Prominent non-mathematical work of mathematicians First of all, sorry if this post is not appropriate for this forum.
I have a habit that every time I read a beautiful article I look at the author's homepage and often find amazing things.
Recently I read a paper of Andrew Hicks and when I opened his homepage I found many links about his invention: Flawless wing mirrors (car mirror).

(Image Source)
I would not be surprised if this invention was made by a non-mathematician. His mirror is an amazing invention to me because every day I see it, but didn't know its inventor is a mathematician! Anyway, I want to ask

Question 1: Are there mathematicians who have done outstanding/prominent non-mathematical work like inventions, patents, solving social/economical/etc. problems, papers in these areas, etc.?

Of course, one can say that almost all technology nowadays is based on the work of mathematicians, but I'm asking for specific contributions/innovations.
I want to ask a similar question (Maybe it will be useful for those who are looking for a job!):

Question 2: Which mathematicians are working in non-mathematical areas/companies?

Note: Please add to your answers the name and the work of the mathematician.
 A: Another politician would be Éamon de Valera, who graduated in mathematics and taught at various schools (and applied for a professorship, but without success), but then became a rather influential Irish politician.
A: René Descartes
Mathematically you're likely to know him for Cartesian geometry.
Philosophers will know him for "Cogito, ergo sum"/"Je pense, donc je suis"
Quoting wikipedia: "Descartes is also widely regarded as one of the founders of modern philosophy." I suspect, as with Russell, people might argue Descartes was a philosopher who did a little mathematics, but ignoring the eponym of Cartesian coordinates is too hard to do.
A: Alexander Grothendieck was a political activist and spiritualist.
A: Raymond Smullyan was a

mathematician, magician, concert pianist, logician, Taoist, and philosopher.

