人工智能和算法的結(jié)合揭示了跨越兩千年來π方程的隱藏結(jié)構(gòu)。

作者：林迪·邱， 編輯：克拉拉·莫斯科維茨

Jeffrey Coolidge/Getty

慶祝圓周率日，并在我們的圓周率日專題頁面 上了解這個數(shù)字在數(shù)學(xué)和科學(xué)領(lǐng)域的應(yīng)用 。

兩千多年來，數(shù)學(xué)家們?yōu)榱烁煊嬎悝?，不斷探索各種方法，積累了越來越多的π方程。這些方程的數(shù)量已達數(shù)千個，算法甚至可以生成無窮多個。每一項發(fā)現(xiàn)都像是孤立的碎片，與其他發(fā)現(xiàn)之間似乎并無關(guān)聯(lián)。但現(xiàn)在，幾個世紀以來積累的π公式首次被揭示為一個統(tǒng)一的、此前不為人知的結(jié)構(gòu)。

用圓的周長除以直徑，就能得到圓周率π。但是，π的具體數(shù)字是多少呢？測量圓的周長并不能告訴你答案——你的工具太笨重，無法揭示π的無窮大數(shù)字。要揭示它的真正值，需要更強大的工具：公式。
這一切都始于阿基米德，他提出了世界上第一個已知的π值的數(shù)學(xué)證明。他將圓視為邊長為零的無限多邊形。處理無窮小的數(shù)學(xué)（微積分）還要再過1900年才會出現(xiàn)，所以他轉(zhuǎn)而將圓的外切面和內(nèi)切面各為96邊形，并利用幾何學(xué)計算它們的周長。他由此確定π的值介于3.140845…和3.142857…之間，將其限制在一個范圍內(nèi)。他的嚴謹性保持了1600年之久。
大約在14世紀，印度數(shù)學(xué)家桑伽瑪格拉瑪?shù)鸟R達瓦給出了第一個精確的π公式，它以無窮級數(shù)的形式表示——一個由無數(shù)項組成的級數(shù)，如果能將它們?nèi)考悠饋?，就能精確地得到π。但問題是：他的級數(shù)收斂速度極其緩慢，需要數(shù)百項才能精確到小數(shù)點后幾位。三百多年后，萊昂哈德·歐拉發(fā)現(xiàn)了另一個收斂速度更快的級數(shù)。到了20世紀初，數(shù)學(xué)家斯里尼瓦薩·拉馬努金提出的公式，至今仍因其高效性而備受推崇。

Amanda Monta?ez；來源：“從歐拉到人工智能：數(shù)學(xué)常數(shù)的統(tǒng)一公式”，作者：Tomer Raz 等人，預(yù)印本于 2025 年 11 月 16 日發(fā)布于https://arxiv.org/pdf/2502.17533（參考文獻）

每個公式看起來都與其他公式毫不相干。但在2025年末，以色列理工學(xué)院（Technion）一個由七名人工智能研究人員組成的團隊發(fā)現(xiàn)了一種此前未知的數(shù)學(xué)結(jié)構(gòu)，這種結(jié)構(gòu)隱藏在數(shù)百個π公式背后，其中包括阿基米德、歐拉和拉馬努金的公式?！澳芤冒⒒椎碌墓娇刹皇敲刻於加械臋C會，”該團隊的博士生邁克爾·沙利特（Michael Shalyt）說道。這種被稱為保守矩陣場（CMF）的結(jié)構(gòu)，就像一個數(shù)學(xué)上的共同祖先，揭示了看似毫不相干的公式實際上是同一個底層對象的不同表達方式。

