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柯尼斯堡七桥问题是怎样改变数学的? – 译学馆
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柯尼斯堡七桥问题是怎样改变数学的?

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

在现在的地图上 你很难找到哥尼斯堡这个城市
You’d have a hard time finding Königsberg on any modern maps,
但是它在地理上奇特之处
but one particular quirk in its geography
使得它在数学上成为最为著名的城市之一
has made it one of the most famous cities in mathematics.
这个中世纪的德国城市坐落于普雷格尔河的两岸
The medieval German city lay on both sides of the Pregel River.
河的中央有两座大的岛屿
At the center were two large islands.
这两座岛屿通过七座桥
The two islands were connected to each other and to the river banks
与河的两岸以及与彼此连接
by seven bridges.
后来成为附近小镇市长的数学家卡尔·戈特利布·埃勒
Carl Gottlieb Ehler, a mathematician who later became the mayor of a nearby town,
对这些桥和岛屿十分着迷
grew obsessed with these islands and bridges.
他一直在考虑一个问题:
He kept coming back to a single question:
哪一条路径可以使人穿过所有这七座桥
Which route would allow someone to cross all seven bridges
并且同一座桥只能经过一次?
without crossing any of them more than once?
思考一下
Think about it for a moment.
7
7
6
6
5
5
4
4
3
3
2
2
1
1
放弃了吗?
Give up?
应该是的
You should.
这是不可能的但是
It’s not possible.
大数学家莱昂哈德·欧拉在试图解释这个数学问题时
But attempting to explain why led famous mathematician Leonhard Euler
开拓了一个新的数学领域
to invent a new field of mathematics.
卡尔向欧拉写信求助开始
Carl wrote to Euler for help with the problem.
欧拉认为这个问题和数学无关 所以不关心这个问题
Euler first dismissed the question as having nothing to do with math.
但是随着他对该问题的思考
But the more he wrestled with it,
他越来越发现该问题有一定的意义
the more it seemed there might be something there after all.
他得出的答案与一类几何学相关
The answer he came up with had to do with a type of geometry
但当时并不存在 他称之为位置几何学
that did not quite exist yet, what he called the Geometry of Position,
就是现在著名的图论
now known as Graph Theory.
欧拉最初的想法
Euler’s first insight
是进入岛屿或河岸和离开岛屿或河岸的路线
was that the route taken between entering an island or a riverbank and leaving it
实际上并不重要这样
didn’t actually matter.
地图上便可以简化为四个岛
Thus, the map could be simplified with each of the four landmasses
用四个简单的点表示
represented as a single point,
我们现在称之为节点
what we now call a node,
它们之间的线或边代表桥这样
with lines, or edges, between them to represent the bridges.
简化的图使我们比较容易计算每个节点的度
And this simplified graph allows us to easily count the degrees of each node.
即连接岛之间桥的数量
That’s the number of bridges each land mass touches.
为什么度很重要呢?
Why do the degrees matter?
试想 根据这个问题的规定
Well, according to the rules of the challenge,
一旦有人想要通过一座桥到达一个岛屿
once travelers arrive onto a landmass by one bridge,
他就必须通过另外的桥离开
they would have to leave it via a different bridge.
也就是说 在任何路线上 通往和离开每个节点的桥
In other words, the bridges leading to and from each node on any route
必须是不同的桥
must occur in distinct pairs,
这意味着连接每个岛的桥的数量
meaning that the number of bridges touching each landmass visited
一定是偶数
must be even.
唯一可能的例外是在出发的位置
The only possible exceptions would be the locations of the beginning
和离开的位置看下图
and end of the walk.
很明显所有四个节点的度都为奇数于是
Looking at the graph, it becomes apparent that all four nodes have an odd degree.
无论选择什么样的路线
So no matter which path is chosen,
在一些点上 一座桥势必会被经过两次
at some point, a bridge will have to be crossed twice.
欧拉用这个证明发展出了一个通用的理论
Euler used this proof to formulate a general theory
适用于存在两个或两个以上节点的图
that applies to all graphs with two or more nodes.
每一个边仅经过一次的欧拉路径
A Eulerian path that visits each edge only once
只在两种情况下有可能第一
is only possible in one of two scenarios.
当仅有两个节点为奇数度时
The first is when there are exactly two nodes of odd degree,
这意味着其它的都是偶数度这样
meaning all the rest are even.
开始点就是奇数度的一个
There, the starting point is one of the odd nodes,
结束点是另外一个第二
and the end point is the other.
当所有的节点都是偶数度时那么
The second is when all the nodes are of even degree.
欧拉路径就从同一个位置开始和结束
Then, the Eulerian path will start and stop in the same location,
这被称为欧拉回路于是
which also makes it something called a Eulerian circuit.
你怎么才能在格尼斯堡找到欧拉路径呢?
So how might you create a Eulerian path in Königsberg?
这很简单
It’s simple.
只要移走任一座桥
Just remove any one bridge.
事实说明 历史创造了欧拉路径
And it turns out, history created a Eulerian path of its own.
二战期间 苏联空军摧毁了两个城市之间的一座桥
During World War II, the Soviet Air Force destroyed two of the city’s bridges,
这便创造出了欧拉路径虽然
making a Eulerian path easily possible.
公平来说 他们的目的不是这样
Though, to be fair, that probably wasn’t their intention.
这些炸弹从地图上抹掉了格尼斯堡
These bombings pretty much wiped Königsberg off the map,
并且这里被重建为之后的俄罗斯加里宁格勒市
and it was later rebuilt as the Russian city of Kaliningrad.
所以尽管格尼斯堡和她的七座桥不再存在
So while Königsberg and her seven bridges may not be around anymore,
但是它们会因这个导致全新数学
they will be remembered throughout history by the seemingly trivial riddle
领域出现的谜团被历史记录下来
which led to the emergence of a whole new field of mathematics.

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