Editing Carl Friedrich Gauss (section)

== Mathematics ==
=== Algebra and number theory ===
==== Fundamental theorem of algebra ====
[[File:DBP 1977 928 Carl Friedrich Gauß.jpg|thumb|upright|German stamp commemorating Gauss's 200th anniversary: the [[complex plane]] or ''Gauss plane'']]
In his doctoral thesis from 1799, Gauss proved the [[fundamental theorem of algebra]] which states that every non-constant single-variable [[polynomial]] with complex coefficients has at least one complex [[root of a function|root]]. Mathematicians including [[Jean le Rond d'Alembert]] had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts led to considerable clarification of the concept of complex numbers.<ref>{{Cite arXiv | last1 = Basu | first1 = Soham | last2 = Velleman | first2 = Daniel J. | date = 21 April 2017 | title = On Gauss's first proof of the fundamental theorem of algebra | class = math.CV | eprint = 1704.06585}}</ref>

==== ''Disquisitiones Arithmeticae'' ====
{{Main|Disquisitiones Arithmeticae}}
In the preface to the ''Disquisitiones'', Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had independently discovered.{{sfn|Bachmann|1922|p=8}} The ''[[Disquisitiones Arithmeticae]]'', written in 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraic [[number theory]]. Therein he introduces the [[triple bar]] symbol ({{math|≡}}) for [[Congruence relation|congruence]] and uses it for a clean presentation of [[modular arithmetic]].{{sfn|Bachmann|1922|pp=8–9}} It deals with the [[unique factorization theorem]] and [[primitive root modulo n|primitive roots modulo n]]. In the main sections, Gauss presents the first two proofs of the law of [[quadratic reciprocity]]{{sfn|Bachmann|1922|pp=16–25}} and develops the theories of [[Binary quadratic form|binary]]{{sfn|Bachmann|1922|pp=14–16, 25}} and ternary [[quadratic form]]s.{{sfn|Bachmann|1922|pp=25–28}}

The ''Disquisitiones'' include the [[Gauss composition law]] for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his [[Legendre's three-square theorem|theorem on three squares]], he proves the triangular case of the [[Fermat polygonal number theorem]] for ''n'' = 3.{{sfn|Bachmann|1922|p=29}} From several analytic results on [[ideal class group|class numbers]] that Gauss gives without proof towards the end of the fifth section,{{sfn|Bachmann|1922|pp=22&ndash;23}} it appears that Gauss already knew the [[class number formula]] in 1801.{{sfn|Bachmann|1922|pp=66&ndash;69}}

In the last section, Gauss gives proof for the [[Constructible polygon|constructibility]] of a regular [[heptadecagon]] (17-sided polygon) with [[Compass and straightedge constructions|straightedge and compass]] by reducing this geometrical problem to an algebraic one.<ref name="Denker">{{cite book | last1 = Denker | first1 = Manfred | last2 = Patterson | first2 = Samuel James | author-link2  = Samuel James Patterson | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 53–62 | chapter = Gauß – der geniale Mathematiker | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> He shows that a regular polygon is constructible if the number of its sides is either a [[power of 2]] or the product of a power of 2 and any number of distinct [[Fermat prime]]s. In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients in [[finite field]]s, which amounts to counting integral points on an [[elliptic curve]].<ref name="Stuhler">{{cite book | last1 = Stuhler | first1 = Ulrich | author-link  = Ulrich Stuhler | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 62–72 | chapter = Arithmetisch-geometrisches Mittel und elliptische Integrale: Gauß und die komplexe Analysis | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> An unfinished chapter, consisting of work done during 1797–1799, was found among his papers after his death.{{sfn|Dunnington|2004|p=44}}<ref name="Frei">{{cite book |last=Frei |first=Günther |author-link=Günther Frei |chapter-url=https://books.google.com/books?id=IUFTcOsMTysC&pg=159 |title=The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae |date=2007 |publisher=Springer |isbn=978-3-540-20441-1 |editor-last1=Goldstein |editor-first1=Catherine |editor-link1=Catherine Goldstein |place=Berlin, Heidelberg, New York |pages=159–198 |chapter=The Unpublished Section Eight: On the Way for Function Fields over a Finite Field |doi=10.1007/978-3-540-34720-0 |editor-last2=Schappacher |editor-first2=Norbert |editor-link2=Norbert Schappacher |editor-last3=Schwermer |editor-first3=Joachim |editor-link3=Joachim Schwermer}}</ref>

