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...
. . . in reply to Inche Abdul Rahman Rashid bin Ali of Kampong Mertang, Negri Sembilan, Tuan Albakri said it was on August 31, 1955, that the Alliance took office as the ruling party and the new Federal Legislative Council was inaugurated . . . The Keeper of the Rulers's Seal, Tuan Haji Mustapha Albakri, provided the definitive explanation for why 31 August was chosen as the date for the proclamation of independence of the Federation of Malaya. On 27 July 1955, Tunku Abdul Rahman led the Alliance to a sweeping electoral victory, winning 51 of the 52 contested seats. The sole exception was the Krian constituency, where Sulaiman Palestin (d. 1996) was defeated by Ahmad Tuan Hussein of PMIP by 8,685 votes to 8,235. Following the election, a meeting with Donald MacGillivray was immediately arranged for 31 July 1955 to discuss the formation of the first cabinet. The list of ministers was finalized and publicly announced on 4 August. On...
The Mandarin word for male and female are nán 男 and nǚ 女, respectively. The design of the glyph for male, nán 男, is rather simple, it is built by stacking blocks of rice field ( 田 ) and labor ( 力 , 力 is abstracted from a picture of an arm) together. Directly translation of the semantic elements gives us: Male 男: worker 力 in the field 田. image/svg+xml The motive force element can also be placed around any position in the field, e.g. on the right hand side 㽖 The Hokkien word for male and female, on the other hand, are completely different. In Hokkien, male is termed tsa-poo while female is called tsa-bóo. And the recommended glyphs by the Ministry of Education of Taiwan are 查埔 and 查某, respectively. For example 這个 旅行團的 查埔 人 比 查某 人 較少 。 We must give our attention to the glyph 查 since it could be describing a common feature found in both male and f...
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...
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