Thursday, November 08, 2012

On the Periodicity of Chinese Sexagenary Indices

I have written a short note on the periodicity of Chinese sexagenary indices (traditionally known as bazi 八字) after I read the book by Hee on the same subject, many years ago. I was motivated primarily by a section of her book. In page 22, Hee wrote:
...because of the numerous possible combinations, it takes 60 years for the same set of year pillars to repeat itself (by comparison, a set of month pillars repeats itself after just five years). Therefore, if you have a certain day and time, the set of four pillars will repeat itself in 60 years. However, since the same day may not appear in exactly the same month - and even if it is in the same month, the day may not be found in the same half month - it takes 240 years before the identical four pillars appear again.
Basically, Hee's claims are as follows:
  1. When considered independently, the periodicity of the year-index is 60 years, which is obviously correct.
  2. When considered independently, the periodicity of the month-index is 5 years, which is also correct.
  3. In general, the complete set of sexagenary indices will repeat itself after 240 years. This claim is incorrect and will be illustrated by a counterexample.
It appeared in the paragraph quoted above that Hee believes that the interval between two successive sexagenary indices is regularly maintained at 240 years. I believe this assertion is flawed when I first read it and I believe I know enough math to check the validity of this claim. First of all, it is easy to formulate an expression for the interval g between two successive sexagenary indices. It is irregular and it is given by

g = 60(365ll2)

As a numerical example, the sexagenary indices (year, month, and day) associated with March 18, 1984 are given by:

甲子, 丁卯, 辛亥

Using calendrical algorithm by Reingold and Dershowitz, we can easily compute other dates with similar sexagenary indices, and they are tabulated in the table below.


In the table, we see that the indices are repeated 60 years after 1984, namely on March 3, 2044. However, after 2044, the next identical indices appear only after 360 years in April 5, 2404. This clearly contradicts Hee's assertion (by the way, Hee is an accountant by training. It is not clear to me as to who is the originator of this theory, probably her mentor, Raymond Lo of Hong Kong?). Another interesting observation is that some entries in the table share identical month and day with March 18, 1984. (i.e. March 18, 4 and March 18, 3964). They appear approximately 1980 years (or exactly 723180 days) before and after March 18, 1984. This is no coincident because suppose f and f + g' are two successive fixed day numbers with identical month and day, then it can be shown that the interval g' is given by

g' = 365l + 366l4

For g and g' to coincide, we need to solve the following Diophantine equation

21900l1 + 60l= 365l + 366l4

of which one of the solution is (l1l2l3l4) = (33, 8, 1575, 402). The following additional constraints apply when solving the Diophantine equation: The ratio of l4  to l3 is approximately 1/4, and that for large l1  we have l= [l2/4], where [...] is the floor function.

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Thursday, September 01, 2011

Why You Should Not Eat Too Many Fruits at One Time

To answer this question, you need to appreciate that the absorption of the frustose molecules in your body is not straightforward, it requires some help from a fructose transmembrane transporter called GLUT5, which is specialized in this task.

Let's begin with our DNA program. The red line in the diagram below marks the location of the SLC2A5 gene in Human Chromosome 1.


The SLC2A5 is the genetic program which carries the instruction to manufacture the GLUT5 in your small intestine. The size of the SLC2A5 program is 53,371 base-pair, approximately 6.5 kilobytes if stored in binary codes (quite a small program!).

The purpose of the GLUT5 transporter is to carry and transfer fructose in intestinal fluid to the blood line across the intestinal epithelial membrane. You can imagine GLUT5 as a truck specially designed to carry fructose molecules. The number of GLUT5 transporter in your intestine is finite and if you eat too many fruits, you will not have enough GLUT5 transporter to carry the fructose molecules and put them into the blood line. The average absorption threshold for one-time fructose consumption is 0.15 mol (about 4 apples)

When this happens, the intestinal osmotic pressure increases and water molecules in the blood line will diffuse into intestine. The fructose molecules will then get flushed to colon and consumed and fermented by bacteria living there.

