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'.
Water disruption at Pangkalan Lumpur On 29 March 1877, Yap Tet Loy 葉德來 wrote a letter to Bloomfield Douglas to relay a water-related grievance at Pangkalan Lumpur tin mine. This letter is important because it contains several important metadata: An Indic title was invoked by Yap Ah Loy and he called himself Kaptan Klang dalam Kuala Lumpur. This wooden stamp (shown in mirrored form) can be found in the Sin Sze Si Ya Temple Pioneers of Kuala Lumpur Museum 吉隆坡師爺廟拓荒博物館. This stamp is likely a replacement copy built later since its impression does not perfectly conform to the Chinese characters found in the letter dated 29 March 1877. In the stamp, the Captain styled his residence/office as Tet Sang of Klang 吉隆德生. The two place names, Kuala Lumpur کوال لمفر and Pangkalan Lumpur فڠکالن ل...
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...
從吉打 Lembah Bujang 布央谷開車回家，經過怡保的時候剛好是晚餐時間。 在麥當勞用過晚餐後，發現開車的我好像也有點睏了，於是決定在怡保睡一覺。我們把車開到 Bandar Meru Raya ，然後跟 Casuarina 要了一間房間。 隔天早我見小兒子辛円睡到很遲還不願意起床，就拿起手機給他照了一張相 1 。 原想給照片標上：阿円睡到日上三竿 2 ，大太陽曬屁股還不起身。但，住在我頭腦裡面的另外一個人問我： 三竿 到底是 : 點？ The timestamp of the photograph was 9:12:09 morning (28 May 2017). The explanation given by National Academy for Educational Research of Taiwan 台灣國家教育研究院 is：太陽已上升到 三根 竹竿相接的 高度 。表示時候不早了。This explanation suggests that ‘三' (three) is to be interpreted vertically and absolutely, instead of umbrally and relative to the physical pole. Given the fact in the phrase 三竿 (three poles) was recorded by an observatory officer, it is unlikely that the former intepretation is correct. 大太陽曬屁股還不起身嗎？ 於是我便上網查了一下該成語的出處：「日上三竿」應該是出自《 南齊書・天文誌上・日光色 》的一段話。《南齊書・天文誌》是南齊政府天文局官員根據每日的觀測日記整...
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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