内容来源：https://www.quantamagazine.org/shark-data-suggests-animals-scale-like-geometric-objects-20251027/

内容总结：

近日，一项发表于《皇家学会开放科学》的研究通过计算机断层扫描和数字建模技术，对54种鲨鱼的体表面积与体积关系展开系统性分析。研究结果显示，尽管鲨鱼在体型、生态位和代谢方式上存在显著差异，其体表面积与体重的缩放比例严格遵循三分之二定律，为生物学中普遍存在的几何缩放规律提供了有力证据。

长期以来，生物学家推测生物体可能遵循与几何物体相似的缩放规律：当体积增长时，表面积以体积的2/3次幂速率增加。但由于测量技术限制，早期对大型动物的研究往往存在误差且涉及伦理争议。本研究团队创新性地采用博物馆标本CT扫描及摄影测量法，构建了从9英寸侏儒鲨到37英尺鲸鲨的三维模型，首次实现无损伤精准测量。

研究涵盖底栖栖息型、远洋捕食型等不同生态类型的鲨鱼，包括头部特化的双髻鲨、尾鳍占体长一半的长尾鲨以及皮肤褶皱复杂的须鲨。令人惊讶的是，尽管这些鲨鱼形态各异，甚至存在部分温血物种（如长尾鲨），所有数据点均紧密贴合三分之二缩放曲线。

科学家推测这种规律可能源于两大进化约束机制：一是胚胎发育过程中的组织分配限制，如同黏土塑形时存在的物理约束；二是热交换效率的优化需求，这也能解释为何北极物种往往具有更大体积以维持体温。尽管研究未计入鲨鱼鳃部等内部结构的表面积，其高度一致的数据模式仍暗示这是适应性进化的结果。

该发现不仅揭示了生命形态背后的数学规律，对兽医学用药剂量计算、临床医学不同体型患者给药方案制定也具有参考价值。研究团队呼吁继续开展跨物种验证，以探索这一几何法则在羽毛、毛发等复杂生物结构中的普适性。

中文翻译：

鲨鱼研究揭示动物体型遵循几何缩放规律

基本几何原理表明，当三维物体（无论是柏拉图球体、细胞还是大象）向各方向均匀生长时，其表面积的增长速度始终慢于体积扩张。只要物体在放大过程中保持几何形状不变，表面积的增长速率约等于体积的2/3次方。几个世纪以来，生物学家始终在探索：形态各异的生命体是否也遵循这一三分之二缩放定律？若成立，则意味着进化过程存在根本性约束机制，这种机制可能影响着生命体与外界环境的互动方式。

近期研究团队通过CT扫描和数字技术，对鲨鱼这一古老而多样的动物类群进行了表面积与体积测算。发表于《皇家学会开放科学》的研究分析了50余种鲨鱼，为动物学尺度规律提供了迄今最坚实的实证依据。研究证实，鲨鱼的体表与体重关系恰如球体般遵循三分之二缩放定律。若该规律普适于其他动物类群，可能反映出制约生物进化的热交换、新陈代谢或发育机制的内在法则。

本研究负责人、澳大利亚詹姆斯·库克大学的鲨鱼生物学家乔尔·盖福德指出，鲨鱼是研究生物尺度规律的理想模型：它们保持着基本形态框架，却呈现出巨大的体型差异、生态位分化和形体多样性。在鲨鱼形态进化研究中，盖福德注意到鳍肢等部位存在明显的尺度关联，这促使他思考鲨鱼形态是否受更深层次的规律制约。

然而针对大型动物的尺度规律研究始终缺乏高质量数据。单细胞生物研究已发现诸多偏离理论值的案例；针对昆虫与蛇类等小型动物的零星研究虽支持三分之二定律，但大型动物研究不仅数量稀少，且多来自数十年前。盖福德指出，受限于19-20世纪的测量技术，当时获取动物体表与体积的尝试"既容易产生误差，又存在伦理争议"。

亚利桑那大学进化生物学家布莱恩·恩奎斯特（未参与本研究）对此深有同感："早期生物学研究最大的瓶颈在于——比如，如何准确测量一头牛的体表面积？"

传统测量方法确实捉襟见肘：研究者或使用计量轮在动物表皮滚动标记，或通过剥制标本手动测量；体积测算则需将动物浸入水槽观察排水量，更甚者直接向新鲜剥取的兽皮内注水测量。

