Issue 028, 2024 SCIENCE FOCUS Not-So-Epic Battles of History: Nikola Tesla vs. Thomas Edison 不存在的宿敵對決：特斯拉大戰愛迪生 Nature’s Palette: The Astonishing Spectrum of Animal Blood Colors 大自然的調色盤：絢爛多彩的動物血液 How Did Music Theorists Decide the Pitch of Each Note? 樂理家如何決定每個音符的音高？ John Horton Conway: The Mathematician Who Made Math Fun John Horton Conway: 讓數學變得有趣的數學家 Q&A with Three Generations of HKUST Math Majors 讀數學系的人：與三代科大學子對談
Dear Readers, Did you recognize the three people on the front cover? They are Thomas Edison, Nikola Tesla and John Conway. In this issue, you will learn not only about their scientific discoveries, but their personal stories. After all, we are all humans: Professional achievements can be linked to personalities, and all aspects of life. Along this line, you will find the first of a new series of articles when we discover personal and professional choices of HKUST students and graduates. First, we will hear from three generations of mathematicians, on their lives before, during and after studying at the HKUST. Last but not least, you will learn about how music and mathematics intertwine and the unexpected colors of animal blood. At the start of a new academic year, let me wish you all the best inside and outside the classroom! Yours faithfully, Prof. Ho Yi Mak Editor-in-Chief 親愛的讀者： 你認得出封面的三位人物嗎？他們是Thomas Edison、Nikola Tesla和John Conway。今期我們不僅會探討他們對科學的貢獻，還 會了解他們背後的故事。最終我們也是人：事業上的成就與個人性格， 還有生命中的種種都息息相關。沿著這道脈絡，我們會向大家呈獻與 科大學生和校友對談的首部曲，了解他們在個人和事業上的選擇。這 次我們會先訪問「三代」讀數學的人，聽聽他們入讀科大前、中、後的 心路歷程。最後，我們會探討音樂與數學密不可分的關係，以及出乎意 料的血液顏色。 新學年伊始，在此祝各位校園內外的生活愉快！ 主編 麥晧怡教授 敬上 Message from the Editor-in-Chief 主編的話 Copyright © 2024 HKUST E-mail: sciencefocus@ust.hk Homepage: https://sciencefocus.hkust.edu.hk Scientific Advisors 科學顧問 Prof. Ivan Ip 葉智皓教授 Prof. Pak Wo Leung 梁伯和教授 Prof. Kenward Vong 黃敬皓教授 Editor-in-Chief 主編輯 Prof. Ho Yi Mak麥晧怡教授 Managing Editor 總編輯 Daniel Lau 劉劭行 Student Editorial Board學生編委 Editors 編輯 Roshni Printer Aastha Shreeharsh Helen Wong 王思齊 Jane Yang 楊靜悠 Social Media Editors 社交媒體編輯 Zoey Tsang 曾鈺榆 Navis Wong 黃諾軒 Graphic Designers 設計師 Coby Ngai 魏敏儀 Winkie Wong 王穎琪 Constance Zhang 張粲璨 Contents Science Focus Issue 028, 2024 What’s Happening in Hong Kong? 香港科技活動 Extinction · Resilience 1 滅絕・新生 China's Lunar and Mars Exploration 中國登月探火 Science in History 昔日科學 Not-So-Epic Battles of History: 2 Nikola Tesla vs. Thomas Edison 不存在的宿敵對決：特斯拉大戰愛迪生 Amusing World of Science 趣味科學 Nature’s Palette: The Astonishing Spectrum of Animal 6 Blood Colors 大自然的調色盤：絢爛多彩的動物血液 How Did Music Theorists Decide the Pitch of 10 Each Note? 樂理家如何決定每個音符的音高？ John Horton Conway: The Mathematician Who 15 Made Math Fun John Horton Conway：讓數學變得有趣的數學家 Who’s Who 科言人語 Q&A with Three Generations of HKUST Math Majors 20 讀數學系的人：與三代科大學子對談
What’s Happening in Hong Kong? 香港科技活動 Fun in Fall Science Activities 秋日科學好節目 Any plans for this fall? Check out the following events! 計劃好這個秋天的好去處了嗎？不妨考慮以下活動！ Extinction · Resilience 滅絕‧新生 The permanent exhibition "Extinction · Resilience" at the Science Museum explores the incredible journey of life on Earth over billions of years. Visitors are invited to learn more about prehistoric biodiversity, mass extinctions, and the remarkable resurgence of life by discovering around 80 sets of remarkable fossil specimens, including the three-billion-year-old stromatolites and the most complete Deinonychus fossil. The lifelike animatronic dinosaurs and engaging interactive exhibits will also amaze visitors by bringing the past to life. After the revamping of exhibits this April, a 423-million-year-old Cooksonia barrandei fossil, one of the earliest land plants on Earth, and 70-million-year-old elongated dinosaur eggs will also be on display. 地點： 香港科學館地下古生物展廳 科學館的常設展覽「滅絕‧ 新生」將帶大家 探索數十億年以來地球生命不可思議的演化旅 程。透過約80組珍貴化石，包括擁有30億年 歷史的疊層石和迄今最完整的恐爪龍化石，館 方希望參觀者能藉此了解更多關於史前生物多 樣性、大滅絕事件和地球生物絕處逢生的故事。 展覽中栩栩如生的機械恐龍和有趣的互動展品 亦能引領大家回到過去。在今年四月更換展品 後，展覽增設四億 2300 萬年前最早陸生植物之 一的頂囊蕨化石，以及7000 萬年前的長形恐龍 蛋。 Venue: G/F Palaeontology Gallery, Hong Kong Science Museum China's Lunar and Mars Exploration 中國登月探火 Explore the exhibition "China's Lunar and Mars Exploration" at the Space Museum, where the wonders of space come to life! This exhibition highlights China's significant achievements in lunar and Mars missions, showcasing full-scale models of the Yutu lunar rover and Zhurong Mars rover. Visitors may discover the evolution of China's space technology, from ancient poetic musings about the Moon to groundbreaking advancements in artificial satellites and manned spacecraft. Period: Now – March 24, 2025 Venue: Foyer, Hong Kong Space Museum Admission fee: Free admission 展期： 現在至2025年3月24日 地點： 香港太空館大堂 入場費： 免費入場 太空館的「中國登月探火」展覽使太空探索 歷程上的奇蹟活現眼前！這個展覽介紹中國月 球和火星探測任務上的偉大成就，當中展示「玉 兔號」月球車和「祝融號」火星車的實物原大模 型。從古人對月亮詩情畫意的聯想，到建造人造 衛星和載人太空船上的突破，參觀人士都可以從 展覽中了解中國航天發展的卓越成就。 1
