Showing posts from 2010

A crude correlation between Hari Raya and Chinese Lunisolar Month

Today is the first day of Ramadhan in Malaysia and about 30 days later, our Muslim friends will celebrate their New Year or Hari Raya Puasa.

In Malaysia, the date of Hari Raya will normally be announced by the Keeper of the Rulers' Seal (Y. M. Engku Tan Sri Dato Sri Ibrahim bin Engku Ngah is the present Keeper) on Hari Raya Eve, who will read the following script in the TV.

It is well known that the Islamic calendar is a lunar calendar and the sighting of new moon will determine the first day of the month. Not too long ago, I tried to tabulate the astronomical new moons (computed using a program by Legrand and Chevalley), and the dates of Hari Raya Puasa celebrated in Malaysia, and the corresponding Chinese dates. The result is the following table.

The following points can be inferenced from the table above:
1. If the astronomical new moon occurs ante meridian, then the Hari Raya Puasa will coincide with the second day in the corresponding Chinese month.

2. If the astronomical new …












McKinney's Generalization of the Birthday Problem: Part II

In Part I of the article, we considered the expression for computing the probability that at least r people will have the same birthday, given n people are selected at random.

In this article, we will consider a special case to illustrate the use of formula numerically: Suppose five people are selected at random, what is the probability that at least three people will share the same birthday?"

In this example, the Frobenius equation n1 + n2 = 5 has three sets of solutions and they can be tabulated as follows:

Following McKinney, we define the general form of the probability P (n; n1, n2) as

It follows that P(E) can be computed as follows:

The required probability that at least three people are sharing their birthdays is therefore:

Alternatively, we could also arrive at the same result if the problem is approached from another direction, but we would require a slight modification on McKinney's formula.

The required probability is thus the summation of the following fractions:

McKinney's Generalization of the Birthday Problem: Part I

In 1966, E. H. McKinney of Ball State University published a paper entitled "Generalized Birthday Problem" in the American Mathematical Monthly, in which he seek to answer two questions related to the famous birthday problem.

The problems considered by McKinney are (a) Suppose n people are selected at random. What is the probability that at least r people will have the same birthday? (b) What is the smallest value of n such that the probability is greater or equal to 1/2 that a least r people to have the same birthday? What follows is the solution by McKinney with some notational adjustment.

Let Xj (= 1, 2, ... , n) be the n random birthdays, let event E' be defined as "r or more birthdays are equal" and the event E be defined as "none of the r random birthdays are equal", then we obviously have P(E)+P(E') = 1, in which P(E) can be computed by summing the probabilities of all ways in which the n random birthdays can take on less than r equal val…

The Mathematics of Honeycomb: Part III

In the first two parts of the article, I have given a historical review of the honeycomb problem and a calculus-based technique to compute the honeycomb rhombic angle. In this article, I will try to give a crude analysis on the results we obtained in Part II.

For a closed curve in two-dimensional space, there exists an inequality to govern its perimeter L and area A.

This inequality is called the isoperimetric inequality. It simply means that the ratio of A/L2 cannot be greater than 1/4π. This result can be generalized to three-dimensional space to give:

This shows that the theoretical maximum value of the ratio of V to S3/2 (when you have the least surface for a given volume) is 1/6π1/2 or 0.0940316. For honeycomb cells, the ratio of V to S3/2 is:

This value is approximately 0.08 for a = 3.7 mm, b = 13.6 mm, which is pretty close to the theoretical maximum. On the other hand, the same ratio is approximately 0.078 for a prism with hexagonal top and hexagonal bottom.

This result is no…

The Mathematics of Honeycomb: Part II

As explained in Part I of the article, instead of building their house the easier way with hexagonal top and bottom, the honeybees are building their cells with hexagonal top and rhombic bottom. In this article, I will try to give a calculus-based demonstration on how the honeybees achieve the economization of their wax resources with a rhombic angle of 70o32'.

