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理
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
The official emblem of the state of Malacca shows two light brown mouse deers on each side of the tree of Malacca. This depiction is rather problematic. Because, if the emblem is meant to commemorate the brave Sang Kancil , as encountered by Parameswara in the Sejarah Melayu, then the color of the animal is probably not correct. If you follow the Malay classic carefully, the Sang Kancil mentioned is not light brown but white . The relatively new The logo was unveiled when Malacca was declared Historical City on April 13, 2003 logo of the City Council of Malacca, however, correctly show the two mouse deers in white . On the other hand, the Sang Kancil story purported to explain the founding of Malacca could be a story modified from a folk-tale from Sri Lanka. This fact was first noted by R. O. Winstedt (1922) Two Legends of Malacca, Journal of Straits Branch of Royal Asiatic Society, Vol. 85, p. 40. I was in the Za'ba Memorial Library , around
論語 的 第一句話 是： 學而時習之，不亦說乎？ 有朋自遠方來，不亦樂乎？ 人不知而不慍，不亦君子乎？ 這句常被拆散來用，例如中間的「 有朋自遠方來 」那一段，你的外地朋友來找你吃飯，你就說：有朋自遠方來，不亦樂乎。 孔子的原話想要帶出的意思其實有點不同。中間的那一段，必須和上段和下端 一起 解讀。 這句話， 黃繼忠 是這樣翻譯的： The Master said: To learn something and regularly practice it—is it not a joy? To have schoolfellows come from distant states—is it not a pleasure? Not to resent when men do no know you—is it not like a gentleman? 孔子時代的六經和六藝 孔子是想講「學」和「學」的應用。在周朝，君子必須學禮儀、學跳舞、學射箭、學駕車、學書法、以及學算術。 論語裡面「 學 」的概念 比較忠於孔子原意的用法是這樣的： 第一句：學校老師教我 拉普拉斯变换 ，然後我在 炸薯片 油鍋溫度控制剛好用上了它，就可以說： 學 拉普拉斯变换而炸薯片 時習之，不亦說乎 ？ 第二句：比利時搞油鍋工藝的 同事 來車間討論薯片油鍋溫度控制的一些實際手段，就可以說： 有朋自 比利時 來，不亦樂乎？ 第三句：車間的 同事 聽不懂我在講什麼，我沒有表現得很不耐煩。就可以說： 同事 不知而不慍，不亦君子乎？
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