内容来源：https://www.quantamagazine.org/first-shape-found-that-cant-pass-through-itself-20251024/

内容总结：

近日，数学界一项突破性研究推翻了延续数百年的猜想：奥地利数学家雅各布·斯坦宁格与谢尔盖·尤尔克维奇首次发现了一种无法让自身复制体穿过的三维立体结构，命名为“诺珀立体”（Noperthedron）。这一成果颠覆了学界对凸多面体“鲁珀特性”的普遍认知。

该研究可追溯至17世纪英国数学家约翰·沃利斯记载的“鲁珀特立方体”问题：若沿立方体空间对角线钻孔，其通道恰好能容纳另一个同等尺寸立方体通过。此后三百年间，数学家陆续发现四面体、八面体等常见凸多面体均具有此特性，逐渐形成“所有凸多面体都具备鲁珀特性”的共识。

斯坦宁格与尤尔克维奇通过理论创新与大规模计算，构建出由150个三角形和2个正十五边形组成的90顶点多面体。研究证明，无论以何种角度在该立体上开凿直线通道，均无法使另一个相同立体通过。其关键突破在于建立了“全局定理”与“局部定理”的双重验证体系，将1800万种空间朝向逐一排除。

谷歌软件工程师汤姆·墨菲评价称，这项研究为几何学开辟了新方向。尽管此前已有学者对某些复杂多面体（如62面体的菱形二十面体）是否具备鲁珀特性存疑，但始终缺乏严格证明。诺珀立体的出现证实了非鲁珀立体确实存在，终结了持续三个世纪的数学探索。

目前，两位研究者正寻求将新方法应用于其他几何体研究。正如斯坦宁格所言：“作为纯粹数学爱好者，我们将继续探索这些迷人的问题。”这项发表于八月的成果，不仅解决了古典几何难题，更展现了跨世纪数学思考的永恒魅力。

中文翻译：

首个无法穿过自身的几何形状被发现

引言
想象你手握两枚尺寸相同的骰子。是否能在其中一枚骰子上钻出足够大的通道，让另一枚骰子顺利穿过？

你的直觉或许会断然否定。有此想法者并非个例。17世纪末，就曾有匿名人士与莱茵河王子鲁珀特就此设下赌局。这位在英国内战中统领保皇党军队的查理一世外甥，晚年隐居于温莎城堡的实验室，潜心研究冶金与玻璃制造工艺。

鲁珀特最终赢得了赌局。数学家约翰·沃利斯在1693年记述此事时，未言明鲁珀特是给出了数学证明，还是直接在立方体上钻出了孔洞。但沃利斯本人通过数学推导证实：若沿立方体内部对角线方向钻取笔直通道，其宽度足以容纳另一相同立方体通过。这种穿行堪称极限操作——只要将穿行立方体尺寸增大4%，便无法通过。

人们自然好奇还有哪些形状具备此特性。"我认为这是个非常经典的问题，"谷歌软件工程师汤姆·墨菲在业余时间对此进行了深入探索，"这个问题注定会被反复发现——就连外星文明也逃不过这个定律。"

马克·贝兰/《量子》杂志

几何形状的多样性令人难以全面掌握，因此数学家通常聚焦于凸多面体：如立方体般具有平整表面、无凸起或凹陷的几何体。当某形状在不同方向上尺寸差异显著时，通常易找到笔直通道容纳另一相同形状通过。但许多著名凸多面体——如十二面体、截角二十面体（足球形状）——具有高度对称性，难以分析。奥地利联邦统计机构的数学家雅各布·斯坦宁格指出："数百年来，我们仅知立方体具备此特性。"

1968年，克里斯托夫·斯克里巴证明四面体与八面体也具有"鲁珀特性"（数学家现今的称谓）。过去十年间，专业数学家与爱好者们相继在众多经典凸多面体中发现了鲁珀特通道，包括十二面体、二十面体及足球形状。

