罗杰·彭罗斯Roger Penrose

罗杰·彭罗斯的父母Lionel Sharples 彭罗斯和Margaret Leathes都受过医学训练。Margaret是一名医生，而Lionel是一名医学遗传学家，被选为皇家学会会士。他参与了一个名为Colchester调查的项目，旨在发现遗传因素还是环境因素在决定某人是否可能患有心理健康问题方面最为重要。彭罗斯出生时，他正在Colchester开展这项工作。彭罗斯的哥哥Oliver Penrose早两年出生。Oliver后来先后成为开放大学的数学教授，然后是苏格兰爱丁堡Heriot-Watt大学的数学教授。彭罗斯还有一个弟弟Jonathan，后来成为心理学讲师。Jonathan在1958年至1969年间十次获得英国国际象棋冠军，许多人认为他是有史以来最有天赋的英国国际象棋棋手。

1939 年，彭罗斯 的父亲携家人前往美国，但由于所有迹象都指向战争爆发，他决定不与家人返回英国，而是接受了加拿大安大略省伦敦市一家医院的任命。彭罗斯 在安大略省伦敦市上学，但尽管正是在这一时期他首次对数学产生兴趣，激发这种兴趣的并不是他的学校教育，而是他的家庭。他写道（[2] 或 [3]）：-

我记得大约十岁时制作了各种多面体……

彭罗斯的父亲成为安大略省伦敦市安大略医院精神病学研究主任，但他对数学，特别是几何学非常感兴趣，而彭罗斯的母亲也对几何学感兴趣。彭罗斯的兄弟Oliver（[2]或[3]）：-

……比我大两岁，但在学校比我高四个年级。他很小的时候就对数学了解很多，并对数学和物理都表现出极大的兴趣。

1945年，第二次世界大战结束后，彭罗斯一家返回英国。彭罗斯的父亲被任命为伦敦大学学院人类遗传学教授，彭罗斯就读于伦敦的大学学院学校。然后他对数学的兴趣开始增加，但他的家人希望他追随父亲的脚步，从事医学事业。然而，正如当时学校的典型情况一样，在大学学院学校，生物学和数学是二选一的，学生必须选择其中之一（[2]或[3]）：-

……我记得有一次我们必须决定最后两年要学哪些科目。我们每个人都会一个接一个地上去见校长，他说：“那么，明年专攻时你想学什么科目？”我说：“我想学生物学、化学和数学。”他说：“不，那不可能——你不能同时学生物学和数学，我们没有那个选项。”由于我不想放弃数学，我说：“数学、物理和化学。”我回家时父母相当恼火；我的医学生涯一下子就消失了。

彭罗斯进入伦敦大学学院，由于他父亲是那里的教授，他有权免交学费。他获得了数学一等荣誉学士学位，然后决定去剑桥从事纯数学研究。他追随他哥哥Oliver的脚步，Oliver也在伦敦大学学院读了本科，并去了剑桥从事研究，但Oliver选择了物理。然而，彭罗斯决心从事数学研究，进入圣弗瑞兹·约翰学院后，他开始在威廉·瓦兰斯·道格拉斯·霍奇的指导下研究代数几何。然而，在剑桥学习一年后，他发现自己的兴趣与威廉·瓦兰斯·道格拉斯·霍奇的兴趣并不特别相关，于是将导师改为约翰·阿瑟·托德。彭罗斯于1957年因在代数和几何方面的工作获得剑桥大学博士学位，但此时他已经对物理产生了兴趣。他描述了在剑桥第一年参加的三门课程如何影响了他（[2]或[3]）：-

我记得参加了三门课程，没有一门与我本应做的研究有关。一门是赫尔曼·邦迪关于广义相对论的课程，非常引人入胜……另一门是Paul 保罗·狄拉克关于量子力学的课程，以完全不同的方式美丽……第三门课程……是Steen关于数理逻辑的课程。我了解了艾伦·图灵机和库尔特·弗雷德里希·哥德尔定理……

