科学与数学acko.net编辑精选
如何折叠朱利亚分形
How to Fold a Julia Fractal
2014 — 2024，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 acko.net 上的一篇交互式长文，用大量可操作的动画图示讲解复数运算，并由此引出 Julia 分形：把复平面上每个点反复平方（加常数）后迭代，观察它逃逸还是收敛，得到的图案就是 Julia 集。作者的重点不只是分形本身，而是让人直观看到「平方＝旋转＋缩放」这类几何直觉，以及复变函数的一些核心概念。
为什么经久不衰：它把通常靠符号推导的复变函数变成了可以动手拖动的图像，十多年来一直是理解复数与分形最直观的入门材料之一；即使今天读，图示思路依然少见。
HN 评论精选
图示让作者第一次真正直观理解复数与 Julia 集。
不知道是因为那些出色的图示，还是因为讲述的方式，这篇文章终于让我对「复数」有了以前从未真正获得的直觉理解。以前我知道复数在某些坐标/向量运算里很好用，但一直没抓住它为什么好用。对坐标系做平方时产生的扭曲大概就是关键。它也把 Julia 集给我讲明白了。
pdkl95查看英文原文
I'm not sure if it was the visual aids (which were outstanding), or simply how the subject was approached and written about, but that article finally gave me an intuitive understanding of "complex numbers" that i've never quite gotten before. Sure, they were nice for some types of coordinate/vector math, but why they were nice always seemed to be something I missed. The warping of the coordinate system that you get when squaring is probably the key. It totally explained Julia sets for me. Now if only I could get the same understanding for the incredible complexity of the Mandelbrot. After seeing[1] what happens down around 1/ 2^1116 (~3E353) - and remember that the set is supposed to be simply-connected - I'm not sure how to begin any kind of real understanding. There is simply too much detail there... [1] http://www.youtube.com/watch?v=PbwaFQ2r2c4 (the entire video is great, but the true insanity starts about halfway through)
文章价值在于用图像化方式呈现数学家的思维方式，暗含分支割线、全纯函数等高级概念。
与其说是 Julia 分形的入门，不如说这篇文章是复数以及复平面函数的好入门。它的呈现方式非常接近我认识的大多数搞数学的人在脑子里想象这些概念的样子——初学者最难理解的正是这一点：多数数学人是用这样的图像而不是符号来思考的。比如第一个可视化大约第 29 张幻灯片，作者实际上画出了「分支割线」却没有用这个术语；最后一个可视化约第 12 张，又暗示了复平面上复数和微分之间的特殊关系，涉及全纯函数、柯西-黎曼方程和柯西积分定理。
jfarmer查看英文原文
More than being an intro to Julia fractals, I think this post is a great introduction to complex numbers and functions of the complex plane[1]. The way this is presented is very similar to how most math-folk I know picture these concepts in their head. This is probably one of the toughest things for beginners, who don't understand that (most) math-folk think in pictures like this and not in symbols. For example, starting at around Slide 29 in the first visualization, the author actually paints a picture of a branch cut[2] without using that term. Likewise, starting at around Slide 12 of the last visualization, the author hints at the special relationship between complex numbers and differentiation in the complex plane. The jargon-y stuff involved here are holomorphic functions[3], the Cauchy-Riemann equations[4], and the very surprising-but-central theorem of complex analysis: Cauchy's integral theorem[5]. [1]: Functions from ℂ to ℂ are "hard" to reason about because there are 4 dimensions involved, at least if you're picturing ℂ as a 2-dimensional plane. [2]: https://en.wikipedia.org/wiki/Branch_point#Branch_cuts [3]: https://en.wikipedia.org/wiki/Holomorphic_function [4]: https://en.wikipedia.org/wiki/Cauchy-Riemann_equations [5]: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem
批评：文中关键一步缺少铺垫，数学基础弱的读者会卡住。
我很喜欢 acko.net 上的可视化，但和这里很多人不一样，我还是没搞懂他想讲的东西。举个例：第一个大可视化第 18 步说「这其实很了不起，因为它意味着我们编造的规则造出了 −1 的平方根，就是 1∠90°」。什么意思？！我确信懂数学的人能明白，但作为数学渣的我完全看不出这句话怎么解释了这个规则造出 −1 的平方根。前面根本没提过平方根，也没说这个向量加旋转跟平方根有什么关系。
greggman查看英文原文
I love the visualization on acko.net but for whatever reason, unlike apparently many of you, I still don't understand the topics he's trying to explain. As just one example on step 18 of the first major visualization it says "And that's actually a remarkable thing, because it means our invented rule has created a square root of −1. It's the number 1∠90∘" WAT?! I'm sure someone skilled at math understands that but as a lamer at math I have no idea how that explains that the rule invented the square root of -1. Nothing before that has mentioned square roots or how this vector + rotation has any relation to them.
页面交互元素很吃性能，老设备上体验很差。
哎，这个页面简直把我浏览器搞废了（Windows 7 上最新版 Firefox）。加载时卡死 20 秒，之后滚动和动画都极不流畅。
Gracana查看英文原文
Ouch, this page absolutely destroys my browser (up-to-date Firefox on Windows 7.) 20 seconds of unresponsiveness on load, followed by very choppy scrolling and animation.
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