Hermann Minkowski (/mɪŋˈkɔːfski, -ˈkɒf-/ ming-KAWF-skee, -KOF-;German: [mɪŋˈkɔfski]; 22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German,Polish,Lithuanian-German, or Russian. He created and developed the geometry of numbers and elements of convex geometry, and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity.
Hermann Minkowski | |
|---|---|
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| Born | 22 June 1864 Aleksotas, Suwałki Governorate, Kingdom of Poland (now in Kaunas, Lithuania) |
| Died | 12 January 1909 (aged 44) |
| Citizenship | Russian Empire or Germany |
| Alma mater | Albertina University of Königsberg |
| Known for |
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| Spouse | Auguste Adler |
| Children | 2 |
| Scientific career | |
| Fields | Mathematics, physics, philosophy |
| Institutions | University of Göttingen and ETH Zurich |
| Doctoral advisor | Ferdinand von Lindemann |
| Doctoral students | Constantin Carathéodory Louis Kollros Dénes Kőnig |
| Signature | |
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Minkowski is perhaps best known for his foundational work describing space and time as a four-dimensional space, now known as "Minkowski spacetime", which facilitated geometric interpretations of Albert Einstein's special theory of relativity (1905).
Personal life and family
Hermann Minkowski was born in the town of Aleksota, the Suwałki Governorate, the Kingdom of Poland, since 1864 part of the Russian Empire, to Lewin Boruch Minkowski, a merchant who subsidized the building of the choral synagogue in Kovno, and Rachel Taubmann, both of Jewish descent. Hermann was a younger brother of the medical researcher Oskar (born 1858). In different sources Minkowski's nationality is variously given as German,Polish, or Lithuanian-German, or Russian.
To escape Jewish persecution in the Russian Empire, the family moved to Königsberg in 1872, where the father became involved in rag export and later in manufacture of mechanical clockwork tin toys (he operated his firm Lewin Minkowski & Son with his eldest son Max).
Minkowski studied in Königsberg and taught in Bonn (1887–1894), Königsberg (1894–1896) and Zürich (1896–1902), and finally in Göttingen from 1902 until his death in 1909. He married Auguste Adler in 1897 with whom he had two daughters; the electrical engineer and inventor Reinhold Rudenberg was his son-in-law.
Minkowski died of appendicitis in Göttingen on 12 January 1909. Max Born delivered the obituary on behalf of the mathematics students at Göttingen.David Hilbert's obituary of Minkowski illustrates the deep friendship between the two mathematicians:
Since my student years, Minkowski was my best, most dependable friend who supported me with all the depth and loyalty that was so characteristic of him. Science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. In it, we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty, and when one of us showed it to the other and we marveled over it together, our joy was complete. He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us.
— Hilbert, 1909
The main-belt asteroid 12493 Minkowski and M-matrices are named in Minkowski's honor.
Education and career
Minkowski was educated in East Prussia at the Albertina University of Königsberg, where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. In 1883, while still a student at Königsberg, he was awarded the Mathematics Prize of the French Academy of Sciences for his manuscript on the theory of quadratic forms. Due to the very young age of 18, which was unheard of in the mathematics community, and his obscurity as a mathematician at the time, his sharing the award with eminent English mathematician Henry Smith (who was certainly a great deal more famous than Hermann and to whom the prize was awarded posthumously) caused severe unrest among English mathematicians. The prize committee, despite the numerous complaints, never changed their decision. He also became a friend of another renowned mathematician, David Hilbert. His brother, Oskar Minkowski (1858–1931), was a well-known physician and researcher.
Minkowski taught at the universities of Bonn, Königsberg, Zürich, and Göttingen. At the Eidgenössische Polytechnikum, today the ETH Zurich, he was one of Einstein's teachers.
Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions. In 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory. He is also the creator of the Minkowski Sausage and the Minkowski cover of a curve.
In 1902, he joined the Mathematics Department of Göttingen and became a close colleague of David Hilbert, whom he first met at university in Königsberg. Constantin Carathéodory was one of his students there.