A: Ruggero Freddi is an Italian mathematics lecturer (holding a PhD) and a former gay pornographic film actor known professionally as Carlo Masi. Here you can read his thesis.
A: Samuel Eilenberg, one of the key mathematicians of the XX Century (co-created category theory, systematized homological algebra, opened new roads in topology, etc), was a good example:
he had (at least) TWO LIVES, with only one thing in common, his short name, Sammy.
In the first one he was the mathematician, whereas in the other he was a formidable expert and collector of Chinese and far-eastern ancient pottery (and other artifacts as well). He was world-famous in his second life just like he was in his first one (when he died he donated his immense collection to NYC, you can still admire it  here).
What is funny (and a bit odd ) is this: Sammy did not like to mix his two lives at all. At his funeral, the two groups (mathematicians and art collectors) collided for the first and last time. Nobody could believe that Prof. Eilenberg, the Math Genius, and Prof. Eilenberg, one of the greatest authorities in ancient eastern arts, were one and the same man.
POST SCRIPT
I have done some research on Sammy's life as art collector: apparently, he was struck by the beauty of indian art during a trip to India. From that point on, he decided that he had to assemble a collection of eastern art and craftmanship, which he did in the next 30 + years. Now I finally see what the two Eilenberg had in common: a passion for aesthetics, for the formal beauty of structures. Alex Heller wrote the following words to honor his Teacher Sammy:
As I perceived it, then, Sammy considered that the highest value in mathematics was to be found, not in specious depth nor in the overcoming of overwhelming difficulty, but rather in providing the definitive clarity that would illuminate its underlying order. This was to be accomplished by elucidating the true structure of the objects of mathematics. Let me hasten to say that this was in no sense an ontological quest: the true structure was intrinsic to mathematics and was to be discerned only by doing more mathematics. Sammy had no patience for metaphysical argument. He was not a Platonist; equally, he was not a non-Platonist. It might be more to the point to make a different distinction: Sammy’s mathematical aesthetic was classical rather than romantic.
A: Jerry McNerney (Wiki page) is a US congressperson from California, with a PhD in differential geometry.
Nowadays he's more known as a congressman than as a mathematician, but every now and then he will give quick floor speeches about math or mathematicians.
See, e.g., his tribute to Mirzakhani.
A: Dutch mathematician Alexander Rinnooy Kan is also a politician and businessman. He used to be a member of the board of directors of ING Group, served as the Chairman of the Social and Economic Council that advises the government, and was a member of the senate.
According to a national newspaper, he was the most influential person in the Netherlands in 2007, 2008, and 2009.
A: I think, one can add Francesco Faa di Bruno to this list, see https://en.wikipedia.org/wiki/Francesco_Fa%C3%A0_di_Bruno. He is to my knowledge the only beatified mathematician.
A: Boris Berezovsky, a Russian oligarch and government official, was a professional mathematician.
A: Tony Scholl is also a bassist with the Cambridge Philharmonic.
A: Daniel Biss received his PhD in mathematics at MIT in 2002, then was an Assistant Professor at the University of Chicago until 2008. He won the 1999 Morgan Prize for outstanding research as an undergraduate. However, in 2007, a serious flaw was discovered that destroyed the main results of papers he had published in the Annals of Mathematics and in Advances. He is now a State Senator in Illinois. In this position, he has worked on legislation to "allow for automatic voter registration," to "elect a number of statewide offices by ranked-choice ballot," and on healthcare, among other things.
A: Sergio Fajardo, the former mayor of the Colombian city of Medellín, wrote a dozen papers in model theory before switching to politics.
A: Charles Lutwidge Dodgson (1832-1898), better known as Lewis Carroll.
A: Claude Elwood Shannon was also an inventor. I recall he also invented a rocket-powered pair of boots, but I cannot seem to find the source anymore.
A: The numerical analyst, Manil Suri, is also an accomplished novelist. He has written a trilogy of novels. The first of which, The Death of Vishnu, was long listed for the Booker
Prize.
A: Sergio Fajardo is a mathematician who was also Mayor of Medellin after the death of Pablo Escobar, was Governor of Antioquia (the state that Medellin is in) from 2012–2016, ran for President of Colombia in 2018, and has announced plans to run again for President. In mathematics, he earned his PhD from UW-Madison and was a professor at the Universidad de los Andes and the Universidad Nacional of Colombia.
In politics, he completely revitalized the city of Medellin, building new parks and libraries, and rebranding the city. The Park of Lights is a great example. That park used to be (figuratively) the darkest place in the city. Escobar had complete control of it and any (non-corrupt) police officer who came nearby would be killed. The park contains two of the oldest buildings in the city, and they were full of squatters. Now, the park is full of life, with Colombian bamboo (Guada kindiana), tons of people, and these giant lightsaber-looking lights that run all night. It's never dark there. It's the safest place in the city. The buildings that used to house squatters now house the department of education. Fajardo also built beautiful libraries to serve the poorest neighborhoods of the city, to give children a chance at a life based around something other than drugs.  He was named "best Mayor of Medellin" in 2007. Regarding his time as governor, according to Wikipedia

During his administration, Antioquia experienced the best national performance in open government, transparency and investment of oil royalties (according to the National Planning Department and the Anti corruption Office of Colombia). He was named the best governor of the country in 2015 by the organization Colombia Líder.