By Aastha Shreeharsh There are many colorful characters, discoveries, and inventions in the annals of scientific history; yet, not much has sparked as much debate and controversy as the infamous rivalry between Nikola Tesla and Thomas Edison. Ranging from episodes in the popular sitcom The Big Bang Theory, the witty illustrations in the comic The Oatmeal, and YouTube videos from channels such as “Epic Rap Battles of History”, the Tesla vs. Edison narrative has become a persistent fixture in popular culture. As reflected in such media, Tesla is hailed by many as the unrecognized genius of the duo, the protégé that surpassed his mentor, Edison. Meanwhile, Edison, who was once hailed by his contemporaries, the media and even our history textbooks as one of the greatest inventors of all time, is now considered a hack by many. Unfortunately for avid Tesla fans (and fortunately for fans of Edison), reality seems to be more nuanced than this trumped-up rivalry. In 1884, the Serbian physicist Nikola Tesla, aged 28, arrived in New York City and took up employment with Edison [1]. Nine years his elder, Edison had already established himself as a success with his invention of a new kind of telegraph, and established the Edison Illuminating Company which furthered his own work in electric light. Tesla, in his recently acquired position, assisted Edison in installing equipment, repairing generators, and designing new machines. Edison’s work relied on the principles of direct current (d.c.), which was the national standard at that time, and Edison was profiting off many patents (Footnote 1) for his electric lighting system that utilized d.c. [1, 2]. However, Tesla saw promise in utilizing alternating current (a.c.), so a year later, he quitted working for Edison and set up his own electric company – one that utilized a.c. – thus setting the scene for the Tesla vs. Edison narrative. Current is the flow of “electricity,” or more accurately, the flow of charged particles like electrons and positive ions. As the name suggests, d.c. flows in one direction while a.c. alternates its direction back and forth in a single second. So, why did Edison and Tesla prefer using different systems of current? Nikola Tesla VS. Thomas Edison 特斯拉大戰愛迪生 Not-So-Epic Battles of History: 不存在的宿敵對決：
3 For electricity to be transmitted over long distances in a city, the major challenge is to overcome power loss. To minimize power loss, which can be represented by the formula P = I2R (where P, I and R denote power loss, current and resistance of the cable respectively), electric energy should be converted to a high-voltage, low-current form before transmission. While d.c. was widely used in the US at the time, it was challenging to convert low-power plant voltages into higher voltages, necessitating many small power plants near users. Tesla's a.c. system resolved this challenge by harnessing the nature of a.c.; a.c. voltage can be easily stepped up and down by a transformer through electromagnetic induction, due to the alternating input current and the induced change in magnetic field. The high efficiency of transmission enables consumers to utilize electricity from power plants miles away [1]. As the party with vested interest, Edison seemed to have motive for waging a smear campaign, play dirty and discredit poor Tesla’s ideas, just as the prevailing narrative suggests. The infamous “current wars” did take place; however, the rivalry between Tesla and Edison was not nearly as personal and epic as people would like to believe. First and foremost, these “current wars” took place after Tesla’s patents were acquired by George Westinghouse [1]. Thus, it was Westinghouse who promoted the a.c. system against Edison, not so much Tesla. Secondly, both Edison and Westinghouse did attempt to discredit each other’s system and promote their own, but Edison lost within a matter of a few years. In 1893, Westinghouse secured the World's Fair electrification bid [1]. By 1896, General Electric, a leading company co-founded by Edison, switched from d.c. to a.c., paving the way for a.c. as the dominant system in the US [1]. Concurrently, Tesla was swiftly