Why Are Rivers So Mathematical?
Rivers, like all “transport networks,” follow a mathematical structure that contributes to their fractal-like appearance.
河流，像所有的“运输网络”一样，遵循一种数学结构，这有助于其呈现出分形的外观。
Ada Zejun Shen/ Quanta Magazine
沉泽君/广达杂志
A river has my heart.
一条河有我的心。
It’s not the austere, black Thames winding through London, where I was born, but a lazy green one 5,000 miles away, where I spent my adolescence: the Blanco River in Texas.
这不是我出生的地方，蜿蜒穿过伦敦的简朴黑色泰晤士河，而是 5,000 英里外一条慵懒的绿色泰晤士河，我在那里度过了青春期：德克萨斯州的布兰科河。
My maternal ancestors have dipped into its waters for generations, as I have on countless summer days.
我的母系祖先世世代代都浸泡在这片水域中，就像我无数个夏日一样。
The Blanco is a tributary of the San Marcos, which flows into the Guadalupe, and on into the Gulf of Mexico.
布兰科河是圣马科斯河的一条支流，流入瓜达卢佩河，然后流入墨西哥湾。
You can probably picture how this looks on a map because all river networks look similar, creeping through the landscape, merging into ever wider and longer channels, downhill to the sea.
您或许可以在地图上想象出这一点，因为所有的河流网络看起来都很相似，蜿蜒穿过景观，汇入更宽更长的河道，下坡入海。
The pattern resembles twigs on branches that connect to trunks of trees (and the branching of their root systems, too), and it likewise resembles the veins of plant leaves, our own systems of blood vessels, and train and highway networks that feed into cities.
这种图案类似于连接到树干的树枝（以及根系的分支），也类似于植物叶子的静脉、我们自己的血管系统以及通往城市的火车和高速公路网络。
There’s something appealing about this ubiquitous pattern, so appealing to me personally that I have it tattooed on my forearm: the silhouette of a tree, with leafless branches reaching upward and roots burrowing downward, almost in mirror image.
这种无处不在的图案有一些吸引人的地方，对我个人来说非常有吸引力，以至于我把它纹在了我的前臂上：一棵树的轮廓，光秃秃的树枝向上延伸，树根向下挖洞，几乎是镜像。
“The shapes of rivers and leaf vasculature and so on — branching networks — you can just about grasp the pattern, but it’s still chaotic, so there’s something fascinating with that,” said Chris Paola , a river scientist at the University of Minnesota.
明尼苏达大学河流科学家克里斯·保拉 (Chris Paola) 表示：“河流和树叶脉管系统等的形状——分支网络——你几乎可以掌握其模式，但它仍然是混乱的，所以这有一些令人着迷的地方。”
Systems that branch in this way are “transport networks”: They transport some fluid substance (water, blood, traffic) from every place to a single place (the sea, a heart, a city center).
以这种方式分支的系统就是“运输网络”：它们将一些流体物质（水、血液、交通）从每个地方运输到一个地方（海洋、心脏、城市中心）。
Of the various examples, rivers are especially revealing, I think, since they arise from neither biological evolution nor urban planning, but rather chaotic Earth processes.
我认为，在各种例子中，河流尤其具有启发意义，因为它们既不是生物进化，也不是城市规划，而是混乱的地球过程。
Yet they obey simple, universal laws.
然而他们遵守简单、普遍的法则。
I n philosophy, “qualia” refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted.
在哲学中，“感受性”指的是我们体验的主观品质：爱丽丝看到蓝色或鲍勃感到高兴是什么感觉。
Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it.
正如已故哲学家丹尼尔·丹尼特所说，感受性是“事物在我们看来的方式”。
In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.
在这些文章中，我们的专栏作家追随他们的好奇心，探索重要但不一定可以回答的科学问题。
A discovery about river networks in 2026 reignited my curiosity about their universal form and mathematical nature.