鲁珀特性看似如此普遍，以致数学家提出通用猜想：所有凸多面体都具备鲁珀特性。始终无人能找出反例——直至今日。

在八月发布的论文中，斯坦宁格与奥地利交通系统公司研究员谢尔盖·尤尔凯维奇描述了一个具有90个顶点、152个面的形状，并将其命名为"诺珀特体"（词源来自墨菲将"鲁珀特"与"否定词"结合的造词）。他们证明无论以何种角度在诺珀特体上钻取直通道，均无法使第二个诺珀特体通过。

这项证明需要理论突破与大规模计算相结合，并依赖于诺珀特体顶点的一种精妙特性。"其成功堪称奇迹，"斯坦宁格感叹道。

穿越投影
要理解立方体互穿原理，可想象将立方体悬于桌面上方观察其投影（假设光源来自上方）。标准放置时投影为正方形，但若将一角垂直向上，投影则呈正六边形。

1693年沃利斯发现：正方形投影能内接于六边形投影并留有微边隙。这意味着当立方体角朝上时，可钻取垂直通道让第二立方体通过。约一世纪后，皮特·尼乌兰发现特定朝向能产生更佳投影——可容纳比通道立方体大6%的立方体。

马克·贝兰/《量子》杂志

后续对复杂形状的分析均沿用此法：旋转物体寻找可相互嵌套的投影。借助计算机，数学家已在众多形状中发现鲁珀特通道。某些穿行条件极为苛刻——例如"三角化四面体"的通道边隙仅约其半径长度的0.000002倍。"计算与离散几何的结合使这类测算成为可能，"史密斯学院荣休教授约瑟夫·奥鲁尔克表示。

编写鲁珀特通道算法的研究者注意到奇特现象：对任意凸多面体，算法要么瞬间找到通道，要么彻底无果。过去五年间，数学家已收集到少量始终未发现通道的顽固形状。

"我让电脑连续运行两周尝试破解菱形二十面体，"约翰斯·霍普金斯大学应用数学家本杰明·格里默提及由62个正三角形、正方形和五边形构成的立体，"它似乎抗拒所有尝试。"

但这种抗拒不足以证明形状属于诺珀特体。物体朝向有无限种可能，计算机仅能检验有限样本。研究者无法判定这些顽固形状是真正的诺珀特体，还是其鲁珀特通道极难发现。

可确定的是，候选诺珀特体极为罕见。自去年起，墨菲开始构建数亿种形状，包括随机多面体、球面顶点多面体、特殊对称多面体，以及通过移动顶点故意破坏原有通道的多面体。他的算法几乎为所有形状都找到了鲁珀特通道。

快速成功与诺珀特候选体的顽固形成鲜明对比，令数学家怀疑真实诺珀特体确实存在。但在八月之前，这仅是猜测。

无路可通
现年30岁的斯坦宁格与29岁的尤尔凯维奇自青少年时期参加数学奥林匹克竞赛便是挚友。尽管两人后来均离开学术界（尤尔凯维奇取得博士学位，斯坦宁格获得硕士学位），他们仍持续共同探索未解难题。

"三小时前我们吃披萨时，几乎全程都在讨论数学，"斯坦宁格告诉《量子》，"这就是我们的日常。"

五年前，两人偶然看到立方体互穿的视频，立刻被深深吸引。他们开发了寻找鲁珀特通道的算法，很快确信某些形状属于诺珀特体。在2021年的论文中，他们猜想菱形二十面体不具备鲁珀特性。"我认为这是首次推测存在不具备该特性的立体，"斯坦宁格表示，这项研究早于墨菲和格里默近期的探索。

要证明某形状是诺珀特体，必须排除两个物体所有可能朝向的鲁珀特通道。每个朝向可表述为旋转角度的集合，这些角度在更高维度的"参数空间"中构成一个点。

假设选择某个朝向，计算机显示第二投影超出第一投影边界，这便排除了参数空间中的一个点。

但实际能排除的范围远不止单点。若第二投影显著超出，需要大幅调整才能将其移入第一投影。换言之，不仅能排除初始朝向，还能排除"邻近"朝向——即参数空间中的整块区域。斯坦宁格与尤尔凯维奇提出"全局定理"，精确定量此种情况下可排除的区域范围。通过测试多个点，有望逐步排除参数空间中的连续区域。

