Cleo (mathematician)

Cleo
Cleo's Math Stack Exchange avatar
Years active	2013–2015
Known for	Providing answers to complex integrals on Stack Exchange

Cleo was the pseudonym of an anonymous mathematician active on the mathematics Stack Exchange from 2013 to 2015, who became known for providing precise answers to complex mathematical integration problems without showing any intermediate steps.

During their active period, Cleo posted 39 answers to advanced mathematical questions, primarily focusing on complex integration problems that had stumped other users. Their answers were characterized by being consistently correct while providing no explanation of methodology, often appearing within hours of the original posts. The account claimed to be limited in interaction due to an unspecified medical condition.

The mystery surrounding Cleo's identity and mathematical abilities generated significant interest in the mathematical community, with users attempting to analyze their solution patterns and writing style for clues. Some compared Cleo to historical mathematical figures like Srinivasa Ramanujan, known for providing solutions without conventional proofs.

History

Background

According to Cleo's mathematics Stack Exchange (also known as Math.SE) profile, they presented themselves as a female mathematician with an undisclosed medical condition that limited their ability to engage in extended discussions or provide detailed explanations. Their profile bio stated:

"My real name is Cleo, I'm female. I have a medical condition that makes it very difficult for me to engage in conversations, or post long answers, sorry for that. I like math and do my best to be useful at this site, although I realize my answers might be not useful for everyone."

Activity on Stack Exchange

Between November 2013 and December 2015, Cleo posted 39 answers to advanced mathematical problems, primarily involving complicated integration. Their first notable contribution came on November 11, 2013, solving a particularly difficult integral that had stumped other users:

Posted November 11, 2013 by Lalia Podlesny

"I need help with this integral:

$I=\int _{-1}^{1}{\frac {1}{x}}{\sqrt {\frac {1+x}{1-x}}}\ln \left({\frac {2x^{2}+2x+1}{2x^{2}-2x+1}}\right)\ \mathrm {d} x$.

[...] The approximate numeric value of the integral:

$I=8.372211626601275661625747121...$

Neither Mathematica nor Maple could find a closed form for this integral, and lookups of the approximate numeric value in WolframAlpha and ISC+ did not return plausible closed form candidates either. But I still hope there might be a closed form for it."

Ron Gordon's solution to the integral involved a contour integral over a keyhole contour and extensive symmetry analysis.

The solution was simply stated by Cleo four and a half hours later as:

$I=4\pi \operatorname {arccot} {\sqrt {\phi }}$

where $\phi$ is the golden ratio. The answer included only a hyperlink defining the golden ratio, with no supporting work.

The Math.SE community initially questioned the value of answers without proofs. Two days later, Ron Gordon, a patent agent and former physicist, provided a comprehensive proof validating Cleo's solution. His approach involved reducing an eighth-degree polynomial to a quadratic equation through symmetry analysis, deriving the golden ratio from the simplified expression, thereby confirming Cleo's original answer. Gordon's detailed solution gained significant recognition, earning over 1,000 upvotes on Stack Exchange and later recognition on the subreddit /r/math as the "Master of Integration". Cleo's posts were noted by editors as mathematically sound despite their unconventional approach of providing only answers.

Cleo's self-presentation on Stack Exchange evolved over time. In her earlier profile from 2013, she quoted Srinivasa Ramanujan's famous description of receiving mathematical insights through dreams:

"While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing."

Following this quote, Cleo added her own philosophical perspective:

"Remember, you are not locked into a single axiom system. You may invent your own, whenever you wish—just use your intuition and imagination."

By the time the account went dormant, the profile had changed to the straightforward biographical statement mentioned above.

Identity

The late mathematician and Fields Medalist Maryam Mirzakhani was among those speculated to be behind the Cleo account.

Some speculated that Cleo was a famous mathematician, like Terence Tao (though Tao himself denied this in an email correspondence), Grigori Perelman,, Stephen Hawking, or Maryam Mirzakhani. Allisan Parshall of Scientific American compared Cleo's posts to the work of Indian mathematician Srinivasa Ramanujan, who similarly came up with complicated mathematical formulas without explanation.

Legacy

Cleo became a notable part of mathematical Internet culture, further popularized by the mystery of the anonymity of the user. One Math.SE user noted after the revelation: "It's the idea of ​​Cleo that matters, not whether she existed... like Jesus."

In a 2023 interview on Scientific American, Gordon said of Cleo's popularity on the Internet:

"I think a lot of people who just hated being told, “Show your work, show your work, show your work...,” here’s someone flaunting not showing their work, and people are cheering behind that."

See also

• List of mathematicians
• Stack Exchange