Fermentation of fructose releases gases like methane, hydrogen and carbon dioxide and propels intestinal peristalsis and, lo and behold, it's toilet time!

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Sunday, August 28, 2011

Should You Migrate to the New Ujrah Scheme for Your PTPTN Loan?

My wife recently asked me whether or not she should participate in the new ujrah scheme as announced by PTPTN. First of all, for new contracts, the introduction of the ujrah scheme at 1% per annum would definitely save the student some interest. For example, for a given loan amount to be repaid in 180 months, the ujrah of 1% is approximately equivalent to an compound interest rate of 1.9%. Thus this scheme save the student 1.1% per annum.

However, for the old contracts signed-off before 2008, the situation is a little bit tricky, and PTPTN offers no guide to the graduated students. To close this gap, I developed a simple criterion to assist her in decision making. And the answer can be yes or no, depending on your current debt level and your remaining repayment periods.

First of all, the total amount of monies to be repaid to PTPTN, if you choose to participate in the Ujrah scheme can be calculated as:


where P'' is your reduced balance as of June 1, 2008 (datum), ΔP is difference between your reduced balance as of now and that of datum.


However, if you you choose to stay with the original compound interest scheme, the total amount of monies to be repaid (3% annual interest over a period of m months) is:


For the migration to be meaningful, the total amount of monies repaid under the Ujrah scheme must be less than that of the original compound interest scheme, thus we must have the following inequality:


From this inequality, we can rearrange and formulate a criterion to decide whether or not you should migrate to the new scheme:


For example, suppose your reduced balance on June 1, 2008 is RM22,500 and your reduced balance as of now is RM16,000. Then λ = (22,500-16,000)/22,500 = 0.289. Suppose you plan to settle your debt within 140 months, then φ(140) = 0.195. Since 0.195 < 0.289, it would be unwise to switch to the Ujrah scheme now.

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Friday, July 22, 2011

Visual Interpolation and Partially Occluded Nude

The human brain evolves to perform visual interpolation, visual extrapolation, and visual completion.


The image on the right depicts an Assam tiger occluded in the Kaziranga National Park, India. When confronted with such a situation in the wild, the human brain must be able compute with the electrical signals transmitted by the photosensors in the eyes, distinguish between different shades of yellow and brown colors, interpolate between the color shades, isolate unimportant information (i.e. grass), and generate a group of electrical signals to represent the predator. When these signals are relayed to the amygdala, the emotional CPU of the brain, a "run" signal will be generated and get cascaded to motor control CPU of the brain.

A human brain which failed to perform the above visual interpolation is bound to become the lunch of another species. So, our brain has the natural tendency to do interpolation and enjoy doing so.

Now, let's turn our attention to the image on the left, which depicts a Chinese model named Zhou Weitong (周韦彤). In this photo, she was captured by a skillful photographer in a natural pose, in which the areola regions were occluded by her arm and hair. This can provide a visual stimulus for the brain to happily perform the visual interpolation and visual completion for the areola regions, and explains why a partially occluded nude is more seductive than a complete nude.

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Monday, July 04, 2011

Puranic Cosmology of the Ancient Indians

The Judeo-Christian time scale as computed by James Ussher in 1650, that the universe was created single-handedly by Yahweh in October 23, 4004 BC (proleptic Julian calendar), is rather simple if you were to put it side by side with the Puranic model conceived by the ancient Indians. The Puranic model of time assumes that the world goes through a manvantara cycle of four yugas or ages, namely, the Krta yuga, the Tretā yuga, the Dvāpara yuga, and the Kali yuga.

The durations of the four yugas is carefully chosen by the ancient Indians so that (a) they form an arithmetic progression; (b) they sum to a multiple of ten; (c) their arithmetic sum is 4.32 million years.