盖福德团队采用了尖端技术：他们测量了54种鲨鱼的尺度数据，从体长23厘米的侏儒鲨（全球最小鲨鱼之一）到现存最大的鱼类鲸鲨。团队并未解剖标本，而是对博物馆精品样本进行CT扫描构建三维模型。对于超出CT机容限的物种，则采用摄影测量法——通过多角度照片合成三维模型（例如佐治亚水族馆那条11米长的鲸鲨）。随后将模型导入原为电子游戏开发的三维处理软件Blender，轻点按钮即可获取精确表面积。

该数据集不仅涵盖从礁栖到底栖、从远洋捕食者到滤食者的各生态位鲨鱼，更收录了丰富形态：包括双髻鲨系列的梭形头冠、尾鳍与体长相当的长尾鲨、如地毯般扁平的须鲨，以及标准流线型鲨鱼。尽管多数鲨鱼为冷血动物，团队仍纳入了包括长尾鲨在内的区域性温血物种。

尽管存在体型、形态、习性及代谢方式的巨大差异，所有鲨鱼数据均完美契合三分之二缩放定律。恩奎斯特评价道："他们证明了该规律的高度稳定性，这非常了不起。"

研究暗示该尺度定律可能具有普适性，但还需在陆生动物（具有羽毛等复杂几何外表）及恒温动物（如哺乳类与鸟类）中进一步验证。盖福德团队正在扩展数据收集，期待同行在更多生物类群中检验该规律。

康奈尔大学生物力学荣休教授卡尔·尼克拉斯坦言，当前表面积测量未包含鳃部结构仍存缺憾。从拓扑学视角看，鳃表虽藏于体内实则属于外表面。他推测若计入鳃部面积，缩放系数可能接近四分之三。但不同鲨种数据的高度一致性表明该规律非偶然现象，尼克拉斯坦言："这必然反映了适应性进化的某种规律。"

虽然具体机制尚未明晰，科学家提出若干假说：其一涉及胚胎发育期的组织分配机制。盖福德用黏土模型类比："在不过度消耗能量的前提下，拉伸黏土塑造不同形态的方式本就有限。"这意味着尺度关系在胚胎期就已确立，进而制约成年个体的形态发展。

另一假说聚焦热交换约束机制。加州大学洛杉矶分校计算生物学家范·萨维奇（未参与研究）解释："体表体积比对热交换至关重要。"当表面积增长慢于体积时，体型越大保温效果越强——这解释了为何北极物种多魁梧厚重，热带物种则纤细灵动。即便对变温动物而言，当鲸鲨在冷暖水域间巡游时，较大体型也有助于维持体温。

这项研究不仅揭示了进化的数学边界，更有实际应用价值：兽医可据此计算从家猫到大丹犬的麻醉剂量，医生能参照尺度规律制定婴幼儿与成人的用药方案。

对盖福德而言，研究凸显了持续实证检验的重要性："人们必须实际验证这些定律，因为它们往往被想当然地视为真理。"