moving onto new inventions [1]. Essentially, the “current wars” were almost entirely a commercial dispute. Then, how did the “Tesla good, Edison bad” narrative prevail? This may be related to Edison’s shock tactics to discredit Tesla’s ideas. To justify the argument that a.c. is more dangerous than d.c., Edison’s West Orange Laboratory has conducted research on electrocution [3, 4], in which various “unwanted” animals like dogs, calves, and a horse were killed [4]. In 1903, despite having essentially lost the a.c. vs. d.c. battle in the US, Edison was associated with the public display of the electrocution of the elephant, Topsy [1, 3, 4], who was sentenced to death after the circus elephant had killed a man and been proved unmanageable (but actually Topsy was physically abused before attacking people in those episodes [3, 4]). As a failing attempt at relevance (both a.c. and d.c. can be dangerous!) and distressing act of animal cruelty, this incident probably does not paint Edison in the best light by today’s standards, although Edison’s supporters would argue that Edison’s research provided a more humane way for Topsy’s execution comparing to hanging, and that Edison might not be personally involved in this incident [3, 4]. The working style of Edison and Tesla varies, too. Famous for his expertise in patenting ideas and dealing with the press, Edison was a businessman through and through. He was also considered a pure empiricist who achieved breakthroughs through endless trials and experiments, while Tesla might favor a different approach as a theorist [5]. We should add that Tesla was not as terrible a businessman as modern-day geeks make him out to be. Being not as commercially minded as Edison, he conceived utopian ideas, such as wireless transmission, with a long-term vision of what technology can do [1]. In today’s capitalistic culture, the Silicon Valley ideals of ingenuity, efficiency, and novel ideas are what make the world go round. Intellectuals proudly wearing the badge of “nerd” in shows like The Big Bang Theory, bemoaning comparisons to Edison and rejoicing in those to Tesla, are reflective of a shift in modern values. However, let us recognize both scientists’ contributions that have and will impact the past, present and future of science. In many cases, truth is not as dramatic as people want to believe. The reality of the Tesla vs. Edison showdown reminds us
科學史冊上記載著各種多姿多彩的偉人、發現和發明， 然而沒有甚麼像尼古拉．特斯拉（Nikola Tesla）和湯馬士． 愛迪生（Thomas Edison）這對知名宿敵之間的較量一樣 能引發激烈的辯論和爭議。從流行處境喜劇《囧男大爆炸》 （The Big Bang Theory），到漫畫《The Oatmeal》中 的詼諧描繪，以及經典饒舌爭霸戰（Epic Rap Battles of History）等 YouTube 頻道的影片，「特斯拉大戰愛迪生」 的故事已成為流行文化中的一個經典戲碼。正如這些媒體 所反映的那樣，許多人將特斯拉譽為寂寂無名的天才，相信 他更勝其師父愛迪生；與此同時，曾被同時代的人、媒體， 甚至我們歷史教科書稱為有史以來最偉大發明家之一的 愛迪生，現在卻被許多人蔑視為一個過譽的庸才。雖說如 此，特斯拉的忠實粉絲未必想聽到的是（或是對愛迪生粉 絲來說值得高興的是），現實中二人關係似乎比傳聞中虛 構的對立來得複雜。 1884 年，當時28歲的塞爾維亞物理學家尼古拉．特 斯拉來到紐約，受聘於愛迪生的公司 [1]。愛迪生比他大九 歲，當時已因發明了一種新型電報取得成功，又創立了愛迪 生照明公司進一步改良自己發明的電燈。特斯拉則在剛入 職的職位中協助愛迪生安裝設備、修理發電機和設計新機 器。愛迪生的系統採用當時的國家標準，依靠直流電運作， 而愛迪生本身亦透過許多直流電照明系統的專利權（註一） 獲取利潤 [1, 2]。然而，特斯拉看到利用交流電的潛力，因 此在一年後辭去工作，成立了使用交流電的電力公司，亦為 「特斯拉大戰愛迪生」的故事寫上序幕。 電流是「電」的流動，更準確地說，是電子或正離子等 帶電粒子的流動。正如字面上的意思，直流電是單向流動 的，而交流電則會在一秒內不斷改變方向。那麼，愛迪生和 特斯拉為甚麼喜歡使用不同的電流系統呢？ 在城市中長距離傳輸電力的主要挑戰是克服功率損 耗。功率損耗可以用公式P = I2R表示（當中P, I 和R分別 為功率損耗、電流和電纜的電阻），因此電能應先被轉換成 高電壓、低電流的形態來進行傳輸以減少功率損耗。雖然 直流電在當時的美國被廣泛使用，但要從發電廠的低電壓 轉為較高的電壓非常具挑戰性，需要在靠近用戶一端興建 大量小型發電站。特斯拉的交流電系統正正利用交流電的 特性解決了這個難題：不斷改變的電流方向和誘導磁場使 電壓能在變壓器中透過電磁感應的原理輕鬆升降，帶來的 高效傳輸使用戶可在距離發電廠數以哩計的家中享用電力 [1]。恰如主流論調所認為的一樣，愛迪生作為既得利益者， 似乎就有了合理動機發起一輪誹謗攻勢，以卑鄙手段抹黑 特斯拉的系統。 著名的「電流大戰」確實就這樣發生，但特斯拉與愛迪 生之間的競爭並沒有人們想像的那麼史詩式，二人也 沒有在個人層面上交鋒。首先，「電流大戰」發生於特斯拉 的專利被喬治‧威斯汀豪斯（George Westinghouse） 收購之後 [1]，因此推廣交流電系統以抗衡愛迪生的是威 斯汀豪斯，而不是特斯拉。其次，無論是愛迪生或威斯汀豪 斯都曾試圖抹黑對方的系統並宣傳自己的系統，但愛迪生 在僅僅幾年就宣告敗北。1893 年，威斯汀豪斯投得為世界 博覽會提供電力的標書 [1]；到1896年，由愛迪生共同創 立的領頭公司通用電氣改用交流電，為交流電成為美國主 要電力系統埋下伏線 [1]。同時，特斯拉很快就改為研究其 他新發明了 [1]。「電流大戰」基本上只是一場商業糾紛。 那麼「特斯拉是好人，愛迪生是反派」的想法是如何興 起的呢？這可能與愛迪生抹黑特斯拉系統的招數有關。為 了證明交流電比直流電更危險，愛迪生的西奧蘭治實驗室 （West Orange Laboratory）進行了關於電刑的研究 [3, that fame is fleeting and so is our recollection of history – which is why, it is all the more important that we do not hastily idolize or demonize historic figures. After all, they were human too. 1. Patent: A legal document that grants the inventor of a new invention the exclusive right to make, use, and sell that invention for a certain period of time, usually 20 years from the date of filing the application. Throughout Edison’s life, he acquired an astonishing number of 1,093 patents [2].
