**INVARIANTS**
Remembering Martin Luther King, I had a dream. I dreamt that
the two worlds of the so called humanities and of technology
had indissolubly melted into a common culture and that scientific
rigour had entered full sail into the extraordinary world of
creativity giving it a much more fascinating connotation of
rigour.

When I say that the field of creativity is much more contiguous
to the scientific rigour than you can imagine, I think the reactions
among professionals, be they creative people (artists, if we
use a term now close to obsolescence) or experts in creativity,
are equal to a wall rather than to a door willingly opened.
The old all in division into two cultures is hard to die. I
speak here of a way of conceiving architecture that can bring
this category of human activities nearer to the field of science
and, consequentially, to the field of logic. Everything begins
with the quite obvious finding that in the course of architectural
design, at any time and anywhere in the world, architects during
the various stages of the design itself, have faced, are facing
and will face a myriad of different problems. These problems,
once somehow solved, will produce an object which sometimes
belongs to architecture, while in most cases simply to construction.
If we try to analyse this large and often confusing set of problems
and needs more carefully and with stricter sequencing we will
come across a very interesting fact, that, in my opinion, deserves
to be investigated by all the people involved in teaching architecture,
by all those experts who give aesthetic judgements - sometimes
wholesale and usually retail - on architecture itself and all
those who of architecture make it their job. I should add that
the argument I am about to expose is fine not only for architects,
but also for people working in the field of the genesis of form
and of its manipulation.

I will start therefore with the statement number 1: in the course
of design, the architect or anyone interested in producing forms
of all kinds, evidently addressing many situations to be solved,
is however and always faced with some problems, that, whatever
the subject of the same design, are always the same, at any
time, in any space, and in any cultural environment.

It is therefore reasonable to say that these issues, for the
fact of always recurring, represent a constant factor easily
traceable in the design of any object of any kind. The indisputable
presence of constant situations/moments in any evolutionary
process entitles us to assume that we are dealing with a category
that could be fairly contiguous to a minimum of scientific rigour.

This contiguity to that field in which the presence of constant
factors, axioms and everything that is constantly repeated in
time and space without change of any kind, suggests that we
are moving in a dimension which is strictly related to scientific
thought. I will begin, therefore, to organize a more rigorous
and more logical analysis of some phenomena that have their
full citizenship in the field of creativity.

So what are those cases, those design situations, which, at
any time, on any point on the face of the earth, in any cultural
circumstance continuously arise for those who have to do with
design?

In the field of architectural practice I have identified some
situations, which I will define invariants, because they come
up so implacably invariant for the designer.

FOREWORD

Of all the ideas current in the various Architecture faculties
in Italian universities on how to teach planning, the most deep-rooted
and widespread is the belief that almost everything can be taught
except design itself.

It is not, however, my intention, in this paper, to try to discover
why this idea is so common (to the extent that it becomes a
pseudo-intellectual screen to hide behind).

It is worth remembering briefly that, while reams of papers
have been published on the above impossibility, discussions
on other subjects have often, inexplicably, stood still. I shall
not speak here of architects who have founded their literary
forces on the strength of reasoning; my point of departure in
what follows is the correct, and above all, logical use of what
has been done up to now, and will continue to be done, in architecture.

**INITIAL CONSIDERATIONS**

We shall start from two established facts:

The existence on the face of the Earth of a myriad of objects
conceived and constructed by man, many of which are firmly
"anchored" to the ground and therefore unlikely
to be moved onto the architect's drawing table or that of
the student architect.

The existence also of an enormous number of publications on
Architecture including drawings, photographs and an immense
amount of words organised in various different ways.

There are, therefore, two artificial universes:

Architectural constructions.

Architecture which has been drawn or written about, or history
of architecture.

ENUNCIATION OF THE PROBLEM

I ask myself:

how it is possible, if it is possible, using these two universes,
to create a way of starting to:

Talk Architecture.