Work on relativity
By 1908 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval 
The mathematical basis of Minkowski space can also be found in the hyperboloid model of hyperbolic space already known in the 19th century, because isometries (or motions) in hyperbolic space can be related to Lorentz transformations, which included contributions of Wilhelm Killing (1880, 1885), Henri Poincaré (1881), Homersham Cox (1881), Alexander Macfarlane (1894) and others (see History of Lorentz transformations).
The beginning part of his address called "Space and Time" delivered at the 80th Assembly of German Natural Scientists and Physicians (21 September 1908) is now famous:
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
Publications
Relativity
- Minkowski, Hermann (1915) [1907]. "Das Relativitätsprinzip". Annalen der Physik. 352 (15): 927–938. Bibcode:1915AnP...352..927M. doi:10.1002/andp.19153521505.
- Minkowski, Hermann (1908). . Nachrichten der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111.
- English translation: "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". In: The Principle of Relativity (1920), Calcutta: University Press, 1–69.
- Minkowski, Hermann (1909). . Jahresbericht der Deutschen Mathematiker-Vereinigung. 18: 75–88. Bibcode:1909JDMaV..18...75M.
- Various English translations on Wikisource: "Space and Time".
- Blumenthal O. (ed): Das Relativitätsprinzip, Leipzig 1913, 1923 (Teubner), Engl tr (W. Perrett & G. B. Jeffrey) The Principle of Relativity London 1923 (Methuen); reprinted New York 1952 (Dover) entitled H. A. Lorentz, Albert Einstein, Hermann Minkowski, and Hermann Weyl, The Principle of Relativity: A Collection of Original Memoirs.
- Space and Time – Minkowski's Papers on Relativity, Minkowski Institute Press, 2012 ISBN 978-0-9879871-3-6 (free ebook).
Diophantine approximations
- Minkowski, Hermann (1907). Diophantische Approximationen: Eine Einführung in die Zahlentheorie. Leipzig-Berlin: R. G. Teubner. Retrieved 28 February 2016.
Mathematical (posthumous)
- Minkowski, Hermann (1910). Geometrie der Zahlen. Leipzig-Berlin: B. G. Teubner Verlag. MR 0249269. Retrieved 28 February 2016.
- Minkowski, Hermann (1911). Gesammelte Abhandlungen 2 vols. Leipzig-Berlin: R. G. Teubner. Retrieved 28 February 2016. Reprinted in one volume New York, Chelsea 1967.
See also
- List of things named after Hermann Minkowski
- Abraham–Minkowski controversy
- Brunn–Minkowski theorem
- Hasse–Minkowski theorem
- Hermite–Minkowski theorem
- Minkowski addition
- Minkowski (crater)
- Minkowski distance
- Minkowski functional
- Minkowski inequality
- Minkowski model
- Minkowski plane
- Minkowski problem
- Minkowski problem for polytopes
- Minkowski's second theorem
- Minkowski space
- Minkowski's bound
- Minkowski's theorem in geometry of numbers
- Minkowski–Hlawka theorem
- Minkowski–Steiner formula
- Smith–Minkowski–Siegel mass formula
- Proper time
- Separating axis theorem
- Taxicab geometry
- World line
Notes and references
- Encyclopedia of Earth and Physical Sciences. New York: Marshall Cavendish. 1998. p. 1203. ISBN 9780761405511.
- "Minkowski". Random House Webster's Unabridged Dictionary.
- "Hermann Minkowski German mathematician". Encyclopædia Britannica. Retrieved 6 January 2021.
- Gregersen, Erik, ed. (2010). The Britannica Guide to Relativity and Quantum Mechanics (1st ed.). New York: Britannica Educational Pub. Association with Rosen Educational Services. p. 201. ISBN 978-1-61530-383-0.
- Bracher, Katherine; et al., eds. (2007). Biographical Encyclopedia of Astronomers (Online ed.). New York: Springer. p. 787. ISBN 978-0-387-30400-7.
- Hayles, N. Katherine (1984). The Cosmic Web: Scientific Field Models and Literary Strategies in the Twentieth Century. Cornell University Press. p. 46. ISBN 978-0-8014-1742-9.
