莫里茨·科内利斯·埃舍尔M. C. Escher

莫里茨·科内利斯·埃舍尔的父母总是叫他Mauk。他由父亲George Escher抚养长大，父亲是一名土木工程师，还有父亲的第二任妻子Sarah，她是一位政府部长的女儿。他和四个哥哥弗拉基米尔·阿诺尔德、Johan、Berend和Edmond一起生活。1912年至1918年间，埃舍尔在阿纳姆先后就读小学和中学，在那里他在许多科目上都不突出，但很早就表现出对音乐和木工的兴趣。

人们认为他拥有数学头脑，但他在求学的任何阶段都从未在这门学科上出类拔萃，并且对这门学科相当不安。他写道[7]：-

在阿纳姆的高中，我的算术和代数极差，因为我过去和现在都对数字和字母的抽象感到非常困难。后来，在立体几何学中，当需要发挥我的想象力时，情况稍好一些，但在学校里我从未在这门学科上出类拔萃。但我们的人生道路可能会有奇怪的转折。

早期的报道详细描述了他对待生活的方法论态度，这被认为是对他工程家庭出身的一种无意识反应。小时候，埃舍尔总是有着极具创造力的一面和一种“敏锐的惊奇感”。他经常声称能在云朵中看到他能联想到的形状。

插图：Escher_puddle.jpeg ↗ Puddle
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埃舍尔和他好友Bas Kist都对印刷技术产生了浓厚兴趣，原因是他们各自的艺术系都给予了良好反馈，鼓励学生进行实验。

家人原本期望埃舍尔能成为建筑师，但他在历史、宪法组织、政治经济学和簿记的期末考试中不及格，令这一期望落空，因此他从未正式毕业。他家搬到奥斯特贝克，荷兰法律的一个漏洞使埃舍尔得以进入代尔夫特高等技术学校（1918—1919年），从而让他能重修一些不及格的科目。由于健康状况不佳，埃舍尔无法也不愿赶上进度，便决定专注于自己的素描和木刻技法。他受到R N Roland Holst的影响，并最初由后者培养：-

他极力建议我做一些木刻，我立刻听从了他的建议……这是极好的工作，但比用油毡工作要困难得多。

1920年9月，埃舍尔搬到哈勒姆，最后一次尝试遵从父亲希望他学习建筑的愿望，并注册了建筑与装饰艺术学校。与图形艺术教师皮埃尔·萨米埃尔 Jesserum de Mesquita的偶然相遇，成为埃舍尔一生中的里程碑事件，他确信图形艺术课程更适合他的技能。De Mesquita将自己所知的木刻印刷技术全部教给了热切的埃舍尔，给他空间进行实验，并鼓励他广泛实验以发展自己的技能。

经常听到埃舍尔抱怨他缺乏天生的绘画能力，因此他的大多数作品需要很长时间才能完成，并且需要多次尝试才能完全满意。在他年轻时，他专注于风景，其中许多是从不寻常的角度绘制的。他还制作了许多植物甚至昆虫的素描，所有这些都经常出现在他后来的作品中。

这些可以在例如图片St Peter's中看到：

插图：Escher_st_peter.jpeg ↗ St Peter's
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从这时起，旅行占据了埃舍尔一生中的大部分时间。1922年4月，他与两位朋友前往佛罗伦萨旅行，整个期间都在写生和饮酒。随后，埃舍尔又独自在意大利各地旅行了一个月，为他的实验性木刻收集素材。

在早期的素描生涯中，埃舍尔只短暂涉及过“填满平面”这一主题，而这一主题的迹象从他年幼时就已显现。许多年后，一位女士[7]：-

……记得这个小男孩[埃舍尔]曾多么仔细地挑选他那些奶酪片的形状、数量和大小，使它们彼此相贴，尽可能精确地覆盖整片面包。这一独特特质从未离开过他……

他第一部以平面的规则分割为主题的作品名为Eight Heads，于1922年完成。

插图：Escher_eight_heads.jpeg ↗ Eight Heads
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埃舍尔于1922年6月乘坐一艘海上货轮前往西班牙，在那里他对规则分割的兴趣短暂地重新燃起。他游历广泛，参观了许多宫殿，从大量建筑和风景中获得了灵感。其中一座对他一生产生巨大影响的建筑是格拉纳达的阿尔罕布拉宫。