2026 年关于河流网络的发现重新激发了我对其普遍形式和数学本质的好奇心。
These were hot topics in the 1980s and ’90s, when rivers were studied as natural examples of “fractals”: mathematical objects whose features repeat in roughly similar forms at many different scales.
这些是 20 世纪 80 年代和 90 年代的热门话题，当时河流被作为“分形”的自然例子来研究：数学对象的特征在许多不同的尺度上以大致相似的形式重复。
Geomorphologists, who specialize in the shape (and continual reshaping) of Earth’s surface, have studied the geometry of river networks far longer, since the late 1800s.
自 1800 年代末以来，专门研究地球表面形状（和不断重塑）的地貌学家对河流网络的几何形状进行了更长时间的研究。
Though many details of river behavior are still being actively studied, the existing mountain of research has yielded explanations that add up to a somewhat satisfying basic understanding.
尽管河流行为的许多细节仍在积极研究中，但现有的大量研究已经给出了一些解释，这些解释加起来形成了一些令人满意的基本理解。
The math is elegant, the geophysics is intuitive, and still my sense of wonder is undiminished.
数学是优雅的，地球物理学是直观的，但我的好奇心仍然没有减弱。
Every square inch of land on Earth’s surface receives precipitation, and much of it drains out, eventually, to an ocean or lake.
地球表面的每一平方英寸土地都会接收降水，其中大部分最终会流入海洋或湖泊。
Rivers are the drainage system.
河流是排水系统。
In 1957, a U.S. Geological Survey scientist named John Hack discovered the most important law of river networks.
1957年，美国地质调查局科学家约翰·哈克发现了最重要的河流网络规律。
In rivers and streams in Virginia and Maryland, Hack measured the length of each stream and the area of the land that slopes toward that stream and therefore drains into it, called its basin or drainage area.
在弗吉尼亚州和马里兰州的河流和溪流中，哈克测量了每条溪流的长度以及向该溪流倾斜并因此流入该溪流的土地面积，称为其流域或流域面积。
What he discovered is now known as Hack’s law: Any stream, from the littlest brook to the mightiest river, has a length that’s proportional to its drainage area raised to the power of 0.6.
他的发现现在被称为哈克定律：任何溪流，从最小的小溪到最大的河流，其长度与其流域面积的 0.6 次方成正比。
(In symbolic form: L ~ A 0.6 .) There’s a bit of variance around that 0.6 value — Earth is, after all, a complicated place — but “the general regularity of the relation is nevertheless remarkable,” Hack wrote .
（符号形式为：L ~ A 0.6。）0.6 的值存在一些差异——毕竟地球是一个复杂的地方——但“这种关系的总体规律性仍然是显着的，”哈克写道。
“Stream lengths tend to increase proportionally to the 0.6 power of the drainage area, regardless of the geological or structural characteristics of the area.”
“无论该地区的地质或结构特征如何，河流长度往往会与流域面积的 0.6 次方成比例增加。”
Dry streambeds in Yemen, as photographed from the International Space Station, show Hack’s law at work.
从国际空间站拍摄的也门干涸的河床显示了哈克定律的作用。
As more and better data has accrued, especially from satellite imagery, Hack’s law has held worldwide.
随着越来越多、更好的数据（尤其是来自卫星图像的数据）的积累，哈克定律在全世界范围内都适用。
Why this is the case is the essential mystery geomorphologists have grappled with ever since.
为什么会出现这种情况，一直是地貌学家一直在努力解决的基本谜团。
“Hack’s law is still the big question,” said Hansjörg Seybold , a geomorphologist at the Institute for Interdisciplinary Mountain Research at the Austrian Academy of Sciences.
“哈克定律仍然是一个大问题，”奥地利科学院跨学科山地研究所的地貌学家 Hansjörg Seybold 说。
It’s not so surprising that the bigger the land area of the basin, the longer the stream that drains it.