- Falconer, K. J. (2013). Fractals: A Very Short Introduction. Oxford University Press. p. 119. ISBN 978-0-19-967598-2.
- Bardon, Adrian (2013). A Brief History of the Philosophy of Time. Oxford University Press. p. 68. ISBN 978-0-19-930108-9.
- Safra, Jacob E.; Yeshua, Ilan (2003). Encyclopædia Britannica (New ed.). Chicago, Ill.: Encyclopædia Britannica. p. 665. ISBN 978-0-85229-961-6.
- А. И. Хаеш (1873). "Коробочное делопроизводство как источник сведений о жизни еврейских обществ и их персональном составе" (in Russian).
...купец Левин Минковский подарил молитвенному обществу при Ковенском казённом еврейском училище начатую им... постройкой молитвенную школу вместе с плацем, с тем, чтобы общество это озаботилась окончанием таковой постройки. Общество, располагая средствами добровольных пожертвований, возвело уже это здание под крышу, но затем средства сии истощились...
- "Kaunas: dates and facts. Electronic directory".
- "Box-Tax Paperwork Records". Archived from the original on 8 January 2015.
Kovno. In 1873 the merchant (kupez), Levin Minkovsky, gave (as a gift) to the prayer association of the Kovno state Jewish school a lot with an ongoing construction of a prayer school that (the construction) he had started so that the association would take care of completing the construction. The association, having some funds from voluntary contributions, had built the structure up to the roof, but then, ran out of money
- "Minkowski biography".
- Oskar Minkowski (1858–1931). Archived 29 December 2013 at the Wayback Machine. The Jewish genealogy site JewishGen.org (Lithuania database, registration required) contains the birth record in the Kovno rabbinical books of Hermann's younger brother Tuvia in 1868 to Boruch Yakovlevich Minkovsky and his wife Rakhil Isaakovna Taubman.
- "Historical note: Oskar Minkowski (1858–1931). An outstanding master of diabetes research". 2006.
- Report of the Federal Security Agency (p. 183); Tyra lithographed tin toy dog; Rudolph Leo Bernhard Minkowski: A Biographical Memoir.
- Greenspan, Nancy Thorndike (2005). The End of the Certain World. The Life and Science of Max Born: The Nobel Physicist Who Ignited the Quantum Revolution. Basic Books. pp. 42–43. ISBN 9780738206936.
- Schmadel, Lutz D. (2007). "(12493) Minkowski". Dictionary of Minor Planet Names – (12493) Minkowski. Springer Berlin Heidelberg. p. 783. doi:10.1007/978-3-540-29925-7_8614. ISBN 978-3-540-00238-3.
- Weisstein, Eric W. "Minkowski Sausage". MathWorld. Retrieved 16 January 2023.
- Weisstein, Eric W. "Minkowski Cover". MathWorld. Retrieved 16 January 2023.
- Dickson, L. E. (1909). "Review: Diophantische Approximationen. Eine Einführung in die Zahlentheorie von Hermann Minkowski" (PDF). Bull. Amer. Math. Soc. 15 (5): 251–252. doi:10.1090/s0002-9904-1909-01753-7.
- Dickson, L. E. (1914). "Review: Geometrie der Zahlen von Hermann Minkowski". Bull. Amer. Math. Soc. 21 (3): 131–132. doi:10.1090/s0002-9904-1914-02597-2.
- Wilson, E. B. (1915). "Review: Gesammelte Abhandlungen von Hermann Minkowski". Bull. Amer. Math. Soc. 21 (8): 409–412. doi:10.1090/s0002-9904-1915-02658-3.