A: Vladimir Voevodsky. He won a Fields Medal for inventing motivic homotopy theory and proving the Milnor Conjecture and in his later years worked on homotopy type theory.
He was a world-class photographer. This is mentioned in his "in memoriam" by Friedlander and you can also see some of his photos hosted by IAS. On that page, it mentions that his photography was included in a 2011 exhibit. I've heard that some of his photos were published by top magazines, like National Geographic, but I don't know all the details. If anyone else does, please add!
A: I have noticed, after reading all previous answers, that no woman mathematician was listed.
My first impression is that most women mathematician born before 1960 (or maybe until today), are outstanding in non mathematics areas as they must have been activist for just having the chance of studying. Many of them were not even recognized or could work as mathematicians, because universities didn't hire women professors. A sad example is that of Emmy Noether who taught for seven year without income from the university. But besides this, there are many women contributing for medicine and biology, social equity, and inclusions of minorities in mathematics. But they are ghosts, as we do not see or know them. Here is a list with some of them: https://en.wikipedia.org/wiki/List_of_women_in_mathematics
Here are two examples:
Sophie- Germain, who, besides being a important mathematician of her time, contributed to elasticity theory and to philosophy - her philosophical work were admired and cited by Auguste Comte. She has struggled to study in her time but were recognized by the great mathematicians Lagrange and Gauss, even after they discovered that she was not Monsieur LeBlanc. ( Her history is worth reading: https://en.wikipedia.org/wiki/Sophie_Germain )
More recently, we have Eugenia Cheng, who is a category theorist and an accomplished pianist. She is also a writer engaged in mathematical popularization and has a column in The Wall Street Journal.
A: Noam D. Elkies, who gave an answer to this question, is himself also an accomplished composer.
A: Nasīr al-Dīn al-Tūsī was a persian polymath of thirteenth century. His contribution to trigonometry includes the plane law of sines. He had a prominent place in the court of Hulagu Khan (a grandson of Genghis Khan and the founder of the Ilkhanate Empire), and benefited from Khan's patronage to found the Maragheh Observatory. The legend even has it that he was influential in persuading Hulagu to siege Baghdad in 1258 which ended the 500 years old Abbasid Caliphate.
A: I've seen a couple of mentions of Emanuel Lasker, a former world chess champion.  Here's his non-mathematical academic works, which include a play "History of Mankind" cowritten with his brother:
Kampf (Struggle), 1906.
Das Begreifen der Welt (Comprehending the World), 1913.
Die Philosophie des Unvollendbar (sic; The Philosophy of the Unattainable), 1918.
Vom Menschen die Geschichte ("History of Mankind"), 1925 – a play, co-written with his brother Berthold.
The Community of the Future, 1940.
In his "Kampf" he foresaw the application of game theory in 20th century social sciences.  He also wrote on other games besides chess as well.  Here's a list of those books.
Encyclopedia of Games, 1929.
Das verständige Kartenspiel (Sensible Card Play), 1929 – English translation published in the same year.
Brettspiele der Völker (Board Games of the Nations), 1931 – includes sections about Go and Lasca.
Das Bridgespiel ("The Game of Bridge"), 1931.
Also, don't forget Max Euwe.  Max Euwe was also world chess champion and president of FIDE (the international chess body).  Dr. Euwe wrote on the Thue-Morse sequence and it implying that, according to the rules of chess at the time, a game could be played as an infinite game without resolution under certain circumstances.  He taught mathematics at one point and was a professor of computer science at the Universities of Rotterdam and Tillberg.
A: Gunnar Carlsson is a professor at Stanford famous for his work in K-theory, and on the Segal Conjecture in homotopy theory. He is also the President and Founder of the company Ayasdi, described as

A machine intelligence software company that offers a software platform and applications to organizations looking to analyze and build predictive models using big data or highly dimensional data sets. Organizations and governments have deployed Ayasdi's software across a variety of use cases including the development of clinical pathways for hospitals, anti-money laundering, fraud detection, trading strategies, customer segmentation, oil and gas well development, drug development, disease research, information security, anomaly detection, and national security applications.