流域的陆地面积越大，排出流域的溪流就越长，这并不奇怪。
But in a purely mathematical sense, one might expect that stream length would follow a slightly different power law.
但从纯粹的数学意义上来说，人们可能会期望流长度会遵循稍微不同的幂律。
Imagine a square patch of land.
想象一块正方形的土地。
You might guess that regardless of slope or size, in idealized form, the land would drain into a stream that’s the length of one of its sides — a vertical line down the middle, for example.
您可能会猜测，无论坡度或大小如何，在理想的形式下，土地都会流入一条与其一侧长度相同的溪流——例如，沿着中间的一条垂直线。
That length is the square root of the area — or A to the power of 0.5.
该长度是面积的平方根，即 A 的 0.5 次方。
Under that circumstance, big river basins would have the same proportions as the small river basins that feed the tributaries within them.
在这种情况下，大河流域将与为其支流提供水源的小河流域具有相同的比例。
Their structure would be the same, regardless of size.
无论大小如何，它们的结构都是相同的。
But that’s not what Hack’s law reveals.
但这并不是哈克定律所揭示的内容。
Precipitation carves fractal-like patterns into the mining landscape through which the Rio Tinto in Spain flows.
降水在西班牙力拓流经的采矿景观中雕刻出类似分形的图案。
Addictive Stock Creatives/Alamy
上瘾的库存创意/Alamy
Instead, as a drainage areas get larger, the length of their streams increases faster.
相反，随着流域面积变大，河流的长度增加得更快。
“A nice way to phrase it would be that small basins are short and squat, and large basins are long and thin,” said Daniel Rothman , a geophysicist at the Massachusetts Institute of Technology.
麻省理工学院地球物理学家丹尼尔·罗斯曼（Daniel Rothman）说：“一个好的说法是，小盆地又短又矮，大盆地又长又薄。”
We unknowingly pick up on this pattern when we look at a network of tributaries on a map; a perfectly self-similar, fractal river network wouldn’t look quite right.
当我们观察地图上的支流网络时，我们会不知不觉地发现这种模式。一个完全自相似的分形河网看起来不太对劲。
Basins and streams become elongated at larger scales, so that river networks have an inherent directionality that stretches toward the sea.
盆地和溪流在更大的尺度上被拉长，因此河流网络具有向大海延伸的固有方向性。
One result of that elongation is that neighboring river networks must lie closer together than they would with a 0.5 power law.
这种伸长的结果之一是相邻河流网络的距离必须比 0.5 幂律下的距离更近。
But where does this pattern come from?
但这种模式从何而来？
There are at least two complementary explanations underlying Hack’s law.
哈克定律至少有两种互补的解释。
One theory, proposed in seminal papers by the Italian hydrologist Andrea Rinaldo and colleagues in the early 1990s, says that river networks are elongated at large scales because this is the most energy-efficient way of transporting water downhill.
意大利水文学家 Andrea Rinaldo 及其同事在 20 世纪 90 年代初在开创性论文中提出了一种理论，该理论认为河流网络大规模拉长，因为这是向山下输送水的最节能的方式。
Imagine designing a drainage system for that square field I mentioned earlier.
想象一下为我之前提到的那个方形场地设计一个排水系统。
Rain falls uniformly on the field, and you need to ferry all the water out through a single point.
雨水均匀地落在场地上，你需要从一个点将所有的水都排出去。
You might try digging many direct channels, leading from every part of the field straight to the outlet.
您可以尝试挖掘许多直接通道，从场地的每个部分直接通向出口。
The problem, though, is that this would require an extremely long channel that must be dug (or, if you are trickling water, eroded), an inefficient process.
但问题是，这需要挖掘一条非常长的通道（或者，如果你滴水，则需要侵蚀），这是一个低效的过程。
The order gives it a feeling of something more than just water and sand.
