文本选自：Scientific American（科学美国人）
作者：Rachel Crowell
原文标题：Mathematician Solves 50-Year-Old Möbius Strip Puzzle
原文发布时间：18 Sept. 2023
Mathematician Solves 50-Year-Old Möbius Strip Puzzle
Möbius strips are curious mathematical objects. To construct one of these single-sided surfaces, take a strip of paper, twist it once and then tape the ends together. Making one of these beauties is so simple that even young children can do it, yet the shapes' properties are complex enough to capture mathematicians' enduring interest.
The 1858 discovery of Möbius bands is credited to two German mathematicians—August Ferdinand Möbius and Johann Benedict Listing—though evidence suggests that mathematical giant Carl Friedrich Gauss was also aware of the shapes at this time, says Moira Chas, a mathematician at Stony Brook University. Regardless of who first thought about them, until recently, researchers were stumped by one seemingly easy question about Möbius bands: What is the shortest strip of paper needed to make one?
Specifically, this problem was unsolved for smooth Möbius strips that are "embedded" instead of "immersed, " meaning they "don't interpenetrate themselves, " or self-intersect, says Richard Evan Schwartz, a mathematician at Brown University. Imagine that "the Möbius strip was actually a hologram, a kind of ghostly graphical projection into three-dimensional space, " Schwartz says. For an immersed Möbius band, "several sheets of the thing could overlap with each other, sort of like a ghost walking through a wall, " but for an embedded band, "there are no overlaps like this."
数学家解决了50年来的莫比乌斯带之谜
精听党背景导读
1790年11月17日，德国数学家和天文学家莫比乌斯出生。
莫比乌斯最著名的成就是发现了三维欧几里得空间中一种奇特的二维单面环状结构——后人称为莫比乌斯带。
莫比乌斯带是用一张长方形纸条扭转半圈并将端头粘接在一起的带子。
文本选自：Scientific American（科学美国人）作者：Rachel Crowell原文标题：Mathematician Solves 50-Year-Old Möbius Strip Puzzle原文发布时间：18 Sept. 2023关键词：数学 莫比乌斯带 谜
精听党带着问题听
1.“单面表面”用英语可以怎么表达？2. 如何理解段二中的“be credited to”？3. 莫比乌斯带是谁先想到的？
精听党选段赏析
标题解读
Mathematician Solves 50-Year-Old Möbius Strip Puzzle
数学家解决了50年来的莫比乌斯带之谜
mathematician n. 数学家；
strip n.（纸、金属、织物等）条，带；带状地带；1. a strip of cloth 一条布料；2. a strip of land 一块土地；3. a strip of paper 一条纸带；
puzzle n. 令人费解的人（或事物），难题；谜；
段一
Möbius strips are curious mathematical objects. To construct one of these single-sided surfaces, take a strip of paper, twist it once and then tape the ends together. Making one of these beauties is so simple that even young children can do it, yet the shapes’ properties are complex enough to capture mathematicians’ enduring interest.
curious adj. 好奇的，求知欲强的；1. Children are naturally curious about the world around them. 孩子们天生对周围的世界充满好奇。
object n. 物体，物品；1. The room was filled with various objects, from books to toys. 房间里摆满了各种各样的物品，从书籍到玩具。
twist vt. 扭曲，拧；
yet adv. 然而，但是；1. She was tired, yet she continued working. 她很累，然而她继续工作。
property n. 性质，特性，属性；
complex adj. 复杂的，复杂性的；1. Solving this complex puzzle required a lot of patience. 解决这个复杂的谜题需要很多耐心。
capture vt. 捕获，俘获；吸引，引起；1. The photographer tried to capture the beauty of the landscape in her photos. 摄影师试图在她的照片中捕捉风景的美丽。
enduring adj. 持久的，持续的；
参考译文
莫比乌斯带是一种神奇的数学对象。要构造这样一个单面表面，取一条纸带，将其扭转一次，然后将两端粘在一起。制作这样一个美妙的形状非常简单，以至于即使是年幼的孩子也能做到，然而这种形状的性质足够复杂，足以吸引数学家们持久的兴趣。
段二
The 1858 discovery of Möbius bands is credited to two German mathematicians—August Ferdinand Möbius and Johann Benedict Listing—though evidence suggests that mathematical giant Carl Friedrich Gauss was also aware of the shapes at this time, says Moira Chas, a mathematician at Stony Brook University. Regardless of who first thought about them, until recently, researchers were stumped by one seemingly easy question about Möbius bands: What is the shortest strip of paper needed to make one?