促使他对物理学产生兴趣的第一位重要影响者是他哥哥的一位物理学家朋友Dennis Sciama。彭罗斯说（[2]或[3]）：-

[Sciama]对我影响很大。他教了我大量物理学知识，做物理学的兴奋感从中透了出来；他就是那种人，能把物理学当前进展的兴奋感传达出来……

在剑桥攻读博士学位期间，他开始发表关于半群和矩阵环的文章。1955年，他在Proceedings of the Cambridge Philosophical Society上发表了A generalized inverse for matrices。在这篇论文中，彭罗斯将复矩形（或可能为方阵且奇异）矩阵$A$的广义逆$X$定义为方程组$AXA = A, XAX = X, (AX)^{T} = AX, (XA)^{T} = XA$的唯一解。他用这个广义逆来解决诸如求解矩阵方程组、寻找一种新型谱分解等问题。他1955年的第二篇出版物是A note on inverse semigroups，发表在同一期刊上，与Douglas Munn合著。逆半群是群的一种推广，至今仍是许多研究论文的主题。这篇早期论文给出了几个等价定义。次年，彭罗斯发表了On best approximation solutions of linear matrix equations，其中利用矩阵的广义逆来求$AX = B$的最佳近似解$X$，其中$A$是矩形非方阵或方阵且奇异。

彭罗斯在1956-57学年担任伦敦贝德福德学院纯数学助理讲师，随后被任命为剑桥圣约翰学院的研究员。这是一个为期三年的职位，在此期间他于1959年与Joan Isabel Wedge结婚。在研究员职位结束前，彭罗斯获得了北约研究奖学金，使他能够在1959-61年期间在美国度过，先是在普林斯顿，然后在锡拉丘兹大学。回到英国后，彭罗斯在接下来的两年1961-63年担任伦敦国王学院的研究助理，然后返回美国，在1963-64年担任德克萨斯大学奥斯汀分校的访问副教授。

1964年，彭罗斯被任命为伦敦伯克贝克学院的准教授，两年后晋升为那里的应用数学教授。1973年，他被任命为牛津大学的Rouse Ball数学讲席教授，并一直担任此职，直到1998年成为荣休Rouse Ball数学讲席教授。同年，他被任命为伦敦格雷沙姆学院的格雷沙姆几何讲席教授。

从1959年开始，彭罗斯发表了一系列关于宇宙学的重要论文。第一篇是The apparent shape of a relativistically moving sphere，1960年他发表了A spinor approach to general relativity。后一篇论文被描述如下：-

对引力理论数学工具的一种优雅而详尽的阐述……重点在于波恩哈德·黎曼张量的几何理论。

除了关于宇宙学的重要论文外，彭罗斯继续发表纯数学论文。他与Henry Whitehead和埃里克·克里斯托弗·齐曼一起在1961年发表了Imbedding of manifolds in euclidean space。除其他结果外，作者在这篇论文中证明了，如果$0 < 2m ≤ n$，那么每个闭$(m-1)$-连通$n$-流形都可以嵌入$\mathbb{R}^{2n-m+1}$中。这一次与阿伯拉罕·伊本·埃兹拉马克斯·纽曼一起，彭罗斯在次年发表了An approach to gravitational radiation by a method of spin coefficients，其中他们证明了：-

……双分量旋量形式导致考虑时空中的一个四元标架，它由两个实零矢量和两个复共轭矢量组成。

1965年，彭罗斯运用拓扑学方法证明了一个重要定理，该定理在他称为“陷获面存在”的条件下，证明了引力坍缩中必然出现奇点。基本上，在这些条件下时空无法延续，经典广义相对论失效。由于量子效应在奇点处占主导地位，彭罗斯寻求一种将相对论与量子理论统一起来的理论。正是由于对“黑洞”存在的这一预言，他在55年后被授予诺贝尔物理学奖。

彭罗斯的主要突破之一是引入扭量理论，试图统一相对论和量子理论。这是一个结合强大代数和几何方法的非凡数学理论。与Wolfgang Rindler一起，彭罗斯于1984年出版了Spinors and space-time的第一卷。这一卷涵盖了双旋量演算和相对论场，而第二卷涵盖时空几何中的旋量和扭量方法，于两年后出版。

