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...
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 proc...
In 1680, Andreas Müller (b. 1630, d. 1694) published a small book containing a collection of the Lord's Prayers in 83 languages. The work was later revised and expanded by other authors in 1700, 1705, and 1713. The following discussion examines the Malay and Chinese texts as presented in the revised edition published in 1713. The Lords Prayer in above a hundred languages, versions, and characters, published in 1713. Oratio Dominica ΠΟΛΥΓΛΩΤΤΟΣ ΠΟΛΥΜΟΡΦΟΣ. Nimirum Plus Centum Linguis, Versionibus, aut Characteribus Reddita & Expressa. Editio Novissima. Speciminibus variis quam priores comitatior. The work is built upon the earlier publication of Müller although the title page of the book did not explicitly mention any editor or author. In the Malay text, the Lord's kingdom 國 was rendered as ra'jat, using an old spelling system devised by Van Ophuijsen. It is...
火者亞三 先生是中古馬六甲時代很出名的一條水。 Khoja Hassan (b. 1473, d. 1521) was a rather colorful mantri in Melaka. He was the maritime mantri (水官 Laxamana 1 ) in Melaka, he took over the role from his father-in-law, a certain mister Tuah in Sejarah Melayu. The role was then passed to his brother-in-law 2 when Melaka was about to be sacked by the Portuguese. It is clear from this example that nepotism is deeply integrated in Asiatic court politics. He was of Southern Chinese descent (probably Fujian or Guangdong), but his name reveals that he was a Muslim Chinese. Khoja is linked the Persian word khvâjəh ( خواجه ) and it is originally used to describe a class of Muslim converts in India. Amur leopard by Kwon Jung-soon Given his familarity with Chinese matters, he was recruited by the Portuguese after the fall of Melaka, as the envoy of the new Portuguese Melaka. When he was in China, he became the play companion of the 10th Ming Emperor (Zh...
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