Suppose we start with a hexagonal prism with flat bottom like the one shown in the figure above. Let AB = a, A''A = b. Then it can be calculated that the surface area S0 and the volume V of this prism are:

respectively. Mark a point B' on the prism so that B'B'' = x and slice a tetrahedron from the prism along the line A''C'' through point B'. Then, flip and rearrange the sliced tetrahedron on top of the prism, as shown below.

If you repeat this procedure for the other two sides of the prism, you will end up with a prism shown below.

Since we are doing slicing and rearrangement o…

Calendrical mathematics of the Ancient Chinese: Part I

To understand how calendrical matters were handled by the ancient Chinese. It is important to learn to count numbers in modulo 60.

It is not known why the ancient Chinese fancied modulo 60, probably the ancestors of the ancient Chinese shared a similar set of extrasomatic knowledge with the ancestors of the Babylonians when they were still in North Africa.

The current norm of handling the grouping of days is to use modulo 7, where we have: Day 1 = Monday, Day 2 = Tuesday, ..., Day 7 = Sunday.

The ancient Chinese, however, used several different styles to count their day of week. One style is to use modulo 10, in which we have: Day 1 = Jia (甲), Day 2 = Yi (乙), ..., Day 10 = Gui (癸). In this way, a Chinese week actually consists of ten days instead of seven.

Sometimes, modulo 12 was preferred over modulo 10 in some occasions. When modulo 12 is used, the days are called differently: Day 1 = Zi (子), Day 2 = Chou (丑), ..., Day 12 = Hai (亥).

Now, to reckon days for a longer period of time, …

The Mathematics of Honeycomb: Part I

The fact that honeycombs is hexagonal is rather well-known. However, this is only a two-dimensional description of the honeycomb. The three-dimensional description of the honeycomb structure is more interesting, but most people are unaware of it.

A real honeycomb has many cells. Each of the cells is a special type of hexagonal prism. Now, the usual hexagonal prism has a flat hexagonal base, but the bottom of the honeycomb cell is not flat, it is a composite structure formed by three rhombuses.

When many of these unit cells are glued together side-by-side, a slab is formed. And when two of these slabs are glued back-to-back, a honeycomb is formed.

The first person who took the trouble to measure the angle of the honeycomb structure is a French astronomer named Giacomo Filippo Maraldi. In 1712, Maraldi measured many samples of honeycomb cells and concluded that the angles of the trapezoidal sides and rhombic bases are always consistent: the smaller angle of the rhombus/trapezium is alway…

Concerning Gohonzon: Part I

The name Gohonzon (ごほんぞん) is the romanization of the Japanese pronunciation of the Chinese word Yu Benzun (御本尊). The word Yu (御) is honorary prefix, while Benzun (本尊) means object of enshrinement in this context. For members of Soka Gakkai (創価学会) and Nichiren Shoshu (日蓮正宗), it may be considered as the object of veneration, at least when they are practicing Gongyo (勤行).

From an artistic point of view, Gohonzon is technically a piece of calligraphy written in mixed Chinese regular script (楷书), semi-cursive scripts (行书) and Siddham scripts, they were first produced by Nichiren (日蓮, 1222-1282), a Japanese monk who lived in the Kamakura period (鎌倉時代). The first Gohonzon (called Yojihonzon, 楊枝本尊) was designed and inscribed by Nichiren in November 19, 1271 (文永8年10月9日), when he was 49 year-old. Between 1271 to 1282, Nichiren actually designed and produced more than 120 mandalas for his disciples, not all of them were identical. But the general trend is that the design of the Nichiren's man…

Concerning Circle and the Constant π: Part II

\(\pi\) is a fundamental constant in mathematics. It is also an important constant in science for it appears naturally in so many applications: in the period of a simple pendulum in mechanics, Maxwell-Boltzmann distribution for gas molecules in thermodynamics, Coulomb's law for electrostatic force in electromagnetic theory, Heisenberg's uncertainty principle in quantum mechanics, Einstein's field equation in general relativity theory. Since \(\pi\) is such an important constant, it should be properly introduced to students, both historically and mathematically. In Malaysian secondary schools, this constant is first taught to a typical 14-year-old when he/she is learning the geometric properties of the circle. In a recent book titled "Essential Mathematics Form 2", published by Longman, the authors wrote, in page 163, the value of \(\pi\) is an estimated value, which is \(\frac{22}{7}\) or 3.142 This remark is technically incorrect. I guess the authors probably…

