先请你做个小实验。

在纸上写一个“由”，然后把纸转半圈——它变成了“甲”。再写一个“士”，转半圈，是“干”。

这种换个方向还能读的字，英文叫 ambigram，中文常译作“双向图”。这个词，是侯世达（Douglas Hofstadter）在 1983 年前后，和几位朋友闲聊时造出来的。

四十多年后的 2025 年 7 月，这位写出《哥德尔、艾舍尔、巴赫》的认知科学家，在八十岁这一年，出版了一整本关于它的书：

他给这本书起了个昵称：ABCD。

规则一句话就能说完：把字母巧妙地变形，让同一幅字换个方向还能读——读成同一个词，或者另一个词。最常见的是旋转 180°，也有镜像对称、首尾相接成链条等玩法。

有些词天生就是双向图，一笔都不用改：

小时候在计算器上敲出 0.7734，再把计算器倒过来看到“hELLO”的人，其实早就玩过双向图了。

可惜，这种“天赐”的词少之又少。绝大多数时候，你得亲自动手。

旋转 180° 时，第一个字母要和最后一个字母配对，第二个和倒数第二个配对……每一对都必须同时满足两种读法。有的配对是字母表里现成的，比如 b 与 q、d 与 p、n 与 u；有的则需要把字母“掰弯”一点，比如 a 与 e、h 与 y。

用上 a↔e、h↔y 这两组配对，就能做出一个小小的双向图：

汉字也能这么玩。日本设计师野村一晟就设计过一幅作品：“令和”倒过来，读作“平成”。

这门手艺的开创者，不止侯世达一人。他的朋友斯科特·金（Scott Kim）早在 1981 年就出版了《Inversions》，这是第一本公开出版的双向图作品集——这次新书的序言，正是他写的。另一位约翰·兰登（John Langdon），则因为给丹·布朗的小说《天使与魔鬼》设计双向图而“出圈”：小说主人公罗伯特·兰登的姓，就是丹·布朗向他致敬的。

《Ambigrammia》由耶鲁大学出版社出版，图文并茂，几乎每一页都有图。先看几个数字：

书中收录了侯世达本人的几百幅作品，外加斯科特·金、约翰·兰登等人的几十幅。它们类型各异，还展示了这门艺术能被延伸到哪些意想不到的方向；据书评介绍，他甚至把这门手艺带进了汉字（kanji）和希伯来字母。

这也是一本相当私人的书。侯世达回忆了自己的人生如何与双向图缠绕在一起：在许多国家，用许多种语言，一个接一个地“创造”——或者说“发现”——它们。

口碑方面，《出版人周刊》（Publishers Weekly）给出星级书评，称它 “Spellbinding and truly one-of-a-kind”（引人入胜，独一无二），称赞它语气亲切、像聊天一样，而且图文并茂；《华尔街日报》也认为这些作品常常颇具魅力；《大西洋月刊》则刊发了书中的节选。

当然也有不同的声音：有读者嫌它节奏偏慢、枝蔓太多，觉得更适合一年读上几篇，而不是一口气读完。说实话，这很“侯世达”——他的书从来不赶路。

副标题 Between Creation and Discovery，是这本书的灵魂。

做过双向图的人大概都有这种体验：盯着一个词反复试，某一刻突然“咔哒”一声，所有字母都对上了。那一刻你很难说清——这个解，是我造出来的，还是它本来就在那里，只是被我找到了？

侯世达的回答是：两者都是。

他为此造了一个新词：discoverativity，姑且译作“发现力”，当作 creativity（创造力）的另一个自我。在他看来，每一个双向图都是一道口袋大小的创造力谜题，创造与发现在其中紧紧交织，难分彼此。

有意思的是，这个问题他已经追问了将近四十年。1987 年，他用意大利文出版过一本《Ambigrammi》，副标题是“研究创造力的理想微观世界”，书中就在讨论“创造与发现”“自由意志与决定论”这两组对立。2025 年的 ABCD，可以看作同一个问题在晚年的回答。