A: Although John Urschel is already mentioned in a comment to the answer about Frank Ryan, I think he deserves his own answer. Urschel was a football player first at Penn State and then with the Baltimore Ravens. He majored in math at Penn State and then, allegedly without the knowledge of the Baltimore Ravens, enrolled in the MIT math PhD program. Urschel will receive his PhD in spring 2021 and already has an impressive list of publications
A: Fields medallist Cédric Villani is a French politician.
A: Omar Khayyam was a Persian mathematician, astronomer, philosopher, and poet. As a mathematician, he is most notable for his work on the classification and solution of cubic equations, where he provided geometric solutions by the intersection of conics. Khayyam also contributed to the understanding of the parallel axiom. As a poet, he gave us the Rubaiyat of Omar Khayyam – "A Jug of Wine, a Loaf of Bread—and Thou."
A: Jim Simons - probably best known to mathematicians for his work on the secondary characteristic classes developed various mathematical models for market trading that have been highly successful. He is an outstanding philanthropist, using much of the income from his financial engineering work to support lots of research activities through e. g. the Simons foundation: the Simons investigators awardees list includes such names as Aaronson, Aganagic, Bhargava, Daubechies, Eskin, Kapustin, Katzarkov, Kitaev, Mirzakhani, Okounkov, Ooguri, Poonen, Rouquier, Seidel, Tao, ...
A: Merely meant as an interesting and amusing fact. People are not born as mathematicians. At the age of 14, long time before his mathematical career and winning the Fields medal, Wendelin Werner played a role in 'The Passerby' at the side of Romy Schneider. She died a few weeks after the movie premiere.
A: Art Benjamin is a professor of mathematics at Harvey Mudd College, with more than 100 publications. He is also an accomplished magician and was the 1997 American Backgammon Tour Player of the Year.
A: Baruch Spinoza was a mathematician, philosopher, and physicist "involved in important optical investigations of the day." His masterpiece, The Ethics, is "written with a forbidding mathematical structure modeled on Euclid's geometry." He was an early Enlightenment thinker of the Dutch Golden Age.
A: Alan Turing is a mathematician famous for his contributions to the foundations of computer science, for being a codebreaker in WW2, and for being persecuted by the UK government for being homosexual. Readers have probably heard of the Turing test in artificial intelligence. I would argue that his contributions to computer science don't count as "non-mathematical work" but that his contributions to biology do. For example, his paper The Chemical Basis of Morphogenesis is highly cited and is the reason the name "Turing patterns" is used to describe zebrafish embryos.
A: Hans Freudenthal was a famous topologist. Indeed, the Freudenthal suspension theorem is the foundational result you need to get stable homotopy theory off the ground. He also invented a language, Lincos, "to make possible communication with extraterrestrial intelligence." And he invented a famous puzzle. And an asteroid is named after him.
A: Patrick Billingsley, author of two well-known books in probability theory, was also a stage and screen actor.
A: Anna Kiesenhofer is a mathematician at Lausanne. She works in PDE, and has nine publications listed at researchgate.
She just won a gold medal in cycling at the Tokyo Olympics.
A: Emanuel Lasker was a great mathematician and is regarded as one of the best chess players ever. If it counts.
A: Bertrand Russell was a mathematician well-known for his philosophical and political work.
A: Tom Lehrer published a couple of papers in mathematical statistics, and taught Mathematics at University of California, Santa Cruz for many years, but is undoubtedly better known for his three albums of humorous songs.
A: From Wikipedia: Hermann Günther Grassmann (German: Graßmann, pronounced [ˈhɛʁman ˈɡʏntɐ ˈɡʁasman]; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguist and now also as a mathematician.
A: The book by French mathematician Jacques Hadamard, The Psychology of Invention in the Mathematical Field.
The book discusses the four stages of thought:

*

*Preparation (conscious work setting up the problem and trying to solve it)


*Incubation (no conscious work on the problem)


*Illumination (the fruits of incubation where an insight is received) and finally


*Verification (consciously testing and verifying the insight)
Gromov's Ergobrain Program for Universal learning program:

*

*Mischa Gromov, Great Circle of Mysteries: Mathematics, the World, the Mind, Springer (2018) doi:10.1007/978-3-319-53049-9 (see also the essay of the same name from 2015).