埃舍尔被这座14世纪摩尔式宫殿的美所震撼，尤其是装饰在建筑许多表面上的马约利卡装饰瓷砖。与摩尔人不同，埃舍尔既热衷于又获准在他临时创作的瓷砖图案中使用可识别的物体。在接下来的几年里，他多次尝试使用这种艺术风格，但对自己这一热情所耗费的时间（由于其试错性质）以及最终作品的质量不佳感到不满，于是他将规则分割搁置了数年。他写道，大约在1924年，[7]：

……我第一次在一块布上印出了一个从木头上刻出的单一动物图案，该图案按照某种系统重复自身，从而遵循了不得出现空白区域的原则。我至少需要三种颜色；我依次用每一种颜色滚动我的印花块，以便使一个图案与其相邻的全等重复形成对比。我把这块布与我的其他作品一起展出，但没有取得成功。

从西班牙返回后，埃舍尔前往意大利居住。他再次广泛旅行，并于1923年在拉韦洛镇逗留期间遇到了他未来的妻子耶塔·乌米克。他们于1924年6月12日结婚，并在罗马郊外的弗拉斯卡蒂安家。他们将有三个孩子：乔治（1926年6月23日生于弗拉斯卡蒂），然后是阿瑟（1928年12月8日出生）和扬（1938年3月6日出生）。

在接下来的十年里，埃舍尔与家人经常在意大利各地度假。随后是多年描绘意大利风景的写生，通常带有不可能实现的透视，直到1935年夏季意大利法西斯政治起义的发展迫使全家离开意大利。他们搬到了瑞士的山村代堡，但耶塔想念意大利，而瑞士的高物价迫使埃舍尔出售更多版画。

起初全家对新环境并不满意，由于缺乏创作灵感，埃舍尔和Jetta出发去地中海旅行。埃舍尔设法与Adria航运公司谈成了一项协议，为自己争取到免费船票和餐食，也为Jetta争取到一张单程票。他用旅途中完成的素描制作的版画支付了费用。旅行于1936年4月26日开始，在接下来的两个月里，两人画了大量素描，供日后创作使用。

插图：Escher_alhambra3.jpeg ↗ An Alhambra sketch
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在1936年这次地中海之旅后，埃舍尔对秩序和对称的迷恋占据了他的生活，此前他第二次参观了阿尔罕布拉宫。埃舍尔评论说这是[7]：-

……我所汲取过的最丰富的灵感来源。

埃舍尔和他的妻子连续数日在阿尔罕布拉宫工作，在那里他们尽可能多地写生，令每天参观的众多游客感到十分有趣。这些写生将成为埃舍尔未来许多作品的基本来源。这次旅行后，埃舍尔痴迷于平面的规则分割这一概念。他写道[7]：-

它仍然是一项极其引人入胜的活动，一种真正的狂热，我已对此上瘾，有时发现很难从中抽身。

埃舍尔觉得他可以在摩尔艺术家的作品基础上加以改进，并用他的素描作为几何网格，据此设计自己的角色来填满平面。他尝试了许多不同的母题，如鸟、举重运动员和狮子，这些都出现在他早期的许多设计中。他在这一时期的全部作品在很大程度上依赖于他自己的想象以及他在西班牙的素描，并且极其耗时。

插图：Escher_fish.jpeg ↗ One of Escher's designs
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1937年10月，埃舍尔向他的兄弟Berend展示了一些新作品，当时Berend已是莱顿大学的地质学教授，两人都在海牙探望父母的家。Berend认识到他兄弟的木刻与晶体学之间的联系，便给兄弟寄去了一份他认为会有帮助的文章清单。这是埃舍尔第一次接触数学。