若这些区域覆盖整个参数空间，即可证明该形状为诺珀特体。但每个区域的大小取决于第二投影超出第一投影的程度，有时超出幅度很小。例如当两形状完全同向时，轻微旋转第二形状仅会使投影略微超出，此时全局定理仅能排除微小区域。这些区域过小无法覆盖整个参数空间，可能遗漏某些对应鲁珀特通道的点。

针对微小旋转，两人提出与全局定理互补的"局部定理"。该定理适用于原始投影边界上存在三个满足特殊要求的顶点（即角点）的情况。例如用这三个顶点构成三角形时必须包含投影中心。研究者证明若满足这些条件，任何微小旋转都会使投影至少将一个顶点推向外侧。因此新投影不可能完全内接于原投影，即无法形成鲁珀特通道。

若形状投影缺乏三个合适顶点，局部定理便不适用。所有已识别的诺珀特候选体都至少存在一个此类投影。斯坦宁格与尤尔凯维奇筛选了包含数百种最对称、最完美凸多面体的数据库，未找到任何完全符合条件的形状，于是决定自行构建合适形状。

他们开发出构建形状并检验三顶点特性的算法，最终生成由150个三角形和两个正十五边形组成的诺珀特体。其外形如丰腴的水晶花瓶，具有宽大的底座与顶部，已有爱好者3D打印成品用作笔筒。

彼得·莱利

研究者将朝向参数空间划分为约1800万个微型区域，检验每个区域中心点对应的朝向是否产生鲁珀特通道，结果全为否定。随后他们证明每个区域都满足局部或全局定理条件，从而排除了整个参数空间。这意味着诺珀特体确实不存在任何鲁珀特通道。

"自然猜想已被证伪，"奥鲁尔克评论道。

数学家能否运用新方法发现更多诺珀特体，或找到能处理菱形二十面体等候选形状的新局部定理，仍有待观察。但既然诺珀特体确实存在，"我们便有了研究其他形状的坚实基础，"墨菲表示。

与斯坦宁格团队同样出于兴趣探索此问题的墨菲，感受到跨越时空的共鸣："鲁珀特王子选择在城堡里用退休时光钻研数理——这种生活令我神往。"

此刻，斯坦宁格与尤尔凯维奇正在寻找新的攻克目标。"我们只是谦卑的数学爱好者——痴迷于解决此类问题，"斯坦宁格说，"我们将继续前行。"