英文来源：

Shark Data Suggests Animals Scale Like Geometric Objects
Introduction
It’s a universal fact that as any 3D object, from a Platonic sphere to a cell to an elephant, grows outward in all directions, its total surface area will increase more slowly than the space it occupies (its volume). If the object’s geometry and shape remain the same as it gets bigger, then its surface area will increase roughly as fast as its volume to the two-thirds power. For centuries, biologists have wondered if life forms, too, follow this two-thirds scaling law, even though they come in a stunning variety of shapes and sizes. If so, it would suggest that there are underlying constraints fundamental to evolution that might influence how life interacts with the world around it.
Recently, researchers used CT scans and digital tools to calculate the surface areas and volumes of an ancient and diverse animal lineage: sharks. The team’s analysis, published in Royal Society Open Science, included more than 50 shark species and provides some of the best empirical evidence to date for some kind of firm scaling rule in zoology. As with a sphere, the surface area and body mass of sharks do indeed follow a two-thirds scaling law, the team found. If this holds true in other animal groups, it probably reflects underlying rules of heat exchange, metabolism or development that constrain evolution.
If you’re looking for an animal group in which to study biological scaling, it’s hard to do better than sharks, according to Joel Gayford, a shark biologist at James Cook University in Australia who led the new study. They share an overall form, but come in many sizes, occupy a plethora of niches, and have huge variations in body shape. In his research into sharks’ morphological evolution, Gayford noticed what appeared to be scaling relationships between their body parts, such as the sizes of their fins. It made him wonder whether there might be more fundamental rules constraining the forms that sharks can take.
However, he could find little high-quality research into scaling in large animals. Research in single cells has uncovered many deviations from expected rules; rare studies in smaller animals such as insects and snakes have found some evidence for two-thirds scaling. But few studies included larger animals, and most of those were conducted decades ago. Plus, Gayford found that existing animal scaling data was somewhat messy. Due to technological limitations in the 19th and 20th centuries, attempts to accurately measure animals’ surface area and volume were “error-prone and also kind of ethically questionable,” he said.
He wasn’t the only one who thought so. “One of the big limitations — especially if you read these early biology studies — is, like, how do you measure the surface area of a cow?” said Brian Enquist, an evolutionary biologist at the University of Arizona who was not involved in the study.
Until recently, options were limited. Researchers could run a measuring wheel across an animal’s hide and mark out units in chalk, or skin the creature and measure its surface area by hand. To calculate its volume, they could drop the animal into a water-filled tub and see how much liquid it displaced; some went one step further and poured water directly into freshly liberated pelts.
Gayford’s team had significantly more advanced technology available. They measured the surface area and volume of 54 different shark species, ranging from a nine-inch pygmy shark, one of the world’s smallest, to a whale shark, the largest living fish. But rather than skinning them, they took CT scans of high-quality museum specimens to create detailed virtual reconstructions. For those species too large to fit in a CT scanner, they used photogrammetry software, which stitches together many photos of an object’s surface to create a 3D model. (In one case, the object in question was a 37-foot-long whale shark that lives at the Georgia Aquarium.) They then loaded the models into 3D processing software called Blender, which was originally developed for rendering objects in video games. To calculate a shark’s surface area, Gayford just had to click a button.
In addition to reflecting a huge range of animal sizes, the dataset also represented sharks that fill a variety of ecological niches, from reef dwellers to bottom feeders to open-ocean predators. They came in “a ton of different unique morphologies,” Gayford said, including several oblong-faced hammerhead species; the common thresher, whose tail fin is nearly as long as the rest of its body; and the flat and frilly wobbegong, along with your more standard shark-shaped sharks. And while most sharks are cold-blooded, or ectothermic, a few species (including great whites) can generate their own heat. Gayford’s team included one of these regionally endothermic sharks — the thresher — in the dataset.
Despite this diversity of size, shape, lifestyle and metabolism, the sharks fit the two-thirds scaling rule almost perfectly. “They showed there’s not a lot of variability in it, so that’s really cool,” Enquist said.
The analysis suggests that this two-thirds scaling rule could be universal for animals. To be certain, more research is needed in other animal groups — including terrestrial animals, which can have complex external geometries such as feathers and hair, and warm-blooded, or endothermic, animals such as mammals and birds. To that end, Gayford’s team is collecting more data; he hopes other researchers will further test biological scaling in the animals they study.
However, the surface area measurements could still be considered incomplete because they only include the sharks’ external features. Even though structures such as gills are tucked away inside animals’ bodies, their surfaces are actually external from a topological perspective, said Karl Niklas, an emeritus professor of biomechanics at Cornell University. If the researchers had analyzed the sharks’ gills as well, Niklas speculated that they would have found a scaling ratio closer to three-fourths. Still, the consistency of the numbers across many different shark species suggests the rule is not incidental. “We’ve got to think of this as some kind of reflection of adaptive evolution,” Niklas said.
The scientists aren’t certain what fundamental mechanisms could be limiting sharks and other animals’ sizes and shapes, but they have hypotheses. One involves tissue allocation during early growth. To visualize this, imagine a developing animal as a ball of clay. “There’s only so many ways you can stretch out the clay to make different shapes without incurring energetic costs,” Gayford said. In that case, the scaling relationship is significant to the embryo, and it goes on to limit the potential forms the adult organism can take.
Alternatively, the ratio could reflect a fundamental constraint on heat exchange. In animals, which can absorb external heat and generate it through metabolism or movement, a scaling principle in which surface area grows more slowly than volume would create an insulating effect as you move from small species to larger ones. “Surface area to volume is really important for heat exchange,” said Van Savage, a computational biologist at the University of California, Los Angeles who was not involved in the study. This may explain why Arctic species tend to be large and bulky, while those living in tropical climates can be svelte: A bigger body is harder to cool down than a smaller one. This applies even to ectothermic animals, which may need to hold on to heat as they move between warm and cool environments — for example, when whale sharks dive into deeper, colder water.
The study offers insight into the mathematical limits of evolution, and could help identify fundamental mechanisms that limit life’s topology. But calculating how organisms scale has practical value as well. It can help veterinarians figure out how much anesthesia to administer to a cat versus a Great Dane, for example, or help doctors determine drug dosages for infants versus adults.
For Gayford, it highlights the need for continued empirical studies of biological scaling. “It’s really important that people actually test these laws,” he said. “Because often they’re just kind of assumed to be correct.”