External links
Hermann Minkowski
- Hermann Minkowski at the Mathematics Genealogy Project
Archival collections
- Hermann Minkowski notebooks, 1882-1906, Niels Bohr Library & Archives
Hermann Minkowski m ɪ ŋ ˈ k ɔː f s k i ˈ k ɒ f ming KAWF skee KOF German mɪŋˈkɔfski 22 June 1864 12 January 1909 was a mathematician and professor at the University of Konigsberg the University of Zurich and the University of Gottingen described variously as German Polish Lithuanian German or Russian He created and developed the geometry of numbers and elements of convex geometry and used geometrical methods to solve problems in number theory mathematical physics and the theory of relativity Hermann MinkowskiBorn 1864 06 22 22 June 1864 Aleksotas Suwalki Governorate Kingdom of Poland now in Kaunas Lithuania Died12 January 1909 1909 01 12 aged 44 Gottingen German EmpireCitizenshipRussian Empire or GermanyAlma materAlbertina University of KonigsbergKnown forGeometry of numbers Minkowski content Minkowski diagram Minkowski s question mark function Minkowski space Work on the Diophantine approximationsSpouseAuguste AdlerChildren2Scientific careerFieldsMathematics physics philosophyInstitutionsUniversity of Gottingen and ETH ZurichDoctoral advisorFerdinand von LindemannDoctoral studentsConstantin Caratheodory Louis Kollros Denes KonigSignature Minkowski is perhaps best known for his foundational work describing space and time as a four dimensional space now known as Minkowski spacetime which facilitated geometric interpretations of Albert Einstein s special theory of relativity 1905 Personal life and familyHermann Minkowski was born in the town of Aleksota the Suwalki Governorate the Kingdom of Poland since 1864 part of the Russian Empire to Lewin Boruch Minkowski a merchant who subsidized the building of the choral synagogue in Kovno and Rachel Taubmann both of Jewish descent Hermann was a younger brother of the medical researcher Oskar born 1858 In different sources Minkowski s nationality is variously given as German Polish or Lithuanian German or Russian To escape Jewish persecution in the Russian Empire the family moved to Konigsberg in 1872 where the father became involved in rag export and later in manufacture of mechanical clockwork tin toys he operated his firm Lewin Minkowski amp Son with his eldest son Max Minkowski studied in Konigsberg and taught in Bonn 1887 1894 Konigsberg 1894 1896 and Zurich 1896 1902 and finally in Gottingen from 1902 until his death in 1909 He married Auguste Adler in 1897 with whom he had two daughters the electrical engineer and inventor Reinhold Rudenberg was his son in law Minkowski died of appendicitis in Gottingen on 12 January 1909 Max Born delivered the obituary on behalf of the mathematics students at Gottingen David Hilbert s obituary of Minkowski illustrates the deep friendship between the two mathematicians Since my student years Minkowski was my best most dependable friend who supported me with all the depth and loyalty that was so characteristic of him Science which we loved above all else brought us together it seemed to us a garden full of flowers In it we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty and when one of us showed it to the other and we marveled over it together our joy was complete He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long Now death has suddenly torn him from our midst However what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us Hilbert 1909 The main belt asteroid 12493 Minkowski and M matrices are named in Minkowski s honor Education and careerMinkowski in 1883 at the time of being awarded the Mathematics Prize of the French Academy of Sciences Minkowski was educated in East Prussia at the Albertina University of Konigsberg where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann In 1883 while still a student at Konigsberg he was awarded the Mathematics Prize of the French Academy of Sciences for his manuscript on the theory of quadratic forms Due to the very young age of 18 which was unheard of in the mathematics community and his obscurity as a mathematician at the time his sharing the award with eminent English mathematician Henry Smith who was certainly a great deal more famous than Hermann and to whom the prize was awarded posthumously caused severe