科学与数学oftenpaper.net
谢尔宾斯基三角形：一个终结其他同类页面的页面
The Sierpinski triangle page to end most Sierpinski triangle pages
2013 — 2015，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
这是一篇关于谢尔宾斯基三角形的超长篇探索页面，用大量 Mathematica 生成的图形、动画和代码，从各种角度展示和讲解这一经典分形，甚至延伸到黎曼球面等话题。作者以幽默的口吻自称这是'终结大多数谢尔宾斯基三角形页面的页面'。
为什么经久不衰：它把一个经典数学主题做到了极致，兼具严谨的数学探索和幽默的呈现方式，多年来反复登上 HN 首页，是'把一件事做透'的典范。
HN 评论精选
页面内容丰富精彩，幽默与深度兼备
太惊艳了，页面往下是一张接一张的精美图形。中途还有一节讲如何理解黎曼球面：作者用'用航天飞机把一头牛空投到生物穹顶上，让牛朝四周 indiscriminately 发射激光'来比喻，甚至还配了带 Mathematica 源码的动画——不过'注意牛不能是球形的，否则会滚下来'。
hansjorg查看英文原文
That's amazing. It just goes on and on with beautiful plots. About halfway down, there's a section on understanding the Riemann sphere: > From what I can tell, one of the settings used to deal with division by 0 is the so-called Riemann sphere, which is where we take a space shuttle and use it to fly over and drop a cow on top of a biodome, and then have the cow indiscriminately fire laser beams at the grass inside and around the biodome. That's my intuitive understanding of it anyway. And there's even animation (with Mathematica source) for this one. However: > (Note the cow cannot be spherical or it will roll off).
标题所言非虚，内容量惊人且令人着迷
好吧，他们确实配得上这个标题。内容太多了，甚至有一头从航天飞机上空投下来发射激光的牛。真得抽时间仔细看看，这以一种'看看数学/编程/Mathematica 能做什么'的方式令人着迷。
Groxx查看英文原文
Uh, wow, I think they earned that claim. There's so much in here, they even have a cow which fires lasers being dropped from a space shuttle. http://www.oftenpaper.net/img/understandingtheriemannsphere.gif seriously going to have to look more closely at this some time, it's fascinating in a "look what you can do with math/programming/mathematica" kind of way.
分享一段可运行的交互式分形探索代码
如果你有 Mathematica，可以试试这段代码，能得到一个平滑高分辨率的交互式分形浏览器（需安装 C 编译器）：[附 Julia 集交互代码]。原帖太棒了，我喜欢那头牛！
taliesinb查看英文原文
For those who happen to have Mathematica, try this to get a smooth, high-resolution interactive fractal explorer (make sure you have a C compiler installed): JuliaFP = Compile[ {{const,_Complex}, {init,_Complex}}, Module[{val=init,n=0}, While[Abs[val] < 5 && n < 25, val = val^2 + const; n++]; {Mod[Arg[val]/(2*3.14159), 1], 1 - Abs[val]/5, 1 - Abs[val]/100} ], CompilationTarget -> "C", RuntimeAttributes -> Listable, RuntimeOptions -> "Speed" ]; {fine, coarse} = Table[Complex[j,i], {dx, {0.008, 0.002}}, {i,-1.5,1.5,dx}, {j,-1.5,1.5,dx} ]; pt = {0.0,0.0}; Graphics[{ Raster[ JuliaFP[Complex @@ pt/2, If[ControlActive[], fine, coarse]], {{-2.0,-2.0}, {2.0,2.0}}, ColorFunction-> Hue ], Locator[Dynamic[pt]]}, ImageSize -> 800 ] // Dynamic P.S. Original post is awesome. I love the cow!
分形作业成为作者编程人生的起点
谢尔宾斯基三角形让我意识到自己有编程天赋，甚至可以说是我职业生涯的功臣之一。80 年代中期我 11 岁时，老师布置了画这个三角形的递归作业。我没有电脑也没资料，卡了很久甚至急哭了，后来洗澡时灵光一闪想出解法，冲回机房写代码成功——第二天发现全班只有我做出来了，那是我一生最骄傲的时刻之一。
dvirsky查看英文原文
The Sierpinski triangle was what made me realise I have some talent as a programmer and is part to blame for my career: When I was about 11 (mid 80s) our school got a shining new computer lab with original IBM 8086 PCs, and one teacher improvised a LOGO class. After a while we were divided into two groups of more advanced kids and the rest of the class. I was in the advanced team, and I think I was the only kid there who didn't have a computer at home and hadn't coded before. Anyway, we were learning about recursion, and the teacher gave us increasingly difficult tasks. Then one day he showed us this triangle thingie, and told us to write an algorithm to draw it for the next class. I was shocked and stuck. Remember, I had no interwebs or even books on the subject. I spent hours in the computer lab banging at it, and couldn't do it. I went home, worried that I'll be the only one in the class who failed to do it - and that I'll be taken out of the advanced group. I was horrified, I was literally crying over it. Then while taking a shower or something like that, the solution suddenly hit me. I ran to the school lab (it was open after hours) wrote the code and it worked. The joy and adrenaline rush of it was something out of this world, let alone the relief that I wouldn't be "downgraded" from the advanced class. The next day in class, the teacher asked us to show him our code. It turned out I was the only kid who figured it out. It was one of the proudest moments of my life I guess, and I've been trying to recreate that rush of solving a hard task ever since.
作者对主题认真、对自己幽默，正是最佳科普姿态
我远比预期中更享受这篇文章。作者自称要创造'互联网被发明出来所等待的那个页面'，而这个页面既是谢尔宾斯基三角形的探索，本身也是'我有多容易忘乎所以'的分形表达。我喜欢这样的数学：作者对主题严肃认真，对自己却不那么严肃。
jwecker查看英文原文
I enjoyed that far, far more than I expected. His intro from the index: "So I was me and I was in math class watching paint dry it was starting to crack when suddenly I realized there was a page for which the internet was invented. I set out to create that page, ultimately succeeding with the sierpinski triangle page to end most sierpinski triangle pages ™. ... So while the sierpinski triangle page to end most sierpinski triangle pages ™ purports to be some kind of exploratory rundown of the Sierpinski triangle, it's also a fractal expression of just how carried away I get..." It's math like I like my music: when the author takes the subject seriously without taking themselves too seriously.
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