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.
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...
如果你或你的公司不還稅， 內陸稅收局 的人就會寄一封公函來罵你並跟你討錢。 在馬來西亞，所有的政府公函，都會印上 Urusan Seri Paduka Baginda 的字樣。「Urusan Seri Paduka Baginda」其實是延續英國人 On His Majesty's Service 的傳統。「 Seri Paduka Baginda 」這三個字的英語是「His Majesty」或「Her Majesty」，而「Urusan」則是「Service」。 現代英國郵遞系統的原型是愛德華四世國王 Edward IV (1461-1483)設計的，他為了加快情報的傳遞速度，在大概每30km就設了一根柱子(post)和一匹新馬。這樣，情報的傳遞便能達到約2 m/s。 J. C. Hemmeon在1912年出版的《The history of the British post office》是這樣寫道的： A somewhat different style of postal service, a precursor of the modern method, was inaugurated by the fourth Edward. During the war with Scotland he found himself in need of a speedier and better system of communication between the seat of war and the seat of government. He accomplished this by placing horses at intervals of twenty miles along the great road between England and Scotland. By so doing his messengers were able to take up fresh horses along the way and his despatches were carried at the rate of a hundred miles a day. 500年前2 m...
For decades, the precise location of Longya Men 龍牙門 remains a subject of dispute in the maritime history of Southeast Asia. The earliest 1 description of the place was given by a Yuan Chinese traveler named Wang Dayuan 汪大淵 (b. 1311, d. 1350?) and the text is usually 2 parsed this way: 門以單馬錫番 兩山 相交若龍牙狀 中有水道以間之 This arrangement is optimizable since Wang's delineation can be made clearer if the two mountains 兩山 are removed from the first line and added to the second line: 門，以單馬錫番 (i) 兩山 相交，若龍牙 (ii) 狀 門 中有水道，以間之 (iii) The first description is relatively easy to handle. Since 門 men 2 is a shorthand for 龍牙門, the first line essentially means: The insular gateway (i) was inhabited by the native people 番 not unlike that of Temasik 單馬錫. This line suggests that in the 14th century, the ancient Srivijaya kingdom remained a maritime power, spanning a vast hydrographic region across S...
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...
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