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.

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

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.

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.

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.

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.

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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