Furthermore, the model also assumes the following conversion factors for dealing with longer time span,

1. 1,000 manvantara cycles is equivalent to one Brahmā day (approx. 4.32 billion years);
2. The next 1,000 manvantara cycles is equivalent to one Brahma night;
3. One Brahma day and one Brahma night is equivalent to one Kalpa (approx. 8.64 billion years)

Surprisingly enough, the value of one Brahma day (or night) is very close to the present estimate of the age of the Earth (4.54 billion years), and that the value of one Kalpa is very close to the present estimate for the age of the solar system (10 billion years).

The ancient Indians were moral absolutists and they assumes that the world moves from the beginning of the Krta yuga (morality = 1) to a progressively more morally degenerate periods until the end of the Kali yuga (morality = 0). The rate of moral degeneration is not explicitly formulated by the Indians, and for computational purposes, we may assume that the morality degenerates linearly with respect to time (model I), or that the inter-yuga morality difference is constant (model II).


The present Kali yuga began with the Mahabhrata war (traditionally dated to 3102 BCE), thus we are currently near the beginning of the Dark Age according to the Puranic time scale.

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Tuesday, May 03, 2011

Why do the words "inflammable" and "flammable" mean the same thing?

I attended an introductory course on the Laws and Regulations on Scheduled Waste in Malaysia recently and I was particularly annoyed when the trainer highlighted a "mistake" in the English version of the Environmental Quality Scheduled Wastes Regulations. She ridiculed the use of the word "inflammable" to mean "easily burst into flame" in the legal document and wondered why such "obvious" mistake can escape the eyes of so many law practioners. I immediately objected to her interpretation and pointed out that the use of the word "inflammable" is correct in its context, but unfortunately she did not believe in me.

First of all, you need to believe that the lawyers are not idiots and the use of correct terms in a legal document is of paramount importance to them. So, the probability of lawyers making such an "obvious" mistake is practically zero. So, if you can accept that lawyers are not stupid, move on to my next paragraph, else you may stop reading here.

If the use of the word "inflammable" is correct, what is the fallacy in the trainer's interpretation of the word? To answer this question in full, you need to have some understanding on the etymology of the words "inflame", "inflammable", "flame", and "flammable".

Contrary to popular belief, the word "inflammable" is not the negative form of "flammable" as the word "inflame" is not formed by the Latin prefix "in-" and the verb "flame". It is, in fact, a word on its own right. The word "inflame" was first used by John Wycliffe in his English translation of the Bible in 1382.

Loo! forsothe the day shal cumme, brennynge as a chymney; and alle proude men, and alle doynge vnpite shuln be stobil; and the day cummynge shal enflawme hem, saith the Lord of oostis, whiche shal not leue to hem rote and buriownyng. (Malachi 4:1)
in which Wycliffe obviously meant to use the word to convey the meaning of "burst into flames" or "to set ablaze". Therefore, naturally, adding a suffix "-able" to the word "inflame" has the following effect:

"inflame" + "-able" = capable of being inflammed
In the history of the English language, this combination was first used in 1605 in a chemistry book.

1605 Timme Quersit. i. xiii. 54 The sulphurous substance and inflamable matter.
Next, I would like to touch on the word "flammable" which is obviously formed as follows:

"flame" + "-able" = capable of emitting flame = "inflammable"
The word "flammable" was first found in a 19th century translation of Lucretius by Thomas Busby, about 200 years after the word "inflammable" was introduced by Timme.

1813 Busby tr. Lucretius I. 731 That igneous seeds, no longer linked To matter flammable, become extinct.
I hope by now you are convinced that the words "inflammable" and "flammable" carry exactly the same meaning.

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Saturday, April 09, 2011

On the Origin of the Chinese Character Representing the Ancestor (祖): Part One

Last Tuesday was officially the Day of Qingming (清明). Calendrically, this day is one of the twenty-four seasonal markers in Chinese lunisolar system, but it was later designated as the "day" to perform ancestral worship (祭祖).

In Malaysia, the Chinese will usually pick the nearest weekends before April 5 to perform the act. Several rounds of grave gatherings are usually required because we tend to have many great grandparents, grandparents and/or parents, both patrilinearly or matrilinearly.

