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.
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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。 大太陽曬屁股還不起身嗎？ 原想給照片標上：阿円睡到日上三竿，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： 三竿到底是幾點？ 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出來的天文異象報告。《日光色》的第三個記錄是這樣寫道的： 永明五年十一月丁亥，日出高三竿， 朱色赤黃，日暈，虹抱珥直背。 這段話有三個部分。第一部分「永明五年十一月丁亥」標的是日期。第二部分「日出高三竿」說的是太陽的高度，也就是當時的時間。第三部分「朱色赤黃，日暈，虹抱珥直背」解釋的當時天文局官員觀察到的天象。 通常我讀文言文的時候都會在頭腦裡面把文字翻譯成英語。因為只有這樣，才能逼我自己將每個字都解讀出來。這段話譯成英語大概是： Sunday. December 7, 487. The shadow cast by the sun is three times the height of the pole. The colour of the sky is between red and yellow. Solar haloes and parhelia are clearly visible. 我們先說日期的那一部分。「永明」指的是南齊的 蕭宣遠 先生。蕭先生是在公元483至493期間做皇帝的。永明五年自然是落在公元487年。後面的「十一月」指的是農曆十一月，而「丁亥」是中國傳統天干地支計日系統的寫法，它是一種 60模 的寫法，例如：甲子 = 1、癸亥 = 60。所以，「丁亥」的值為24。 接著我們用 Reingold與Dershowitz 的方法把農曆日期換算成格列高曆。其結果是：公元487年12月7日（既該年的第341天，\(d = 341\)）。 根據日影我
Water disruption at Pangkalan Lumpur In March 1877, Captain Yap Ah Loy wrote a letter to Captain W. B. Douglas (r. 1875 - 1882, Acting Resident of Selangor) to relay a water-related grievance at Pangkalan Lumpur tin mine. This letter is important because it contains several important data points: The toponyms ‘Kuala Lumpur' and ‘Pangkalan Lumpur' can be seen together in the same document, Captain Yap's usage appears to show that Pangkalan Lumpur was a Pangkalan (فڠکالن) is a derivative of pangkal (terminal point in Malay). One effect of adding the an-suffix to a Malay word m is the production of m -an, which is a noun representing an object/place closely associated with m . ‘Pangkal + an' is normally invoked in land/sea travel to mean disembarkation point , e.g. Pangkalan Batu. Another example is empang + an = empangan (the end result when a river is dammed). Empang (امفڠ barrier/blockade) was borrowed to denote tin mine . T
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
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