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Surfaces and Topological Forms 
An Introduction to the fascinating patterns of Visual Math

Minimal
Surfaces and Geodesic Forms
Minimal
Surfaces are the most economical connections between loops
or lines in 3d 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 minimalsurface 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


 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:

Relationship Tree
of names and classes of minimal surfaces
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 ... 










This Last one was from a site in Japanese,
i have no idea what is represented here.
Maybe a complex minimal surface seen in 3D crosssection
which
looks like an organic highrise apartment of the future.
If you know what this is .... email
me ;)  Miqel

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