5 References 參考資料： [1] Edwards, P. (2015, Jul 21). Tesla vs. Edison — and what the never-ending battle says about us. Vox. https://www.vox.com/2015/7/21/8951761/tesla-edison [2] Library of Congress. (n.d.). Life of Thomas Alva Edison. https://www.loc.gov/collections/edisoncompany-motion-pictures-and-sound-recordings/ articles-and-essays/biography/life-of-thomas-alvaedison/#:~:text=In%20his%2084%20years%2C%20 he,his%20inventions%20to%20the%20public. [3] School of Arts and Sciences, Rutgers, The State University of New Jersey. (n.d.). Myth Buster-Topsy the Elephant. https://edison.rutgers.edu/life-ofedison/essaying-edison/essay/myth-buster-topsythe-elephant [4] Eschner, K. (2017, January 4). Topsy the Elephant Was a Victim of Her Captors, Not Thomas Edison. Smithsonian Magazine. https://www. smithsonianmag.com/smart-news/topsy-elephantwas-victim-her-captors-not-really-thomasedison-180961611/ [5] Tesla Says Edison Was An Empiricist. (1931, October 19). The New York Times. https://teslauniverse. com/nikola-tesla/articles/tesla-says-edison-wasempiricist 4]，過程中處決了各種「沒有人想收養」的動物，例如狗、小 牛和一匹馬 [4]。儘管「交流電對直流電」一戰於1903年 在美國基本上大局已定，但愛迪生卻在此時與公開處決馬 戲團大象Topsy一事扯上關係 [1, 3, 4]，當中大象因殺害 一名男子，以及被馬戲團認定為無法控制，因而被判處極 刑（但其實Topsy每次攻擊人類前均遭受虐待 [3, 4]）。 這次殘酷對待動物的行為不但未能證明交流電比較危險 （事實上兩者也可以很危險！），更令人痛心疾首，以現今 標準來看這件事可謂將愛迪生的形象毀於一旦，儘管其支 持者會辯稱愛迪生的研究為Topsy提供了比絞刑更人道 的處決方式，而愛迪生亦可能沒有親身參與事件 [3, 4]。 愛迪生和特斯拉在工作風格上也有所不同。愛迪生以 擅於將想法轉化成專利權和應對傳媒的手腕聞名，是個不 折不扣的生意人。他也是一位純經驗主義者，靠無止境地 不斷嘗試和進行實驗來尋求突破，反之特斯拉作為理論主 義者很可能會採取截然不同的策略 [5]。另一方面，雖然特 斯拉作為商人並不像現代科技宅所說的那麼糟糕，但他也 沒有像愛迪生那樣側重商業考慮，因此能構想出像無線傳 輸這些具烏托邦色彩的概念，以獨特的遠見向世人展現科 技在未來的各種可能性 [1]。 在現今的資本主義文化中，矽谷所追求的足智多謀、高 效和嶄新概念才是推動世界運轉的原動力。從《囧男大爆 炸》等節目中可以看到現今知識分子對「書呆子」這些本為 貶意的稱號毫不在乎；他們一方面抗拒人們拿愛迪生和自 己比較，另一方面渴望擁有特斯拉的特質，這些都反映現 代價值觀的轉變。然而，我們都不能否認這兩位科學家的 貢獻對科學都有著深遠的影響，除了改變科學的過去和現 在，影響力還會持續至未來。很多時候，現實並不像人們所 想的那樣戲劇化。特斯拉與愛迪生對決的真相提醒了我們 名利是短暫的，人們對歷史的記憶也是如此 — 這就是為 什麼我們不應盲目地將歷史人物神話化或妖魔化，而他們 終究也是人。 1 專利：授予發明者獨家權利在一定時間內（通常從提交申請一天起計的20 年）製造、使用和銷售該發明的法律文件。愛迪生一生中擁有過高達 1,093 項專利 [2]。
What is the color of blood? While the instinctive response for many of us may be “red,” the truth is that nature’s artistry extends far beyond our imagination. Let’s delve into the astonishing world of blood's vibrant hues – where blues, purples, greens, and even the absence of color thrive [1, 2]. Determinant of Blood Color: Respiratory Pigment The key to this breathtaking diversity of animal blood colors lies in the type(s) of respiratory pigment in our blood. Respiratory pigments are metal-containing proteins that help transport respiratory gasses such as oxygen and carbon dioxide (footnote 1). The wellknown hemoglobin in our blood is just one example of such a pigment, along with lesser-known ones like chlorocruorin, hemocyanin, and hemerythrin. These pigments differ in their chemical structures and use different metal complexes to bind oxygen molecules. Together, these subtle variations cause each pigment to absorb and reflect unique wavelengths of light, resulting in a vast array of blood colors. Be careful: When we talk about blood color, we are referring to the color of the oxygenated pigments rather than the deoxygenated ones. When oxygen binds to the metals within these pigments, it alters the threedimensional structures of the whole pigments and, in some cases, changes the oxidation states of the metals [2]. This, in turn, results in a shift in their light absorption and reflection spectra, changing the blood colors we see. Blue Blood In certain invertebrates such as squid, octopus, lobster, and horseshoe crab, the presence of the respiratory pigment hemocyanin gives their blood a distinct blue color [2]. Unlike hemoglobin, which utilizes iron (Fe2+) to bind oxygen, hemocyanin relies on copper (Cu2+) for oxygen transportation in these marine animals. The copper (II) ion strongly absorbs red light while reflecting blue light, resulting in the characteristic blue appearance of their blood. These blue-blooded invertebrates have evolved to use hemocyanin for two reasons [3]. Firstly, the effectiveness of hemoglobin to transport oxygen decreases at low temperatures, such that hemocyanin outperforms hemoglobin in the deep sea. Secondly, hemoglobin exhibits superior efficiency in binding oxygen compared to hemocyanin but only in oxygenrich environments, because the binding of every new oxygen molecule can facilitate that of the next oxygen molecule, until the four vacancies in the hemoglobin are taken up. However, in oxygen-deprived settings, hemoglobin’s oxygen-binding efficiency diminishes, and hemocyanin proves to be more effective. Hence, the switch to hemocyanin gives these marine animals an edge in obtaining oxygen in the