How the Königsberg bridge problem changed mathematics - Dan Van der Vieren

Prevoditelj: Tamara Rabuzin Recezent: Ivan Stamenković

You'd have a hard time finding Königsberg on any modern maps,

Königsberg ćete teško pronaći na današnjoj zemljopisnoj karti,

but one particular quirk in its geography

ali jedna dosjetka vezana za njegov tlocrt

has made it one of the most famous cities in mathematics.

učinila ga je jednim od najslavnijih gradova vezanih uz matematiku.

The medieval German city lay on both sides of the Pregel River.

Srednjovjekovni njemački grad ležao je na obje strane rijeke Pregel.

At the center were two large islands.

U središtu su bila dva velika otoka.

The two islands were connected to each other and to the river banks

Dva otoka bila su povezana međusobno i s obalama rijeke

by seven bridges.

pomoću sedam mostova.

Carl Gottlieb Ehler, a mathematician who later became the mayor of a nearby town,

Carl Gottlieb Ehler, matematičar koji je postao gradonačelnik obližnjeg grada,

grew obsessed with these islands and bridges.

postao je opsjednut ovim otocima i mostovima.

He kept coming back to a single question:

Stalno se vraćao na isto pitanje:

Which route would allow someone to cross all seven bridges

na koji način netko može prijeći svih sedan mostova

without crossing any of them more than once?

tako da svaki most prijeđe samo jednom?

Think about it for a moment.

Razmislite o tome na trenutak.

7

7

6

6

5

5

4

4

3

3

2

2

1

1

Give up?

Odustajete?

You should.

Trebali biste.

It's not possible.

To nije moguće učiniti.

But attempting to explain why led famous mathematician Leonhard Euler

Ali pokušaj objašnjavanja zašto je tako vodio je matematičara Leonharda Eulera

to invent a new field of mathematics.

do stvaranja novog područja matematike.

Carl wrote to Euler for help with the problem.

Carl je pisao Euleru moleći ga za pomoć.

Euler first dismissed the question as having nothing to do with math.

Euler je najprije odbacio problem jer je vjerovao da nema veze s matematikom.

But the more he wrestled with it,

Ali što je više razmišljao o njemu,

the more it seemed there might be something there after all.

činilo se više mogućim da se u njemu nešto krije.

The answer he came up with had to do with a type of geometry

Odgovor do kojeg je došao imao je veze s vrstom geometrije

that did not quite exist yet, what he called the Geometry of Position,

koja još nije postojala, a koju je on nazvao Geometrija položaja,

now known as Graph Theory.

a danas je poznata kao Teorija grafova.

Euler's first insight

Eulerova prva spoznaja

was that the route taken between entering an island or a riverbank and leaving it

bila je da ruta između stupanja na otok ili obalu rijeke i napuštanja istog

didn't actually matter.

zapravo nije važna.

Thus, the map could be simplified with each of the four landmasses

Prema tome, karta se može pojednostavniti tako da se svaki od četiri kopnena čvora

represented as a single point,

prikaže pomoću točke,

what we now call a node,

koju ćemo zvati vrh,

with lines, or edges, between them to represent the bridges.

a linije koje prikazuju mostove, zvat ćemo bridovi.

And this simplified graph allows us to easily count the degrees of each node.

Na ovom jednostavnom grafu lako možemo odrediti stupanj svakog vrha.

That's the number of bridges each land mass touches.

To je broj mostova kojim je svaki kopneni čvor povezan.

Why do the degrees matter?

Zašto je stupanj važan?

Well, according to the rules of the challenge,

Prema pravilima izazova,

once travelers arrive onto a landmass by one bridge,

kad putnik stigne na kopneni čvor pomoću jednog mosta,

they would have to leave it via a different bridge.

mora ga napustiti prelazeći preko drugog.

In other words, the bridges leading to and from each node on any route

Drugim riječima, mostovi koji vode do vrha i s njega na bilo kojoj ruti

must occur in distinct pairs,

moraju se pojavljivati u parovima,

meaning that the number of bridges touching each landmass visited

što znači da broj mostova koji dodiruju svaki prijeđeni čvor

must be even.

mora biti paran.

The only possible exceptions would be the locations of the beginning

Jedine moguće iznimke bile bi

and end of the walk.

početak i kraj šetnje.

Looking at the graph, it becomes apparent that all four nodes have an odd degree.

Gledajući graf, postaje očito da sva četiri vrha imaju neparan stupanj.

So no matter which path is chosen,

Pa koji god put odaberemo,

at some point, a bridge will have to be crossed twice.

u jednom trenutku, jedan od mostova moramo prijeći dvaput.

Euler used this proof to formulate a general theory

Euler je pomoću ovog dokaza oblikovao opću teoriju

that applies to all graphs with two or more nodes.

koja se odnosi na sve grafove s dva i više vrha.

A Eulerian path that visits each edge only once

Eulerova staza kod koje se svaki vrh prelazi jednom

is only possible in one of two scenarios.

moguća je jedino u dva slučaja.

The first is when there are exactly two nodes of odd degree,

Prvi je kad postoje točno dva vrha neparnog stupnja,

meaning all the rest are even.

pa su svi ostali parni.

There, the starting point is one of the odd nodes,

Tada je početna točka jedan od dva neparna vrha,

and the end point is the other.

a kraj šetnje je drugi.

The second is when all the nodes are of even degree.

Drugi slučaj je kada su svi vrhovi parnog stupnja.

Then, the Eulerian path will start and stop in the same location,

Tad će Eulerova staza započeti i završiti u istom vrhu,

which also makes it something called a Eulerian circuit.

što se u teoriji grafova zove Eulerova tura.

So how might you create a Eulerian path in Königsberg?

Dakle, kako kreirati Eulerovu stazu u Königsbergu?

It's simple.

Jednostavno je.

Just remove any one bridge.

Samo treba ukloniti jedan most.

And it turns out, history created a Eulerian path of its own.

Dogodilo se da je povijest sama stvorila Eulerovu stazu.

During World War II, the Soviet Air Force destroyed two of the city's bridges,

U II. svjetskom ratu Sovjetske zračne sile uništile su jedan od dva gradska mosta,

making a Eulerian path easily possible.

pa je Eulerova staza postala moguća.

Though, to be fair, that probably wasn't their intention.

Doduše, to vjerojatno nije bila njihova namjera.

These bombings pretty much wiped Königsberg off the map,

Bombardiranje je gotovo izbrisalo Königsberg s karte,

and it was later rebuilt as the Russian city of Kaliningrad.

te je poslije ponovo izgrađen kao ruski grad Kaliningrad.

So while Königsberg and her seven bridges may not be around anymore,

Iako Königsberg i njegovih sedam mostova više ne postoje,

they will be remembered throughout history by the seemingly trivial riddle

bit će zapamćeni u povijesti zbog naizgled trivijalne zagonetke

which led to the emergence of a whole new field of mathematics.

koja je vodila do stvaranja potpuno nove grane matematike.