該項目源于小組負責(zé)人伊多·卡米納 (Ido Kaminer) 于 2019 年開發(fā)的“拉馬努金機器”(Ramanujan Machine)，這是一個人工智能機器人，旨在尋找計算數(shù)學(xué)常數(shù)的新猜想。任何人都可以免費下載該軟件，許多人已經(jīng)利用它找到了新的 π 公式，加入了拉馬努金的行列。盡管數(shù)學(xué)家們并未完全接受，但該機器人的非傳統(tǒng)方法卻取得了巨大的成功。“當(dāng)我們開始在這個數(shù)學(xué)領(lǐng)域進行人工智能研究時，”卡米納說，“這被認為是一個邊緣化的想法。”

但隨著機器和其他數(shù)學(xué)家不斷推導(dǎo)出各種公式，最終這個問題變得不可避免：它們之間是否有聯(lián)系？

這個團隊成員都擁有物理和數(shù)學(xué)等領(lǐng)域的背景，他們像實驗學(xué)家一樣研究這個問題，決定收集數(shù)據(jù)集。當(dāng)時在以色列理工學(xué)院攻讀碩士學(xué)位的托默·拉茲編寫了一段代碼，用于下載所有上傳到預(yù)印本服務(wù)器arXiv.org的數(shù)學(xué)論文。他每周七天、每天24小時不間斷地運行筆記本電腦，持續(xù)六周，最終以足夠慢的速度下載了455,050篇論文，以避免超出網(wǎng)站的下載限制。

隨后，該團隊將 GPT-4o 與專門的算法結(jié)合使用，以檢測與 π 相關(guān)的公式，將其轉(zhuǎn)換為可執(zhí)行代碼，并去除無關(guān)的重復(fù)項。他們從近 50 萬篇論文中提取了 385 個獨特的公式，其中約 10% 源自拉馬努金機器。

接下來，他們將這385個方程改寫成相同的形式——一種特殊的無窮級數(shù)。但這些表達式仍然全部收斂于π，因此沒有明顯的比較方法。他們需要更深入的分析。

那個東西就是CMF，是卡米納團隊的一些成員在2023年提出的。沙利特稱它為數(shù)學(xué)界的瑞士軍刀?！八梢越y(tǒng)一兩千年前的公式，并為數(shù)學(xué)中的常數(shù)建立層級關(guān)系，我們希望用它來證明一些與黎曼猜想相關(guān)的無理性性質(zhì)，”他說。

可以將CMF想象成定義在網(wǎng)格上的引力。每個 π 公式在網(wǎng)格上都描繪出一條不同的路徑。正如引力場保證兩點之間的能量差與路徑無關(guān)一樣，CMF 保證只有終點才重要。從這一約束條件出發(fā)，一個非凡的結(jié)論浮現(xiàn)出來：當(dāng)兩個 π 公式在同一個 CMF 網(wǎng)格上描繪出平行路徑時，它們是等價的（一個可以轉(zhuǎn)化為另一個），無論它們表面上看起來多么不同。

研究小組推導(dǎo)出了π的CMF（關(guān)鍵矩陣），然后利用算法確定每個公式在網(wǎng)格中的位置，從而找到相似公式的集合。算法正式證明了一組公式是否屬于該CMF。結(jié)果顯示：所有已知的π公式中，43%源自同一個CMF。另有51%屬于更廣泛的公式集合。（研究人員仍在研究它們之間的具體關(guān)系。）只有6%的公式是孤立的，沒有證據(jù)表明它們與其他任何公式存在關(guān)聯(lián)。

卡米納表示，更復(fù)雜的CMF能否涵蓋所有方程組，目前尚無定論。另一個懸而未決的問題是，CMF生成的每個方程是否都是π公式——到目前為止，團隊嘗試過的所有方程都有效。

大衛(wèi)·貝利是一位退休的計算機科學(xué)家，曾就職于勞倫斯伯克利國家實驗室，他沒有參與這項研究（盡管π公式以他的名字命名，而且該小組使用了他的一個算法），他說，該項目的結(jié)果就好像17世紀的化學(xué)家們一直在逐一發(fā)現(xiàn)原子元素，“然后突然之間，有人發(fā)布了一個計算機程序，自動構(gòu)建了整個元素周期表”。

