《数学与人类文明》Mathematics & Civilizations

《数学与人类文明》数学与现代文明:

1. 哲学:希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴, 达芬奇, Golden Ratio

3. 工业革命:Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .

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Symmetry Music Scores : 9 “Eigen” Notes 特征音符

This is a good mathematical symmetry trick to remember all musical notes thru 3 notes (1,4,5) in 9 positions.

Remember these 9 “Eigen“ Notes (C1, F4, G5) which are symmetric in position on the 5 lines, the other notes can be derived easily.

“Eigen” (German : characteristics 特征) as in Eigen vectors / Eigen values.

Green 4 (top ) & 5 (bottom)

Blue 5 (2nd line bottom up ) & 4 (2nd line top down)

Orange 1 (3rd space bottom up ) & 1 (3rd space top down)

https://m.toutiaoimg.cn/a6761794168806179331/?app=news_article_lite&is_hit_share_recommend=0

Music = Math + History

To really appreciate music, you must understand this Math & history :
5 notes to 7 notes to 12 notes.
& fractions: 3/2, 3/4, 2/1 (octave)

Music = Math + History

1. Ancient China 600 BC 管仲 (5 notes 宫商角徵\zhǐ羽)

2. Pythagoras 450 BC (7 notes, Chords 和弦)

3. Modern Music: 十二平均律 (明朝 1600 AD 朱载堉- > Bach 1700 AD)

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Pythagoras did not believe in Irrational numbers (sqroot 2) , so only 7 integer notes.

17CE Chinese 明朝 Prince invented 12 notes music, using a giant 81-row Abacus to compute the12th root of 2 (picture below) :

https://chinoiseries2014.wordpress.com/2019/02/07/%e5%8d%81%e4%ba%8c%e5%b9%b3%e5%9d%87%e7%8e%87/

十二平均律 12-tone Equal Temperament

《十二平均律》 是明朝 "布衣王子" 朱载堉 (1536年-1610年) 发明的,由当时在中国传教的意大利人 利马窦 (Matteo Ricci, 1552 – 1610), 传给欧洲的法国数学家 Marin Mersenne (Mersenne Prime ) 。现代音乐之父 巴哈 Bach 第一个采用,制作世界第一架钢琴有12黑白键,并作曲 《Bach 12-tone Equal Temperament》。

朱载堉把 “1” 到 “i” 的八度 (Octave) 分为等比 (ratio) 距离的12个半音 (half note), 每个音是前音的 \sqrt[12] {2 },第8高音频率 (frequency) “i” 巧好是第一音 “1”的2倍。他用81档的大算盘算出:先开立方根,后开平方根 2 次

\boxed { \displaystyle \sqrt[12] {2} = 2^{\frac {1}{12} }= 2^{{\frac {1}{3}}.{\frac {1}{2}}.{\frac {1}{2}}} = \sqrt {\sqrt {\sqrt[3]{2}}}}

Side Note:

1977年法国大学数学教授在课堂好奇地问我,你们祖先如何解代数?是用算盘吗?当时计算机还不流行,复杂的算法只能用Log Table 或 计算尺 (Slide Rule) 。

朱载堉的算盤算法就是个例子。

Bach 12-tone Equal Temperament

https://zh.m.wikipedia.org/wiki/%E5%8D%81%E4%BA%8C%E5%B9%B3%E5%9D%87%E5%BE%8B

倍大吕:

Music and Mathematics (12 Tone Equal Temperament)- Dr. Eugenia Cheng (Prof Math in Category Theory)

Arrow’s Theorem

The Axioms for a fair voting system (eg. Political Election) :

1. Non-dictatorship:
◇ Outcome decided not by one ‘dictator’, but more than one person.

2. Unanimity
◇ If everyone votes that A is better than B, then A will be ranked higher than B in the final result.

3. Independence of irrelevant alternatives
◇ The ranking of A relative to B should not be affected by someone changing their mind about C.

Arrow’s Theorem says that if there are more than 2 people (parties) to vote for, then there is no fair voting system.

Note:
Most democratic voting systems violate the 3rd axiom (independence of irrelevant alternatives).