ocean. Purple Blood Blood naturally takes on a purple color in lamp shells and certain marine worms [1, 2]. These marine invertebrates use neither hemoglobin nor hemocyanin, but hemerythrin. While hemerythrin, like hemoglobin, uses iron as the oxygen-binding material, it gives a violet-pink color instead of bright red in its oxygenated state, and is colorless when no oxygen is bound. Green Blood For some earthworms and leeches, green blood is the norm [1, 2]. These invertebrates contain chlorocruorin, another iron-based pigment that makes their blood look green. While chlorocruorin is typically linked to green hues, its color is concentrationdependent – lower amounts appear green, but higher concentrations cause the pigment to take on red coloration. By Helen Wong 王思齊 絢爛多彩的動物血液 The Astonishing Spectrum of Nature’s Palette 大自然的調色盤: Animal Blood Colors
7 At the molecular level, chlorocruorin closely resembles hemoglobin. In fact, its chemical composition diverges from hemoglobin in just one respect: Chlorocruorin contains an aldehyde group (-CHO) whereas hemoglobin has a vinyl group (-CH=CH2). Nevertheless, it is worth noting that chlorocruorin does not contain chlorine as its name may otherwise suggest. Here comes an interesting twist: Green blood does not solely rely on the presence of chlorocruorin. Like most vertebrates, green-blooded skinks from New Guinea use hemoglobin to carry oxygen. However, their blood and tissues are green [4]. This peculiar phenomenon is related to how these lizards recycle hemoglobin. In humans, the recycling of hemoglobin involves two steps, first by breaking the pigment down into a green chemical named biliverdin, then by converting biliverdin to a yellow compound called bilirubin. However, the lizards lack the ability to further metabolize biliverdin, leading to an accumulation of the green pigment in their blood [1, 2]. The color is so intense that it overshadows the natural red color of hemoglobin. Colorless Blood Perhaps the strangest “blood color” is having no color at all. The Antarctic icefishes stand out as one of the most unusual vertebrates as they lack any respiratory pigment [2, 3]. As a result, icefish blood is colorless with only blood plasma. Despite the absence of hemoglobin, scientists have discovered remnants of hemoglobin genes in icefish genomes, suggesting that these genes were lost during the course of evolution [5]. But the question remains: How do these Antarctic fishes survive without such an important oxygen carrier? Icefishes have developed several adaptations to compensate for that [2]. They have a larger blood volume than related fish species, and lead a relatively sedentary lifestyle, which helps reduce their oxygen demands. On the other hand, the cold water of the Southern Atlantic also promote icefish survival by maintaining higher concentrations of dissolved oxygen than warmer seas. Conclusion Now back to the same question: What is the color of blood? This time, you might be tempted to respond with a range of colors we have covered – red, blue, purple, green, and even colorless – and they are all right! For other creative answers that you may think of, why not? Scientific knowledge is tentative and subject to change when new evidence appears – so who knows, perhaps there are yet undiscovered ones waiting to join the evergrowing palette of blood colors! 1. Editor’s note: While most of the carbon dioxide is transported through blood plasma in the form of bicarbonate ions in humans (as mentioned in high school textbooks), 10% of the gas is actually carried by hemoglobin [6]. Hemoglobin (Heme B) Hemocyanin Chlorocruorin Hemerythrin
血是甚麼顏色的呢？ 大多數人可能會首先想到紅色，但事實上，大自然的創 造力遠超我們想像。讓我們一起探索血液色彩的奇妙世界 — 那裡有藍色、紫色、綠色，甚至是透明 [1, 2]！ 決定血液顏色的因素：呼吸色素 血液顏色之所以擁有如此驚人的多樣性，關鍵在於動 物血液中呼吸色素的種類。呼吸色素是一種含有金屬的蛋 白質，作用是運送氧和二氧化碳等呼吸氣體（註一）。我們 熟悉的血紅蛋白只是其中一種，其他較少人熟悉的包括血 綠蛋白、血藍蛋白和蚯蚓血紅蛋白。這些呼吸色素各有不 同的化學結構，並使用不同的金屬絡合物與氧分子結合。 正是這些微妙的差異使每種色素吸收和反射獨特波長的光 線，從而產生各種各樣的血液顏色。 值得留意的是，在討論血液顏色時，我們指的是含氧色 素而非缺氧色素的顏色。當氧與呼吸色素中的金屬結合時， 會改變色素的三維結構，在某些情況下金屬的氧化態亦會 產生變化 [2]。這會改變色素的吸收和反射光譜，進而改變 我們所看到的血液顏色。 藍色血液 在某些無脊椎動物中，如魷魚、章魚、龍蝦和鱟，血藍 蛋白的存在使牠們的血液呈現明顯的藍色 [2]。與血紅蛋 白利用鐵（Fe2+）與氧結合不同，這些海洋生物中的血藍蛋 白依靠銅（Cu2+）來運送氧。銅（II）離子強烈吸收紅光並 反射藍光，使這些海洋動物的血液呈現獨特的藍色。 進化最終使這些無脊椎動物使用血藍蛋白的原因有兩 個 [3]。首先，血紅蛋白在低溫下運送氧的效率會下降，因 此血藍蛋白在深海的表現比血紅蛋白更佳。另外，血紅蛋白 僅在富氧環境下與氧結合的能力較強，因為每個氧分子與 血紅蛋白結合後都會促進下一個分子的結合，直至四個位置 被佔滿為止；然而在缺氧環境中，血紅蛋白與氧的結合在缺 乏這種加乘下，效率將大幅降低，使血藍蛋白反而更勝一籌。 因此，使用血藍蛋白能使這些海洋生物在獲取氧方面取得 優勢。 紫色血液 腕足動物和一些海洋蠕蟲的血液呈現紫色 [1, 2]。這些 海洋無脊椎動物均沒有血紅蛋白或血藍蛋白，而是含有蚯蚓 血紅蛋白。儘管蚯蚓血紅蛋白和血紅蛋白都使用鐵作為氧 結合物質，但前者在含氧狀態下呈現粉紫色而非鮮紅色，在 缺氧狀態下則不呈現任何顏色。 綠色血液 對某些蚯蚓和水蛭來說，綠色的血再正常不過 [1, 2]。 這些無脊椎動物含有血綠蛋白，一種使血液呈綠色的鐵基 色素。雖然血綠蛋白通常被認為是綠色，但其顏色其實取決 於濃度：較低含量時呈現綠色，但較高濃度時呈現紅色。 從分子層面來看，血綠蛋白與血紅蛋白非常相似。事實 上，其化學結構僅在一個位置與血紅蛋白不同：血綠蛋白 在那個位置含有醛基（-CHO），而血紅蛋白則含有乙烯基 （-CH=CH2）。然而值得注意的是，儘管血綠蛋白的英文名 稱「chlorocruorin」以「chloro」開首，但它並不像其含義 一樣含有氯（chlorine）。 然而事情並不就此作結：綠色血液有時與血綠蛋白無 關。和大多數脊椎動物一樣，新幾內亞的綠血石龍子（屬 於石龍子科的蜥蜴）以血紅蛋白運送氧，但牠們的血液和 組織卻呈綠色 [4]。這個奇特的現象與這些蜥蜴如何回 收血紅蛋白有關。人類中血紅蛋白的回收涉及兩個步驟： 首先是將血紅蛋白分解成一種稱為膽綠素的綠色化學物 質，然後是將膽綠素轉化為另一種名為膽紅素的黃色化 合物。然而，這些蜥蜴缺乏代謝膽綠素的能力，導致該色 素在血液中積聚 [1, 2]。由於膽綠素的顏色過於強烈，以 至掩蓋了血紅蛋白的天然紅色。 透明血液 也許最奇怪的「血色」是沒有顏色。南極冰魚是最不尋 常的脊椎動物之一，因為牠們沒有任何呼吸色素 [2, 3]，因
