Showing posts from April, 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 \(A\): $$A \le \tfrac{1}{4\pi}L^2$$ This inequality is called the isoperimetric inequality. It simply means that the ratio of \(A/L^2\) cannot be greater than \(\frac{1}{4\pi}\). This result can be generalized to three-dimensional space to give: $$V \le \tfrac{1}{6\sqrt{\pi}}S^{3/2}$$ This shows that the theoretical maximum \(V/S^{3/2}\) (when you have the least surface for a given volume) is \(\frac{1}{6\sqrt{\pi}} = 0.0940\). For honeycomb cells, the ratio of \(V\) to \(S^{3/2}\) is: $$\frac{V_{\rm honeycomb}}{(S_{\rm honeycomb})^{3/2}} = \frac{\frac{3\sqrt{3}}{2}a^2 b}{\left(6ab + \frac{6}{\sqrt{8}}a^2\right)…

The Mathematics of Honeycomb: Part II

As explained in Part I of the article, instead of building their house with the easy method of hexagonal top + hexagonal bottom, honeybees are building their cells with hexagonal top + 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 70°32'. 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 shown that the surface area \(S_0\) and the volume \(V\) of this prism are: $$S_0= 6ab + \frac{3\sqrt{3}}{2}a^2, \quad V = \frac{3\sqrt{3}}{2}a^2b$$ 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 …

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. The base of a honeycomb unit cell is a composite surface formed by three rhombuses A real honeycomb has many cells. Each cell is a special type of hexagonal prism. The usual hexagonal prism has a flat hexagonal base, but the bottom of the honeycomb cell is not flat. The base of a honeycomb unit cell is a composite surface 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. When the bases of many honeycomb unit cells are glued together 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 sam…

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…