埃舍尔读了乔治·波利亚1924年关于平面对称的论文群。尽管他不理解乔治·波利亚论文中讨论的抽象群概念，但他确实理解了文中描述的17种平面对称群。随后他自学了这17个群各自运作的原理。1937年至1941年间，埃舍尔研究可能的周期性密铺，创作了43幅具有多种对称类型的彩色图画。他采用了高度数学化的方法，使用自己发明的记号进行系统研究。埃舍尔还研究了F Haag在1923年写的一篇文章，并在对该主题进一步研究之后，最终对文献中表达的一些观点提出了质疑。

1937年底，埃舍尔一家移居比利时，那里成为他们的家，直到1941年2月20日入侵的德军迫使他们逃往荷兰的巴恩。第二次世界大战对埃舍尔来说是一段情感上极为沉重的时期，使他无法专注于工作。

在随后的几年里，埃舍尔利用全部17个对称群制作了大量木刻版画。随着实践，他的技艺自然提高了，因此他设计和完成每件作品的速度远快于早年。他的艺术成为家庭生活不可分割的一部分，埃舍尔每天上午8点到下午4点会在他的书房里工作。新的构想可能需要数月甚至数年才能成熟，然后才会讨论完成的作品并向家人解释。他的一个孩子写道[7]：-

循环的终点，即制作第一张印张，给父亲带来了喜悦与悲伤交织的感受。第一次从沾满油墨的木版上揭起纸张，看着完成的印张随着纸张被小心掀起而逐渐在纸边显现，清晰而完美，这既令人兴奋又令人满足。但父亲总有一种失望感，觉得自己未能充分描绘出他的思想。毕竟经过他所有的努力，这一结果与最初那如此清晰却又误导性地简单的想法相比，差得多么远啊！

广泛的研究和调查在1941年以他的第一本笔记Regular Division of the plane with Asymmetric congruent Polygons达到顶点。这本笔记在接下来的一年里得到扩充和改进，当时纳入了从广泛的基于颜色的分割研究中获得的结果。这些书从未打算出版——只是为了提供背景信息，让他能够继续作为一名有远见的艺术家。

这些笔记本证明了这样一个事实：无论埃舍尔个人在数学上多么缺乏安全感，他已成为最高级别的研究型数学家。他发展出了自己的分类系统，涵盖了形状、颜色和对称性质的所有可能组合。因此，他在不知不觉中研究的晶体学领域，比任何从事这一领域的专业数学家都早了好几年。他写道[7]：-

很久以前，我在一次漫游中偶然发现了这个领域[平面的规则分割]……然而，另一方面我却陷入了一片荒野……我来到了数学的敞开的大门。有时我以为我已经覆盖了整个领域……然后我突然发现了一条新路径，体验到了新的喜悦。

埃舍尔被来自世界各地的演讲邀请所淹没。在1953年的一次演讲中，埃舍尔说[3]：-

……我常常觉得与从事科学工作的人（尽管我自己肯定不是这样）比与我的艺术家同行更亲近。

到1956年左右，埃舍尔的兴趣再次改变，将平面的规则分割提升到了一个新的水平，通过在固定的二维平面上表现无限。在他职业生涯的早期，他曾使用闭合回路的概念来尝试表达无限，如Horseman所示。

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20世纪40年代，他曾将自己的设计应用于各种三维物体，如柱体和球体，再次试图赋予作品一种无尽的透视感。后来他尝试运用相似性的概念，使用尺寸递减的相同图案，排列成一系列同心圆，但正如他的许多作品一样，他对最终的质量并不满意。

1954年，埃舍尔遇到了哈罗德·斯科特·麦克唐纳·考克斯特，他们成为终身朋友。埃舍尔偶然读到哈罗德·斯科特·麦克唐纳·考克斯特写的一篇文章，同样虽然无法理解文字，他仅凭论文中的图表就能确定关于双曲镶嵌的规则。埃舍尔为感谢哈罗德·斯科特·麦克唐纳·考克斯特，寄给他一幅自己的新作Circle Limit I。埃舍尔继续发展和完善这一领域，创作了更多以圆形和方形为框架的版画作品。