References 參考資料： [1] Brunning, A. (2014, October 28). The Chemistry of the Colours of Blood. Compound Interest. https://www.compoundchem.com/2014/10/28/ coloursofblood/ [2] Lutz, D. (2010, February). The Many Colors of Blood. ChemMatters. Retrieved from https:// teachchemistry.org/chemmatters/february-2010/ the-many-colors-of-blood [3] Rennie, J. (2019, April 22). Icefish Study Adds Another Color to the Story of Blood. Quanta Magazine. https://www.quantamagazine.org/ icefish-study-adds-another-color-to-the-story-ofblood-20190422/ [4] Rodriguez, Z. B., Perkins, S. L., & Austin, C. C. (2018). Multiple origins of green blood in New Guinea lizards. Science Advances, 4(5). https:// doi.org/10.1126/sciadv.aao5017 [5] Cocca, E., Ratnayake-Lecamwasam, M., Parker, S. K., Camardella, L., Ciaramella, M., di Prisco, G., & Detrich, H. W. (1997). Do the hemoglobinless icefishes have globin genes? Comparative Biochemistry and Physiology Part A: Physiology, 118(4), 1027–1030. https://doi.org/10.1016/s03009629(97)00010-8 [6] Doyle, J., & Cooper, J. S. (2023, July 4). Physiology, Carbon Dioxide Transport. In: StatPearls. https:// www.ncbi.nlm.nih.gov/books/NBK532988/ 9 此只有血漿的冰魚血液是無色的。儘管現代冰魚沒有血紅 蛋白，但科學家在牠們的基因組中還是發現了血紅蛋白基因 的殘骸，意味著血紅蛋白基因可能是在進化過程中丟失 [5]。 這衍生出一個耐人尋味的問題：沒有了如此重要的氧載 體，南極冰魚到底是如何生存的呢？原來冰魚為此發展出多 種適應機制 [2]：它們擁有比近親魚類更大的血液體積，並 通過相對靜態的生活模式來減少自身的氧需求。另一方面， 南大西洋的寒冷水域相比其他溫暖海域有更高的溶氧度，這 亦有利於冰魚生存。 結論 現在回到最初的問題：血是甚麼顏色的呢？ 這次，你可能會給出我們所討論過的一系列顏色 — 紅 色、藍色、紫色、綠色，甚至是透明 — 它們全都是正確的！ 至於其他你可能想到極具創意的答案，又有何不可呢？科學 知識是暫時性的，意味著當有新證據出現時，我們的認知亦 可能隨之改變 — 説不定還有一些未被發現的顔色正等待加 入這個燦爛多彩的血液大家庭呢！ 血紅蛋白（血基質 B） 血藍蛋白 血綠蛋白 蚯蚓血紅蛋白 1 編按：雖然大部分二氧化碳在人體中都以碳酸氫鹽離子的形式透過血漿運 輸（正如高中教科書所提到的一樣），但實際上10% 的二氧化碳是由血紅 蛋白攜帶的 [6]。
Introduction Known as the temperament or scale in music theory, we learn to sing “do-re-mi-fa-sol-la-ti” as early as kindergarten. We also learn that all sounds are essentially produced by the vibrations that hit our eardrum, whose frequency decides the pitch. Then, have you ever thought about how music theorists chose a pitch for each note in “do-re-mifa-sol-la-ti” from an infinite number of options on a number line? In this article, we will introduce you to “Pythagorean temperament,” an early musical scale often attributed to an ancient Greek mathematician, Pythagoras [1, 2]. We will also delve into how music theorists and mathematicians later developed “equal temperament,” which has become the most widely used musical scale in western music since the 19th century [1]. How Did Music Theoris Decide the Pitch of 樂理家如何決定每個音 By Jane Yang 楊靜悠 Pythagorean Temperament First of all, let's understand the concept of an "octave." Mathematically, two sounds are considered an octave apart if their frequencies have a ratio of 2:1. For example, the standardized “middle A” has a frequency of 440 Hz (Footnote 1), or vibrations per second, while the next A which is an octave higher has a frequency of 880 Hz. When they are played at the same time, they sound so consonant that the human brain perceives them as the “same” note but the latter in a higher pitch. This phenomenon is called “octave equivalence [1, 2].” Therefore, to create a musical scale, we only need to consider an octave, or one cycle of “do-remi-fa-sol-la-ti”. We can then multiply or divide the frequencies of the notes in an octave by any power of two to obtain a higher or lower octave because of octave equivalence. Pythagoras also discovered two notes that are a "fifth" apart, meaning their frequencies have a ratio of 3:2, also sound pleasant when played together. Hence, he decided that the task was to create as many ratios of 3:2 and 2:1 as possible to provide convenience for composers. Obviously, Pythagoras should have no access to the accurate frequency of each note, so the tuning was probably completed by hearing the pitch and comparing its relative distance to the base note. However, for a better understanding, let’s unveil the ancient method based on our modern understanding. To decide a frequency to each of the notes in an octave, Pythagoras started with the note A at 440 Hz and multiplied its frequency by 3/2 to obtain the note at 660 Hz. By multiplying 3/2 again, he got 990 Hz.