Concerning Circle and the Constant π: Part I

In most of the secondary school level mathematics reference books found in local bookstores, the mathematical object "circle" is introduced as follows:
A circle is a set of points that are equidistant from a fixed point.There is something wrong with this description. It is rather vague because a set of points could mean the following set of 8 points {P1, P2, P3, P4, P5, P6, P7, P8}:

Although these points are equidistant from a fixed point, this particular set does not form or look like a circle.

This problematic description of "circle" is not entirely the fault of the reference book authors, because it actually originates from page 45 of the Curriculum Specifications for Form 2 Mathematics, compiled in 2002, by the Curriculum Development Centre, Ministry of Education Malaysia.

I must say that it is rather difficult to describe "circle" or any other mathematical objects with words. A better version (but still imprecise) would be:A circle is a closed curve t…

Concerning Hikayat Hang Tuah

The nucleus of the story of Hikayat Hang Tuah was born during the Malaccan period in the 15th century, but its cytoplasm contains elements from Sejarah Melayu and the Johore Sultanate in mid 17th century. The final form of Hikayat Hang Tuah as we now know it was last edited probably in 1710s, 250 years away from the Malaccan period, 300 years away from now.

Some web sources suggest that Hang Tuah was actually a Chinese, and recently a friend of mine raised the same question to me. I personally do not think Mr. Hang is a Chinese, although it is very inviting to think of "Hang" as a word of chinese origin, like what I postulated in the case of Hang Li Po. In fact, my family name was spelt as "Hang" instead of "Ang" during the time of my grandfather's father when he followed his boss Tan Kah Kee (陈嘉庚) to Nanyang.

Now back to Hang Tuah, he is definitely a pseudo-character, probably loosely modelled after a real person - Laksamana Abdul Jamil. Abdul Jamil …

Hexagram No. 31 in the Yi Jing (易经)

The icon and slogan found on the paperbag of Eu Yan Sang (余仁生) chinese drug store:
(咸因)有心才会感恩 (approximately: Be grateful (to your parents), if you have heart)reminds me of an hexagram in the Chinese divinational classic. For the uninitiated, the slogan is a clever manipulation of Xian Yin (咸因) and Gan En (感恩). In the former, the hearts (心) are removed, and does not carry any meaning in modern Chinese. My guess is that the creator of the slogan did not know that the character Xian (咸) is actually a primitive form of the character Gan (感), a verb meaning sense or feel.

Now back to the hexagram business, there are altogether 64 hexagrams documented in Yi Jing (易经, Wade-Giles romanization: I-Ching). One of them is very interesting from a human reproductive perspective because it contains the instructions on how to carry out foreplay before penile-vaginal penetration, it is Hexagram No. 31, known as Hexagram of Xian (咸卦).

Given below is my translation of the interpretation of the hexagramat…

Tunku Halim's New Book

Some days ago, I chanced on a new book by Tunku Halim at MPH. The book title is "History of Malaysia: A Children's Encyclopedia". As a check on the factual accuracy of the book, I turned to the Parameswara section to see what Tunku has to say about him. To my horror, I read: Parameswara was married to Hang Li Po.

Tunku's statement is incorrect for the following reasons.

1. According to Sejarah Melayu, Hang Li Po was a Chinese princess married to Sultan Mansur Shah, not Parameswara.

2. Reputable historians in general doubt the historical existence of Hang Li Po, because the name of the princess cannot be found in the Chinese or Portuguese sources. It is only mentioned in Sejarah Melayu. And Sejarah Melayu is full of historical inaccuracies.