对侯世达来说，双向图从来不只是字体游戏。

他一辈子追问的核心问题是：我们是怎么“认出”一样东西的？同一个字母 A，可以写得千奇百怪，我们为什么总能认出它？“A 之为 A”，本质到底是什么？

早在 1982 年，他就写过一篇长文《Metafont, Metamathematics, and Metaphysics》，论证字母的“神韵”无法被几个参数穷尽：每个字母都是一个边界模糊、不断生长的概念。后来，他带领的“流体类比研究组”（FARG）做了 Copycat、Letter Spirit 等项目，研究的都是类比、风格与“同一性”。

双向图，恰好是这些问题的极限测试：一个字母要扭曲到什么程度，才会“叛变”成另一个字母？在那条模糊的边界上，一个 a 可以同时是一个 e。每做一个双向图，都是在用手摸一摸人类认知里“范畴”的边缘。

也正因如此，书里谈到了人工智能。据《出版人周刊》的书评，侯世达坦言，他并不确定 AI 能否复制这种在他看来本质上属于人类的认知。

熟悉《集异璧》的读者，对这种“形式的戏法”不会陌生。

1979 年出版的《哥德尔、艾舍尔、巴赫》，封面上悬着两块木头：在三个方向的光照下，它们分别投下 G、E、B 和 E、G、B 的影子。这两块红木，是侯世达亲手用带锯做出来的，他管它们叫 trip-let。

那本书的英文副标题 An Eternal Golden Braid，首字母 EGB 恰是 GEB 的重新排列。到了中文版，译者在作者本人的参与下，把副标题译成“集异璧之大成”——“集异璧”，正是 G、E、B 的谐音。

从 GEB 到 ABCD，从木块的影子到一张倒过来的纸，四十多年里，他始终是那个会为一个字母的形状琢磨一下午的人。

在 AI 一秒钟就能生成一千张图的年代，读一位八十岁的学者讲他如何花几十年，把字母一笔一笔掰弯、再掰回来，是一件有点奢侈的事。

但也许正是这种慢，让双向图保有一点神秘：

“你以为你在创造，其实你在发现；
你以为你在发现，其实你也在创造。

下次你随手把一张纸转半圈的时候，也许会想起他。

First, a little experiment.

Write 由 on a sheet of paper, then turn the paper halfway round: it has become 甲. Now write 士 and turn it halfway round: it is 干.

Writing that can still be read from another direction is called an ambigram in English; in Chinese it is usually rendered 双向图 (“two-way picture”). The word was coined around 1983 by Douglas Hofstadter, in a casual conversation with a few friends.

More than forty years later, in July 2025, the year he turned eighty, the cognitive scientist who wrote “Gödel, Escher, Bach” published an entire book about them:

He gave the book a nickname: ABCD.

The rule fits in one sentence: reshape the letters cleverly, so that the same piece of writing can still be read from another direction, as the same word or as a different one. The most common kind works under a 180° rotation, but there are also mirror ambigrams, chains that join up end to end, and other variations.

Some words are ambigrams by nature, without a single stroke changed:

If you ever typed 0.7734 into a calculator as a kid and turned it upside down to see “hELLO”, you were already playing with ambigrams.

Sadly, such “heaven-sent” words are few and far between. Most of the time you have to do the work yourself.

Under a 180° rotation, the first letter has to pair up with the last, the second with the second-to-last, and so on. Every pair must work for both readings at once. Some pairs come ready-made in the alphabet, such as b and q, d and p, n and u; others require you to “bend” the letters a little, such as a and e, or h and y.

With the two pairs a↔e and h↔y, you can make a small ambigram:

Chinese characters can play this game too. The Japanese designer Issei Nomura made a piece in which the era name 令和 (Reiwa), turned upside down, reads 平成 (Heisei).

Hofstadter was not the only pioneer of the craft. His friend Scott Kim published Inversions back in 1981, the first published collection of ambigrams, and it is he who wrote the foreword to the new book. Another, John Langdon, broke into the mainstream by designing the ambigrams for Dan Brown’s novel “Angels & Demons”: the surname of its hero, Robert Langdon, is Brown’s tribute to him.