11 ts Each Note? 符的音高？ However, this exceeded the desired octave range (i.e. greater than 880 Hz), so he divided it by two to get the note equivalent to it at 495 Hz. Pythagoras repeated this process of multiplying by 3/2, and dividing by two if the resulting frequency exceeds 880Hz, until he obtained a musical scale consisting of seven nonequivalent notes which is enough to play simple melodies [1]. He rearranged those frequencies in order, creating a musical temperament very similar to the one we use today (Table 1). Equal Temperament However, the seven notes in Pythagorean temperament are just enough for playing simple melodies. Before we examine the problem of Pythagorean temperament, let’s look at the modern system called "equal temperament". This temperament Table 1 The frequencies of notes and their ratios with respect to the note A in Pythagorean temperament. The values are rounded off to the nearest integer. divides an octave into 12 equal musical intervals. Keep in mind that our brain perceives the distance of musical interval by ratio instead of difference. Therefore, the frequencies of each note in a scale should have an exponential relation, with a ratio r between each pair of adjacent notes satisfying r12 = 2, i.e. r = 21/12. By multiplying the starting frequency by the ratio r = 21/12 for 12 times, we obtain the frequencies of all the notes within an octave (Table 2). Key Change So, why is equal temperament preferred over Pythagorean temperament? You may have heard of a musical jargon called “key change” before. Actually, the mathematical implication of key change is to multiply the frequency of each note of a melody by a constant number. After performing this trick, human brains will still perceive the two melodies as the same since the musical interval (i.e. the frequency ratios) between any two adjacent notes are retained [1]. For example, a melody that plays 440Hz, 660Hz, and 733.3Hz in order is considered equivalent to a melody that plays 550Hz, 825Hz and 916.6Hz. Key change in music usually helps musicians express their feelings: Changing to a higher key in the midway of a piece of music can express excitement or encouragement, Table 2 The frequencies of notes and their ratios with respect to the note A in equal temperament. The values are rounded off to the nearest integer. The frequencies in shaded cells are played by the black keys of a piano. Ratio 1 9/8 81/64 4/3 3/2 27/16 243/128 2 Frequencies(Hz) 440 495 557 587 660 743 835 880 Ratio 1 21/12 22/12 23/12 24/12 25/12 26/12 Frequencies(Hz) 440 466 494 523 554 587 622 Ratio 27/12 28/12 29/12 210/12 211/12 2 Frequencies(Hz) 659 698 740 784 831 880
1. Middle A: In the case of C major (one of the easiest modes in music), “do-re-mi-fa-sol-la-ti” corresponds to C, D, E, F, G, A, B respectively in representation. Chosen as a standard note for tuning musical instrument, the “middle A” corresponds to “la” in C major. Although it should have a frequency of 440 Hz by the ISO 16 standard [5], the tune is sometimes set at 442 Hz in some wind bands to cater the wind instruments. 2. Editor’s note: The number 1.0136 is given by 312 / 219, i.e. taking the fifths 12 times while reducing the octaves seven times. while lowering the key may convey sorrow or tranquility. In addition, by lowering the key of a song, a singer whose voice range is too low to cover the high pitch can now sing the song. After understanding the concept of key change, you would discover that the equal temperament adapts to key change perfectly because the ratio of the frequencies between any adjacent notes is a constant [1, 2]. Pianists, for example, only need to move up every note for one key on a piano keyboard tuned with equal temperament to complete the key change, and the finite number of keys on the keyboard is sufficient to cover all notes required for any key changes. On the other hand, the seven notes in Pythagorean temperament don’t suffice. Instead of having a constant ratio, adjacent notes in Pythagorean temperament have a ratio of either 9:8 or 256:243 [2]. We have to continue Pythagoras’ calculation to create more and more notes so that key changes can be performed perfectly from any note. By extending his calculation beyond the first octave, we wish the value will return to the starting point 440 Hz at some point, so that we can get a finite number of notes. Nevertheless, this has been proved impossible, due to the fact that (3/2)n is never a power of two, so we will need an infinite number of black keys for a musical instrument to perform key changes, which is simply not practical [1]. Although Pythagoras was able to get close to the desired frequency 440 Hz, there was still a small discrepancy known as the "Pythagorean comma" [2]. This slightly higher frequency ratio of 1.0136:1 posed challenges for musicians and mathematicians until the invention of equal temperament (Footnote 2). Historical