插图：Escher_circle_limit_1.jpeg ↗ Circle limit I
点击图片查看大图

这种艺术风格需要极大的奉献精神，因为需要精心的规划和试验性草图，再加上必要的手工和雕刻技巧，但这给埃舍尔带来了巨大的满足感。他写道[7]：-

我再次发现，只要眼睛能足够清楚地看到手在做什么，人手就能够执行微小而又完全受控的动作。

1995年，哈罗德·斯科特·麦克唐纳·考克斯特发表了一篇论文，证明埃舍尔在他的一幅蚀刻版画中达到了数学上的完美。Circle Limit III仅使用简单的绘图工具和埃舍尔的卓越直觉创作而成，但哈罗德·斯科特·麦克唐纳·考克斯特证明了[8]：-

……[埃舍尔]精确到毫米，绝对精确到毫米……遗憾的是，他未能活到亲眼见证我的数学证明。

插图：Escher_circle_limit_3.jpeg ↗ Circle limit 3
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这个证明凸显了埃舍尔惊人的天赋，他能够将自己的艺术技巧和从他人那里学到的技术结合起来，创造出数学上完美的设计。

插图：Escher_circle_limit_4.jpeg ↗ Circle limit IV （Heaven and Hell）
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到1958年，埃舍尔已经获得了非凡的声誉。他继续讲学，并与渴望向他学习的人通信。他举办了自己的第一次重要作品展，还登上了Time杂志。埃舍尔在其职业生涯中获得了无数奖项，包括奥兰治杰森·约翰·拿骚骑士团勋章（1955年），并经常受委托为世界各地的显贵设计艺术品。

1958年，他发表了Regular Division of the Plane，在这部作品中他说：-

起初，我完全不知道有可能系统地构建我的图形。我不知道……对于一个未受过数学训练的人来说这是可能的，尤其是因为我提出了自己的外行理论，这迫使我思考各种可能性。

Regular Division of the Plane 埃舍尔 再次写道：-

在数学界，平面的规则分割已在理论上得到考虑。……[数学家们]打开了通往广阔领域的大门，但他们自己并未进入这个领域。就其本性而言，他们更感兴趣的是门如何被打开，而不是门后的花园。

埃舍尔 一生的工作涵盖了多种主题。他早年对肖像、罗马和意大利风景以及自然的喜爱，最终让位于平面的规则分割。他的许多作品从不同寻常的视角绘制，从而创造出神秘的空间效果。他精通多种不同的版画技术，如木刻、石版画和美柔汀。超过150件色彩丰富且易于辨认的作品证明了 埃舍尔 的独创性以及对平面规则分割的兴趣。他成功地在固定的二维平面上捕捉了双曲空间的概念，并将规则分割的原理转化到球体、柱体和立方体等若干三维物体上。他的许多版画将二维和三维图像结合在一起，产生了惊人的效果，例如在 Reptiles 中所展示的那样。

插图：Escher_reptiles.jpeg ↗ Reptiles
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他写道[7]：-

当平面分割的某个元素向我暗示某种动物的形态时，我立刻想到一个立体。这种“扁平形状”让我恼火——我感觉自己仿佛在对我的形象们喊叫：“你们对我来说太虚构了；你们只是静止而冻结地躺在那里；做点什么吧，从那里出来，让我看看你们能做什么！”于是，我让它们从平面中出来。但它们真的做到了吗？相反，我故意不一致，通过光和影在平面上暗示可塑性。

他着迷于拓扑学，这一领域在他有生之年才开始被研究，奥古斯特·费迪南德·莫比乌斯条带即为例证。晚年他从英国数学家罗杰·彭罗斯那里学到很多，并运用这些知识设计了许多“不可能”的版画，如Waterfall或Up and down。

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埃舍尔在其Metamorphosis系列设计中用图像讲述故事。这些设计汇集了埃舍尔的许多技艺，并展示了通过一系列对平面中规则图案的细微改变，从一个独特物体向另一个独特物体的转变。

尤其是1937年印制的Metamorphosis 1，让人得以洞见埃舍尔一生中此时发生的艺术风格变化。一段意大利海岸线通过一系列凸多边形转变为平面中的规则图案，直到最终浮现出一个独特的、彩色的、人物母题，标志着他的视角从风景作品转向平面的规则分割。
插图：Escher_metamorphosis_1.jpeg ↗ Metamorphosis I
点击图片查看大图

