|Died||27 January 1860 (aged 57)|
|Education||TherMilAk ( diploma, 1822)|
|Known for||Non-Euclidean geometry|
|Academic advisors||Farkas Bolyai|
János Bolyai (Hungarian: [ˈjaːnoʃ ˈboːjɒi]; 15 December 1802 – 27 January 1860) or Johann Bolyai,  was a Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consistent alternative geometry that might correspond to the structure of the universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world. 
By the age of 13, he had mastered calculus and other forms of analytical mechanics, receiving instruction from his father. He studied at the Imperial and Royal Military Academy (TherMilAk) in Vienna from 1818 to 1822.
Bolyai became so obsessed with Euclid's parallel postulate that his father, who had pursued the same subject for many years, wrote to him in 1820: "You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone...Learn from my example." 
János, however, persisted in his quest and eventually came to the conclusion that the postulate is independent of the other axioms of geometry and that different consistent geometries can be constructed on its negation. In 1823, he wrote to his father: "I have discovered such wonderful things that I was amazed...out of nothing I have created a strange new universe."   Between 1820 and 1823 he had prepared a treatise on parallel lines that he called absolute geometry. Bolyai's work was published in 1832 as an appendix to a mathematics textbook by his father.
Carl Friedrich Gauss, on reading the Appendix, wrote to a friend saying "I regard this young geometer Bolyai as a genius of the first order."  To Bolyai, however, Gauss wrote: "To praise it would amount to praising myself. For the entire content of the work...coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."    In 1848 Bolyai learned that Nikolai Ivanovich Lobachevsky had published a similar piece of work in 1829. Though Lobachevsky published his work a few years earlier than Bolyai, it contained only hyperbolic geometry. Working independently, Bolyai and Lobachevsky pioneered the investigation of non-Euclidean geometry.
In addition to his work in geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers. Although he never published more than the 24 pages of the Appendix, he left more than 20,000 pages of mathematical manuscripts when he died. These can now be found in the Teleki-Bolyai Library in Marosvásárhely (today Târgu Mureş), where Bolyai died.
It is related of him that he was challenged by thirteen officers of his garrison, a thing not unlikely to happen considering how differently he thought from everyone else. He fought them all in succession—making it his only condition that he should be allowed to play on his violin for an interval between meeting each opponent. He disarmed or wounded all his antagonists. It can be easily imagined that a temperament such as his was not one congenial to his military superiors. He was retired in 1833. 
No original portrait of Bolyai survives. An unauthentic picture appears in some encyclopedias and on a Hungarian postage stamp. 
The Babeş-Bolyai University in Cluj-Napoca, that was established in 1959, bears his name, as does the crater Bolyai on the Moon  and the János Bolyai Mathematical Institute at the University of Szeged. Furthermore, 1441 Bolyai, a minor planet discovered in 1937, is named after him; and many[ quantify] primary and secondary schools in the Carpathian Basin bear his name, e.g. Bolyai János Műszaki Szakközépiskola in Budapest, Bolyai János Gyakorló Általános Iskola és Gimnázium in Szombathely, Bolyai János Általános Iskola in Debrecen, etc.[ citation needed] A street in Budapest, Hungary and another two in Kolozsvár  and Temesvár (now Cluj-Napoca and Timișoara), Romania are also named after him. The professional society of Hungarian mathematicians also bears his name. Bolyai is a minor character in the 1969 science-fiction/fantasy story "Operation Changeling", where his unique abilities allow the protagonists to navigate the non-Euclidean geometry of Hell.
There is also a mathematical award given out every five years named the Bolyai Prize.
His grave lies in the Lutheran Cemetery in Târgu Mureș, Romania. 
- "Appendix scientiam spatii absolute veram exhibens; a veritate aut falsitate axiomatis XI Euclidei, a priori haud unquam", appendix to
Farkas Bolyai, Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi (An Attempt to Introduce Studious Youths to the Elements of Pure Mathematics), 1832.
- English translation: "The Science Absolute of Space: Independent of the Truth or Falsity of Euclid's Axiom XI (Which Can Never Be Decided A Priori)", The Neomon, Austin, 1896.
- Dénes, Tamás (January 2011). "Real Face of János Bolyai" (PDF). Notices of the American Mathematical Society. 58 (1): 41–51. Retrieved 2011-06-18.
- Tucker McElroy. A to Z of Mathematicians
- Encyclopædia Britannica: János Bolyai
- Ellenberg, Jordan (May 2014). How Not to Be Wrong. New York, NY: Penguin Group. p. 395. ISBN 978-0-14-312-753-6.
- Tóth, László Fejes (1965). "1.3.4. Anmerkungen". Reguläre Figuren (in German) (1 ed.). B. G. Teubner Verlagsgesellschaft, Leipzig, East-Germany / Akadémiai Kiadó, Budapest, Hungary. pp. 97–98. ES-No. 19 83. Kenn-No. 375. Lizenz-No. 294. (2+316+6 pages, 12 anaglyph cards, 1 red-green glasses)
- "Bolyai Biography". University of St Andrews. March 2004. Retrieved 2018-04-03.
- Elemér, Kiss, "Matematikai kincsek Bolyai János kéziratos hagyatékából"
- Hinton, Charles Howard (1912) . The Fourth Dimension. London: G. Allen & Unwin Ltd. p. 46. Sources cited, p. 41. Entire chapter V, "The Second Chapter in the History of Four Space", pp. 41–60, provides accessible, illustrated introduction to his life and work.
- NASA website Archived 23 October 2005 at the Wayback Machine
- Martin Gardner (2001) Non-Euclidean Geometry, Chapter 4 of The Colossal Book of Mathematics, W. W. Norton & Company ISBN 0-393-02023-1
- Marvin Greenberg (1994) Euclidean and Non-Euclidean Geometries: Development and History, 3rd edition, W. H. Freeman
- Elemér Kiss (1999) Mathematical Gems from the Bolyai Chests. János Bolyai's discoveries in number theory and algebra as recently deciphered from his manuscripts. Translated by Anikó Csirmaz and Gábor Oláh. Akadémiai Kiadó, Budapest; TypoTeX, Budapest, ISBN 963-05-7563-9;
- Tibor Weszely (2013) János Bolyai. Die ersten 200 Jahre, Birkhäuser, (translated from Hungarian by Manfred Stern), ISBN 978-3-0346-0046-0
- A. Todea, F. Maria, M. Avram (2004) Oameni de știință mureșeni - Dicționar biobibliografic, CJ Mureș Biblioteca Județeană Mureș, tipografia Mediaprint SRL (in Romanian)
- Silva Oliva (2018) Janos Bolyai. Uno sguardo psicoanalitico su genio matematico e follia, ed Mimesis.