彭罗斯也许最为人所知的是他撰写的一些杰出科普著作。他于1989年出版了The Emperor's New Mind : Concerning computers, minds, and the laws of physics。次年，该书获得了罗纳-普朗克科学图书奖。Sklar在评论该书时写道，其目的是：-

……阐述并批判性地抨击一种关于心智本质的近期观点……该观点将心智活动归结为执行一种算法过程，并提出一种更充分的心智理论必须建立在一种迄今尚不存在的、足以说明物质世界已知性质的物理理论之上。在论证过程中，以适合未受过专门训练但相当成熟的读者的水平，对广泛的主题作了优雅的阐述，从算法和抽象可计算性的本质，到不可判定性和不完全性的结果，经典物理学的基本结构，量子力学的基本结构和哲学谜题，熵不对称的基本特征及其与宇宙学结构的关系，对充分的量子引力理论的探索，直到神经解剖学的一些结果以及关于大脑功能的研究。

1994年，彭罗斯出版了Shadows of the mind : A search for the missing science of consciousness，该书继续发展The emperor's new mind的主题。1996年，彭罗斯和史蒂芬·霍金出版了The nature of space and time。这本书记录了1994年两人在剑桥大学艾萨克艾萨克·牛顿数学科学研究所进行的一场辩论。两人各作了三场讲座，交替进行，以便各自能回应对方的论点，然后在最后一场会议中，两人之间进行了一场辩论。我们引用彭罗斯的发言，因为他清楚地陈述了自己的立场以及史蒂芬·霍金的立场：-

在这场辩论开始时，Stephen说他认为自己是一个实证主义者，而我是柏拉图主义者。我乐于让他做实证主义者，但我认为这里的关键点毋宁说在于我是一个实在论者。此外，如果人们将这场辩论与大约七十年前尼尔斯·玻尔和阿尔伯特·爱因斯坦的著名辩论相比较，我认为Stephen扮演的是尼尔斯·玻尔的角色，而我扮演的是阿尔伯特·爱因斯坦的角色！因为阿尔伯特·爱因斯坦主张应当存在某种类似真实世界的东西，不一定由波函数来代表，而尼尔斯·玻尔则强调波函数并不描述一个“真实的”微观世界，而只描述对作出预言有用的“知识”。

还有彭罗斯工作的另一个方面我们必须提及。这就是他在非周期镶嵌方面的工作，这是他在剑桥读研究生时开始感兴趣的一个课题。他最初的尝试取得了成功，但用了大量的瓷砖。经过多年的进一步工作，彭罗斯发现他可以用仅六块瓷砖找到非周期镶嵌，最终他以仅两块瓷砖找到了看似不可能的非周期镶嵌。所谓非周期，我们指的是这些镶嵌在任何平移下都不变。以下是该镶嵌的一些性质：在任何有限的镶嵌区域内，只有一种镶嵌是可能的；在平面的无限镶嵌中，任何出现的区域镶嵌都会在平面其他地方无限次重复出现，并且必须在你首次发现它的地方的两倍区域直径内再次出现。事实上，任何有限区域的镶嵌最终都会出现在每一个彭罗斯镶嵌中。

除了我们上面提到的彭罗斯的主要职位外，他还担任过一些访问和兼职职位。1966-67年和1969年，他在叶史瓦、普林斯顿和康奈尔担任访问职位。从1983年到1987年，他是休斯顿莱斯大学的Lovett讲席教授。然后他成为纽约锡拉丘兹大学的物理学和数学杰出教授，直到1993年，他成为宾夕法尼亚州立大学的Francis and Helen Pentz物理学和数学杰出教授。

彭罗斯因其贡献获得了许多荣誉。他于1972年当选为伦敦皇家学会会士，1998年当选为美国国家科学院外籍院士。我们提到了他因《皇帝新脑》获得的科学图书奖（1990年），但这只是众多奖项之一。其他奖项包括剑桥大学的约翰·柯西·亚当斯奖；沃尔夫物理学基金会奖（与Stephen史蒂芬·霍金共同获得，表彰他们对宇宙的理解）；美国物理学会和美国物理联合会的Dannie Heinemann奖；皇家学会皇家奖章；英国物理学会的保罗·狄拉克奖章和奖章；Royal Astronomical Society的亚瑟·爱丁顿奖章；London Mathematical Society的Naylor奖；以及阿尔伯特·爱因斯坦学会的阿尔伯特·爱因斯坦奖和奖章。1994年，他因对科学的贡献而被授予爵士称号。