Controversies Over the Invention of Equal Temperament One interesting coincidence is that the equal temperament was invented by the Chinese mathematician, physicist and music theorist Zhu Zaiyu in 1584, and given a mathematical definition by the Flemish mathematician Simon Stevin around the period between 1585–1608 [3]. There are still controversies on who should receive the credit and whether the development was independent [3, 4], but we may never know the truth. Nevertheless, one thing you can take away is that anything we take for granted today may have been the outcome of the struggle of our predecessors for thousands of years, and there may actually be a scientific reason behind it. From Pythagoras' exploration to the invention of equal temperament, these mathematicians have shaped the music we enjoy. So next time you sing "do-re-mi-fa-sol-la-ti", remember the mathematical journey that led to these familiar notes. 引言 我們從幼稚園時期就學會唱「do-re-mi-fa-sol-la-ti」， 而在樂理中，這個概念被稱為音律或音階。我們還學過所有 聲音本質上都是衝擊耳膜的振動，這些振動的頻率決定了 音高。那麼，你有否思考過樂理家是如何從數軸上無窮多的 選擇中，為「do-re-mi-fa-sol-la-ti」裡的每個音符選定音 高呢？在這篇文章裡，我們將介紹「畢氏音律」─ 一種普遍 認定為由古希臘數學家畢達哥拉斯提出的早期音樂音階 [1, 2]。我們還會探討樂理家和數學家後來如何發展出「十二平 均律」，這種音階自19 世紀以來一直都是西方音樂最廣泛 使用的音階 [1]。 畢氏音律 首先讓我們了解「八度」這個概念：在數學上，如果兩個 音符的頻率比為2:1，它們就是相差一個八度，例如標準「中 音A」的頻率為440 Hz（註一），亦即每秒振動440次，那 麼比它高一個八度的A的頻率則為880 Hz。當同時演奏這 兩個音時，重疊的聲音聽起來會和諧得使人腦認為它們是 「相同」的音符，只是後者的音高較高而已，這種相似性被 稱為「等價八度」[1, 2]。 因此，就創造一個音階而言，我們只需要考慮一個八度，
亦即是循環裡的一次「do-re-mi-fa-sol-la-ti」。然後我們 可以藉等價八度的特性，將八度內的音符頻率乘或除以任 何二的次方數，以獲取更高或更低的八度。畢達哥拉斯還發 現當兩個音符的頻率比為 3:2，即是相隔一個「五度」時，同 時演奏這兩個音也會有非常悅耳的效果。因此，他決定音階 裡要儘可能包含最多的 3:2 和 2:1 比例，以便作曲家創作。 顯然，畢達哥拉斯當時應該無法得知每個音符的準確頻 率，因此調音可能只是透過聆聽音高來估計一個音與基準 音之間的相對距離來完成。然而，為了方便理解，讓我們從 現代角度揭示這古老的調律方法。 為了決定八度內每個音的頻率，畢達哥拉斯先從440 Hz的A音入手，將其頻率乘以3/2以獲得660 Hz這個音。 透過再乘以3/2，他得到990 Hz，但這超出了八度範圍（即 大於880 Hz），因此他將其除以二以獲得等價的495 Hz。 畢達哥拉斯重複這個乘以 3/2，然後如果得出頻率超過 880 Hz 則把其除以 2 的過程，直至得到一個由七個不等價音符 組成的音階為止，這音階足以演奏簡單旋律 [1]。他把這些 音按頻率順序排列，創造出一個與現代版本非常相似的音律 （表一）。 十二平均律 然而，畢氏音律中的七個音符僅僅足以演奏簡單旋律。 在檢視畢氏音律的不足之前，讓我們先了解現代的「十二平 均律」。這種音律將一個八度分為12個相等的音程。但要 注意的是，我們大腦是以頻率間的比例而非差異來判斷兩 個音之間的距離，因此音階中每個音符的頻率應具有指數關 係，每對相鄰音符之間的比例r滿足r12 = 2，即r = 21/12。 透過將起始頻率乘以比例r = 21/12 12 次，我們就可以獲得 八度內所有音符的頻率（表二）。 調性變換 那麼，為甚麼十二平均律比畢氏音律更受青睞呢？你可 能聽過一個音樂術語叫「轉調」，它在數學上是指將旋律中 每個音符的頻率乘以一個常數，這樣做的話人腦仍會將新 旋律視為與舊旋律相同，因為任何兩個相鄰音符之間的音 程（即頻率比）保持不變 [1]，例如一段由440 Hz、660 Hz 和733.3 Hz 音符組成的旋律聽起來與由550 Hz、825 Hz 和916.6 Hz組成的是同一個旋律。音樂中的調性變換為 音樂家提供表達情感的途徑：在樂曲中途轉換到更高調性 13 表一 畢氏音律中各音符的頻率以及它們與 A音頻率之比。數值四捨五入至最接近的整數。 表二 十二平均律中各音符的頻率以及它們與 A音頻率之比。數值四捨五入至最接近的整數。表中棕色方格的頻率在鋼琴中由黑 鍵演奏。 比例 1 9/8 81/64 4/3 3/2 27/16 243/128 2 頻率(Hz) 440 495 557 587 660 743 835 880 比例 1 21/12 22/12 23/12 24/12 25/12 26/12 頻率(Hz) 440 466 494 523 544 587 622 比例 27/12 28/12 29/12 210/12 211/12 2 頻率(Hz) 659 698 740 784 831 880
可以表達振奮和鼓舞，而降低調性則可以傳達悲傷 或平靜的感覺。此外，通過降低歌曲的調性，音域 較低的歌手也能夠演唱原本包含高音的歌曲。 了解調性變換的概念後，你會發現轉調能完 美地在十二平均律中執行，因為任何相鄰音符之間 的頻率比是一個常數 [1, 2]，譬如鋼琴手只需在以 十二平均律調音的鋼琴上將旋律中每個音符順移一 個鍵，就能完成調性變換，而鍵盤上有限數量的鍵 足以涵蓋調性變換所需的任何音符。 另一方面，畢氏音律中的七個音符並不足夠進 行調性變換，因為相鄰音符之間的頻率比並不恆 定，可以是9:8或256:243 [2]。我們必須延續畢達 哥拉斯的計算以創造出更多的音符，才能從任何音 符起進行調性變換。透過將他的計算擴展到第一 個八度之外，我們希望頻率會在某處返回起點的 440 Hz，這樣我們就不會製造出無限多的音符。然 而，這已被證明是不可能的，因為 (3/2)n 永遠不會 是二的次方數，因此我們需要製造無限多的黑鍵來 讓樂器進行調性變換，但這是不切實際的 [1]。雖 然畢達哥拉斯的計算在某處已非常接近所需的頻 率440 Hz，但仍存在一個被稱為「畢氏音差」的 小差距 [2]。這個略高於一的頻率比1.0136:1使 音樂家和數學家苦惱不已，直到十二平均律出現 為止（註二）。 發明十二平均律的歷史爭議 有個有趣巧合，就是十二平均律由中國數學家、 物理學家及音樂理論家朱載堉於1584 年提出，然 後在 1585至 1608 年間由佛蘭德數學家 Simon Stevin給予數學上的定義 [3]。對於二人當中是誰 發明了十二平均律，以及輾轉間二人到底是否得知 對方研究這件事至今仍然存在爭議 [3, 4]，但也許 我們永遠無法得知真相。 儘管如此，我們能從中學到的是今天我們視為 理所當然的任何事物，可能都是前人經過幾千年努 力所取得的成果，背後可能也有科學上的根據。從畢達哥拉 斯的早期研究到十二平均律的發明，數學家塑造了我們今 時今日所聆聽的音樂。因此下次當你唱「do-re-mi-fa-solla-ti」時，希望你能回想起這些熟悉音符背後的數學故事， 感慨音樂世界中的數學之美。 References 參考資料： [1] Formant. (2022, August 12). The Mathematical Problem with Music, and How to Solve It [Video]. YouTube. https:// www.youtube.com/watch?v=nK2jYk37Rlg [2] Benson, D. (2008, December 14). Music: A Mathematical Offering. Cambridge University Press. https://homepages. abdn.ac.uk/d.j.benson/pages/html/music.pdf [3] Yung, B. (1981). A Critical Study of Chu Tsai-yü's Contribution to the Theory of Equal Temperament in Chinese Music. By Kenneth Robinson. Additional Notes by Erich F. W. Altwein; Preface by Joseph Needham. Wiesbaden: Franz Steiner Verlag (Sinologica Coloniensia Band 9), 1980. x, 136 pp. Figures, Appendixes, Bibliography. N.p. The Journal of Asian Studies, 40(4), 775–776. [4] Kuttner, F. A. (1975). Prince Chu Tsai-Yü's life and work: A re-evaluation of his contribution to equal temperament theory. Ethnomusicology, 19(2), 163–206. [5] International Organization for Standardization. (1975). ISO 16:1975 Acoustics — Standard tuning frequency (Standard musical pitch). https://www.iso.org/ standard/3601.html 1 中音A：在C大調（音樂中最簡單的調之一），「do-re-mi-fa-sol-la-ti」分 別對應 C、D、E、F、G、A、B。作為樂器調音的標準音符，中音A 對應 C 大 調中的「la」。根據 ISO 16 標準，它的頻率應為 440 Hz，但某些管樂團有時 為了迎合管樂器會將它調成 442 Hz [5]。 2 編按：1.0136是由312 / 219 所得，即是升12次五度和降七次八度。
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