A: Per Enflo is famous for solving Banach's basis problem, Grothendieck's approximation problem, and the invariant subspace problem for general Banach spaces, and has other fundamental research in linear  and non linear geometric functional analysis. He also has done research in population dynamics. A child prodigy as a pianist, he continues to give concerts in Europe and the United States, but you can hear him at home by creating a Per Enflo station on Spotify.
A: Danica McKellar may not qualify as a mathematician, but (all quotes below are from Wikipedia)
McKellar studied at the University of California, Los Angeles where she earned a Bachelor of Science degree summa cum laude in Mathematics in 1998. As an undergraduate, she coauthored a scientific paper with Professor Lincoln Chayes and fellow student Brandy Winn titled "Percolation and Gibbs states multiplicity for ferromagnetic Ashkin–Teller models on ${\bf Z}^2$." Their results are termed the "Chayes–McKellar–Winn theorem". Later, when Chayes was asked to comment about the mathematical abilities of his student coauthors, he was quoted in The New York Times, "I thought that the two were really, really first-rate." For her past collaborative work on research papers, McKellar is currently assigned the Erdős number four, and her Erdős–Bacon number is six.
Also, she
wrote six non-fiction books, all dealing with mathematics: Math Doesn't Suck, Kiss My Math, Hot X: Algebra Exposed, Girls Get Curves: Geometry Takes Shape, which encourage middle-school and high-school girls to have confidence and succeed in mathematics, Goodnight, Numbers, and Do Not Open This Math Book.
Her acting career, in brief:
She played Winnie Cooper in the television series The Wonder Years from 1988–1993, and since 2010 has voiced Miss Martian in the animated superhero series Young Justice.
In 2015, McKellar was cast in the Netflix original series Project Mc2. She appears in several television films for Hallmark Channel. She is the current voice of Judy Jetson from The Jetsons since 2017 following Janet Waldo's death in 2016.
A: Glen E. Bredon is the author of the programs  DOS.MASTER for Apple II computers, Merlin (a macro assembler) and ProSel for Apple machines. He was the professor of mathematics at Berkley and IAS and author of worthwhile math books like
Bredon, Glen E., Topology and geometry., Graduate Texts in Mathematics. 139. Berlin: Springer. xiv, 557 p. (1997). ZBL0934.55001.
and
Bredon, Glen E., Introduction to compact transformation groups, Pure and Applied Mathematics, 46. New York-London: Academic Press. XIII,459 p. $ 21.00 (1972). ZBL0246.57017.
A: Not so prominent I think but I'd like to mention that the MathTime Professional 2 (MTPro2) fonts were designed by Michael Spivak of Publish or Perish Inc., see here.
A: It has been noted already that Noam Elkies is an accomplished composer.
What I find at least as extraordinary about him is that he can hum-whistle some of Bach's two-part inventions. (Anecdotal "evidence", will remove this answer if it is false.)
A: Marcel-Paul Schuetzenberger (https://en.wikipedia.org/wiki/Marcel-Paul_Sch%C3%BCtzenberger) is also an interesting example (he started as a physician).
A: James Garfield submitted an original proof of Pythagoras's theorem to the  New-England Journal of Education and which was published in the April 1, 1876 issue.
He taught the liberal arts and practised law, he was also a brigadier-general in the Civil War.
He was also the 20th President of the USA, inaugurated in 1881. He developed the proof whilst a member of Congress.
A: Felix Hausdorff wrote philosophical works, essays, poems and plays under the pseudonym Paul Mongré. Let me quote from the information of the Hausdorff Center for Mathematics in Bonn:
https://www.hcm.uni-bonn.de/about-hcm/felix-hausdorff/about-felix-hausdorff/

Hausdorff pursued, especially during the early years in Leipzig, a
kind of double identity: as Felix Hausdorff, the productive
mathematician, and as Paul Mongré. Under this pseudonym, Hausdorff
enjoyed remarkable recognition within the German intelligentsia at the
end of the 19th century as a writer, philosopher and socially critical
essayist....Between 1897 and 1904, Hausdorff reached the peak of
his literary-philosophical accomplishment: during this period, 18 of a
total of 22 works were published under his pseudonym. These included
the volume of aphorisms Sant’ Ilario: Thoughts from Zarathrustra’s
Country, his critique Das Chaos in kosmischer Auslese, a book of poems
entitled Ekstases, the farce Der Arzt seiner Ehre, as well as numerous
essays....The play was Hausdorff’s greatest literary success, as it was
performed over 300 times in 31 cities.