Ambigrammia is published by Yale University Press and richly illustrated, with pictures on almost every page. First, a few numbers:

The book collects several hundred of Hofstadter’s own pieces, plus a few dozen by Scott Kim, John Langdon and others. They span many different types and show the unexpected directions in which the art can be stretched; according to reviews, he even carries the craft into Chinese characters (kanji) and the Hebrew alphabet.

It is also a very personal book. Hofstadter recalls how his life became entwined with ambigrams: in many countries and in many languages, “creating” them, or perhaps “discovering” them, one after another.

As for reception, Publishers Weekly gave it a starred review, calling it “Spellbinding and truly one-of-a-kind” and praising its warm, conversational tone and its wealth of illustrations; The Wall Street Journal found the pieces often charming; and The Atlantic ran an excerpt from the book.

There are dissenting voices too: some readers find it slow and full of digressions, better taken a few pieces a year than in one sitting. Honestly, that is very “Hofstadter”: his books are never in a hurry.

The subtitle, Between Creation and Discovery, is the soul of the book.

Anyone who has made an ambigram probably knows the feeling: you stare at a word, trying again and again, and suddenly there is a “click” and all the letters fall into place. In that moment it is hard to say: did I make this solution, or was it there all along, and I simply found it?

Hofstadter’s answer: both.

He coined a new word for it, discoverativity, as an alter ego of creativity. In his view, every ambigram is a pocket-sized puzzle in creativity, in which creation and discovery are so tightly intertwined that they cannot be told apart.

Interestingly, he has been pursuing this question for nearly forty years. In 1987 he published a book in Italian, Ambigrammi, subtitled “an ideal microcosm for the study of creativity”, which already discusses two oppositions: creation versus discovery, and free will versus determinism. The 2025 ABCD can be read as the answer he has reached, late in life, to the same question.

For Hofstadter, ambigrams have never been just a typographic game.

The central question he has pursued all his life is this: how do we “recognize” something? The same letter A can be written in a thousand strange ways, so why can we always recognize it? What, at bottom, makes an A an A?

As early as 1982 he wrote a long essay, Metafont, Metamathematics, and Metaphysics, arguing that the “spirit” of a letter cannot be captured by a handful of parameters: each letter is a concept with blurry edges that keeps on growing. Later, the Fluid Analogies Research Group (FARG) he led built projects such as Copycat and Letter Spirit, all of them about analogy, style and “sameness”.

Ambigrams are the ultimate stress test for these questions: how far can a letter be distorted before it “defects” and becomes another letter? On that blurry boundary, an a can be an e at the same time. Every ambigram you make is a way of feeling, with your own hands, the edges of the “categories” in human cognition.

That is also why the book touches on artificial intelligence. According to the Publishers Weekly review, Hofstadter admits that he is not sure AI can replicate this kind of cognition, which he sees as essentially human.

Readers of “Gödel, Escher, Bach” will find this kind of “sleight of form” familiar.

On the cover of “Gödel, Escher, Bach”, published in 1979, two blocks of wood hang in the air: lit from three directions, they cast the shadows G, E, B and E, G, B. Hofstadter cut these two red wooden blocks himself on a band saw, and called each of them a trip-let.

That book’s subtitle, An Eternal Golden Braid, has the initials EGB, a rearrangement of GEB. For the Chinese edition, the translators, working with the author himself, rendered the subtitle as 集异璧之大成 (roughly, “a grand gathering of rare jades”), and 集异璧, pronounced jí yì bì, is a sound-alike of G, E, B.

From GEB to ABCD, from the shadows of wooden blocks to a sheet of paper turned upside down, for more than forty years he has remained a man who will spend an afternoon pondering the shape of a single letter.

In an age when AI can generate a thousand images in a second, reading an eighty-year-old scholar describe how he spent decades bending letters, stroke by stroke, and bending them back again, is something of a luxury.

But perhaps it is exactly this slowness that lets ambigrams keep a little of their mystery:

“You think you are creating, but you are discovering;
you think you are discovering, but you are creating too.

The next time you idly turn a sheet of paper halfway round, perhaps you will think of him.