英文来源：

First Shape Found That Can’t Pass Through Itself
Introduction
Imagine you’re holding two equal-size dice. Is it possible to bore a tunnel through one die that’s big enough for the other to slide through?
Perhaps your instinct is to say “Surely not!” If so, you’re not alone. In the late 1600s, an unidentified person placed a bet to that effect with Prince Rupert of the Rhine. Rupert — a nephew of Charles I of England who commanded the Royalist forces in the English Civil War — spent his sunset years studying metallurgy and glassmaking in his laboratory at Windsor Castle.
Rupert won the bet. The mathematician John Wallis, recounting the story in 1693, didn’t say whether Rupert wrote a proof or bored a hole through an actual cube. But Wallis himself proved mathematically that, if you drill a straight tunnel in the direction of one of the cube’s inner diagonals, it can be made wide enough to allow another cube through. It’s a tight squeeze: If you make the second cube just 4% larger, it will no longer fit.
It’s natural to wonder which other shapes have this property. “I think of this problem as being quite canonical,” said Tom Murphy, a software engineer at Google who has explored the question extensively in his free time. It “would have gotten rediscovered and rediscovered — aliens would have come to this one.”
Mark Belan/Quanta Magazine
The full menagerie of shapes is too diverse to get a handle on, so mathematicians tend to focus on convex polyhedra: shapes, like the cube, that have flat sides and no protrusions or indentations. When such a shape is much wider in some directions than others, it’s usually easy to find a straight tunnel that will allow another copy of the shape to pass through. But many famous convex polyhedra — for instance the dodecahedron, or the truncated icosahedron, the shape that forms a soccer ball — are highly symmetric and difficult to analyze. Among these, “for hundreds of years we only knew of the cube,” said Jakob Steininger, a mathematician at Statistics Austria, Austria’s federal statistics organization.
Then, in 1968, Christoph Scriba proved that the tetrahedron and octahedron also have the “Rupert property,” as mathematicians now call it. And in a burst of activity over the past decade, professional mathematicians and hobbyists have found Rupert tunnels through many of the most widely studied convex polyhedra, including the dodecahedron, icosahedron and soccer ball.
The Rupert property appeared to be so widespread that mathematicians conjectured a general rule: Every convex polyhedron will have the Rupert property. No one could find one that didn’t — until now.
In a paper posted online in August, Steininger and Sergey Yurkevich — a researcher at A&R Tech, an Austrian transportation systems company — describe a shape with 90 vertices and 152 faces that they’ve named the Noperthedron (after “Nopert,” a coinage by Murphy that combines “Rupert” and “nope”). Steininger and Yurkevich proved that no matter how you bore a straight tunnel through a Noperthedron, a second Noperthedron cannot fit through.
The proof required a mix of theoretical advances and massive computer calculations, and relies on a delicate property of the Noperthedron’s vertices. “It’s a miracle that it works,” Steininger said.
Passing Through the Shadows
To see how one cube can pass through another, imagine holding a cube over a table and examining its shadow (assuming it’s illuminated from above). If you hold the cube in the standard position, the shadow is a square. But if you point one of the corners directly upward, the shadow is a regular hexagon.
In 1693, Wallis showed that the square shadow fits inside the hexagon, leaving a thin margin. That means that if you point a cube’s corner upward, you can bore a vertical tunnel that’s big enough for a second cube to pass through. About a century later, Pieter Nieuwland showed that a different orientation casts an even better shadow — one that can accommodate a cube more than 6% larger than the cube with the tunnel.
Mark Belan/Quanta Magazine
Every subsequent analysis of more complicated shapes has relied on this process of turning the shape in different directions and looking for one shadow that fits inside another. With the aid of computers, mathematicians have found Rupert passages through a wide variety of shapes. Some are incredibly tight fits — for instance, the passage in a “triakis tetrahedron” has a margin that’s only about 0.000002 times the length of the shape’s radius. “The world of mixing computation and discrete geometry has flowered to make these kinds of calculations possible,” said Joseph O’Rourke, an emeritus professor at Smith College.
Researchers who have written algorithms to find Rupert passages have noticed a curious dichotomy: For any given convex polyhedron, the algorithm seems to either find a passage almost immediately, or not find one at all. In the past five years, mathematicians have accumulated a small collection of holdout shapes for which no passage has been found.
“I’ve had my desktop churn for two weeks on trying the rhombicosidodecahedron,” said Benjamin Grimmer, an applied mathematician at Johns Hopkins University, referring to a solid made of 62 regular triangles, squares and pentagons. “That one just seems to resist any attempt.”
But such resistance doesn’t prove that a shape is a Nopert. There are infinitely many ways to orient a shape, and a computer can only check finitely many. Researchers don’t know whether the holdouts are true Noperts or just shapes whose Rupert passages are hard to find.
What they do know is that candidate Noperts are incredibly rare. Starting last year, Murphy began to construct hundreds of millions of shapes. These include random polyhedra, polyhedra whose vertices lie on a sphere, polyhedra with special symmetries, and polyhedra in which he moved one vertex to intentionally mess up a previous Rupert passage. His algorithm easily found Rupert tunnels for nearly every one.
