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为什么大自然偏爱六边形 – 译学馆
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为什么大自然偏爱六边形

Why Nature Loves Hexagons

美国公共电视网 数字工作室
Is nature a mathematician?
大自然是位数学家吗?
Patterns and geometry are everywhere.
各种图案和几何图形随处可见
But nature seems to have a particular thing for the number 6.
但大自然似乎对数字6特别青睐
Beehives.
蜂房
Rocks.
岩石
Marine skeletons.
海洋生物的骨骼
Insect eyes.
昆虫的眼睛
It could just be a mathematical coincidence.
这究竟是数学上的巧合
Or could there be some pattern beneath the pattern.
亦或是图案的背后隐藏着某种规律?
Why nature arrive at this geometry?
自然界为何会产生这种六边形结构?
We’re going to figure that out with some bubbles.
要解答这一问题 我们将借助一些气泡
And some help from our favorite mathematician:
以及我们最喜爱的数学家
Kelsey, from Infinite Series.
来自《无穷级数》的Kelsy的帮助
Happy to help.
乐意效劳
《聪明刷》
A bubble is just some volume of gas, surrounded by liquid.
气泡是指由液体包裹着的一定体积的气体
It can be surrounded by a lot of liquid,
包裹气泡的液体既可能数量很大
like in champagne,
例如香槟
or just a thin layer, like in soap bubbles.
也可能像肥皂泡般只有薄薄的一层
So why do these bubbles have any shape at all?
那这些气泡为什么会有形状呢?
Liquid molecules are happier wrapped up on the inside,
液体分子更乐意呆在液体内部
where attraction is balanced, than they are at the edge.
因为在内部受到的引力比在边缘更加平衡
This pushes liquids to adopt shapes with the least surface.
液体由此向表面积最小的形状靠拢
In zero g, this attraction pulls water into round blobs.
重力为零时 这种引力使水变成圆形水滴
Same with droplets on leaves or a spider’s web.
树叶或蜘蛛网上的水滴同样如此
Inside thin soap films,
在薄皂膜内部
attraction between soap molecules shrinks the bubble
肥皂分子间的相互引力使气泡收缩
until the pull of surface tension is balanced by the air pressure pushing out.
直到表面张力与气压的向外推力达到平衡
It’s physics.
这是物理学的解释
Physics is great,
物理学的解释很棒
but mathematics is truly the universal language.
但数学才是真正的普世语言
Bubbles are round
气泡之所以成圆形
because if you want to enclose the maximum volume with the least surface area,
是因为若要同时实现表面积最小和体积最大
a sphere is the most efficient shape.
球体是最有效的形状
Yeah. That’s another way of putting it.
好吧 这是另一种解释方法
What’s cool is if we deform that bubble,
有趣的是 如果我们把气泡戳变形
the pull of surface tension always evens back out
表面张力总会把气泡重新拉回
to the minimal surface shape.
表面积最小的球体状
This even works when soap films are stretched between complex boundaries,
这一原理同样适用于在复杂的边界间拉伸的皂膜
they always cover an area using the least amount of material.
它们总能用最少的材料覆盖某一区域
That’s why German architect Frei Otto
这也是为何德国建筑师弗雷·奥托
used soap films to model ideal roof shapes
在建造他的奇特建筑时会利用皂膜
for his exotic constructions.
为理想中的屋顶形状打样
Now let’s see what happens when we start to pack bubbles together.
接下来我们看看把气泡聚在一起会怎样
A sphere is a three-dimensional shape,
球体是三维的
but when we pack bubbles in a single layer,
但当我们把气泡挤放在一层中时
we really only have to look at the cross-section: a circle.
我们只需关注气泡的横截面:一个圆形平面
Rigid circles of equal diameter can cover, at most
直径相等的规则圆形最大可覆盖
90% of the area on a plane,
一个平面面积的90%
but luckily, bubbles aren’t rigid.
不过幸好 气泡的形状并不总是规则的
Let’s pretend for a moment
我们假定在某一刻
these bubbles were free to choose any shape they wanted.
这些气泡可以自由变成任意形状
If we want to tile a plane with cells of equal size
如果要用同一形状铺满一个平面
and no wasted area,
且任何地方都被覆盖到的话
we only have three regular polygons to choose from:
我们只能从三种正多边形中选择:
triangles, squares, or hexagons.
三角形 正方形和六边形
So which is best?
那么哪种最合适呢?
We can test this with actual bubbles.
我们用真正的气泡来测试一下
Two equal-sized bubbles?
两个相同的气泡连在一起会怎样呢?
A flat intersection.
连接面成了一个平面
Three,
三个在一起
and we get walls meeting at 120˚.
连接面就以120˚的角度交汇于一处
But when we add a fourth,
但当添加第四个气泡时
instead of a square intersection,
连接面并不会变成方形
the bubbles will always rearrange themselves
气泡总会自行重新排列
so their intersections are 120˚,
以使连接面呈现出120°角
the same angle that defines a hexagon.
和六边形的内角一致
If the goal is to minimize the perimeter for a given area,
若目标是实现给定区域的边界最小化
it turns out that hexagonal packing beats triangles and squares.
六边形组合比三角形和正方形更满足要求