埃舍尔于1964年在北美进行一系列讲座时首次病倒。结果他被迫大幅削减日程，后来将大部分时间用于与朋友通信。在[11]中，他的最后几年被描述如下：-

埃舍尔的世界观转向内省，他创作了他最著名的令人费解的版画。这些作品，撇开艺术不谈，确实充满智识上的趣味，但他本人却并非如此。他的生活转向内省，他切断了与外界的联系，朋友寥寥无几。……他在久病之后去世……

他最后的版画作品，一幅木刻，Snakes花了六个月才完成，最终于1969年7月揭幕。这幅非凡的蚀刻版画在画面的中心和边缘都通向无限。在进一步的手术之后，埃舍尔搬到了拉伦的罗莎·斯皮尔之家，后来在医院去世。

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[所有 M C 埃舍尔 作品 © 2001 Cordon Art - Baarn - Holland。保留所有权利。经许可使用。]

Maurits Escher was always referred to by his parents as Mauk. He was brought up by his father, George Escher, who was a civil engineer, and his second wife Sarah who was the daughter of a government minister. He lived with his four older brothers, Arnold, Johan, Berend, and Edmond. Maurits attended both elementary and secondary school in Arnhem between 1912 and 1918, where he failed to shine in many of his subjects, but exhibited an early interest in both music and carpentry.

People expressed the opinion that he possessed a mathematical brain but he never excelled in the subject at any stage during his schooling and treated the subject with some considerable unease. He wrote [7]:-

At high school in Arnhem, I was extremely poor at arithmetic and algebra because I had, and still have, great difficulty with the abstractions of numbers and letters. When, later, in stereometry [solid geometry], an appeal was made to my imagination, it went a bit better, but in school I never excelled in that subject. But our path through life can take strange turns.

Early reports detailed his methodological approach to life which was taken to be an unconscious reaction to his engineering family upbringing. As a child, Maurits always had an intensely creative side and an 'acute sense of wonder'. He often claimed to see shapes that he could relate to in the clouds.

插图：Escher_puddle.jpeg ↗   Puddle
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Maurits, and his good friend Bas Kist both developed a deep interest in printing techniques as a consequence of receiving good reports from their respective art departments who had encouraged their student to experiment.

Family aspirations that Maurits would train as an architect were disappointed when he failed his final exams in history, constitutional organisations, political economies and book keeping, and as a result he never officially graduated. His family moved to Oosterbeek where a loophole in Dutch law allowed Maurits to enrol at the Higher Technical School in Delft (1918-1919) and thus allowed him to repeat some of the subjects he had failed. Unable and unwilling to catch up following poor health, Maurits decided to concentrate on his drawing and his woodcut techniques. He was influenced and initially trained by R N Roland Holst:-

He strongly advised me to do some woodcuts, and I immediately followed his advice ... It is wonderful work but far more difficult than working with linoleum.

In September 1920 Maurits moved to Haarlem in a final attempt to try follow his father's wish that he study architecture and he enrolled at the School for Architecture and Decorative Arts. A chance meeting with Samuel Jesserum de Mesquita, a graphic arts teacher, proved a landmark event in Escher's life and he became convinced that a graphic arts programme would be better suited to his skills. De Mesquita taught the eager Escher all he knew of woodcut printing techniques, gave him space to experiment, and encouraged him to experiment widely in order to develop his skills.

Escher was regularly heard to complain about his lack of natural drawing ability and as a result most of his pieces took a long time to complete, and required numerous attempts before he was completely happy. In his youth he concentrated on landscapes, many of which were drawn from unusual perspectives. He also made numerous sketches of plants and even insects, all of which regularly appear in his later work.

These can be seen in, for example in the picture St Peter's:

插图：Escher_st_peter.jpeg ↗   St Peter's
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Travelling took up a large part of Escher's life from this point on. He made a trip with two friends to Florence in April 1922 and spent the whole time sketching and drinking. Escher then spent a further month travelling alone around Italy gathering material to use in his experimental woodcuts.