2000年，他获得了功绩勋章。2004年，London Mathematical Society授予他奥古斯塔斯·德摩根奖章。部分引文如下：-

他在广义相对论方面的深入工作是我们理解黑洞的一个主要因素。他发展的扭量理论为数学物理的经典方程提供了一种优美而富有成效的方法。他的平面镶嵌是新发现的准晶体的基础。

Royal Society于2005年授予彭罗斯他们的Copley奖章。公告如下：-

彭罗斯，OM，FRS因其对几何学和数学物理的杰出贡献而被授予皇家学会的Copley奖章，这是世界上最古老的科学成就奖。牛津大学数学荣休Rouse Ball讲席教授彭罗斯爵士在广义相对论和宇宙学方面做出了杰出贡献，最著名的是他在黑洞和大爆炸方面的工作。

Royal Society主席彭罗斯解释了彭罗斯的杰出贡献，这些贡献使他获得了该奖项：-

彭罗斯半个世纪以来一直在提出原创且重要的科学思想。他的工作以非凡的几何和物理洞察力为特征。他将新的数学技术应用于阿尔伯特·爱因斯坦的理论，并引领了20世纪60年代引力理论的复兴。他关于空间和时间的新颖思想以及他的‘扭量’概念正日益具有影响力。甚至他的娱乐活动也产生了智力影响：例如，在莫里茨·科内利斯·埃舍尔的艺术作品中普及的‘不可能图形’，以及‘彭罗斯铺砌’的永不重复图案。他通过讲座以及他的畅销且内容广泛的书籍影响和激励了广大公众。

在接受该奖项时，彭罗斯说：-

获得皇家学会的科普利奖章对我来说完全是个惊喜。这是一项非凡的荣誉，这是皇家学会最古老、最杰出的奖项，首次颁发恰好在我出生前200年。我的名字被添加到那份极其杰出的往届获奖者名单中，我感到非常谦卑。

多所大学授予彭罗斯荣誉学位，包括新不伦瑞克大学（1992年）、萨里大学（1993年）、巴斯大学（1994年）、伦敦大学（1995年）、格拉斯哥大学（1996年）、埃塞克斯大学（1996年）、圣安德鲁斯大学（1997年）、圣蒂尼克坦大学（1998年）、华沙大学（2005年）、鲁汶天主教大学（2005年）和约克大学（2006年）。他因预测黑洞的存在而获得2020年诺贝尔物理学奖。

Roger Penrose's parents, Lionel Sharples Penrose and Margaret Leathes, were both medically trained. Margaret was a doctor while Lionel was a medical geneticist who was elected a Fellow of the Royal Society. He was involved with a project called the Colchester survey which aimed to discover whether inherited factors or environmental factors were the most significant in determining if someone would be likely to suffer from mental heath problems. He was in Colchester carrying out this work at the time Roger was born. Roger's brother, Oliver Penrose, had been born two years earlier. Oliver went on to become professor of mathematics first at the Open University, then at Heriot-Watt University in Edinburgh, Scotland. Roger also had a younger brother Jonathan who went on to become a lecturer in psychology. Jonathan was British Chess Champion ten times between 1958 and 1969 and, many argue, was the most naturally talented British chess player of all time.

In 1939 Roger's father went to the United States with his family but as all the indications pointed towards the outbreak of war, he decided not to return to England with his family but accepted an appointment in a hospital in London, Ontario, Canada. Roger attended school in London, Ontario but although it was during this period that he first became interested in mathematics it was not his schooling which stimulated this interest, rather it was his family. He writes ([2] or [3]):-

I remember making various polyhedra when I was about ten ...