unrest among English mathematicians The prize committee despite the numerous complaints never changed their decision He also became a friend of another renowned mathematician David Hilbert His brother Oskar Minkowski 1858 1931 was a well known physician and researcher Minkowski taught at the universities of Bonn Konigsberg Zurich and Gottingen At the Eidgenossische Polytechnikum today the ETH Zurich he was one of Einstein s teachers Minkowski explored the arithmetic of quadratic forms especially concerning n variables and his research into that topic led him to consider certain geometric properties in a space of n dimensions In 1896 he presented his geometry of numbers a geometrical method that solved problems in number theory He is also the creator of the Minkowski Sausage and the Minkowski cover of a curve In 1902 he joined the Mathematics Department of Gottingen and became a close colleague of David Hilbert whom he first met at university in Konigsberg Constantin Caratheodory was one of his students there Work on relativityBy 1908 Minkowski realized that the special theory of relativity introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincare could best be understood in a four dimensional space since known as the Minkowski spacetime in which time and space are not separated entities but intermingled in a four dimensional space time and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval x2 y2 z2 c2t2 displaystyle x 2 y 2 z 2 c 2 t 2 see History of special relativity The mathematical basis of Minkowski space can also be found in the hyperboloid model of hyperbolic space already known in the 19th century because isometries or motions in hyperbolic space can be related to Lorentz transformations which included contributions of Wilhelm Killing 1880 1885 Henri Poincare 1881 Homersham Cox 1881 Alexander Macfarlane 1894 and others see History of Lorentz transformations The beginning part of his address called Space and Time delivered at the 80th Assembly of German Natural Scientists and Physicians 21 September 1908 is now famous The views of space and time which I wish to lay before you have sprung from the soil of experimental physics and therein lies their strength They are radical Henceforth space by itself and time by itself are doomed to fade away into mere shadows and only a kind of union of the two will preserve an independent reality PublicationsRelativity Minkowski Hermann 1915 1907 Das Relativitatsprinzip Annalen der Physik 352 15 927 938 Bibcode 1915AnP 352 927M doi 10 1002 andp 19153521505 Minkowski Hermann 1908 Die Grundgleichungen fur die elektromagnetischen Vorgange in bewegten Korpern Nachrichten der Gesellschaft der Wissenschaften zu Gottingen Mathematisch Physikalische Klasse 53 111 English translation The Fundamental Equations for Electromagnetic Processes in Moving Bodies In The Principle of Relativity 1920 Calcutta University Press 1 69 Minkowski Hermann 1909 Raum und Zeit Jahresbericht der Deutschen Mathematiker Vereinigung 18 75 88 Bibcode 1909JDMaV 18 75M Various English translations on Wikisource Space and Time Blumenthal O ed Das Relativitatsprinzip Leipzig 1913 1923 Teubner Engl tr W Perrett amp G B Jeffrey The Principle of Relativity London 1923 Methuen reprinted New York 1952 Dover entitled H A Lorentz Albert Einstein Hermann Minkowski and Hermann Weyl The Principle of Relativity A Collection of Original Memoirs Space and Time Minkowski s Papers on Relativity Minkowski Institute Press 2012 ISBN 978 0 9879871 3 6 free ebook Diophantine approximations Minkowski Hermann 1907 Diophantische Approximationen Eine Einfuhrung in die Zahlentheorie Leipzig Berlin R G Teubner Retrieved 28 February 2016 Mathematical posthumous Minkowski Hermann 1910 Geometrie der Zahlen Leipzig Berlin B G Teubner Verlag MR 0249269 Retrieved 28 February 2016 Minkowski Hermann 1911 Gesammelte Abhandlungen 2 vols Leipzig Berlin R G Teubner Retrieved 28 February 2016 Reprinted in one volume New York Chelsea 1967 See alsoList of things named after Hermann Minkowski Abraham Minkowski controversy Brunn Minkowski theorem Hasse Minkowski theorem Hermite Minkowski theorem Minkowski addition Minkowski crater Minkowski distance Minkowski functional Minkowski inequality Minkowski model Minkowski plane Minkowski problem Minkowski problem for