About two weeks ago, I went to the graves of my great grandparents (Hong Longwang 洪隆往 and Xie Minniang 谢敏娘) in Bukit Kangkar. Both of my grandparents died on the same year. My great grandfather died on February 10, 1949 (己丑年正月十三日酉时), while his wife died approximately nine months later on October 23 (己丑年九月二日未时). Nobody knows when they were born as the birth dates are not engraved on the tombstones. And possibly neither of them did not know their birthdays. I was told that my great-grandfather was in his sixties when he died. He was once a cook to Tan Kah Kee (陈嘉庚) before he saved enough money to settle down in Muar with some rubber plantations, probably in the 1930s.

As Chinese, we all roughly know that grave gathering on Day of Qingming is part of our unique culture coded based on Confucian obedience and filial piety. I guess that is the basic understanding for the subject for an average Chinese. I would like to, in this article, shed some new lights on the tradition of our ancestral worship, as it was practiced eons ago.

I would like to do this by analyzing the chinese character representing the word "ancestor" itself, that is, the word Zu (祖). The character is actually a union of two pictographs. On the left is the word Shi (示), etymologically, it means the altar or table for ancestral worship.


The pictograph on the right is the word Qie (且), it means the ancestral tablets stylized in the form of human male external reproductive organ, that is, the penis. The word Zu, is therefore, a reminiscence of an ancient practice called phallic worship. Similar arguments purported to explain the two characters can also be found here.

Traces of this ancient practice can still be found in places like remote regions that are rather isolated from mainland, such as Bhutan and Japan. Both of them were probably once shared the same set of beliefs as the ancient Chinese in the course of the evolution of their civilizations.

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Tuesday, August 10, 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 moon occurs post meridian, then the Hari Raya Puasa will coincide with the third day in the corresponding Chinese month.
The two observations hold true for the last 20 years (1991 - 2009), but it is not known whether or not they can be applied to other years.

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Friday, August 06, 2010

关于“司空见惯”这一成语的出处(二)

《天南地北谈华语》是AIFM高级主播黄华敏制作的电台节目,两天前这个节目的主题是“司空见惯”这一句成语。这就碰巧跟我之前的一篇贴文的主题一样。

在这里我想针对黄华敏对“司空见惯”这一成语的出处和历史背景的理解做出一些点评。

节目里头说李绅请了一个漂亮的歌妓来给陪刘禹锡唱歌劝酒。这里用了“请”字,想必黄华敏是认为李绅是从外面请来了一名歌妓来娱宾。其实该名歌妓不是从外头请回来的,她是住在李绅的家里的家妓。所以李绅才能“以妓赠之”。

节目里头只给出了刘禹锡当时作的七言律诗的后面两句,并解释说这是刘禹锡作来送给主人家的。其实那一首诗的前面两句“鬟髯梳头宫样妆,春风一曲杜韦娘”是用来称赞该名妓女的美丽及歌艺。后面两句“司空见惯浑闲事,断尽江南刺史肠”是刘禹锡因为羡慕李绅家里有那么多的家妓的一时感触。就这样,刘禹锡只用了二十八的字就得到了他心仪的女人。

假设黄华敏见过《本事诗》的原文,那我只能整理出以下的结论:
(一)她不了解唐人的性道德观。之前我说过,中国在唐代以前,对性和男女关系所持的态度是高度开放及不受礼教约束的。
(二)她不想讨论故事中“以妓赠之”的这一部分。但是如果不讨论的话,我认为我们没有办法真正的理解刘禹锡的“司空见惯浑闲事”到底指是什么。

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Tuesday, July 27, 2010

关于“司空见惯”这一成语的出处

“司空见惯”是一句很普通的成语。据《新华成语词典》,这一成语出自唐代孟棨所编的《本事诗》。它通常用以形容某事常见,不足为奇。有鉴于《新华成语词典》给出的典故不是很完整,我想在这里补充一下。