这种秩序给人一种不仅仅是水和沙子的感觉。
Chris Paola, University of Minnesota
克里斯·保拉，明尼苏达大学
A better approach is a treelike system of channels with tiny streams merging into larger tributaries and, eventually, one main trunk leading to the outlet.
更好的方法是建立一个树状的河道系统，其中细小的溪流汇入较大的支流，最终形成一条通向出水口的主干。
Individual raindrops may travel farther than they did before, but the network as a whole dissipates less energy (through friction) to drain the field, because much more of the infrastructure is shared.
单个雨滴可能比以前传播得更远，但整个网络消耗的能量（通过摩擦）更少，因为更多的基础设施是共享的。
When large streams are elongated, more of the shorter tributaries can share those major channels, and this reduces the overall transport cost.
当大河流被拉长时，更多较短的支流可以共享这些主要河道，这降低了总体运输成本。
The pioneers of this “optimal channel network” theory simulated a bunch of river networks on a computer and showed that those that abide by Hack’s law, with an exponent of 0.6, dissipate the least energy.
这种“最佳渠道网络”理论的先驱者在计算机上模拟了一堆河流网络，结果表明，遵守指数为 0.6 的哈克定律的河流网络消耗的能量最少。
This scaling law is the most efficient and least resistant way to transport all that rain to the sea.
这种比例定律是将所有雨水输送到海洋的最有效且阻力最小的方式。
Implicit in this argument, though, is the assumption that rivers naturally reconfigure themselves to minimize total energy expenditure.
然而，这个论点隐含着这样一个假设：河流会自然地重新配置自己，以最大限度地减少总能量消耗。
But how?
但如何呢？
That’s where landscape evolution models come in.
这就是景观演化模型的用武之地。
Rivers do shift their layouts all the time.
河流确实一直在改变它们的布局。
In the 1990s, in parallel with the work on optimal channel networks, geomorphologists developed powerful landscape evolution models to capture this constant adjustment and show the mechanism by which Hack’s law etches itself on the landscape.
20 世纪 90 年代，在研究最佳河道网络的同时，地貌学家开发了强大的景观演化模型来捕捉这种不断的调整，并展示哈克定律在景观中蚀刻自身的机制。
These computer simulations start with water flowing downhill, eroding rock as it goes.
这些计算机模拟从水流下坡开始，沿途侵蚀岩石。
Tiny, random irregularities in the topography cause some channels to capture more runoff than others.
地形上微小的、随机的不规则性导致一些河道比其他河道捕获更多的径流。
Those channels in turn erode faster and deepen, which causes them to attract still more water.
这些渠道反过来侵蚀得更快、更深，导致它们吸引更多的水。
One streambed might grow toward its neighbor, and thereby intercept some of its runoff.
一条河床可能会向邻近的河床延伸，从而拦截部分径流。
The victorious stream grows longer and carries more water, while the losing stream shrinks or disappears.
获胜的河流会变长并携带更多的水，而失败的河流会缩小或消失。
These sorts of local adjustments like these route water along ever more efficient paths.
这些局部调整，例如沿着更有效的路径浇水。
As the entire drainage network gradually reorganizes over thousands of years or more, it attains and then continues to tweak a configuration that transports water downhill with minimal energy dissipation.
随着整个排水网络在数千年或更长时间内逐渐重组，它实现并继续调整一种以最小的能量耗散将水输送到山下的配置。
The Yukon Delta in Alaska formed where the Yukon River flows into the Bering Sea.
阿拉斯加的育空三角洲是育空河流入白令海的地方形成的。
Recent research found that deltas also follow Hack’s law.
最近的研究发现，三角洲也遵循哈克定律。
Contains modified Copernicus Sentinel data (2017), processed by ESA
包含修改后的哥白尼哨兵数据 (2017)，由 ESA 处理
Gravity and friction are the driving forces of this process.
重力和摩擦力是这个过程的驱动力。
Gravity supplies potential energy to flowing water.
