Frei Otto
MINIMAL NETS

The following reports represent the authors’ notes which
have been coordinated only as far as absolutely
necessary. This field of work is still too new to be
shown uniformly as a whole. The reader will therefor
notice different interpretations, sometimes even
contradictions .
History of this work
Nhen the Institute for Light Weight Structures (IL)
started its work at the Technische Hochschule Stuttgart
in 1964, the following functions which had been dealt
with before at the Entwicklungsstätte für den Leichtbau
3erlin, were taken over among others:
Arrangement and classification of Structures,
Minimal Line Systems,
Minimal Surface Systems,
Minimal Structures.

The most difficult field are minimal structures whereas
the most simple are the minimal line systems, short
minimal nefs.
From the basic considerations on constructive lightweight
 structures continually carried on since 1946 the
course led from "fish-shape" roof trusses with a small
total slat length to cable nefs and membranes and thus
automatically to the problems of minimal surfaces.
Regarding our works (1946 - 52 in Chartres and Berlin)
on roof trusses as well as on compression-stressed treelike
 constructions (1960 - 64 in Berlin and New Haven,
Yale-University) and with regard to all of our works or
two or three-dimensional nets (starting 1951) the sum
of all products from individual slat lengths multiplied
with individual slat force played an important part in
valuing the efficiency.

The sum of all individual path lengths multiplied with
the respective traffic loads represents the key for the
valuation of traffic systems. There is a close connection
 between topographic and traffic problems.
1 1 (1949)

In the years of 56 - 64 many soap bubble tests have
oeen made in Berlin. The observation of the most
highly complicated minimal surface formations, which
can easily be produced experimentally, entertained
he wish to form minimal line systems consisting of
one-dimensional elements instead of minimal surface
systems composed of two-dimensional elements. If, in
the model test, surfaces can be generated with soap
solution, a respective "thread-forming” liquid, i. e
viscose substances, such as honey or soft cheese is
required. Up to now the experiments were not successful.
 The tension of the thread is in relation to the
dead weight too small, The forms dropped down or
rardened too early so that no equality in tension
~vould be guaranteed.

More practicable possibilities for solution were investigated
 (1960). Threads on rolls were stressed with
weights, rubber threads or measure springs. The method
s suitable for both plane and dimensional systems,
n case of unequal loads (tensions) nets occur with un-2qual
 values of the paths.
"he result is another form.

In lack of usable thread-forming liquids the simple
idea was successful, namely to use liquids, which form
surfaces. The experience gained with soap skins was
helpful. (see lit.)
First, rubber plugs were inserted between two plane
parallel glass plates. The resulting minimal surface
was a minimal path system of the same width (1962).
These problems have been discussed in a seminary in
1964 where the method as described here was demonstrated
 ,
The instrument built and refined with high precision
oy Jochen Schilling resulted quite unexpectedly also
in a method for determining the form of plane memosranes
 of the same surface tension between cables.
Up to now, if was not possible with lots of points to
prove exactly as to whether always that system having
the smallest path length would definitely arise from
the different possible path systems.
We assume that the glass plate method developed at
the IL which can certainly be further improved is the
axactest up to now. It can be applied in practice.
The test series were carried out by Gernot Minke and
Gunther Schsfl with temporary assistance of Ekkehard
3ertram and Rainer Gaupp.
oth Eberhard Haug (chain lines at diagonal plates)
and Ekkehard Bertram took many efforts with regard to
the mathematical principles.
The documentation of the work has been made by the
authors of this number in collaboration with the editor
(IL).
Only literature studies (1967) draw our attention to the
works of R. Courant who published the following
drawing. (see lit.)

It is intended to further improve the procedure as described
 here and especially to investigate three-dimensional
 minimal path nets and nets of unequal path
loads, furthermore closed nets as well as the effect anc
application on the fields of topography and construc=~
tion.