Roger's father became Director of Psychiatric Research at the Ontario Hospital in London Ontario, but he was very interested in mathematics, particularly geometry, while Roger's mother was also interested in geometry. Roger's brother Oliver ([2] or [3]):-

... was two years older than I was, but four years ahead in school. He knew a lot about mathematics at a young age and took a great interest in both mathematics and physics.

In 1945, after the World War II ended, the Penrose family returned to England. Roger's father was appointed as Professor of Human Genetics at University College London and Roger attended University College School in London. Then his interest in mathematics began to increase but his family saw him following in his father's footsteps and taking up a medical career. However, as was typical in schools at this time, biology and mathematics were alternatives at the University College School with pupils having to choose one or the other ([2] or [3]):-

... I remember an occasion when we had to decide which subjects to do in the final two years. Each of us would go up to see the headmaster, one after the other, and he said "Well, what subjects do you want to do when you specialise next year". I said "I'd like to do biology, chemistry and mathematics" and he said "No, that's impossible - you can't do biology and mathematics at the same time, we just don't have that option". Since I had no desire to lose my mathematics I said "Mathematics, physics and chemistry". My parents were rather annoyed when I got home; my medical career had disappeared in one stroke.

Penrose entered University College London which he was entitled to do without paying fees since his father was professor there. He was awarded a B.Sc. degree with First Class Honours in Mathematics and then decided to go to Cambridge to undertake research in pure mathematics. He was following in the footsteps of his older brother Oliver who had also taken his undergraduate degree at University College London and had gone to Cambridge to undertake research but Oliver had chosen physics. Roger, however, was set on research in mathematics and on entering St John's College he began research in algebraic geometry supervised by Hodge. However, after one year of study at Cambridge, finding that his interests were not particularly central to those of Hodge, he changed his supervisor to John Todd. Penrose was awarded his Ph.D. for his work in algebra and geometry from the University of Cambridge in 1957 but by this time he had already become interested in physics. He described how three courses which he attended during his first year at Cambridge influenced him ([2] or [3]):-

I remember going to three courses, none of which had anything to do with the research I was supposed to be doing. One was a course by Hermann Bondi on general relativity which was fascinating ... Another was a course by Paul Dirac on quantum mechanics which was beautiful in a completely different way ... And the third course ... was a course on mathematical logic by Steen. I learnt about Turing machines and Gödel's theorem ...

The first major influence prompting his interest in physics had been Dennis Sciama, a physicist friend of his brother. Penrose said ([2] or [3]):-

[Sciama] was very influential on me. He taught me a great deal of physics, and the excitement of doing physics came through; he was that kind of person, who conveyed the excitement of what was currently going on in physics ...

While at Cambridge working towards his doctorate he began to publish articles on semigroups, and on rings of matrices. In 1955 he published A generalized inverse for matrices in the Proceedings of the Cambridge Philosophical Society. In this paper Penrose defined a generalized inverse $X$ of a complex rectangular (or possibly square and singular) matrix $A$ to be the unique solution to the equations $AXA = A, XAX = X, (AX)^{T} = AX, (XA)^{T} = XA$. He used this generalized inverse for problems such as solving systems of matrix equations, and finding a new type of spectral decomposition. His second publication of 1955 was A note on inverse semigroups published in the same journal and co-authored with Douglas Munn. An inverse semigroup is a generalisation of a group and continues to be the subject of many research papers. This early paper gave several alternative definitions. In the following year Penrose published On best approximation solutions of linear matrix equations which used the generalized inverse of a matrix to find the best approximate solution $X$ to $AX = B$ where $A$ is rectangular and non-square or square and singular.

Penrose spent the academic year 1956-57 as an Assistant Lecturer in Pure Mathematics at Bedford College, London and was then appointed as a Research Fellow at St John's College, Cambridge. This was a three year post and during its tenure he married Joan Isabel Wedge in 1959. Before the fellowship ended Penrose had been awarded a NATO Research Fellowship which enabled him to spend the years 1959-61 in the United States, first at Princeton and then at Syracuse University. Back in England, Penrose spent the following two years 1961-63 as a Research associate at King's College, London before returning to the United States to spend the year 1963-64 as a Visiting Associate Professor at the University of Texas at Austin.

