科学与数学aleph.se
甜甜圈地球：环形世界里的物理与生活
Torus-Earth
2014 — 2022，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这篇博文做的是一个“甜甜圈形地球”的思想实验：如果行星是环面，引力场、表面重力、截面形状和稳定性会怎样。作者从物理约束出发推导环形世界可能的样子，并讨论这种结构能否长期存在。
为什么经久不衰：它把熟悉的引力与行星物理推到极端几何，既展示硬核推导，又为科幻/世界构建提供灵感，所以多年后仍值得读。
HN 评论精选
指出文章对引力的讨论有误：环面内侧引力可为 0，并用空心球壳定理作直觉解释。
（先说清楚：这当然是一篇很酷的文章。）至少它对引力的讨论是错的。反直觉的是，环面内侧（最靠近中心、在轴平面上的表面）受到的引力为 0。典型例子是致密物质构成的空心球：球外看起来像实心行星，球内则完全没有引力场，无论你离表面多近；而且球越大也一样。直觉解释是，当你稍微减小半径时，对面物质的总量（球表面积）按 r^2 增加，抵消了 r^-2 的减弱。
srl查看英文原文
(Lest I come across as a complete ass: this is a cool piece, of course.) At least the discussion of gravitation is wrong. Somewhat unintuitively, the gravitational force on the inner part of the torus (the surface closest to the center, on the plane of the axes) is 0. The pathological example is a hollow sphere of dense material. Outside the sphere, it "looks like" (if you just measure g) a solid planet. Inside the sphere, there is no gravitational field whatsoever, no matter how close you come to the surface. Surprisingly, this holds no matter how large the sphere is. Suppose you're sitting on the inside of the surface of the sphere, and you decrease the radius a bit. Now the gravitational pull /away/ from this surface decreases like r^-2, so you would expect the gravity towards the surface (which is essentially unchanged) to increase. The issue is that the amount of material opposite you - the surface area of a sphere, really - also increases like r^2. (This is informal, but the best I can do for an intuitive explanation.) Tough to find a good explanation of this online. http://physics.stackexchange.com/questions/364/gravity-on-a-doughnut-shaped-mobius-planet
补足地质学视角：环形地球的板块构造/地幔对流未被讨论，且质量再分布可能让行星解体，地磁发电机也难成立。
这很酷！但重要的问题是，环形地球上的地幔对流以及由此产生的构造活动是什么样，文章没有讨论……对于表面重力相差 3 倍的“甜甜圈”地球，可以设想在南北极环处出现俯冲系统（可能是双板片型），在内、外赤道处出现扩张系统。我不知道更轻、更分异出来的弧岩会怎样……它们会聚集在双侧俯冲系统的中间吗，即使那里是重力高点？鉴于这类行星对扰动不稳定，构造活动和地壳分异造成的质量再分布可能足以把它撕碎。另外，建立地磁发电机也会很难……
cossatot查看英文原文
This is cool! But the important question of what mantle convection, and therefore tectonics, looks like on a toroidal earth is not addressed... for the 'donut' earth where surface gravity varies by a factor of 3 on the surface, one could envision subductions systems, possibly double (two-slab) types, setting up at the northern and southern polar rings, while spreading systems would set up at the inner and outer equators. I have no idea what would happen to lighter, more differentiated arc rocks... would they gather in the middle of a two-sided subduction system, even though that is a gravitational high? Given the instability of these planets to perturbations, it's possible that sufficient redistribution of mass due to tectonics and crustal differentiation would be enough to rip them apart. Also, setting up a geodynamo would be tough...
赞赏分析，尤其截面形状非圆形；好奇能否在轴向平面绕单臂或沿双纽线轨道运行。
多么迷人的前提。作者做了如此周到的分析，值得称赞。他推导截面形状的方法尤其有趣。我从未想过最优截面形状会是圆形以外的任何东西。我想知道是否可能在轴向平面内绕环运行，绕环面的单臂运行？甚至沿双纽线轨道，穿过内侧拉格朗日点？
chairmanwow查看英文原文
What a fascinating premise. Kudos to the author for doing such a thoughtful analysis. His technique for arriving at the cross-sectional shape is especially interesting. I would have never considered that the optimal cross-sectional shape to have been anything other than a circle. I wonder if it would be possible to orbit the ring in the axial plane, around a single arm of the torus? Or even in a lemniscate, intersecting the inner Lagrange point?
推荐硬科幻《Heavy Planet》/Mesklin 作为类似设定。
这很有趣。如果你喜欢这类东西，《Heavy Planet》是一本不错的硬科幻小说，背景是 Mesklin——一颗虚构的超级地球，自转快到让赤道重力显著降低。如果你喜欢 1950 年代科幻，它也是一个好冒险故事。
roywiggins查看英文原文
That was fun. If you like this sort of thing, "Heavy Planet"[1] is a good hard scifi book set on Mesklin, a fictional super-Earth which spins fast enough to make gravity significantly less at the equator. It's also a good adventure story, if you like 1950s scifi. [1] http://en.wikipedia.org/wiki/Mesklin
历次讨论
同类还值得读
如果月球只有 1 像素：一个极其精确的太阳系模型（2014）
If the moon were only 1 pixel: A tediously accurate solar system model (2014)
一个网页版太阳系模型，以“如果月亮只有 1 像素”的真实比例呈现：你必须一路横向滚动，才能体会行星之间空旷到何等程度。
科学与数学上首页 7 次 · 2014–2025joshworth.com编辑精选沉浸式线性代数（2016）
Immersive Linear Algebra (2016)
《Immersive Linear Algebra》是一本完全免费、开放在线的线性代数“超文本教科书”，其最大特色是大量可交互的图形：读者可以直接拖动、操作图中的向量、矩阵和变换…
科学与数学上首页 5 次 · 2015–2024immersivemath.com编辑精选如何将贝果切成两个相连的半环（2009）
How to Slice a Bagel into Two Linked Halves (2009)
数学家 George Hart 的经典教程，演示如何用一刀把贝果切成两个相互扣住的环，并附上切割步骤与数学原理。
科学与数学上首页 5 次 · 2009–2022georgehart.com“犹太问题”：一组用于歧视的数学题
Jewish problems
一篇 arXiv 论文，整理并讨论苏联时期犹太学生在申请大学数学系时被问到的特殊数学难题：这些题往往有简单解法却极难想到，被用作面试中拒绝犹太考生的手段。
科学与数学上首页 5 次 · 2011–2021arxiv.org你的身体并非为长久而造：人类死亡率数据的一课
Your body wasn’t built to last: a lesson from human mortality rates
文章从人类死亡率随年龄变化的统计规律出发，解释身体并非为长久使用而设计：年轻时死亡风险较低，之后随年龄近似指数上升，像复杂系统在不断累积损伤。
科学与数学上首页 5 次 · 2009–2018gravityandlevity.wordpress.com