be credited to 归功于；
August Ferdinand Möbius 奥古斯特·费迪南德·莫比乌斯（德国数学家）；
Johann Benedict Listing 约翰·本尼迪克特·利斯廷（德国数学家）；
though conj. 虽然，尽管；
Carl Friedrich Gauss 卡尔·弗里德里希·高斯（德国数学家）；
regardless of 不管，不顾；
until recently 直到最近；
stump vt. 困扰，难住；
seemingly adv. 表面上，看似；
参考译文
斯托尼布鲁克大学的数学家莫伊拉·查斯说，1858年莫比乌斯带的发现归功于两位德国数学家奥古斯特·费迪南德·莫比乌斯和约翰·本尼迪克特·利斯廷，尽管证据表明，数学巨人卡尔·弗里德里希·高斯当时也知道这些形状。不管是谁首先想到它们，直到最近，研究人员都对一个看似简单的关于莫比乌斯带的问题感到困惑：需要多短的纸带才能制作一个？
段三
Specifically, this problem was unsolved for smooth Möbius strips that are “embedded” instead of “immersed,” meaning they “don't interpenetrate themselves,” or self-intersect, says Richard Evan Schwartz, a mathematician at Brown University. Imagine that “the Möbius strip was actually a hologram, a kind of ghostly graphical projection into three-dimensional space,” Schwartz says. For an immersed Möbius band, “several sheets of the thing could overlap with each other, sort of like a ghost walking through a wall,” but for an embedded band, “there are no overlaps like this.”
specifically adv. 具体地，明确地；1. I need you to specifically outline the key points of your proposal. 我需要你明确列出你提案的关键点。
unsolved adj. 未解决的；
embedded adj. 嵌入的，内置的；1. The microchips are embedded in the circuit board. 这些微芯片嵌入电路板中。
immersed adj. 专注的；浸入的；1. She was fully immersed in her research project. 她全身心投入她的研究项目。
interpenetrate vt.（互相）贯穿，渗透；混合，交融；
self-intersect vi. 自相交，自交叉；
hologram n. [激光] 全息图；全息摄影，全息照相；
ghostly adj. 幽灵般的，鬼魅似的；1. The old mansion had a ghostly atmosphere at night. 这座老宅夜晚有一种幽灵般的氛围。
graphical adj. 图形的，图像的；
projection n. 投影，投射，投影图；1. The projection of the film on the big screen was impressive. 电影在大银幕上的投影令人印象深刻。
overlap vi.（使）重叠，部分地共用；
sort of like 有点像，有几分类似；1. The new restaurant is sort of like the one we used to go to, but the menu is different. 这家新餐馆有点像我们过去常去的那家，但菜单不同。
参考译文