==== Further investigations ====
One of Gauss's first results was the empirically found conjecture of 1792 – the later called [[prime number theorem]] – giving an estimation of the number of prime numbers by using the [[Logarithmic integral function|integral logarithm]].<ref>{{cite book | last1 = Koch | first1 = H. | author-link1 = Herbert Koch | last2 = Pieper | first2 = H. | title = Zahlentheorie | publisher = VEB Deutscher Verlag der Wissenschaften | place = Berlin | date = 1976 | pages = 6, 124}}</ref>{{efn|Gauss told the story later in detail in a letter to [[Johann Franz Encke|Encke]].{{sfn|Bachmann|1922|p=4}}}}

In 1816, [[Heinrich Wilhelm Matthias Olbers|Olbers]] encouraged Gauss to compete for a prize from the French Academy for a proof for [[Fermat's Last Theorem]]; he refused, considering the topic uninteresting. However, after his death a short undated paper was found with proofs of the theorem for the cases ''n'' = 3 and ''n'' = 5.<ref>{{cite journal | last1 = Kleiner | first1 = I. | year = 2000 | title = From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem | journal = [[Elemente der Mathematik]] | volume = 55 | pages = 19–37 | url = http://math.stanford.edu/~lekheng/flt/kleiner.pdf | doi = 10.1007/PL00000079
 | s2cid = 53319514 | url-status = dead | archive-url = https://web.archive.org/web/20110608052614/http://math.stanford.edu/~lekheng/flt/kleiner.pdf | archive-date = 8 June 2011}}</ref> The particular case of ''n'' = 3 was proved much earlier by [[Leonhard Euler]], but Gauss developed a more streamlined proof which made use of [[Eisenstein integers]]; though more general, the proof was simpler than in the real integers case.{{sfn|Bachmann|1922|pp= 60&ndash;61}}

Gauss contributed to solving the [[Kepler conjecture]] in 1831 with the proof that a [[Close-packing of equal spheres|greatest packing density]] of spheres in the three-dimensional space is given when the centres of the spheres form a [[Cubic crystal system|cubic face-centred]] arrangement,<ref>{{Cite journal | last = Hales | first = Thomas C. | author-link = Thomas Callister Hales | title = Historical overview of the Kepler conjecture | doi = 10.1007/s00454-005-1210-2 | mr = 2229657 | year = 2006 | journal = [[Discrete & Computational Geometry]] | issn = 0179-5376 | volume=36 | issue = 1 | pages = 5–20| doi-access = free}}</ref> when he reviewed a book of [[Ludwig August Seeber]] on the theory of reduction of positive ternary quadratic forms.<ref>{{Cite book | last = Seeber | first = Ludwig August | year = 1831 | title = Untersuchungen über die Eigenschaften der positiven ternaeren quadratischen Formen | place = Mannheim  | url = https://books.google.com/books?id=QKJGAAAAcAAJ}}</ref> Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements.<ref>{{cite journal | date = July 1831 | title = Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber | url = https://babel.hathitrust.org/cgi/pt?id=mdp.39015064427944&seq=387 | journal = [[Göttingische gelehrte Anzeigen]] | issue = 108 | pages = 1065–1077}}</ref>

In  two papers on [[Quartic reciprocity|biquadratic residue]]s (1828, 1832) Gauss introduced the [[ring theory|ring]] of [[Gaussian integers]] <math>\mathbb{Z}[i]</math>, showed that it is a [[unique factorization domain]],<ref name="Kleiner1998">{{cite journal | url = https://ems.press/journals/em/articles/664 | title = From Numbers to Rings: The Early History of Ring Theory | first1 = Israel | last1 = Kleiner | author-link = Israel Kleiner (mathematician) | journal = [[Elemente der Mathematik]] | volume = 53 | number = 1 | doi = 10.1007/s000170050029 | year = 1998 | pages = 18–35 | zbl = 0908.16001 | doi-access = free}}</ref> and generalized some key arithmetic concepts, such as [[Fermat's little theorem]] and [[Gauss's lemma (number theory)|Gauss's lemma]]. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity<ref name="Kleiner1998" /> – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.<ref>{{cite book | last1 = Lemmermeyer | first1 = Franz | title = Reciprocity Laws: from Euler to Eisenstein | series = Springer Monographs in Mathematics | publisher = Springer | place = Berlin | year = 2000 | page = 15 | isbn = 3-540-66957-4 | doi= 10.1007/978-3-662-12893-0}}</ref>

In the second paper, he stated the general law of biquadratic reciprocity and proved several special cases of it. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claimed the techniques of these proofs ([[Gauss sum]]s) can be applied to prove higher reciprocity laws.{{sfn|Bachmann|1922|pp= 52, 57&ndash;59}}