The contrast between these quick results and the stubbornness of the Nopert holdouts made some mathematicians suspect that true Noperts do exist. But until August, all they had were suspicions.
No Passage
Steininger, now 30, and Yurkevich, 29, have been friends since they participated together as teenagers in mathematics Olympiad competitions. Even though both eventually left academia (after a doctorate for Yurkevich and a master’s for Steininger), they have continued to explore unsolved problems together.
“We just had pizza three hours ago, and we talked about math almost the whole time,” Steininger told Quanta. “That’s what we do.”
Five years ago, the pair happened upon a YouTube video of one cube passing through another, and they were instantly smitten. They developed an algorithm to search for Rupert tunnels and soon became convinced that some shapes were Noperts. In a 2021 paper, they conjectured that the rhombicosidodecahedron is not Rupert. Their work, which preceded Murphy’s and Grimmer’s recent explorations, was, “I think, the first to conjecture that there might be solids that don’t have this property,” Steininger said.
If you want to prove that a shape is a Nopert, you must rule out Rupert tunnels for every possible orientation of the two shapes. Each orientation can be written down as a collection of rotation angles. This collection of angles can then be represented as a point in a higher-dimensional “parameter space.”
Florentina Stadlbauer; Courtesy of Jakob Steininger
Suppose you choose an orientation for your two shapes, and the computer tells you that the second shadow sticks out past the border of the first shadow. This rules out one point in the parameter space.
But you may be able to rule out much more than a single point. If the second shadow sticks out significantly, it would require a big change to move it inside the first shadow. In other words, you can rule out not just your initial orientation but also “nearby” orientations — an entire block of points in the parameter space. Steininger and Yurkevich came up with a result they called their global theorem, which quantifies precisely how large a block you can rule out in these cases. By testing many different points, you can potentially rule out block after block in the parameter space.
If these blocks cover the entire parameter space, you’ll have proved that your shape is a Nopert. But the size of each block depends on how far the second shadow sticks out beyond the first, and sometimes it doesn’t stick out very far. For instance, suppose you start with the two shapes in exactly the same position, and then you slightly rotate the second shape. Its shadow will at most stick out just a tiny bit past the first shadow, so the global theorem will only rule out a tiny box. These boxes are too small to cover the whole parameter space, leaving the possibility that some point you’ve missed might correspond to a Rupert tunnel.
To deal with these small reorientations, the pair came up with a complement to their global theorem that they called the local theorem. This result deals with cases where you can find three vertices (or corner points) on the boundary of the original shadow that satisfy some special requirements. For instance, if you connect those three vertices to form a triangle, it must contain the shadow’s center point. The researchers showed that if these requirements are met, then any small reorientation of the shape will create a shadow that pushes at least one of the three vertices further outward. So the new shadow can’t lie inside the original shadow, meaning it doesn’t create a Rupert tunnel.
If your shape casts a shadow that lacks three appropriate vertices, the local theorem won’t apply. And all the previously identified Nopert candidates have at least one shadow with this problem. Steininger and Yurkevich sifted through a database of hundreds of the most symmetric and beautiful convex polyhedra, but they couldn’t find any shape whose shadows all worked. So they decided to generate a suitable shape themselves.
They developed an algorithm to construct shapes and test them for the three-vertices property. Eventually, the algorithm produced the Noperthedron, which is made of 150 triangles and two regular 15-sided polygons. It looks like a rotund crystal vase with a wide base and top; one fan of the work has already 3D-printed a copy to use as a pencil holder.
Peter Lely
Steininger and Yurkevich then divided the parameter space of orientations into approximately 18 million tiny blocks, and tested the center point of each block to see if its corresponding orientation produced a Rupert passage. None of them did. Next, the researchers showed that each block satisfied either the local or global theorem, allowing them to rule out the entire block. Since these blocks fill out the entire parameter space, this meant that there is no Rupert passage through the Noperthedron.
The “natural conjecture has been proved false,” O’Rourke said.
It remains to be seen whether mathematicians can use the new method to generate other Noperts, or if they can find a different local theorem that can handle candidates like the rhombicosidodecahedron. But now that mathematicians know that Noperts do exist, “we’re on sound footing to study other shapes,” Murphy said.
Murphy, who like Steininger and Yurkevich has been exploring the question for its own sake, independent of his day job, feels a kinship across the centuries with Prince Rupert. “I like that he chose to use his retirement to do math and science in his castle,” he said.
Meanwhile, Steininger and Yurkevich are on the lookout for new questions to tackle. “We’re just humble mathematicians — we love working on such problems,” Steininger said. “We’ll keep doing that.”