In other words, more filling with fewer edges.
就是说 以最短的边长达到更大的填充面积
In the late 19th century,
19世纪后期
Belgian physicist Joseph Plateau calculated
比利时物理学家尤瑟夫·普拉托计算出
the junction of 120° are also most machenically stable arangement,
120°接角几乎是力学上最稳定的结构
where the forces on the films are all in balance.
因为薄膜中的所有力在此角度达到平衡
That’s why bubble rafts form hexagon patterns.
这就是气泡群形成六边形的原因
Not only does it minimize the perimeter,
六边形不仅边长最小
the pull of surface tension in each direction is most mechanically stable.
各方向上的表面张力在力学上也最稳定
So let’s review:
那么我们来回顾一下:
The air inside a bubble wants to fill the most area possible.
气泡内的空气想要填充尽可能大的空间
But there’s a force, surface tension, that wants to minimize the perimeter.
但是表面张力又想使气泡边界最小化
And when bubbles join up,
而当气泡聚集时
the best balance of fewer edges and mechanical stability is
能用最少的边界达到力学稳定性的是
hexagonal packing.
六边形组合
Is this enough to explain some of the six-sided patterns we see in nature?
这能否解释自然界的一些六边形图案呢?
Basalt columns like Giant’s Causeway, Devil’s Postpile,
巨人堤道 魔鬼柱石之类的玄武岩柱
and the Plains of Catan
甚至游戏中的卡坦岛平原
form from slowly cooling lava.
都是火山熔岩缓慢冷却后形成的
Cooling pulls the rock to fill less space,
冷缩效应使得岩石收缩 填充空间更小
just like surface tension pulls on a soap film.
和表面张力对皂膜的作用如出一辙
Cracks form to release tension,
岩石上形成裂缝以释放张力
to reach mechanical stability,
并达到力学上的稳定
and more energy is released per crack
若裂缝间互成120°角
if they meet at 120˚.
则每条裂缝释放的张力越大
Sounds pretty close to the bubbles.
这与气泡的排列方式很相似
The forces are different,
尽管两者涉及的力不同
but it’s using similar math to solve a similar problem.
但所用的数学原理和所解决的问题是相似的
What about the facets of insect’s eye?
昆虫的复眼为何也呈六边形呢?
Here, instead of a physical force, like in the bubble or the rock,
和气泡 岩石所受的物理上的力不同
evolution is the driver.
昆虫复眼呈六边形是受进化的驱动
Maximum light-sensing area?
这是为了达到最大感光面积吗?
That’s good for the insect,
这当然对昆虫有益
but so is minimizing the amount of cell material around the edges.
同时也能使边缘处的细胞物质最少化
Just like the bubbles,
如同气泡一样
the best shapes are hexagons.
最适合昆虫复眼的形状也是六边形
What’s even cooler,
更酷的是
if you look down at the bottom of each facet?
当你往下看六边形的底部
There’s a cluster of four cone cells,
会发现四个一组的锥形细胞
packed just like bubbles are.
它们的组合方式和气泡组合一样
Bubbles can even help explain honeycomb.
气泡原理还可以解释蜂窝结构
It would be nice to imagine
这么想象一定很有趣
a number of crunching bees, experimenting with triangles and squares
一群蜜蜂先用三角形和正方形做实验
and realizing hexagons are
最后发现六边形才是
the most efficient balance of wax to area.
平衡蜡量与面积的最有效的方法
but with a brain the size of a poppy seed
但靠着小如罂粟种子般的脑袋
They’re no mathematicians.
他们可当不上数学家
It turns out honeybees make round wax cells at first.
这说明蜜蜂一开始造的是圆形的蜂蜡细胞
And as the wax is softened by heat from busy bees,
随着蜂蜡被忙碌的蜜蜂加热软化
it’s pulled by surface tension into stable hexagonal shapes.
它被表面张力拉成稳定的六边形
Just like our bubbles.
就像我们的气泡那样
So is nature a mathematician?
那么大自然真的是位数学家吗?
Some scientists might say nature loves efficiency.
有的科学家或许认为大自然喜欢高效率
Or maybe that nature seeks out the lowest energy.
或是可能在寻求低能耗
And some people might say nature follows the rules of mathematics.
有的人或许认为大自然只是在遵循数学规律
However you look at it,
无论你秉持何种看法
nature definitely has a way
大自然一定有自己的方式
of using simple rules to create elegant solutions.
能用最简单的法子得到最完美的结果
Stay curious.
永葆好奇心吧
So that’s how nature arrives at
这就是大自然如何
the optimal solution for three-dimensional bees,
为三维空间的蜜蜂找到最佳解决方案的
but you know
但你知道
how mathematicians love to take things to the next level.
数学家总喜欢把问题提升到另一个层次
What would the honeycomb look like for a four dimensional bee?
四维空间的蜜蜂会做出什么样的蜂巢呢?
Follow me over to Infinite Series
请跟随我到《无穷级数》栏目
and me and Joe will comb through the math.
我和Joe会通过数学 为你搭建蜂巢

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视频概述

科普关于大自然中出现的很多六边形的现象。

听录译者

收集自网络

翻译译者

何不秉烛游

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视频来源

https://www.youtube.com/watch?v=Pypd_yKGYpA

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