During his early drawing career Escher touched only briefly on the subject of 'filling the plane', signs of which had been visible from an early age. Many years later a lady [7]:-

... remembered the care with which this little boy [Escher] had selected the shape, quantity and size of his slices of cheese, so that, fitted one against the other, they would cover as exactly as possible the entire slice of bread. This particular trait never left him ...

His first work featuring regular division of the plane was named Eight Heads, and was completed in 1922.

插图：Escher_eight_heads.jpeg ↗   Eight Heads
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Escher visited Spain in June 1922, making the voyage on a sea freighter, and there his interest in regular division was briefly revitalised. He travelled widely and visited many palaces and was inspired by a great number of both buildings and landscapes. One building which was to have an immense influence on his life was the Alhambra Palace in Granada.

Escher was overwhelmed by the beauty of the 14th century Moorish palace and in particular, by the decorative majolica tilings which decorated many of the surfaces of the building. Unlike the Moors, Escher was both keen and permitted to use recognisable objects in his ad-hoc versions of the tilings. He made a number of attempts at using this style of artwork over the next couple of years but was unhappy about both the length of time this passion was taking (due to its trial and error nature) and the poor quality of his final work, and he left aside regular division for a number of years. We wrote that, in about 1924, [7]:-

... for the first time I printed on a cloth a single animal motif cut out of wood which repeats itself according to a certain system, thereby adhering to the principle that no blank spaces may occur. I needed at least three colours; with each in turn I rolled my stamping block in order to contrast one motif with its adjoining congruent repetitions. I exhibited this cloth together with my other work, but I did not have any success with it.

Following his return from Spain, Escher went to live in Italy. Again he travelled widely and in 1923, whilst staying in the town of Ravello, he met his future wife Jetta Umiker. They married on 12 June 1924 and made their home in Frascati, just outside Rome. They would have three children, George (born 23 June 1926 in Frascati), then Arthur (born 8 December 1928) and Jan (born 6 March 1938).

Escher with his family took frequent holidays around Italy during the next decade. Years of sketching Italian landscapes, usually with impossible perspectives, followed before the family were forced to leave Italy as a result of the Fascist political uprising which developed in Italy during the summer months of 1935. They moved to the mountain village of Chateau-d'Oex in Switzerland but Jetta missed Italy and the high Swiss prices forced Escher to sell more prints.

The family was unhappy at first in their new surroundings and, lacking inspiration for his work, Maurits and Jetta set out on a Mediterranean excursion. Escher managed to negotiate a deal with the Adria Shipping Company which gave free passage and meals for himself and also a one way ticket for Jetta. He made payment with prints which he completed using sketches made on the journey. The trip began on the 26 April 1936, and during the next two months the pair made volumes of sketches from which to work from in the future.

插图：Escher_alhambra3.jpeg ↗   An Alhambra sketch
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Escher's fascination with order and symmetry took over his life after this Mediterranean journey in 1936 after he made his second visit to the Alhambra. Escher remarked that it was [7]:-

...the richest source of inspiration I have ever tapped.

Escher and his wife spent days on end working at the Alhambra Palace, where they sketched as much as they could, much to the amusement of the numerous tourists who visited each day. These sketches were to become a fundamental source for much of Escher's future work. After this trip Escher became obsessed with the concept of regular division of the plane. He wrote [7]:-

It remains an extremely absorbing activity, a real mania to which I have become addicted, and from which I sometimes find it hard to tear myself away.

Escher felt that he could improve upon the work of the Moorish artists and used his sketches as a geometric grid from which to design his own characters to fill the plane. He experimented with many different motifs such as birds, weightlifters and lions, all of which appear in many of his early designs. All of his work during this time period relied heavily on his own imagination along with his Spanish sketches, and was immensely time consuming.

插图：Escher_fish.jpeg ↗   One of Escher's designs
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In October 1937 Escher showed some of his new work to his brother Berend, by then a professor of geology at Leiden University, when both were visiting their parents home in The Hague. Recognising the connection between his brother's woodcuts and crystallography, Berend sent his brother a list of articles that he felt would be of assistance. This was Escher's first contact with mathematics.