In 1964 Penrose was appointed as a Reader at Birkbeck College, London and two years later he was promoted to Professor of Applied Mathematics there. In 1973 he was appointed Rouse Ball Professor of Mathematics at the University of Oxford and he continued to hold this until he became Emeritus Rouse Ball Professor of Mathematics in 1998. In that year he was appointed Gresham Professor of Geometry at Gresham College, London.

Beginning in 1959, Penrose published a series of important papers on cosmology. The first was The apparent shape of a relativistically moving sphere while in 1960 he published A spinor approach to general relativity. This latter paper was described as follows:-

An elegant and detailed exposition ... of the mathematical apparatus of gravitation theory, with emphasis on the geometrical theory of the Riemann tensor.

As well as important papers on cosmology, Penrose continues to publish papers on pure mathematics. Together with Henry Whitehead and Christopher Zeeman he published Imbedding of manifolds in euclidean space in 1961. Among other results, the authors prove in this paper that, if $0 < 2m ≤ n$, then every closed $(m-1)$-connected $n$-manifold can be imbedded in $\mathbb{R}^{2n-m+1}$. This time with Ezra Newman, Penrose published An approach to gravitational radiation by a method of spin coefficients in the following year in which they show that:-

... the two-component spinor formalism leads to the consideration of a tetrad in space-time consisting of two real null-vectors and two complex conjugate ones.

In 1965, using topological methods, Penrose proved an important theorem which, under conditions which he called the existence of a trapped surface, proved that a singularity must occur in a gravitational collapse. Basically under these conditions space-time cannot be continued and classical general relativity breaks down. Penrose looked for a unified theory combining relativity and quantum theory since quantum effects become dominant at the singularity. It was for this prediction of the existence of "black holes" that he would be awarded the Nobel Prize for Physics 55 years later.

One of Penrose's major breakthroughs was his introduction of twistor theory in an attempt to unite relativity and quantum theory. This is a remarkable mathematical theory combining powerful algebraic and geometric methods. Together with Wolfgang Rindler, Penrose published this first volume of Spinors and space-time in 1984. This volume covered two-spinor calculus and relativistic fields while the second volume covering spinor and twistor methods in space-time geometry appeared two years later.

It is for a number of outstanding popular books that Penrose is perhaps best known. He published The Emperor's New Mind : Concerning computers, minds, and the laws of physics in 1989. In the following year the book was awarded the Rhone-Poulenc Science Book Prize. Sklar, reviewing the book, writes that its aim is:-

... to expound and critically attack one recent view of the nature of mind ... taken as reducing mental activity to the carrying out of an algorithmic process, and to propose that a more adequate theory of mind will have to be founded on an as yet not existing physical theory adequate to the known nature of the material world. In the process of the argument elegant expositions, at a level suitable for the unlearned but reasonably sophisticated reader, are given of a wide variety of topics ranging from the nature of algorithms and abstract computability, through results on undecidability and incompleteness, the basic structures of classical physics, the basic structures and philosophical puzzles in quantum mechanics, the basic features of entropic asymmetry and its relation to cosmological structure, the search for an adequate quantum theory of gravity, to some of the results of neuro-anatomy and research into the functioning of the brain.

In 1994 Penrose published Shadows of the mind : A search for the missing science of consciousness which continues to develop the topic of The emperor's new mind. In 1996 Penrose and Hawking published The nature of space and time. This book is a record of a debate between the two at the Isaac Newton Institute of Mathematical Sciences at the University of Cambridge in 1994. Each of the two gave three lectures given alternately so that each could respond to the other's arguments, and then, in a final session, there is a debate between the two. We quote from Penrose's contribution since he states clearly his own position, and that of Hawking:-

At the beginning of this debate Stephen said that he thinks that he is a positivist, whereas I am a PlatoniSt I am happy with him being a positivist, but I think that the crucial point here is, rather, that I am a realiSt Also, if one compares this debate with the famous debate of Bohr and Einstein, some seventy years ago, I should think that Stephen plays the role of Bohr, whereas I play Einstein's role! For Einstein argued that there should exist something like a real world, not necessarily represented by a wave function, whereas Bohr stressed that the wave function doesn't describe a "real" microworld but only "knowledge" that is useful for making predictions.