在《本事诗》的第一章里头,孟棨记录了以下的文字:
刘尚书禹锡罢和州,为主客郎中、集贤学士。李司空罢镇在京,慕刘名,尝邀至第中,厚设饮馔。酒酣,命妙妓歌以送之。刘于席上赋诗曰:“鬟髯梳头宫样妆,春风一曲杜韦娘。司空见惯浑闲事,断尽江南刺史肠。”李因以妓赠之。
在唐代,一般有钱人都有自己的歌舞团和家妓,客人到访时,主人家就会安排她们出来表演及陪酒。有一次诗人刘禹锡到司空大人(大概相等于现在的工程部长)李绅的家里做客,对其中一名歌妓有好感,随即便做了“断尽江南刺史肠”这样肉麻的诗句来泡妞。主人李绅一听马上知晓其意,因为仰慕刘禹锡的才华已久,便把该名歌妓送给了刘禹锡带回家享用。

这个故事给出了唐代歌妓的定位:基本上她们只是主人娱乐客人的工具。只要客人喜欢,主人就可以将她们当礼物来送给客人,其中无需考虑歌妓自身的感受。

更重要的是,这个故事说明在唐代(和唐以前),中国人的性观念的很自由和高度开放的,所谓礼教基本上只是理论上的东西,没有人会把礼教应用在男女关系和性方面。高流动的性关系乃属司空见惯的事。唐以后,宋儒开始大力提倡礼教,才导致中国人性文化倒退和性观念彻底改观。

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Sunday, May 23, 2010

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:


This is approximately 0.007 percent. From the probability table above, we noted that if we were to select five people at random, chances are 97.3% that they will all have unique birthday, this is consistent with our intuition.

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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 values. Suppose we use the following notation:


Then it is obvious that nj are governed by the following Frobenius equation (also called the Diophantine equation of Frobenius):


In general, there are m sets of solutions to the Frobenius equation and we denote the i-th solution as


The general term of the summation representing P(E) is the probability that there are exactly n1 unique birthdays, n2 pairs of equal birthdays, n3 triples of equal birthdays, ... , nr-1 (r-1)-tuples of equal birthdays, which is given by


Therefore, the probability that r or more birthdays are equal is given by


McKinney was using an IBM 7090 computer to compute the probabilities and he stopped at r = 4. For r = 5, the computation for one case of n is estimated by McKinney to take more than 2 hours. Thus it would take him 26 days to arrive at the solution if the computation was attempted.

I tried to repeat McKinney's calculation for second problem in Mathematica 7 and detected that the third/fourth decimal place of the probabilities given in original paper was not correct, possibly due to roundoff errors in his IBM 7090.


It took only 22 seconds to compute the probabilities in Mathematica for the case of r = 5. Not surprisingly, I ran into memory problem when the case of r = 6 is attempted as the number of terms in the summation is expected to exceed 15 millions. In Part II of the article, a numerical example will be given to illustrate the use of the formula for a simple case.

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Thursday, April 15, 2010

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 not too surprising from an evolutionary perspective. Suppose initially all of the ancestors of the honeybees were building their cells the easier way with hexagonal top and hexagonal bottom.

Then owing to genetic mutation, a new group of honeybees evolved the ability to construct cells with rhombic bottom. Over time, this group of honeybees gained evolutionary advantage for able to economize their resources, they were able to reproduce more efficiently with the same amount of food resources, and slowly displaced all of the old honeybee groups.

Note that a small difference of 0.02 or 2.5% in this ratio is sufficient for Nature to decide which of the honeybee species were to be eliminated. Nature is very calculative in her rule of natural selection.

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Tuesday, April 13, 2010

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 of parts on the prism, the volume of this rearranged prism is conserved. However, the surface area S1 of this reformed prism no longer the same as S0. It is not difficult to show that the S1 is


The difference between the two surface areas (i.e. the amount of wax saved) is thus:


To minimize the amount of wax used, the honeybees are interested in maximizing the value of ΔS. This is achieved by setting the first derivative of ΔS to zero:


The solution to this equation is: x = a/√8. If you plug this value into the honeycomb cell geometry, you will understand why the honeybees choose the rhombic angle to be 70o32'.