賓夕法尼亞州立大學(xué)榮譽退休數(shù)學(xué)教授喬治·安德魯斯（他曾因發(fā)現(xiàn)拉馬努金遺失的大量筆記而聞名）此前曾批評該團隊以拉馬努金的名字命名他們的機器。但他對目前的工作贊不絕口。“這是以嚴謹?shù)姆绞竭M行的嚴肅數(shù)學(xué)研究，”他說?！拔磥響?yīng)該會有更多令人驚訝的結(jié)果出現(xiàn)?！?/p>

LYNDIE CHIOU是一位科學(xué)家、科學(xué)作家，也是科學(xué)會議網(wǎng)站ZeroDivZero的創(chuàng)始人。她的文章也曾發(fā)表在《天空與望遠鏡》雜志上。您可以在Xbox上關(guān)注她： @lyndie_chiou

Mathematicians find one pi formula to rule them all

A mixture of AI and algorithms uncovered a hidden structure spanning 2,000 years of equations for pi

BY LYNDIE CHIOU EDITED BY CLARA MOSKOWITZ

Jeffrey Coolidge/Getty

Math

Celebrate Pi Day and read about how this number pops up across math and science on our special Pi Day page.

For more than two millennia, mathematicians have produced a growing heap of pi equations in their ongoing search for methods to calculate pi faster and faster. The pile of equations has grown into the thousands, and algorithms now can generate an infinitude. Each discovery has arrived alone, as a fragment, with no obvious connection to the others. But now, for the first time, centuries of pi formulas have been shown to be part of a unified, formerly hidden structure.

Divide any circle’s circumference by its diameter and you get pi. But what, exactly, are its digits? Measuring physical circles won’t tell you—your tools are too clunky to discover pi’s endless numerals. Uncovering its true value requires something much more powerful: a formula.

It all started with Archimedes, who developed the world’s first known mathematical proof for pi’s value. He thought of a circle as an infinite-sided polygon with sides of zero length. The math to handle infinitesimals (calculus) wouldn’t arrive for another 1,900 years, so instead he circumscribed 96-sided polygons on the outside and inside of a circle and used geometry to calculate their perimeters. He was able to determine that pi fell somewhere between 3.140845... and 3.142857..., trapping it in a range. His rigor stood for 1,600 years.

Then, around the 14th century, Indian mathematician Madhava of Sangamagrama provided the first exact formula, expressed as an infinite series—a sum of endlessly many terms that, if you could somehow add them all up, would yield pi exactly. The catch: his series converged agonizingly slowly, requiring hundreds of terms just to nail down a few decimal places. More than three hundred years later Leonhard Euler discovered another series that converged faster. And in the early 1900s, the mathematician Srinivasa Ramanujanproduced formulas that are still revered for their efficiency today.

Amanda Monta?ez; Source: “From Euler to AI: Unifying Formulas for Mathematical Constants,” by Tomer Raz et al. Preprint posted November 16, 2025 to https://arxiv.org/pdf/2502.17533 (reference)

Each equation seemed unrelated to the others. But in late 2025, a team of seven AI researchers at the Technion–Israel Institute of Technology found a previously unknown mathematical structure underlying hundreds of pi formulas, including those of Archimedes, Euler and Ramanujan. “It’s not every day that you get to cite Archimedes,” says Ph.D. student Michael Shalyt, part of the team. The structure, called a conservative matrix field, or CMF, acts as a kind of mathematical common ancestor, showing how formulas that look nothing alike turn out to be different expressions of the same underlying object.

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The project grew out of group head Ido Kaminer’s 2019 Ramanujan Machine, an AI bot that seeks out new conjectures for calculating mathematical constants. Anyone can download the software for free, and many have used it to find new pi formulas to join the heap. The bot’s unconventional approach was a viral success, if not taken entirely seriously by mathematicians. “When we started doing AI research in this area of math,” Kaminer says, “it was seen as a fringe idea.”