polytopes Minkowski s second theorem Minkowski space Minkowski s bound Minkowski s theorem in geometry of numbers Minkowski Hlawka theorem Minkowski Steiner formula Smith Minkowski Siegel mass formula Proper time Separating axis theorem Taxicab geometry World lineNotes and referencesEncyclopedia of Earth and Physical Sciences New York Marshall Cavendish 1998 p 1203 ISBN 9780761405511 Minkowski Random House Webster s Unabridged Dictionary Hermann Minkowski German mathematician Encyclopaedia Britannica Retrieved 6 January 2021 Gregersen Erik ed 2010 The Britannica Guide to Relativity and Quantum Mechanics 1st ed New York Britannica Educational Pub Association with Rosen Educational Services p 201 ISBN 978 1 61530 383 0 Bracher Katherine et al eds 2007 Biographical Encyclopedia of Astronomers Online ed New York Springer p 787 ISBN 978 0 387 30400 7 Hayles N Katherine 1984 The Cosmic Web Scientific Field Models and Literary Strategies in the Twentieth Century Cornell University Press p 46 ISBN 978 0 8014 1742 9 Falconer K J 2013 Fractals A Very Short Introduction Oxford University Press p 119 ISBN 978 0 19 967598 2 Bardon Adrian 2013 A Brief History of the Philosophy of Time Oxford University Press p 68 ISBN 978 0 19 930108 9 Safra Jacob E Yeshua Ilan 2003 Encyclopaedia Britannica New ed Chicago Ill Encyclopaedia Britannica p 665 ISBN 978 0 85229 961 6 A I Haesh 1873 Korobochnoe deloproizvodstvo kak istochnik svedenij o zhizni evrejskih obshestv i ih personalnom sostave in Russian kupec Levin Minkovskij podaril molitvennomu obshestvu pri Kovenskom kazyonnom evrejskom uchilishe nachatuyu im postrojkoj molitvennuyu shkolu vmeste s placem s tem chtoby obshestvo eto ozabotilas okonchaniem takovoj postrojki Obshestvo raspolagaya sredstvami dobrovolnyh pozhertvovanij vozvelo uzhe eto zdanie pod kryshu no zatem sredstva sii istoshilis Kaunas dates and facts Electronic directory Box Tax Paperwork Records Archived from the original on 8 January 2015 Kovno In 1873 the merchant kupez Levin Minkovsky gave as a gift to the prayer association of the Kovno state Jewish school a lot with an ongoing construction of a prayer school that the construction he had started so that the association would take care of completing the construction The association having some funds from voluntary contributions had built the structure up to the roof but then ran out of money Minkowski biography Oskar Minkowski 1858 1931 Archived 29 December 2013 at the Wayback Machine The Jewish genealogy site JewishGen org Lithuania database registration required contains the birth record in the Kovno rabbinical books of Hermann s younger brother Tuvia in 1868 to Boruch Yakovlevich Minkovsky and his wife Rakhil Isaakovna Taubman Historical note Oskar Minkowski 1858 1931 An outstanding master of diabetes research 2006 Report of the Federal Security Agency p 183 Tyra lithographed tin toy dog Rudolph Leo Bernhard Minkowski A Biographical Memoir Greenspan Nancy Thorndike 2005 The End of the Certain World The Life and Science of Max Born The Nobel Physicist Who Ignited the Quantum Revolution Basic Books pp 42 43 ISBN 9780738206936 Schmadel Lutz D 2007 12493 Minkowski Dictionary of Minor Planet Names 12493 Minkowski Springer Berlin Heidelberg p 783 doi 10 1007 978 3 540 29925 7 8614 ISBN 978 3 540 00238 3 Weisstein Eric W Minkowski Sausage MathWorld Retrieved 16 January 2023 Weisstein Eric W Minkowski Cover MathWorld Retrieved 16 January 2023 Dickson L E 1909 Review Diophantische Approximationen Eine Einfuhrung in die Zahlentheorie von Hermann Minkowski PDF Bull Amer Math Soc 15 5 251 252 doi 10 1090 s0002 9904 1909 01753 7 Dickson L E 1914 Review Geometrie der Zahlen von Hermann Minkowski Bull Amer Math Soc 21 3 131 132 doi 10 1090 s0002 9904 1914 02597 2 Wilson E B 1915 Review Gesammelte Abhandlungen von Hermann Minkowski Bull Amer Math Soc 21 8 409 412 doi 10 1090 s0002 9904 1915 02658 3 External linksWikisource has original works by or about Hermann Minkowski German Wikisource has original text related to this article Hermann Minkowski Wikiquote has quotations related to Hermann Minkowski Wikimedia Commons has media related to Hermann Minkowski Hermann Minkowski at the Mathematics Genealogy ProjectArchival collections Hermann Minkowski notebooks 1882 1906 Niels Bohr Library amp Archives