Nature is a stingy mathematician.

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Sunday, April 11, 2010

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 ancient Chinese would combine their nomenclatures of modulo 10 and modulo 12 to form a system in modulo 60. During Zhou Dynasty (and possibly during Xia Dynasty), modulo 60 is the common system used for recording their dates when they carried out divinational activities with oracle bones. An example of the oracle bones is available here.


Following Dershowitz and Reingold, the conversion between the moduli of 10, 12, and 60 is pretty straightforward, and it is handled by the following set of equations:


where amod is the adjusted modulo function. For example x amod 12 can be easily implemented in Microsoft Excel as:


In Part II of the article, the modulo 60 system will be extended to describe year, month, and hour.

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Friday, April 09, 2010

The Mathematics of Honeycomb: Part I

The fact that honeycombs are 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 is unaware of it.

Technically, the basic unit of a honeycomb is special type of hexagonal prism bound by six trapeziums, the familiar hexagonal top and a base formed by three rhombuses. When many of these basic units 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 always about 70o. By postulating that the rhomboidal angle and trapezoidal angle are exactly equal, Maraldi was able to compute this angle exactly, that is, 70o32'.

Several years later, a French biologist named René Antoine Ferchault de Réaumur took up the same problem and postulated that the angle of the rhombic base is related to the minimization of the construction material of honeycomb. This make sense from an evolutionary point of view for nature will select and favor bee species that is able to economize its resources. Réaumur asked this question to his Swiss friend Johann Samuel König. Being a mathematician, König was able to utilize his calculus skill to solve the problem. König's result was 70o34', which disagrees with Maraldi's result by 2'.

In 1743, the Scot mathematician Colin Maclaurin gave the problem a fresh shot and solved the problem by geometric method, and concluded that the rhomboidal angle is 70o32'. This result is similar to that of Maraldi's. Maclaurin also pointed out that there was a mistake in König's earlier computation. König's results was off by 2' because there was a misprint in the logarithm table he used.

In Part II of the article, I will give a calculus-based demonstration on why the rhombic angle of the honeycomb is 70o32' or cos-1(1/3).

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Wednesday, April 07, 2010

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 mandala was getting more complicated as time progressed.

As the lay arm of Nichiren Shoshu, Soka Gakkai and its members were originally worshiping the Mandala inscribed by Nikken (日显), possibly based on official wood version enshrined by Nichiren Shoshu in their head temple (Taisekiji, 大石寺) at the foot of Fuji Mountain.

Unfortunately, when Soka Gakkai was excommunicated by Nichiren Shoshu in November 29, 1991, they were prohibited from using Mandala No. 67 as their Gohonzon. Luckily, in less than 2 years, Soka Gakkai found a solution to their Gohonzon in 1993, when Senda Narita (成田宣道), chief priest of a temple called Joen-ji (浄円寺) in Oyama City, Tochigi Prefecture (栃木県小山市), offered Soka Gakkai to use the mandala enshrined in his temple.

However it should be noted that the Joen-ji's version is actually not produced by Nichiren himself, but copied by Nichikan (日宽) in July 17, 1720 (享保5年6月13日) from one of the Nichiren's originals. Nichikan was the 26th High Priest of Taisekiji.

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Sunday, April 04, 2010

Concerning Circle and the Constant π: Part II

The constant π is a fundamental constant in mathematics. It is also an important constant in science for it arises naturally in so many applications such as 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 π is such an important constant, I think it should be properly introduced to students, both historically and mathematically. In Malaysian secondary schools, this constant is taught when a typical 14-year-old in Form 2 is learning the properties of circle. In a recent book titled "Essential Mathematics Form 2", published by Longman, the authors wrote, in page 163,
the value of π is an estimated value, which is 22/7 or 3.142
This sentence is very odd from both grammatical/mathematical point of view. I guess the authors probably intend to say: the commonly used approximate value of π is 22/7 or 3.142.