Escher read Pólya's 1924 paper on plane symmetry groups. Although he did not understand the abstract concept of groups discussed in Pólya's paper he did understand the 17 plane symmetry groups described there. He subsequently taught himself the principles by which each of the 17 groups operated. Between 1937 and 1941 Escher worked on possible periodic tilings producing 43 coloured drawings with a wide variety of symmetry types. He adopted a highly mathematical approach with a systematic study using notation which he invented himself. Escher also studied an article written by F Haag in 1923 and he eventually challenged some of the views expressed in the literature following further research into the topic.

Near the end of 1937 the Escher family moved to Belgium which became their home until the 20 February 1941 when the invading German army forced them to flee to Baarn in Holland. World War II was a deeply emotional time for Escher and prevented him from concentrating on his work.

Over the years that followed Escher made numerous woodcuts utilising each of the 17 symmetry groups. With practice his skills naturally improved and as a result he could design and complete each piece far quicker than in his earlier years. His art formed an integral part of family life, and Escher would work in his study between 8 am and 4 pm every day. New concepts could take months or even years to come to fruition before the finished work was discussed and explained to the family. One of his children wrote [7]:-

The end of the cycle, making the first print, gave father a mixture of joy and sadness. It was exciting and satisfying to lift the paper from the inked wood for the first time, to see the finished print, crisp and immaculate, gradually appearing around the edge of the paper as it was carefully raised. But father had always a feeling of disappointment, of not having been able to depict adequately his thoughts. After all his efforts, how far short of the originally so lucid and misleading simple idea did this result fall!

Extensive research and investigation culminated in 1941 with his first notebook Regular Division of the plane with Asymmetric congruent Polygons. This notebook was extended and improved over the course of the following year, when the results obtained from extensive colour based division investigations were included. These books were never meant for publication - only for background information to allow him to continue as a visionary artist.

The notebooks were evidence of the fact that Escher had become a research mathematician of the highest order, regardless of his personal feeling of mathematical insecurity. He had developed his own categorisation system which covered all the possible combinations of shape, colour and symmetrical properties. As such he had unknowingly studied areas of crystallography years in advance of any professional mathematician working in this field. He wrote [7]:-

A long time ago, I chanced upon this domain [of regular division of the plane] in one of my wanderings... However, on the other side I landed in a wilderness.... I came to the open gate of mathematics. Sometimes I think I have covered the whole area ... and then I suddenly discover a new path and experience fresh delights.

Escher was inundated with requests to give lectures all over the world. In a lecture in 1953 Escher said [3]:-

... I have often felt closer to people who work scientifically (though I certainly do not do so myself) than to my fellow artists.

By around 1956 Escher's interests changed again taking regular division of the plane to the next level by representing infinity on a fixed 2-dimensional plane. Earlier in his career he had used the concept of a closed loop to try to express infinity as demonstrated in Horseman.

Click the picture to see a larger version

He had put his designs on to a variety of three-dimensional objects such as columns and spheres during the 1940s, again in an attempt to impart an endless perspective to his work. He later tried working with the concept of similarities, using identical motifs of diminishing size, arranged in a series of concentric circles, but as with much of his work, he was unhappy about the final quality.

In 1954 Escher met Coxeter and they became life-long friends. Escher came across an article written by Coxeter, and again whilst unable to understand the text, he was able to determine the rules regarding hyperbolic tessellations using only the diagrams in the paper. Escher paid thanks to Coxeter by sending him a copy of one of his new works Circle Limit I. Escher continued to develop and enhance this field and produced many more prints using both circles and squares as the frames for his works.

插图：Escher_circle_limit_1.jpeg ↗   Circle limit I
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This style of artwork required enormous dedication because of the careful planning and trial sketches required, coupled with the necessary hand and carving skill, but was an enormous source of satisfaction to Escher. He wrote [7]:-

I discovered once again that the human hand is capable of executing small and yet completely controlled movements, on the condition that the eye sees sufficiently clearly what the hand is doing.

In 1995 Coxeter published a paper which proved that Escher had achieved mathematical perfection in one of his etchings. Circle Limit III was created using only simple drawing instruments and Escher's great intuition, but Coxeter proved that [8]:-

... [Escher] got it absolutely right to the millimetre, absolutely to the millimetre .... Unfortunately he didn't live long enough to see my mathematical vindication.