重力为流动的水提供势能。
Friction, the cause of erosion, dissipates that energy.
摩擦是腐蚀的原因，会消耗能量。
A channel configuration that wastes energy by forcing water along inefficient routes tends to erode rapidly and change.
迫使水沿着低效路线流动而浪费能源的渠道配置往往会迅速侵蚀和改变。
A configuration that routes water more effectively is stabler and therefore more persistent.
更有效地输送水的配置更稳定，因此更持久。
The network becomes optimal through this dynamic evolution, eventually arriving at a form that adheres to Hack’s law.
网络通过这种动态演化变得最优，最终达到遵守哈克定律的形式。
That explanation of river network geometry hangs together for me, though geomorphologists still have many questions.
尽管地貌学家仍然有很多问题，但对河网几何学的解释对我来说仍然是一致的。
Some study rivers that deviate from Hack’s law.
有些研究偏离哈克定律的河流。
Others organize transport networks that follow Hack’s law into one class of optimal transport networks, among a whole family of them.
其他人则将遵循哈克定律的传输网络组织成一类最佳传输网络，并属于整个网络家族。
Trees, which branch in three dimensions instead of two, would be in a different class from rivers and follow different optimal scaling laws, for instance.
例如，树木在三个维度而不是两个维度上分支，与河流属于不同的类别，并且遵循不同的最佳缩放法则。
Now, geomorphologists have a new finding to explain.
现在，地貌学家有一个新发现需要解释。
In April 2026, Tian Dong of the University of Texas, Rio Grande Valley and co-authors made the cover of Science for discovering that Hack’s law holds not only for rivers’ tributary networks, but also for their deltas, the fanlike structures that form where a river meets the sea.
2026 年 4 月，德克萨斯大学里奥格兰德河谷分校的田冬和合著者登上了《科学》杂志的封面，因为他们发现哈克定律不仅适用于河流的支流网络，也适用于河流的三角洲，即河流与大海交汇处形成的扇形结构。
Rivers essentially hit a brick wall when they reach the (nonflowing) ocean.
当河流到达（不流动的）海洋时，河流基本上会撞上砖墙。
The sudden deceleration of the water causes it to drop the sediments it carries.
水流的突然减速导致其携带的沉积物掉落。
These pile up to form new land.
这些堆积起来形成新的土地。
In the process, the river’s water splits into a different kind of network of channels, which shift locations constantly as sediments build up and wash away.
在此过程中，河水分裂成不同类型的河道网络，随着沉积物的堆积和冲走，河道的位置不断变化。
Scientists told me that they’ve long wondered about the organization of channels in river deltas, but they are hard to study.
科学家告诉我，他们长期以来一直想知道河流三角洲河道的组织结构，但它们很难研究。
Unlike the upstream river network, where slope and elevation differences make it easy to calculate the area of land that drains into any given tributary, deltas are flat and especially dynamic.
与上游河网不同的是，上游河网的坡度和高程差异可以轻松计算流入任何给定支流的土地面积，三角洲是平坦的，而且特别动态。
But through a sophisticated analysis of satellite data that allowed them to distinguish land from water, Dong and his collaborators determined that the length of a channel in a river delta scales with the size of its nourishment area — the area that it supplies with sediments — raised to the power of 0.6.
但通过对卫星数据的复杂分析，董和他的合作者能够区分陆地和水域，确定河流三角洲河道的长度与其滋养区域（提供沉积物的区域）的大小成正比，增加到 0.6 次方。
Rivers’ tributary networks and distributary networks are opposites — sediments are transported away from one end and deposited at the other — yet they abide by the same math.
河流的支流网络和分流网络是相反的——沉积物从一端运走并沉积在另一端——但它们遵循相同的数学原理。
Geomorphologists are now considering why Hack’s law should apply in this inverse context.
