语言学提示：
沿虚轴扩展国际音标元音平面，由此生成"复元音"
——一种凡人无法理解的晦气声音。

国际音标中的元音可以按发音位置(前-后)和口型(闭-开)两个维度分类, 即所谓’VOWEL PLANE’. 如下图:

作者借用了数学上的解析延拓概念, 设想在元音平面基础上再加上虚轴, 即得到了"复元音". 这里的"复"是复数的复. (双元音的英文是’diphthong’.)

视角直径拐点
(图中用手机代替星系，亮度和红移已调整到可见范围)
近大远小，离我们越远的物体看起来越小，但非常非常远的物体看起来反而会变大！
因为当它们的光线刚刚发射出来的时候宇宙还很小，当时它们距离我们（现在的空间位置）更近。

由于光速有限, 眺望远方其实也是眺望过去(字面意义上的). 所以整个夜空其实是展示给我们的一部宇宙历史, 有意思的是这历史的最外层其实是宇宙早期的显微结构(参见解耦). 设想, 从地球向外, 把夜空(过去光锥)划分成一圈圈包围我们的天球球层, 在地球附近这些球层半径不断变大, 最后又变得很小, 那么中间一定会经历一个拐点. 我有点好奇这拐点具体发生在什么时间, 从这篇xkcd的漫画图中来看, 大概是发生在再电离事件之前的.

好久没有更新博客了, 去年至今其实有一些酝酿中的想法, 没时间整理成paper, 只能再往后推了. 最近打算先翻译一些有趣的xkcd漫画作品.

非常怀念过去在科学松鼠会上看Ent^*翻译xkcd等各种科学漫画的时光.

来自：YouTube 梁文道：聊聊知识是什么
上一集：YouTube 梁文道：聊聊反智主义是如何兴起的

To describe a chess game state, you don’t need to mention every chess piece’s details. Essentially, a chess piece is just a symbol. Ignoring appearance detail helps us to grab the game gist. Till now, a popular way to represent a Rubik’s cube state is the facelets expanding graph, like this:

By this way, you can’t tell which facelets are adjacent each other straightway, also it’s hard to imagine what the cube will change into after a twist applied. That because it comes from the appearance, but not the essential. Furtherly, as my preivous blog wrote, Rubik’s Cube solver programs who construct cube state from facelet color is clumsy. Rubik’s Cube is a game about cubes’s rotation and permutation (but not painting color), matrix is the most proper math tool here.

However, the fact revealed by this method is not easy to see through, and that is just what I will tell you in this blog.

Some days after the former post of StyleGAN Mapping Network Geometry Visualization, I realized that there are some canonical dimension reduction methods for data visualization, such as PCA, t-SNE. These ways may be more intuitive to show data characteristics. So I did some attempts on this.

StyleGAN^{[1] [2]} generator network has two parts: full-connected mapping network (named mapping), and pyramid CNN synthesis network (named g). Mapping is a transformation from dimension 512 to 512, and g is a transformation from dimension 512 to 1024×1024×3. The design of mapping is intended to disentangle the manifold mapping from latent space to feature variation space. I’m interested in how the shape of learned mapping in network warps exactly, so this is my experiment.

Motivation

Canonical Rubik’s Cube solver algorithm¹ constructs cube state from face colors and a lot of permutation rules. That may waste too many coding :) Face color is merely appearance, cubies’ orientation is essential. Since Rubik’s Cube seems already been used as the avatar of group theory (check this wikipedia entry), it’s better to clarify all details of the cube rotation group structure, and construct the whole Rubik’s Cube representations based on cubies’ orientation.

Regardless of Rubik’s Cube, orthogonal rotation in 3D space is usual and connected with interesting problems. E.g. how to quickly tell if 2 orthogonal Euler angles are the same rotation, purely by algebra without experiment? Quaternion calculus may be a short answer, but when you do that, irrational numbers are inevitable, and that seems wasting and precise problematic.

Programmers prefer easy implementation, which based on a set of simple representation and rules applied to them. No float numbers, no redundancy. All you need is a multiplication table, and the table is highly symmetric, so let’s begin from analyzing the R³ orthogonal rotation group structure.

This is just a page embedding test. Have fun.

Blog open, this is the first post.