插图：Escher_circle_limit_3.jpeg ↗   Circle limit 3
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This proof serves to highlight Escher's amazing natural ability of being able to combine both his artistic skills and the techniques that he learned from others, into mathematically perfect designs.

插图：Escher_circle_limit_4.jpeg ↗   Circle limit IV (Heaven and Hell)
Click the picture to see a larger version

By 1958 Escher had achieved remarkable fame. He continued to give lectures and correspond with people who were eager to learn from him. He had given his first important exhibition of his works and had also been featured in Time magazine. Escher received numerous awards over his career including the Knighthood of the Oranje Nassau (1955) and was regularly commissioned to design art for dignitaries around the world.

In 1958 he published Regular Division of the Plane and in this work he says:-

At first I had no idea at all of the possibility of systematically building up my figures. I did not know ... this was possible for someone untrained in mathematics, and especially as a result of my putting forward my own layman's theory, which forced me to think through the possibilities.

Again in Regular Division of the Plane Escher writes:-

In mathematical quarters, the regular division of the plane has been considered theoretically. ... [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature they are more interested in the way in which the gate is opened than in the garden lying behind it.

Escher's work covered a variety of subjects throughout his life. His early love of portraits, Roman and Italian landscapes and of nature, eventually gave way to regular division of the plane. Many of his pieces were drawn from unusual perspectives thus creating enigmatic spatial effects. He was skilled in the art of a number of different printing techniques such as woodcuts, lithographs and mezzotints. Over 150 colourful and recognisable works testify to Escher's ingenuity and interest in regular division of the plane. He managed to capture the notion of hyperbolic space on a fixed 2-dimensional plane as well as translating the principles of regular division onto a number of 3-dimensional objects such as spheres, columns and cubes. A number of his prints combine both 2 and 3-dimensional images with startling effect as demonstrated for example in Reptiles.

插图：Escher_reptiles.jpeg ↗   Reptiles
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He wrote [7]:-

When an element of plane division suggests to me the form of an animal, I immediately think of a volume. The "flat shape" irritates me - I feel as if I were shouting to my figures, "You are too fictitious for me; you just lie there static and frozen together; do something, come out of there and show me what you are capable of!" So I make them come out of the plane. But do they really do that? On the contrary, I am deliberately inconsistent, suggesting plasticity in the plane by means of light and shadow.

He was fascinated by topology, which only began to be studied during his lifetime, as illustrated by the Möbius strip. In his later years he learned much from the British mathematician Roger Penrose and used this knowledge to design many "impossible" etchings such as Waterfall or Up and down.

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Escher used pictures to tell a story in his Metamorphosis series of designs. These designs brought together many of Escher's skills and show the transformation from one distinct object to another, by means of a series of slight changes to a regular pattern in the plane.

Metamorphosis 1 in particular, printed in 1937, yields an insight into the change of artistic style which occurred in Escher's life at this time. An Italian coastline is transformed through a series of convex polygons into a regular pattern in the plane until finally a distinct, coloured, human motif emerges, signifying his change of perspective from landscape work to regular division of the plane.
插图：Escher_metamorphosis_1.jpeg ↗   Metamorphosis I
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Escher fell ill initially in 1964 whilst delivering a series of lectures in North America. As a result he was forced to cut down his schedule substantially, later devoting most of his time to correspondence with friends. In [11] his last years are described as follows:-

When Escher's view of the world turned inward he produced his best known puzzling prints. which, art aside, were truly intellectually playful, yet he was not. His life turned inward, he cut himself off and he had few friends. ... He died after a protracted illness...

His final graphic work, a woodcut, Snakes took six months to complete and was finally unveiled in July 1969. This exceptional etching heads off to infinity at both the centre and the edges of the picture. Following further operations Escher moved to the Rosa Spier house in Laren and later died in hospital.

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A list of all the pictures available from this page is at THIS LINK.

[All M C Escher works © 2001 Cordon Art - Baarn - Holland. All rights reserved. Used by permission.]