Concerning the origin of the name of the Chinese princess, I offer the following explanation.

The name "Hang" is easy, it is simply a transliteration of the word "汉", meaning Chinese, or the word could possibly deriv…

Concerning Mycophobia

A friend of mine is mycophobic. She is afraid of seeing living mushrooms and couldn't handle them as cooking ingredients, but she can eat them when they are cooked.

I am curious about the exact etiology of mycophobia, but it is not well-documented in the literature. I rarely find any scholarly discussion on mycophobia.

Many suggests maybe mycophobia has a cultural basis. My guess is that mycophobia should not be memetic. It should have a genetic basis. It should be hardwired in the DNA program and is transferrable to the next generation. Although this function is no longer very useful in the modern world.

In the long history of human evolution, some groups of human, possibly living in a region surrounded by many poisonous species of fungi, must have evolved the defense mechanism to avert fungi species, for the fear of consumption of deadly mushroom species. This DNA trait is important for the survival of that particular human groups.

I offer two hypotheses on the mechanism of mycopho…

The Sang Kancil Story of Malacca

The official emblem of the state of Malacca depicts two normal brown-colored chevrotains (kancil in Malay) on each side of the tree of Malacca. However, if the emblem meant to show the chevrotain as encountered by Parameswara in the Sejarah Melayu, then the color of the animal is probably not correct. (The emblem is explained official portal of the state government.)

If you were to follow the Malay classic carefully, the chevrotain mentioned is white in color. The relatively new logo of the City Council of Malacca (the logo was unveiled when Malacca was declared Historical City on April 13, 2003), however, correctly show the two chevrotains in white.

On the other hand, the familiar kancil story purported to explain the founding of Malacca could be a story modified from a folk-tale from Sri Lanka. This fact was first noted by R. O. Winstedt in 1922. Dr. Winstedt was the pioneer in the systematic study of Malay history. He served one term as the general advisor to the Sultan Ibrahim of …

Wuxing and Compatibility Coefficient: Part IV

In Chinese fate calculation, there exists a mapping between the wuxing (五行) and sexagenary components (celestial stem, γ = tiangan 天干 and terrestrial branches, ζ = dizhi 地支). When the birthdate of a person is stated in sexagesimal form, one is able to deduce the xing associated with the eight sexagenary components or bazi (八字).

Normally this mapping is in table form:

A useful set of numerical formulas to handle the mapping is to use

The basic principle to compute the mutual compatibility of two sets of sexagenary birth data is to pair the corresponding sexagenary components of the two individuals and compute their compatibility coefficients.

We can then construct a normalized rank based on the eight compatibility coefficients computed, and "foretell" whether the two individuals can live in harmony.


今天早上八点一条Allianz的保险代理跑来跟我讲关于他们公司的风险分析(risk analysis)和医疗核保(medical underwriting)的程序。有时候一些医疗报告及医疗概率常会被错误诠释,这对受保人是有欠公平的。


1999年11月,35岁的英国妈妈莎丽克拉克(Sally Clark)被指分别在1996年11月29日和1998年1月26日谋杀她两个只有几个月的男婴。这两名男婴去世时分别只有82和58天大。

案子的“专家证人”是一个叫罗伊梅都(Roy Meadow)的小儿科教授,来自英国利兹大学(University of Leeds)。罗伊教授认为这个案子不属于婴儿突然死亡候群症,因为两个连续婴儿猝死的或然率是1/8543 x 1/8543 = 1/72982849,也就是一对7千3百万。



关于罗伊教授概率算法,英国皇家统计学会(Royal Statistical Society)曾在2001年发表过一份声明,表示罗伊教授的此项推导是没有统计学根据的。英国皇家统计学会主席更亲自写了一封信给英国大法官表示对此事的遗憾。

一名叫玛丽莲的律师觉得事有蹊跷,她挖出了一份之前没曝光的病理化验报告。报告里表明莎丽的二儿子是感染了金黄色葡萄球菌(Staphylococcus aureus),应属自然死亡。