A: I think that the name of Archimedes immediately springs to mind. Among its inventions are the block-and-tackle pulley systems based on the lever, the screw, the parabolic reflectors used to burn Roman ships attacking Syracuse, the mechanical planetarium.
A: Paul Painlevé was briefly prime minister of France on two separate occasions, as well as holding many other government posts including Minister of Defense. In mathematics, he is best remembered for his contributions to nonlinear differential equations.
A: Gauß contributed to the development of the telegraph.
Gauß was also an astronomer, metrology engineer and land surveyor.
A: Frank Ryan earned a math PhD from Rice for the dissertation "Characterization of the set of asymptotic values of a function holomorphic in the unit disc." He published two fundamental papers on the set of asymptotic values of a function holomorphic in the unit disc in Duke Mathematical Journal. [the papers, not the unit disc, were in the journal] He was an assistant professor at the Case Institute of Technology, 1967-71. He has an Erdős number of 3. He was also a lecturer in mathematics at Yale University, 1977-86.
But he is best-known as a quarterback in the National Football League, 1958-70. He led the Cleveland Browns to an NFL title in 1964.
A: Persi Diaconis is an accomplished magician, in addition to his mathematical career.
A: Martin Hairer has written a widely used professional audio editing software "Amadeus Pro" (https://www.hairersoft.com/).
A: Harald Bohr was a member of the Danish national football team for the 1908 Summer Olympics, where he won a silver medal. The semi-final against France was 17-1.
A: I'm surprise no-one has mentioned Isaac Newton.  He spent almost half his career outside academia fighting against forgery at the Royal Mint.  He was also an MP and wrote a huge amount on biblical chronology, alchemy, and theology.
A: Doctor Ahmed Chalabi earned a PhD from the University of Chicago in 1969, founded Petra Bank in 1977, was sentenced in absentia in Jordan for bank fraud in 1992, and (allegedly) very successfully lobbied for the 2003 Invasion of Iraq.
A: Andrew Blumberg, in addition to being an accomplished homotopy theorist, a full professor at UT Austin, and having published more than 60 papers, also works on issues related to geolocation data and privacy. For example, he worked on an amicus brief for a Supreme Court case.
A: I learned this answer from another thread where the OP said it didn't quite fit.
Frank Garside did important work in the 1960s related to braid groups. He also invented what is now known as the Garside element, leading to Garside groups. Later, he was the mayor of Oxford.
In a related vein, George Reid was an algebraist who was later mayor of Cambridge in 1990-91.
A: Ron Graham's mathematical work is probably familiar to most users of this website.
In his youth, he won a title as California state trampoline champion. In 1972 he was elected president of the International Jugglers' Association.
A: I think Greg Egan (sci-fi author) qualifies. He has a few papers on arxiv, and made the news for his work on superpermutations, which motivates the title 'Mathematician'. He is also an occasional MO-user.
A: Nikolai Durov, who launched Telegram with his brother Pavel Durov, has some works on Arakelov geometry.
A: Joseph Fourier was also an Egyptologist.
A: I am on the fence regarding whether this question should stay open, but while it does, I thought the following example might be of interest:
Peter Rosenthal, perhaps best known to mathematicians for his work on subspace lattices of operator algebras (see e.g. his book with Heydar Radjavi), developed a 2nd career/mission as a lawyer.
Wikipedia link: https://en.wikipedia.org/wiki/Peter_Rosenthal
Link from comments: 2014 profile in the Toronto Star
A: Per Enflo, a Swedish mathematician known for solving the so called ``basis problem'', one of the problems from Scottish book (about the existence of Schauder basis). Besides being a mathematician he is also known as a talented pianist.
A: Mathematician Eric Temple Bell

President of the MAA, 1931-32
Author of the book Men of Mathematics
Bell numbers, Bell polynomials
etc.

