# 书法”九宫格” 的”均” ＝黄金分割

1, 1, 2, 3, 5, 8,… (Fibonacci Series)

Note : Golden Ratio 黄金比率 ＝1.618 （长：短）

Fibonacci Number 1 1 2 3 5 8 13 21…

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

[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/

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

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

# The Genius of the East (2/4)

Note:
China: 九章算术， 秦九韶
Persia : Algebra, Algorithm
Italy: Fibonacci

# Sanskrit Poetry Math

By Manjul Bhargava
Professor of Mathematics
Princeton University

In the rhythms of Sanskrit poetry, there are 2 kinds of syllables – long with 2 beats (^^) and short with 1 beat (*)

Question: how many rhythms can we construct having n beats, using long and short syllables ?
(Example n=8: ***^^*^^)

Solution:

Case: n=1 beat
*
=> 1 rhythm

Case: n= 2 beats
**
^^
=> 2 rhythms

Case: n=3 beats
***
*^^
^^*
=> 3 rhythms

Case: n= 4 beats
****
^^^^
*^^*
**^^
^^**
=> 5 rhythms

This is the Fibonacci sequence;
1 2 3 5 8 13 21 34 …

For n = 5 beats, 8 rhythms
For n = 6 beats, 13rhythms
For n = 7 beats, 21 rhythms
For n = 8 beats, 34 rhythms