=== Analysis ===
One of Gauss's first discoveries was the notion of the [[arithmetic-geometric mean]] (AGM) of two positive real numbers.{{sfn|Schlesinger|1933|p=41–57}} He discovered its relation to elliptic integrals in the years 1798–1799 through [[Landen's transformation]], and a diary entry recorded the discovery of the connection of [[Gauss's constant]] to [[lemniscatic elliptic functions]], a result that Gauss stated  "will surely open an entirely new field of analysis".<ref name="Cox">{{cite journal | last = Cox | first = David A. | author-link = David A. Cox | date = January 1984 | title = The Arithmetic-Geometric Mean of Gauss | url = https://www.researchgate.net/publication/248675540 | journal = [[L'Enseignement mathématique]] | volume = 30 | issue = 2 | pages = 275–330}}</ref> He also made early inroads into the more formal issues of the foundations of [[complex analysis]], and from a letter to Bessel in 1811 it is clear that he knew the "fundamental theorem of complex analysis" – [[Cauchy's integral theorem]] – and understood the notion of [[residue (complex analysis)|complex residues]] when integrating around [[pole (complex analysis)|poles]].<ref name="Stuhler" /><ref>Letter Gauss to Bessel from 18 December 1811, partly printed in the [https://gdz.sub.uni-goettingen.de/id/PPN236010751?tify=%7B%22pages%22%3A%5B96%5D%2C%22pan%22%3A%7B%22x%22%3A0.553%2C%22y%22%3A0.327%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.893%7D ''Collected Works'', Volume 8, pp. 90&ndash;92].</ref>

[[Pentagonal number theorem|Euler's pentagonal numbers theorem]], together with other researches on the AGM and lemniscatic functions, led him to plenty of results on [[Jacobi theta functions]],<ref name="Stuhler" /> culminating in the discovery in 1808 of the later called [[Jacobi triple product identity]], which includes Euler's theorem as a special case.<ref>{{cite book | last = Roy | first = Ranjan | author-link = Ranjan Roy | date = 2021 | edition = 2 | title = Series and Products in the Development of Mathematics | place = Cambridge | pages = 20–22 |volume=2 | publisher = Cambridge University Press | url = https://assets.cambridge.org/97811087/09453/frontmatter/9781108709453_frontmatter.pdf |isbn=9781108709378}}</ref> His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808.{{sfn|Schlesinger|1933|pp=185-186}}{{efn|Later, these transformations were given by Legendre in 1824 (3th order), Jacobi in 1829 (5th order), [[Ludwig Adolf Sohncke|Sohncke]] in 1837 (7th and other orders).}}{{efn|In a letter to Bessel from 1828, Gauss commented: "Mr. Abel has [...] anticipated me, and relieves me of the effort [of publishing] in respect to one third of these matters ..."{{sfn|Schlesinger|1933|p=41}}}}

Several mathematical fragments in his [[Nachlass]] indicate that he knew parts of the modern theory of [[modular forms]].<ref name="Stuhler" /> In his work on the [[Multivalued function|multivalued]] AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM and its two "simplest values".<ref name="Cox" /> In his unpublished writings he recognized and made a sketch of the key concept of [[fundamental domain]] for the [[modular group]].{{sfn|Schlesinger|1933|pp=101–106}}<ref name="Houzel">{{cite book | last = Houzel | first = Christian | editor-last1 = Goldstrein | editor-first1 = Catherine | editor-last2 = Schappacher | editor-first2 = Norbert | editor-last3 = Schwermer | editor-first3 = Joachim | editor-link1 = Catherine Goldstein | editor-link2 = Norbert Schappacher | editor-link3 = Joachim Schwermer | title = The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae | publisher = Springer | place = Berlin, Heidelberg, New York| date = 2007 | page = 293 | chapter = Elliptic Functions and Arithmetic | doi = 10.1007/978-3-540-34720-0 | isbn = 978-3-540-20441-1 | url = https://link.springer.com/book/10.1007/978-3-540-34720-0}}</ref> One of Gauss's sketches of this kind was a drawing of a [[tessellation]] of the [[unit disk]] by "equilateral" [[hyperbolic triangle]]s with all angles equal to <math>\pi/4</math>.<ref>Printed in the [https://gdz.sub.uni-goettingen.de/id/PPN236010751?tify=%7B%22pages%22%3A%5B110%5D%2C%22view%22%3A%22info%22%7D ''Collected Works'', Volume 8, p. 104].</ref>