布朗大学的数学家理查德·埃文·施瓦茨说，具体来说，对于“嵌入式”的而非“沉浸式”的光滑莫比乌斯带，这个问题没有得到解决，这意味着它们“不会穿越自身”，或者说不会自我交叉。施瓦茨说，想象一下，“莫比乌斯带实际上是一个全息图，一种投射到三维空间中的幽灵般的图形投影”。对于一个沉浸式的莫比乌斯带，“几层物体可能会彼此重叠，就像一个幽灵穿过墙壁一样”，但对于一个嵌入式的莫比乌斯带，“就没有这样的重叠”。
精听党每日单词
mathematician
/ˌmæθəməˈtɪʃn/ n. 数学家；
strip
/strɪp/ n.（纸、金属、织物等）条，带；带状地带；
puzzle
/ˈpʌz(ə)l/ n. 令人费解的人（或事物），难题；谜；
curious
/ˈkjʊriəs/ adj. 好奇的，求知欲强的；
object
/ˈɑːbdʒekt/ n. 物体，物品；
twist
/twɪst/ vt. 扭曲，拧；
yet
/jet/ adv. 然而，但是；
property
/ˈprɑːpərti/ n. 性质，特性，属性；
complex
/kəmˈpleks/ adj. 复杂的，复杂性的；
capture
/ˈkæptʃər/ vt. 捕获，俘获；吸引，引起；
enduring
/ɪnˈdʊrɪŋ/ adj. 持久的，持续的；
be credited to
归功于；
August Ferdinand Möbius
奥古斯特·费迪南德·莫比乌斯（德国数学家）；
Johann Benedict Listing
约翰·本尼迪克特·利斯廷（德国数学家）；
though
/ðoʊ/ conj. 虽然，尽管；
Carl Friedrich Gauss
卡尔·弗里德里希·高斯（德国数学家）；
regardless of
不管，不顾；
until recently
直到最近；
stump
/stʌmp/ vt. 困扰，难住；
seemingly
/ˈsiːmɪŋli/ adv. 表面上，看似；
specifically
/spəˈsɪfɪkli/ adv. 具体地，明确地；
unsolved
/ˌʌnˈsɑːlvd/ adj. 未解决的；
embedded
/ɪmˈbedɪd/ adj. 嵌入的，内置的；
immersed
/ɪˈmɜːrst/ adj. 专注的；浸入的；
interpenetrate
/ˌɪntərˈpenɪtreɪt/ vt.（互相）贯穿，渗透；混合，交融；
self-intersect
/self ˌɪntərˈsekt/ vi. 自相交，自交叉；
hologram
/ˈhoʊləɡræmˌˈhɑːləɡræm/ n. [激光] 全息图；全息摄影，全息照相；
ghostly
/ˈɡoʊstli/ adj. 幽灵般的，鬼魅似的；
graphical
/ˈɡræfɪkl/ adj. 图形的，图像的；
projection
/prəˈdʒekʃ(ə)n/ n. 投影，投射，投影图；
overlap
/ˌoʊvərˈlæp/ vi.（使）重叠，部分地共用；
sort of like
有点像，有几分类似；
精听党文化拓展
“莫比乌斯带”有点神秘，一时又派不上用场，但是人们还是根据它的特性编出了一些故事，据说有一个小偷偷了一位很老实的农民的东西，并被当场捕获，将小偷送到县衙，县官发现小偷正是自己的儿子。
于是在一张纸条的正面写上：小偷应当放掉，而在纸的反面写了：农民应当关押。县官将纸条交给执事官由他去办理。聪明的执事官将纸条扭了个弯，用手指将两端捏在一起。然后向大家宣布：根据县太爷的命令放掉农民，关押小偷。县官听了大怒，责问执事官。执事官将纸条捏在手上给县官看，从“应当”二字读起，确实没错。仔细观看字迹，也没有涂改，县官不知其中奥秘，只好自认倒霉。
县官知道执事官在纸条上做了手脚，怀恨在心，伺机报复。一日，又拿了一张纸条，要执事官一笔将正反两面涂黑，否则就要将其拘役。执事官不慌不忙地把纸条扭了一下，粘住两端，提笔在纸环上一划，又拆开两端，只见纸条正反面均涂上黑色。县官的毒计又落空了。
现实可能根本不会发生这样的故事，但是这两个故事却很好地反映出“莫比乌斯带”的特点。
精听党每日美句
"Life can only be understood backwards; but it must be lived forwards." - Søren Kierkegaar
“人生只能向后理解，但必须向前生活。”——索伦·克尔凯郭尔尔