There is one further aspect of Penrose's work which we must mention. This is his work on non-periodic tilings, an interest which he took up while a graduate student at Cambridge. His first attempts led to success but with a large number of tiles. Further work over many years led to Penrose discovering that he could find non-periodic tilings with only six tiles, then finally he achieved the seemingly impossible with finding non-periodic tilings with only two tiles. By non-periodic we mean that the tilings are not invariant under any translation. Here are some properties of the tiling: in any finite tiled region, only one tiling is possible; in an infinite tiling of the plane, any tiling of a region that occurs is repeated infinitely often elsewhere in the plane and must reoccur within twice the diameter of the region from where you first found it. In fact the tiling of any finite region will eventually appear in every Penrose tiling.

In addition to Penrose's main appointments which we have mentioned above, he also held a number of visiting and part-time posts. He held visiting positions at Yeshiva, Princeton and Cornell during 1966-67 and 1969. From 1983 until 1987 he was Lovett Professor at Rice University in Houston. He then became Distinguished Professor of Physics and Mathematics at Syracuse University in New York until 1993 when he became Francis and Helen Pentz Distinguished Professor of Physics and Mathematics at Pennsylvania State University.

Penrose has received many honours for his contributions. He was elected a Fellow of the Royal Society of London (1972) and a Foreign Associate of the United States National Academy of Sciences (1998). We mentioned the Science Book Prize (1990) which he received for The Emperor's New Mind but this is only one of many prizes. Others include the Adams Prize from Cambridge University; the Wolf Foundation Prize for Physics (jointly with Stephen Hawking for their understanding of the universe): the Dannie Heinemann Prize from the American Physical Society and the American Institute of Physics; the Royal Society Royal Medal; the Dirac Medal and Medal of the British Institute of Physics; the Eddington Medal of the Royal Astronomical Society; the Naylor Prize of the London Mathematical Society; and the Albert Einstein Prize and Medal of the Albert Einstein Society. In 1994 he was knighted for services to science.

In 2000 he received the Order of Merit. He was awarded the De Morgan Medal by the London Mathematical Society in 2004. Part of the citation reads:-

His deep work on General Relativity has been a major factor in our understanding of black holes. His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics. His tilings of the plane underlie the newly discovered quasi-crystals.

The Royal Society awarded Penrose their Copley Medal in 2005. The announcement reads:-

Sir Roger Penrose, OM, FRS has been awarded the Royal Society's Copley medal the world's oldest prize for scientific achievement for his exceptional contributions to geometry and mathematical physics. Sir Roger, Emeritus Rouse Ball Professor of Mathematics at the University of Oxford, has made outstanding contributions to general relativity theory and cosmology, most notably for his work on black holes and the Big Bang.

Martin Rees, President of the Royal Society, explained Penrose's exceptional contributions which led to the award:-

Roger has been producing original and important scientific ideas for half a century. His work is characterised by exceptional geometrical and physical insight. He applied new mathematical techniques to Einstein's theory, and led the renaissance in gravitation theory in the 1960s. His novel ideas on space and time and his concept of 'twistors' are increasingly influential. Even his recreations have had intellectual impact: for instance the 'impossible figures' popularised in Escher's artwork, and the never-repeating patterns of 'Penrose tiling'. He has influenced and stimulated a wide public through his lectures, and his best-selling and wide-ranging books.

On receiving the award, Penrose said:-

The award of the Royal Society's Copley Medal came as a complete surprise to me. It is an extraordinary honour, this being the Royal Society's oldest and most distinguished award, first given just 200 years before I was born. I feel most humbled for my name to be added to that enormously distinguished list of previous recipients.

Several universities have awarded Penrose an honorary degree including New Brunswick University (1992), the University of Surrey (1993), the University of Bath (1994), the University of London (1995), the University of Glasgow (1996), Essex University (1996), the University of St Andrews (1997), Santiniketon University (1998), Warsaw University (2005), Katholieke Universiteit Leuven (2005) and the University of York (2006). He was awarded the 2020 Nobel Prize in Physics for his prediction of the existence of black holes.