Also was a successful science fiction author under the pseudonym John Taine.
See the fascinating book
Reid, Constance, The Search for E.T. Bell, MAA spectrum, The Mathematical Association of America, 1993
A: Otto Iulievich Schmidt is now best remembered for his polar expeditions and a geophysical institute in Russia bears his name. The English wikipedia does not mention it, but he was also one of the founders of the modern theory of groups and had a strong influence on A.G. Kurosh.
A: Richard Garfield is a mathematician and former math professor who is nowadays famous as the inventor of the wildly successful card game Magic: The Gathering, and many other card and board games. Here is what Wikipedia says about his math background:

After college, he joined Bell Laboratories, but soon after decided to continue his education and attended the University of Pennsylvania, studying combinatorial mathematics for his PhD. Garfield studied under Herbert Wilf and earned a Ph.D. in combinatorial mathematics from Penn in 1993. His thesis was On the Residue Classes of Combinatorial Families of Numbers. Shortly thereafter, he became a professor of mathematics at Whitman College in Walla Walla, Washington.

A colleague of mine who is somewhat knowledgeable about Magic: The Gathering once claimed that the game became as successful as it was because it was invented by a mathematician who systematically set out to create the most addictive game possible. I cannot say how much truth there is to that claim.
A: Gian-Carlo Rota in addition to being an influential combinatorialist was a philosopher, and his philosophical writing was not in the tradition often thought of as being closest to math ('analytic philosophy') but was rather inspired by phenomenology. Apparently this heterodoxy caused some consternation from e.g. his colleagues in the philosophy department at MIT.
A: In addition to his work in number theory, Carl Størmer made important contributions to the study of the aurora borealis. Here he is conducting an experiment rather far from his blackboard:

A: Alexander Esenin-Volpin was a mathematician well-known for his work as a Soviet political dissident.
EDIT: I'm sorry for writing the above so glibly. Among other things, Esenin-Volpin was repeatedly imprisoned or else confined to mental institutions for political reasons -- according to wikipedia he spent 6 years cumulatively in either of those situations. I'd love if somebody more competent than I would write something more informative and fitting.
A: Emily Riehl is a professor of mathematics at Johns Hopkins. She does transformative work in abstract homotopy theory, and has won many grants. She is also a professional Australian rules football player.
A: Benjamin F. Logan was primarily an electrical engineer who spent his career at Bell Labs, but he has 37 publications listed in Mathematical Reviews. His best known mathematical work is his 1977 paper with Larry Shepp, A variational problem for random Young tableaux, in which they proved that if $L_n$ is the expected length  of the longest increasing sequence in a randomly chosen permutation of $\{1,\dots,n\}$ then $\lim_{n\to\infty} L_n/\sqrt{n}\ge 2$.
Long ago when I used to go to bluegrass festivals, I sometimes saw a bluegrass fiddler named Tex Logan who played with Peter Rowan then, but had earlier played with such greats as Mike Seeger, Bill Monroe, and The Lilly Brothers & Don Stover. Tex Logan also wrote songs recorded by Johnny Cash, Emmylou Harris, and Bob Dylan.
It wasn't until many years later that I learned that B. F. Logan and Tex Logan were the same person.
A: Here is some mathematical work by someone better known for other work.
A: Abu Rayhan al-Biruni (973 – after 1050) was an Iranian polymath,  physicist, astronomer, natural sciences, historian, chronologist and linguist. One of the best his achievements is a method proposed and used by him to estimate the radius and circumference of the Earth almost 1000 years ago.
He was also inventor of minutes and seconds of time.