Because of the way these two approximations are represented in textbooks and reference books available in Malaysia, they are usually misunderstood by students and teachers of mathematics as the value of π. My guess is that if you ask this question
Of the two values, which one would you choose to better represent π? (A) 3.142 (B) 3.14159
Most of the students and teachers will choose option (A), which is wrong.

In another local book titled "Focus Goal Additional Mathematics", published by Pelangi, the authors wrote the following in page 151:
The constant π was introduced by a German mathematician Gottfried Wilhem von Leibniz. The value of π is approximately 3.142... The degree of accuracy of π can be determined by the formula: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9...
This sentence is problematic for the following reasons:

1. First, Leibniz's name was never associated with the introduction of the notation of π in the history of mathematics. The first person who use π in its present meaning was a Welsh mathematician named William Jones, when he wrote π = 3.14159&c in 1706. But it was the Swiss mathematician Euler who made the notation popular when he used it in 1737.

2. Second, the Nilakantha-Gregory-Leibniz series for π is one of slowest convergence series for π. One would require 1 million terms to compute π correct to 5 decimal places. I would rather use the Zu-ratio 355/113 to obtain a numerical value of π (If you divide 355 by 113, you get 3.141593, this is accurate up to 6 decimal places). Thus it is very inefficient to use the Nilakantha-Gregory-Leibniz series to compute π accurately. However, through a certain manipulation, the series can be made to converge faster.

As the web is full of inaccurate stuffs, it is pretty easy for the uninformed to obtain wrong information and put them in their books.

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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 traced by the complete set of points equidistant from a fixed point.
You may also want to find out how "circle" is described in Wikipedia and MathWorld. For me, the best way to describe "circle" is still to employ the mathematical language, such as:
r = a, 0 ≤ θ ≤ 2π (in polar coordinates)

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Saturday, April 03, 2010

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 was once the most powerful man in the Johore Sultanate. A lot of the story narrated in Hikayat Hang Tuah was actually modelled after the real events which took place between 1650 - 1680.

Our local expert on the Hikayat Hang Tuah is Kassim Ahmad. Kassim studied in University of Malaya, Singapore and graduated in 1959. His B. A. final year paper, supervised by J. C. Bottoms, was a research on the characterization in Hikayat Hang Tuah. This research work was later chosen to be published by Dewan Bahasa dan Pustaka in 1966. Between 1963 to 1966, Kassim was a lecturer in the School of Oriental and African Studies (SOAS), London.

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Thursday, April 01, 2010

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 hexagramatic lines of the diagram.

初六:咸其拇。
Yin line No. 1: (As a first check of her sexual receptivity), touch her big toe.

六二:咸其腓。凶,居吉。
Yin line No. 2: Caress her calves. This may cause her to react violently (if she does not like you). But if she abide, then you are lucky and may proceed to the next step.

九三:咸其股,执其随,往吝。
Yang line No. 3: Caress her thigh and continue the movement upwards along her thigh. Going forward in this way may cause you to regret in future.

九四:贞吉,悔亡,憧憧往来,朋,从尔思。
Yang line No. 4: However, if you are firm in your decision, you will be happy and will have no regret. Morever, if she seems sexually perturbed, you may continue to do whatever you wish to do.

九五:咸其脢,无悔。
Yang line No. 5: Caress her back and her upper body (After this step, she should be fully aroused and receptive. At this point, your testosterone level will become high enough that you are not able to control your action consciously), there will be no occasion for repentance and there is no turning back.

上六:咸其辅、颊、舌。
Yin line No. 6: Kiss her lips, cheek, and tougue. (This will be the final step before you perform vaginal penetration)

This shows that, contrary to popular belief, the Chinese peoples in the Zhou Dynasty were already equipped with knowledge in reproductive psychology.

A different english translation of the same hexagram given by the Scotish sinologist named James Legge is available here.

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