An example of Gauss's insight in analysis is the cryptic remark that the principles of circle division by compass and straightedge can also be applied to the division of the [[lemniscate curve]], which inspired Abel's theorem on lemniscate division.{{efn|This remark appears at article 335 of chapter 7 of ''Disquisitiones Arithmeticae'' (1801).}} Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of [[quadratic Gauss sums]], in which he solved the main problem by introducing [[Gaussian binomial coefficient|q-analogs of binomial coefficient]]s and manipulating them by several original identities that seem to stem from his work on elliptic function theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic function theory, and only the later work of mathematicians such as [[Carl Gustav Jacob Jacobi|Jacobi]] and [[Hermite]] has exposed the crux of his argument.{{sfn|Schlesinger|1933|pp=122&ndash;123}}

In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general [[hypergeometric function]] <math>F(\alpha,\beta,\gamma,x)</math>, and shows that many of the functions known at the time are special cases of the hypergeometric function.{{sfn|Schlesinger|1933|pp=136–142}} This work is the first exact inquiry into [[Convergent series|convergence]] of infinite series in the history of mathematics.{{sfn|Schlesinger|1933|p=142}} Furthermore, it deals with infinite [[continued fraction]]s arising as ratios of hypergeometric functions, which are now called [[Gauss continued fraction]]s.{{sfn|Schlesinger|1933|pp=136–154}}

In 1823, Gauss won the prize of the Danish Society with an essay on [[conformal mapping]]s, which contains several developments that pertain to the field of complex analysis.{{sfn|Stäckel|1917|pp=90&ndash;91}} Gauss stated that angle-preserving mappings in the complex plane must be complex [[analytic function]]s, and used the later-named [[Beltrami equation]] to prove the existence of [[isothermal coordinates]] on analytic surfaces. The essay concludes with examples of conformal mappings into a sphere and an [[ellipsoid of revolution]].{{sfn|Bühler|1981|p=103}}

==== Numerical analysis ====
Gauss often deduced theorems [[Inductive reasoning|inductively]] from numerical data he had collected empirically.{{sfn|Schlesinger|1933|p=18}} As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions to [[numerical analysis]], such as the method of [[Gaussian quadrature]], published in 1816.<ref>{{cite book | last = Gautschi | first = Walter | author-link = Walter Gautschi | edition = 1 | title = E.B. Christoffel. The Influence of his Work on Mathematics and the Physical Science | editor-last1 = Butzer | editor-first1 = Paul B. | editor-last2 = Fehér | editor-first2 = Franziska | editor-link1 = Paul Butzer | chapter = A Survey of Gauss-Christoffel Quadrature Formulae | place = Birkhäuser, Basel | year = 1981 | pages = 72–147 | publisher = Springer | doi = 10.1007/978-3-0348-5452-8_6 | isbn = 978-3-0348-5452-8 | chapter-url = https://link.springer.com/chapter/10.1007/978-3-0348-5452-8_6}}</ref>

In a private letter to [[Christian Ludwig Gerling|Gerling]] from 1823,<ref>{{Cite book|url=https://gdz.sub.uni-goettingen.de/id/PPN335994989?tify=%7B%22pages%22:%5B316%5D,%22view%22:%22info%22%7D|title=Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling|publisher=Elsner}}</ref> he described a solution of a 4x4 system of linear equations with the [[Gauss-Seidel method]] – an "indirect" [[iterative method]] for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than two equations.<ref>{{cite arXiv | author = Yousef Saad | author-link = Yousef Saad | title = Iterative Methods for Linear Systems of Equations: A Brief Historical Journey | date = 2 August 2019 | class = math.HO | eprint = 1908.01083v1}}</ref>

Gauss invented an algorithm for calculating what is now called [[discrete Fourier transform]]s when calculating the orbits of Pallas and Juno in 1805, 160 years before [[James Cooley|Cooley]] and [[John Tukey|Tukey]] found their similar [[Cooley-Tukey FFT algorithm|Cooley–Tukey algorithm]].<ref>{{cite journal | last1 = Cooley | first1 = James W. | first2 = John W. | last2 = Tukey | title = An algorithm for the machine calculation of complex Fourier series | journal = [[Mathematics of Computation]] | volume = 19 | issue = 90 | pages = 297–301 | year = 1965 | doi = 10.2307/2003354 | jstor = 2003354 | doi-access=free }}</ref> He developed it as a [[trigonometric interpolation]] method, but the paper ''Theoria Interpolationis Methodo Nova Tractata'' was published only posthumously in 1876,<ref>{{cite book | last = Gauss | first = C.F. | date = 1876 | title = Theoria Interpolationis Methodo Nova Tractata | place = Göttingen | language = la | pages = 265–327 | url = https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B273%5D%2C%22pan%22%3A%7B%22x%22%3A0.524%2C%22y%22%3A0.333%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.856%7D | publisher = K. Gesellschaft der Wissenschaften zu Göttingen}}</ref> well after [[Joseph Fourier]]'s introduction of the subject in 1807.<ref>{{cite journal | last1 = Heideman | first1 = Michael T. | last2 = Johnson| first2 = Don H. | last3 = Burrus | first3 = C. Sidney | author-link3 = C. Sidney Burrus | title = Gauss and the history of the fast Fourier transform | journal = IEEE ASSP Magazine | year = 1984 | volume = 1 | issue = 4 | pages = 14–21 | doi = 10.1109/MASSP.1984.1162257|s2cid=10032502 | url = http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf | archive-url=https://web.archive.org/web/20130319053449/http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf | archive-date = 19 March 2013 | url-status=live}}</ref>