地貌学家现在正在考虑为什么哈克定律应该适用于这种相反的情况。
Reflecting on my own question, I think it’s the coexistence of simplicity and determinism with chaos and randomness that makes the optimal structure of rivers so captivating.
反思我自己的问题，我认为正是简单性和决定论与混乱和随机性的共存使得河流的最佳结构如此迷人。
Natural efficiency is, perhaps, innately appealing to us.
自然效率也许天生就对我们有吸引力。
Paola agreed.
保拉同意了。
“There’s something about an artery of water moving through the landscape in this great big channel that we can’t stay away from,” he said.
他说：“在这条巨大的河道中，有一条水流在景观中流动，我们无法远离它。”
“It has so much structure.
“它有这么多的结构。
…
……
The chaos is sort of what you expect from nature.
这种混乱是你所期望的自然现象。
You could think, water runs around on a landscape and makes puddles and stuff.
你可能会想，水在景观中流动并形成水坑之类的东西。
And the order gives it a feeling of something more than just water and sand.
这种秩序给人一种不仅仅是水和沙子的感觉。
I’m not saying that it actually is anything more than that.
我并不是说它实际上不止于此。
I don’t mean to be metaphysical.
我并不是想成为形而上学的人。
I think it strikes us that way and draws us in.”
我认为它以这种方式打动我们并吸引我们。”
N etwork shape isn’t the only attribute of rivers that generates geometric regularity.
网络形状并不是河流产生几何规律性的唯一属性。
In work that began back in the early 2010s, Daniel Rothman from MIT, Hansjörg Seybold, now at the Austrian Academy of Sciences, and their collaborators modeled stream evolution as a physical process called Laplacian growth, which posits, simply put, that “growth begets growth.” Through this they predicted and subsequently observed that wherever a stream forms two branches, the bifurcation angle between them will be around 72 degrees.
在 2010 年代初开始的工作中，麻省理工学院的 Daniel Rothman、现供职于奥地利科学院的 Hansjörg Seybold 及其合作者将流演化建模为一种称为拉普拉斯增长的物理过程，简单地说，该过程假设“增长引发增长”。通过这一预测，他们随后观察到，只要一条溪流形成两个分支，它们之间的分叉角就会在 72 度左右。
If the split were any wider, an influx of water from the land between the two branches would cause erosion that would make the channels migrate toward one another.
如果裂缝再宽一些，水从两个分支之间的土地涌入就会造成侵蚀，从而使河道向彼此迁移。
If it were narrower, excess water would seep in from their outer banks and draw them apart.
如果它更窄，多余的水就会从外岸渗入并将它们分开。
Seventy-two degrees is one-fifth of the way around a circle, so this splitting angle appears to give river networks a kind of fivefold symmetry.
七十二度是绕圆一周的五分之一，因此这种分裂角度似乎赋予河流网络一种五重对称性。
It’s impossible for regular, repeating patterns to have fivefold symmetry: Imagine trying to tile a space with regular pentagons — you can’t.
规则的重复图案不可能具有五重对称性：想象一下尝试用规则的五边形平铺一个空间 - 你做不到。
Interestingly, though, pentagonal arrangements of atoms can fill space in almost-but-never-quite-repeating patterns with fivefold symmetry, forming materials called quasicrystals .
但有趣的是，原子的五边形排列可以以几乎但绝不完全重复的五重对称图案填充空间，形成称为准晶体的材料。
These exotic forms of metal alloys are not as regular as crystals, but not amorphous like glass, which gives them unusual properties.
这些奇特形式的金属合金不像晶体那样规则，但也不像玻璃那样无定形，这赋予了它们不寻常的特性。
“There has been a lot of speculation about fivefold symmetry in a branching network,” such as a river, Seybold said, and how it might be related to the geometric and physical strangeness of quasicrystals.
塞博尔德说，“对于河流等分支网络中的五重对称性，人们有很多猜测，以及它与准晶体的几何和物理奇异性之间的关系。”
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