Maximum Equalangular lines in N-dim Space


2 Dimensional plane, max 3 lines to cross to have opposite angles equal (cross at 90° or 60°).

3 Dim space, max 6 (eg COVID virus icosahydron, all diagonal lines)

N dim ?

Application: code the photo imaging data points sent from Mars in N dim Matrix, verify the codes accuracy with this “Max” Theorem.



网购 当当网:

Vedic Factorisation of Polynomial E(x,y,z)

Any Algebra Theory to explain this Vedic Polynomial Factorization ?

Why it works by setting y=0, z=0 ?

E'(x,y,0)= 0 when {x= -2y} or {x=y}

E'(x,0,z)= 0 when {x= -3z} or {x=z}

Why ? Combine them:

E(x,y,z) = 0 when {x=-2y-3z} or {y+z}

which implies:

E(x,y,z) = (x+2y+3z)(x-y-z)

Note: The 2 Ideals of Polynomial Ring E are therefore :



Singapore Modeling Math (PSLE 2021) sans Algebra

[No Algebra Method , please]

Please try it yourself. PSLE Exams is for all 12 year-old Singapore pupils.

This question is tough for adults, yet easy if the pupils with logical mind trained in the “Singapore Modeling Math”.

A Tutor’s advices on coping with tough PSLE Math:

. [Solution: scrolled far far below… ]




























Hilbert Problem : “Polynomial equation of degree 7”


Galois had proved Polynomial equations of degree 5 & above have no ‘radical ‘ ((+*/, nth root) roots

Hilbert Problem : “Polynomial equation of degree 7” : Algebraic Geometry + Topology + Number Theory

How to Study University Math 如何学好高等数学

Different from Secondary (O-level) or High school (A-level) Math, University Math (aka Higher Math 高等数学) is proof-oriented (证明) based on student’s understanding of the abstract concepts (概念) 。

Interesting to note the last 2 examples 2.4 & , 2.6 how tricky (astucieux) by ‘高级规则’ to find N in terms of expression (ε) +1. Doesn’t matter any expression (ε) bcos N is not UNIQUE, as long as you could magnify (放大) to get a simple expression (ε) eg ε/2, ε/3…

Galois Theory explains Quintic Equation sans radical root

总结:Field Extension 扩域

还是解释得很快,听得蒙是正常,因为 Cauchy, Fourier, Gauss, Poisson 等世纪大师都听不懂 20岁 Galois 讲’西咪’ ?

Get to appreciate the reasoning outline is enough, “the devil is in the details” for the Math grads in the 4th year Hons degree.

科普: Icosahydron(正20面体)

第一集:Abel’s & Galois’s tragic life

第二集:Solving equations of degree 2, 3, 4

第三集:Introduction to Group, Equation’s roots and their relation with the coefficients.

第四集:Group secret in a game (odd & even parity)

第五集:Icosahydron unsolvable

The Math Power of Number 4

Despite Chinese & Japanese don’t like ‘4’ for the same sound as \si (death), ‘4’ is the magic number for all ideal compositions:

1) All TCM 药方 has 4 compositions : 君臣佐使
eg. “银翘解毒片” (Chinese Panadol) :
银花 (君药)有毒, 但可以杀flu virus,
连翘(臣药),中和neutralise 银花的毒。
(佐药 Assistant ) 甘草,..
(使药 Smoothening, sweetening ) 蜂蜜

2)烹饪 Cuisine :
eg. 炒饭
臣:蛋 x2
佐:mix veggies, 腊肠,烧肉,…
使:香料 (葱花,胡椒粉,garlics,… ), 麻油/鱼露

3) Music
eg. Quartet
佐:flute, guitar,…
使:Cello,… Double base

Eg. “4-Color” Problem : the only one Math impossible by mathematicians but proven only by computer.

5) Exemplary Nation Racial Harmony
Eg. Switzerland / Singapore : 4 races & official languages.

6) War against Evils:
Eg. WW2 Allied :
USA, UK, France, China, against evils Japan & German (+Italy).

7) Four seasons :

8) Healthy Meal:
Fill 1/4 plate with wholegrains.
• Fill 1/4 plate with good sources of protein.
• Fill 1/4 plate with fruit and 1/4 plate of vegetables.

That’s why the 19-year-old Math genius Galois (1832 AD) proved by his invention Abstract Algebra the “Group Theory” that Polynomial equations with radical solution (+ – */,nth root) ONLY possible up to maximum degree 4.
That explains why the “A5-Group” Icosahydron (二十面体) structured SARs-like viruses (eg. COVID-19) WILL NEVER have “radical” drug SOLUTION, only possble killed by our own Antibody pre-trained by Vaccines (eg. mRNA or Sinovac… ).

Also humans impossible to have more than 四代同堂。

The 5th Generation Computer “Prolog Machine” by Japan in 1980s failed miserably.

Anything more than 4G has only “particular case” solution with physical limit beyond which we have to shift paradigm / technology to restart on another track.
Eg 5G by Huawei facing so much difficulty by Western sabotage, only by jumping to new “Quantum Communication Technology” to overcome these physical & artificial human resistance, whether they like it or not by Western countries.

IMO 2020 (6th Question)

2020 IMO (6th problem) only one Chinese 李 student scored full 7 marks.

This question is similar to Analysis finding epsilon-delta value of N:

(Epsilon-Delta Analysis, aka Advanced Calculus ) definition for “Limit of Series” :
It exists N such that, for all n > N…

[Solution] :

[See also] IMO 2020 (4th Question) :

3 Math Foundation Skillsets for Engineering / Finance Students


理工/金融科Engineering/Finance 的三个扎实数学基础 :
1) Calculus
2) Linear Algebra
3) Probability / Statistics

Prerequisites for the 3 above in
1) O/A level : Trigonometry, Algebra, Geometry
2) Vector Space (with some basic Algebraic Structures : Group, Ring, Field )
3) O/A level : Permutation/Combinatorics

Leibniz’s original proof : Integration by Parts


Leibniz’s original proof : Integration by Parts (分部积分公式”)


金庸武术的道理 和学数学一样:
1)华山正邪二派 : 气宗 (正) vs 剑宗 (邪)
数学:数学理论 vs 刷题技巧
2)虚竹 :忘掉以前的少林功夫才能学 消遥派
数学: 忘掉A-level 前的思维 (concrete) ,才能学好大学抽象(Abstract ) 数学.

3) 少林寺僧好高鹜远:还没学精本派“一阳指”,就想去换 印度鸠摩罗的功夫。
数学: 先打好 大学基础数学 (Epsilon-Delta, Abstract Algebraic Structures) ,才去学其他的高深东西 (Category, Differential Geometry, Algebraic Topology,… )

金庸的武林江湖 宗师 vs 数学江湖

东邪(孤僻冷漠) :黄药师 vs 德国. Gauss

西毒 (妒忌,狠毒) :欧阳锋 vs 法国. Cauchy (迫害 年轻天才 Abel, Galois )

南僧(避世隐士): 大理国王 一灯大师 vs 法国.Fermat

北丐 (流浪天涯):洪七公 vs 匈牙利. Paul Erdos