Talk about Architecture ?

**PRESENTATION OF THE PROBLEM**

It must be considered that, despite temporal and cultural
changes, and although affected by varied forms of conditioning
and evolution, the architect, in certain parts of the designing
process has always been faced by certain situations which
have never changed trough the years.

I shall explain myself better: the design of any objects (no
matter what its dimensions may be) can be compared to the
articulate organisation of material and forms.

Within this organisation, however complex, there are situations
which are always unchanged, as follows:

The putting together of different materials which are on the
same plane but have different directions.

The change of plane of two different materials.

The change of plane of a single material.

The confines of any material.

The treatment of spaces with the same material around them.

At this point we have two terms on our drawing board:

An organisation which is always complex.

The five unchanging situations.

Let us now consider the above mentioned complex organisation
as a set and the unchanging situations as a few elements of
the set.

Since we have introduced two terms taken from mathematical
terminology, that is set and elements of the set, the transfer
of the problem from a conceptual area to an area belonging
to mathematical logic seems inevitable. The term inevitable
has been used in the light of the following considerations:

1st consideration: "If any discourse is presented in
such a way that it consists of symbols and precise rules on
the operation of these symbols, which are only subject to
the condition of having an internal coherence, that discourse
is mathematical." C. Boyer, demonstrating this important
concept expressed in "The Mathematical Analysis of Logic"
by George Boole (1847).

2nd consideration: "Mathematics can be compared to an
exquisitely made mill which grinds materials of any grade
of fineness; but, nevertheless, what comes out depends on
what goes in; and just as the greatest mill in the world will
not extract flour from pea pods, so pages and pages of formulae
will not produce precise results from incoherent data!"
T.H. Huxley (1825 - 1895).

3rd consideration: In 1872 George Cantor defined a set as
" The joining in a whole of objects of our intuition
or of our thoughts, which are well defined and different each
from the other."

4th consideration: "Pure mathematics is the class of
all the propositions which have the form "p implies q"
where "p" and "q" are propositions containing
one or more variables, which are the same in each proposition,
and neither "p" nor "q" contains a single
constant except the logical constants." Bertrand Russel
"The Principles of Mathematics" 1903.

5th consideration: In the 1939 first edition of "Théorie
des Ensembles", Nicolas Bourbaki wrote: "A set is
made up of elements which have certain properties and which
have a certain relationship with each other and with elements
of other sets".

6th consideration: In 1950, N. Bourbaki wrote in "The
Architecture of Mathematics", American Mathematical Monthly:
"From the axiomatic point of view, mathematics is presented
as a storehouse of abstract forms: mathematical structures;
and it happens that, without us knowing why, certain aspects
of empirical reality adapt themselves to these forms, almost
through a sort of predisposition."

7th consideration: In 1950 André Weil wrote in "The
future of Mathematics", American Mathematical Monthly:
"It is through unexpected combinations……that
the mathematician of the future will lay out in a

new way and resolve the problems that we have left him to
inherit."

FINAL NOTES

From a didactic point of view, and to make things clearer
for the reader, it is worth remembering that the five unchanging
situations can be rewritten in the following way, with the
addition of useful comments:

Study how the problem: when two different materials, having
the same planes and different directions, are in contact with
each other has been resolved structurally and/or formally
or how it can be resolved.

Study how the problem: when two different materials, having
two different planes and joining at the same point, are in
contact with each other has been resolved structurally and/or
formally or how it can be resolved.

Study how the problem: when a single material changes plane
has been resolved structurally and/or formally or how it can
be resolved.

Study how the problem: when a material ends has been resolved
structurally and/or formally or how it can be resolved.

Study how the problem: when two different materials ends,
a space begins and the same material begins again has been
resolved structurally and/or formally or how it can be resolved.

Any correspondence should be addressed to:

prof. arch. Franz Falanga

via Campaner 1

31034 CAVASO (TV)

ITALIEN