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:
$$\frac{V_{\rm honeycomb}}{(S_{\rm honeycomb})^{3/2}} = \frac{\frac{3\sqrt{3}}{2}a^2 b}{\left(6ab + \frac{6}{\sqrt{8}}a^2\right)^{3/2}}= \frac{\sqrt{2}\lambda}{(\sqrt{2}+4\lambda)^{3/2}}$$
where \(a = \lambda b\). For honeycomb, the value of \(V/S^{3/2}\) is approximately \(0.0901\) if \(\lambda_{\rm drone} = 2.64\), which is pretty close to the theoretical maximum. On the other hand, the same ratio is approximately \(0.0866\) for a prism with hexagonal top and hexagonal bottom.
Honeybees and their honeycomb unit cells
This result is not too surprising from an evolutionary perspective. Suppose initially all of the ancestors of the honeybees were building their cells the easier way with hexagonal top and hexagonal bottom. Owing to genetic mutation, a new group of honeybees evolved the ability to construct cells with rhombic bottom. Over time, this group of honeybees gained evolutionary advantage for able to economize their resources, they were able to reproduce more efficiently with the same amount of food resources, and slowly displaced all of the old honeybee groups.
Note that a small difference of 4 percent in this ratio is sufficient for Nature to decide which honeybee species were to be eliminated. Nature is very calculative in her rule of natural selection.
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...
. . . 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...
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