A: Charles Kalme, a mathematician (Contributions to the Theory of Discontinuous Groups of Mobius Transformations, Ph.D. thesis, NYU, 1967) and pioneering chess programmer. His achievements as a chessplayer include winning the United States Junior Championship and playing twice in the United States Chess Championship. He was also a master of contract bridge.
A: There is Bertrand Russell, a logician, who had another career as a philosopher as well as an anti-war protestor. He helped publicise what was happening in Vietnam - in fact, according to Paul McCartney of the Beatles, at one point he lived on the same square or street with Russell and he had knocked on his door to introduce himself and Russell proceeded to tell him all about the war. This is, according to Paul McCartney, how the Beatles became involved in the anti-war protest movement, in particularly against the war of aggression by the USA on Vietnam.
Laurent Schwartz, a french mathematician known for his work was on distributions, was also an anti-war activist and who focused on labour activism as well on the colonial war in Algeria.
Grothendieck, who needs no introduction, was also an ati-war activist, focusing on what was then occuring Vietnam. His fater was a revolutionary socialist.
Smale was also another anti-war activist; also Solzhenitsyn and many others.
Sofia Kovalevskaya was one of the first handful of female mathematicians. She was also a writer, having written a well-received autobiography, A Russian Childhood in where she reveals she met Dostoyevsky as a young woman and did not think much of him.
Likewise, Lewis Carroll, also a logician and who wrote the best-selling Alice's Adventures in Wonderland.
Also Abdus Salam, who was a mathematical physicist and had a second career setting up the Triest Centre of Mathematical Physics to help te Third World get on to its feet.
And a special mention for Walter Sisulu, who studied a science degree, but after joining the ANC, was imprisoned several times, and was finally imprisoned on Robbens Island with Nelson Mandela and sentenced to twenty-five years - the same sentence as Nelson. He rose to become the ANC's Secretary-General and Deputy President of the organisation.
A: I think, John F. Nash fits in this category,  because of 1994  Nobel Memorial Prize in Economic Sciences. (I am not aware the details of his work, I just know this because of "A beautiful mind" movie!)
A: Although it is difficult to say exactly what Thales accomplished, if you take what he's given credit for it is immense.  By your criteria he would qualify as an answer.
Although he lacked the rigor of proof, he was the first mathematician we attribute a mathematical result to - namely Thales' Theorem used to get the distances of ships at sea.
He was also the first philosopher.  With his contention that everything is water we have the first theory that says everything is one.
Also with his contention that everything is water we have a testable claim.  This makes him the first scientist.
He also was an astronomer, predicting the eclipse of May 28th, 585 BC.
He even dabbled in business, by legend.  There is a famous story attributed to Aristotle, and others, about him.  If true, Thales predicted the weather and foresaw a good olive harvest for the coming year.  He then bought all the olive presses at a discount, and rented them out during the harvest.  This would be the first recorded use of futures to turn a profit.
Legend has it that Thales was an engineer.
A: Albert Einstein's contribution to differential geometry was significant. One could mention also Einstein's tensor notation.
On the other hand, Einstein was an inventor, he had several inventions (patents) to his credit.
A: Professionally, Pierre de Fermat was a lawyer; he had also contributed to physics.
Henry Poincare, Solomon Lefschetz, and Raoul Boot were into engineering when they were young.
Hugo Steinhaus was serious about applications, e.g. about the different feet shapes in order to help shoe designers. The famous Steinhaus slogan was
                    A mathematician will do it better.
In particular, Steinhaus patented longimeter.
Karol Borsuk invented a successful game during WWII, and it helped him to survive in those hardship years.
Stan (Stanisław) Ulam was the main inventor of the H bomb. He has also invented cellular automata.
Israel Gelfand at his mature stage turned his interests toward biology and medicine.
Rene Thom in his later years became seriously interested in too many things to mention here.
Several American mathematicians, including John Milnor, worked on DOD contracts.
Topologist James Munkres made a significant contribution to the assignment problem -- it's even called Munkres assignment algorithm or Kuhn–Munkres algorithm.
A parallel processor invented by a mathematician in the US, and the technology that followed, had an impact on the fall of communism.
Greg Kuperberg was an early pioneer in the field of computer games when he was in his teens; this helped him to pay his tuition at MIT.
A: Stan Wagon, whose work was featured in several Mathoverflow posts (e.g. an unexpected image, Gaussian prime spirals), has also an entry in Ripley's Believe It or Not due to his Square Wheel Bike.
A: I see nobody mentioned about John Forbes Nash, who obtained Nobel prize in Economic science for his contribution to economy. Further German Mathematician Carl Fredric Gauss, Von Neumann etc both were not only prominent mathematicians but also prominent physicist as well.
