i suppose because they
are such fundamental forms in nature, math and mind ~ Miqel

Minimal Surface with symmetry axes marked side view

same surface, Surprising Top view

Another Minimal Surface with symmetry lines marked. (right) Same surface, Surprising Top view
Klein Surface Variation

"Below is an example with genus 1 and 4 ends
which is not embedded as a complete surface"




Minimal Surfaces Exhibit Exotic Natural Beauty

Below
is a surface with two catenoidal ends and one planar end.
The parts of a surface which extend infintely far away are called the ends of the surface.
There are special important ends named after the most simple surface where
they appear.



This one looks like DNA; twisted scherk surface - compare to simple Helicoid at right






They don't all look like futuristic jellyfish
here is A More Linear Minimal Surface with a Surprising Checkerboard Top View

"Karcher observed that one can add handles to the
doubly periodic Scherk surface. One can even add more handles."




"Below The singly periodic Scherk surfaces approaches two orthogonal planes.
Here is a variation where the two planes are not orthogonal."

- A Formation Worthy of Deep Meditation and Study -

"minimal Klein bottle with one end -- there is only one boundary curve!"

Like a jellyfish creature from Outer Space:

2, 3 and 5 Saddles

A Few Animation of the Minimal Surfaces

A few of the Categories of Minimal Surfaces include the Cartenoid (far left), Saddle shaped forms, Helixes, and combined or transitional forms and more ...