书法”九宫格” 的”均” ＝黄金分割

May 10, 2020Chinoiseries2014 Chinese, Elementary Math Leave a comment

书法是”字如人品” 。

大自然的美表现在 “黄金比率” (Golden Ratio) ＝道，暗藏在唐初欧阳询发明的楷书”九宫格” : 一个比率 “均” (即：均称，对称 Symmetry) 。

1, 1, 2, 3, 5, 8,… (Fibonacci Series)

黄金分割= 0.618 ~ 1:2 ¦ 2:3 ¦ 3:5 ¦ 5:8

Note : Golden Ratio 黄金比率＝1.618 （长：短）

…

欧阳询楷书《九成宫》

晋/隋/唐的行书, 草书是在汉朝的楷书 (aka 真书) 的”均“基础上”速度化” : “正” (角度)，变 (笔画突) ，尚保存”均” 的美。

东晋. 王羲之《兰亭序》行书

唐.张旭草书：

毛泽东草书《沁园春· 雪》

https://m.toutiaocdn.com/group/6821832651129750030/?app=news_article_lite&timestamp=1589048685&req_id=202005100224450100140481400C466DDA&group_id=6821832651129750030

Math Paradox : Toroflux

October 7, 2018Chinoiseries2014 Math Games 1 Comment

Fibonacci Number 1 1 2 3 5 8 13 21…

BM Category Theory II 8: F-Algebra, Lambek’s Lemma , Catamorphism, Coalgebra, Anamorphism

May 9, 2017ChefCouscous IT, Modern Math 1 Comment

[Continued from previous BM Category Theory …]

$\boxed { \text {type Algebra f a = f a} \to \text {a} }$

Intuition: [Artificial Intelligence] You teach the computer like to a Primary 6 kid, that Algebra is a type of expression (f) which, after evaluation, returns a value.

If a = i (initial) [or u (terminal)],
$\boxed { \text {(f i} \to \text {i )} \implies \text {f = Fix-point} }$

Intuition: Fix-point because, the Initial “i”, after evaluating the expression f, returns itself “i”.

Lambek’s Lemma
$\boxed { \text {Initial Algebra is an Isomorphism} }$

Note: Endo-functor is a functor (equivalent to function in Set Theory) within the same Category (Endo = Self = 自)

Video 8.1 F-Algebras & Lambek’s Lemma

Video 8.2 Catamorphism & Anamorphism

foldr ~ catamorphism (浅层变质) of a Fix-point endo-functor on a List.

Examples: Fibonacci, Sum_List

Remark: Cool Math! the more advanced concept it is, the more closer to Nature (eg.Geology, Biology) : Catamorphism 浅层(风化)变质, or “thin-layer change in nature” (in Functional Programming languages: foldr or map) eg : add1 to a list (1 5 3 8…)
= (2 6 4 9 …)

$\boxed { \text {type Coalgebra f a = a} \to \text {f a} }$

Intuition: Reverse of Algebra, given a value, Coalgebra returns an expression (f).

Anamorphism (合成变质) ~ unfoldr

Example: Prime numbers

Remark: Anamorphism (合成变质) or “synthesised change in nature“: eg. Start from a “seed” prime number “2” generates all other infinite prime numbers (3 5 7 9 11 13 17 …)

Note: In Haskell, no difference between Initial and Terminal Fix-points. However, since Fix-point is not unique, in Category Theory there is the Least Fix-point (Initial) and Greatest Fix-point (Terminal).

Ref:

Reading “Understanding F-Algebra ” by BM: https://bartoszmilewski.com/2013/06/10/understanding-f-algebras/

Catamorphism (下态) : https://www.zhihu.com/question/52180621/answer/129582557

Anamorphism : https://zhuanlan.zhihu.com/cofree/21354189

F-Algebra & F-coalgebra: http://stackoverflow.com/a/16022059/5822795