But as the machine and other mathematicians kept churning out formulas, eventually the question became unavoidable: Were any of them connected?

The group, who also have backgrounds in areas such as physics and math, approached the problem like experimentalists and decided to gather a dataset. Tomer Raz, then a master’s student at Technion, wrote code to download every math paper that had ever been uploaded to the preprint server arXiv.org, running his laptop seven days a week, 24 hours a day, for six weeks to download 455,050 papers at a slow enough rate to respect the website’s limit.

The group then deployed GPT-4o in combination with specialized algorithms to detect pi-related equations, translate them into executable code, and remove trivial duplicates. From nearly half a million papers, they extracted 385 unique formulas, including about 10 percent that originated from the Ramanujan Machine.

For the next step, they recast the 385 equations into the same format—a special type of infinite series. But the expressions still all converged to pi, leaving no obvious way to compare them. Something deeper was needed.

That something was the CMF, which some members of Kaminer’s group had introduced in 2023. Shalyt calls it a Swiss army knife for mathematics. “It can unify 2,000-year-old formulas [and] give hierarchy for constants in math, and we hope to [use it to] prove some properties of irrationality related to the Riemann hypothesis,” he says.

Think of the CMF like gravity defined on a grid. Each pi formula traces a different path across the grid. Just as a gravitational field guarantees that the energy difference between two points is the same, regardless of route, the CMF guarantees that only the destination matters. From this single constraint, something remarkable emerges: when two pi formulas trace parallel paths through the same CMF grid, they are equivalent (one can be transformed into the other), however mismatched they appear on the surface.

The group derived the CMF of pi, then used algorithms to see where each formula fit inside the grid, finding clusters of similar equations. An algorithm formally proved whether a cluster of equations belonged to the CMF. The result: 43 percent of all known pi formulas descend from a single CMF. Another 51 percent belong to broader clusters. (The researchers are still working out their precise relationships.) Only 6 percent of the formulas remain orphans, with no proven connection to anything else.

It’s an open question whether a more complex CMF could capture the entire set, Kaminer says. Another open question is whether every single equation generated from the CMF is a pi formula—so far, all the equations the team has tried have worked.

David Bailey, a retired computer scientist formerly at Lawrence Berkeley National Laboratory, who wasn’t involved in the study (though a pi formula bears his name and the group used one of his algorithms), says the project’s results are as if 17th-century chemists had been discovering atomic elements one by one “and then all of a sudden, someone let loose a computer program that constructed the whole periodic table automatically.”

Mathematician George Andrews, a professor emeritus at the Pennsylvania State University (who famously uncovered a lost trove of Ramanujan’s notes) had previously criticized the group for naming their machine after Ramanujan. But he had nothing but praise for the current work. “This is serious mathematics done in a serious way,” he says. “More and more surprising things should emerge.”

Join the discussion: What is the nerdiest or most unusual way you or someone you know has celebrated Pi Day?

This year people in Scientific American’s New York office brought in pies ahead of Pi Day, but we know there must be stranger and more interesting ways to celebrate the iconic number. How have you or people you know celebrated Pi Day? What are the most interesting ways pi comes up in your work or everyday life? Is there another number you think deserves more attention?

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What is the nerdiest or most unusual way you or someone you know has celebrated Pi Day?

This year people in Scientific American’s New York office brought in pies ahead of Pi Day, but we know there must be stranger and more interesting ways to celebrate the iconic number. How have you or people you know celebrated Pi Day? What are the most interesting ways pi comes up in your work or everyday life? Is there another number you think deserves more attention?

Join the Discussion

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LYNDIE CHIOUis a scientist, a science writer and founder of ZeroDivZero, a science conference website. Her writing has also appeared in Sky & Telescope. Follow her on X @lyndie_chiou

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