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 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 70°. By postulating that the rhomboidal angle and trapezoidal angle are exactly equal, Maraldi was able to compute this angle exactly, that is, 70°32'.
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 70°34', which disagrees with Maraldi's result only 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 70°32'. 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 70°32'.
1957/0118652W In the Court of the Senior Magistrate at Kuala Lumpur Civil Suit No. 138 of 1898. In the matter of the Estate of Yap Ah Loy, deceased, between Yap Hon Chin (28) Yap Loong Shin (23) Yap Leong Soon (18) Yap Kim Neo Yap Leong Sem by his next friend Ong Chi Siew Yap Leong Fong Plaintiffs and Kok Kang Keow (48), otherwise called Kok Ngeo Nga who is sued as Administratrix of the Estate and Effects of Yap Ah Loy, deceased. On the tombstone of Yap Ah Loy, we were given the following list: 1 隆興 (b. 29 December 1869, d. 5 January 1933), 2 隆盛 (Loong Shin, b. 4 April 1875, d. <1925?), 3 隆顺 (Leong Soon, b. 6 March 1880, d. 8 December 1907, died when he was only 27.8), 4 隆發 (b. 10 August 1882, d. 21 September 1900, died when he was only 18...
. . . I pointed out how the Bandar, in refusing to accept terms and in fighting against his Sovereign, was a rebel, and that all who assisted him were rebels. That, as he knew well, two out of his three captains, China, had sided with the Bandar, and had given him (the Bandar) more than moral support in his recent operations. (I here showed him the Chinese document attached, which is an order of the head of the Macaos, Wong Hin, to any of his people to render prompt assistance at any time and in any way to the Bandar) . . . . after some more conversation we parted, but shortly afterward he brought to our quarters the two captains, China, I had been advising him about, Wong Hin and Ugoli Kim, and said that, as the Governor had sent me to assist him, he had made up his mind to put the management of everything into my hands . . . C. B. Plunket, Inspector-General of Police of the Straits Settlements November 30, 1874 Pa...
Ramadan is the month in which the Quran was revealed as guidance to man and clear proof of guidance, and criterion (of falsehood and truth). So when you see the new moon you should fast the whole month. Surah al-Baqarah, verse 185. English translation by Ahmed Ali (1993) Al-Quran: A comtemporary translation, Princeton University Press, Priceton. The original text reads: شَهْرُ رَمَضَانَ ٱلَّذِىٓ أُنزِلَ فِيهِ ٱلْقُرْءَانُ هُدًۭى لِّلنَّاسِ وَبَيِّنَـٰتٍۢ مِّنَ ٱلْهُدَىٰ وَٱلْفُرْقَانِ ۚ فَمَن شَهِدَ مِنكُمُ ٱلشَّهْرَ فَلْيَصُمْهُ ۖ وَمَن كَانَ مَرِيضًا أَوْ عَلَىٰ سَفَرٍۢ فَعِدَّةٌۭ مِّنْ أَيَّامٍ أُخَرَ ۗ يُرِيدُ ٱللَّهُ بِكُمُ ٱلْيُسْرَ وَلَا يُرِيدُ بِكُمُ ٱلْعُسْرَ وَلِتُكْمِلُوا۟ ٱلْعِدَّةَ وَلِتُكَبِّرُوا۟ ٱللَّهَ عَلَىٰ مَا هَدَىٰكُمْ وَلَعَلَّكُمْ تَشْكُرُونَ ١٨٥ The first day of the 10th month in Islamic lunar calendar, \(s\), is know as Eid al-Fitr عيد الفطر . A...
Sunday, 3 February 1822 1 I walked this morning round the walls and limits of the ancient town of Singapore, for such in reality had been the site of our modern settlement. It was bounded to the east by the sea ( BR ), to the north by a wall ( RMN ), and to the west by a salt creek or inlet of the sea ( NCB ). The inclosed space is a plain, ending in a hill of considerable extent, and 150 feet in height. The whole is a kind of triangle, of which the base is the sea-side, about a mile 2 in length The wall, which is about 16 feet in breadth at its base, and at present about 8 or 9 in height, runs very near a mile from the sea coast to the base of the hill, until it meets a salt marsh. As long as it continues in the plain, it is skirted by a little rivulet running at the foot of it, and forming a kind of moat; and where it attains the elevated side of the hill, there are apparent the remains of a dry ditch. Since ...
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