YOU ARE

HERE

FAW 7

Stockholm - October 1994

 

- An Exercise in ALGEBRAIC TOPOLOGY -

(After Brouwer's Fixed Point Theorem)

 

You may say that you are there, just where you are. But... Can you prove it?

Brouwer give's you a handy tool:

 

Brouwer's fixed-point theorem

Under a continuous mapping f : S-> S of an n- dimensional simplex into itself there exists at least one point x E S such that f (x)=x.

This theorem was proved by L. E. J. Brouwer [1]. An equivalent theorem had been proved by P. G. Bohl [2] at a somewhat earlier date. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an n-dimensional topological vector space and is extensively employed in proofs of theorems on the existence of solutions of various equations. Brouwer's theorem can be generalized to infinite-dimensional topological vector spaces.

References

[1] Brouwer L.E J.: 'Ueber eineindeuitinger sterige Transformationen von Flächen in sich'. Math. Ann. 69 (1910). 176-180

[2] Bohl. P.: 'Ueber die Beweging eines mechanischen Systems in der Nähe einer Gleichgewichtslage'. J. Reine Angew. Marh.127 (1904), 179-276.

V.I. Sobolev

Editorial comments: There are many different proofs of the Brouwer fixed-point theorem. The shortest and conceptually easiest, however, use algebraic topology. Completely-elementary proofs also exist. Of. eg. [A1], Chapt. 4. In 1886, H. Poincare proved a fixed-point result on continuous mappings f : En -> En which is now known to be equivalent to the Brouwer fixed-point theorem, [A2]. There are effective ways to calculate approximate Brouwer fixed points and these techniques are important in a multitude of applications including the calculation of economic equilibria. [A1]. The first such algorithm was proposed by H. Scarf, [A3]. Such algorithms later developed in the so-called homotopy or continuation methods for calculating zeros of functions, cf. e.g. [A4], [A5].

References

[A1] Istratescu. V.I. Fixed point theory. Reidel 1981.

[A2] ointcare. H.'Sur les courbes definies par les équations differentielles', J. de Math. 2 (1886).

[A3] Scarf, H.: 'The approximation of fixed points of continuous mappings', SIAM J. Appl. Math 15 (1967) 1328-1343.

[A4] Karamadian, S. (ED.):Fixed points, Algorithms and applications, Acad. Press,1977.

 

OK?

 
.

ALGEBRAIC TOPOLOGY

(A dangerous simplification of BROUWERS Theorem, for you, to be able to understand it)

What Brouwer wants to prove with his theorem (everything in the abstract magic mathematic, that in the end of the chain decides our daily life), is that there is at least one point where an "object" (the mass, you) confronts inevitably it self (it contains it self). This, ladies and gentlemen, does have serious consequences in our lives.

Brouwer's theorem in the hands of an economist (dangerous people) can be helpful to calculate a certain economical balance. Meaning that after endless lines of zeros (homothopy, says J. Scarf) a national economist can come to conclusions like that of supply and demand (always in abstract baundarys as mass /quantity (s) and functions (f)) meet each other exactly on that steady point - or like the late parrot said: the market adjust itself. In other worlds: The market actually sometimes developes products that the consumers actually needs (and off course buy) and not only the opposite.

In difference from (or just like) economy, when one uses the theorem in algebraic topology (continuos mapping f : S -> S) it proves that a subject (f : function) definitely can be situated on the same place where it says to be situated, the one that draws a map includes oneself in its own drawing. In other words, if you walk any street in whatever city it is proved, thanks to Brouwer, that there is a 100% possibility that in all reality you are situated exactly there.

So just take it easy, you can stop at any spot and say f (x)=x , without any doubt.

Of course it can be very complicated, and not everybody can reach that label of knowledge. To make easy for your function (f) to be the same as your self (x) we have placed red dots in different places around town to inform you that, thanks to x = S, you are obviously there, that means:

YOU ARE

HERE

Believe it.

 

For this little incomprehensible exercise in Mathematics, Brouwer got the Nobel Price.

 

 

.

MAPPING ON

When we were already busy placing the 50 cm big red YOU ARE HERE dots in the verry non-tourist places: Stockholm City most boring and ugliest streets, we received the invitation from Gallery Söderlund to use their space for an installation. This gallery (they own a small but beautiful collection from Beartling, Mondrian and Hilma af Klint) is definitely a No Hype Gallery because of its love for early constructivist art. Actually post-modern-hype-artist does not exhibit in such gallery, it has no status, prestige and it is politically incorrect. And exactly that was what made it interesting, and also because the spirit of Mondrian is completely in harmony with Brouwer's, not to mention with a continuos mapping analysis and similar thoughts. We extend our street project into the gallery room and re-developed it with the possibilities of being indoors.

 

Follow me:

We installed a white wall-to-wall carpet on the floor and on to it, we draw the map of the 300-sq. meters surrounding area, the Gallery's block in the centre of the room.

the map of the area

the map on the floor

 

On the map's block on the floor, where the gallery was situated, we placed a red dot (like those around the city), indicating the exact location of the gallery on the map.

That red dot was placed exactly in the middle of the room.

Now: Standing on that red dot you happened to be in a multiple mapping. You where standing exactly in the centre of the real room, in the real gallery. At the same time you where standing on the map's centre indicating and overlaying the actual position of the map, the block, the gallery, the room, the dot, and you. At the same time you where standing on the real map, the 1:1 scale map of the city, the City itself, indicating, overlaying and including the exact position of the map, the block, the gallery, the room, the dot, and you. So, you where standing on the centre of the dot, the room, the map, the city, the world, the Universe.

:mapping on mapping on mapping:

Algebraic topology.

 

 

The gallery's windows where cover with silk paper exept for the center, that shown a transparent map of the city superpost to the real city behind

The map was showing the situation of the other red dots around the city

Almost Pascal, isn't it? You know... the sphere, where the centre is everywhere and the circumference nowhere...