=== Geometry ===
==== Differential geometry ====
{{Main|Theorema Egregium}}
The geodetic survey of [[Hanover]] fuelled Gauss's interest in [[differential geometry]] and [[topology]], fields of mathematics dealing with [[curve]]s and [[Surface (topology)|surfaces]]. This led him in 1828 to the publication of a work that marks the birth of modern [[differential geometry of surfaces]], as it departed from the traditional ways of treating surfaces as [[Cartesian coordinate system|cartesian graphs]] of functions of two variables, and that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. As a result, the [[Theorema Egregium]] (''remarkable theorem''), established a property of the notion of [[Gaussian curvature]]. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring [[angle]]s and [[distance]]s on the surface, regardless of the [[embedding]] of the surface in three-dimensional or two-dimensional space.{{sfn|Stäckel|1917|pp=110–119}}

The Theorema Egregium leads to the abstraction of surfaces as doubly-extended [[manifold]]s; it clarifies the distinction between the intrinsic properties of the manifold (the [[metric tensor|metric]]) and its physical realization in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a [[sphere]] or an [[ellipsoid]] cannot be transformed to a plane without distortion, which causes a fundamental problem in designing [[map projection|projection]]s for geographical maps.{{sfn|Stäckel|1917|pp=110–119}} A portion of this essay is dedicated to a profound study of [[geodesic]]s. In particular, Gauss proves the local [[Gauss–Bonnet theorem]] on geodesic triangles, and generalizes [[Legendre's theorem on spherical triangles]] to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the surface curvature at the vertices of the triangle, regardless of the behaviour of the surface in the triangle interior.{{sfn|Stäckel|1917|pp=105–106}}

Gauss's memoir from 1828 lacks the conception of [[geodesic curvature]]. However, in a previously unpublished manuscript, very likely written in 1822–1825, he introduced the term "side curvature" (German: "Seitenkrümmung") and proved its [[invariant (mathematics)|invariance]] under isometric transformations, a result that was later obtained by [[Ferdinand Minding]] and published by him in 1830. This Gauss paper contains the core of his lemma on total curvature, but also its generalization, found and proved by [[Pierre Ossian Bonnet]] in 1848 and known as the [[Gauss–Bonnet theorem]].{{sfn|Bolza|1921|pp=70–74}}

==== Non-Euclidean geometry ====
{{Main|Non-Euclidean geometry}}
[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|upright|Lithography by [[Siegfried Bendixen]] (1828)]]
During Gauss' lifetime, the [[Parallel postulate]] of [[Euclidean geometry]] was heavily discussed.{{sfn|Stäckel|1917|pp=19–20}} Numerous efforts were made to prove it in the frame of the Euclidean [[axiom]]s, whereas some mathematicians discussed the possibility of geometrical systems without it.{{sfn|Bühler|1981|pp=100–102}} Gauss thought about the basics of geometry from the 1790s on, but only realized in the 1810s  that a non-Euclidean geometry without the parallel postulate could solve the problem.{{sfn|Klein|1979|pp=57–60}}{{sfn|Stäckel|1917|pp=19–20}} In a letter to [[Franz Taurinus]] of 1824, he presented a short comprehensible outline of what he named a "[[non-Euclidean geometry]]",<ref name=":0">{{Cite journal | last = Winger | first = R. M. | date = 1925 | title = Gauss and non-Euclidean geometry | url = https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-31/issue-7/Gauss-and-non-euclidean-geometry/bams/1183486559.full | journal = Bulletin of the American Mathematical Society | volume = 31 | issue = 7 | pages = 356–358 | doi = 10.1090/S0002-9904-1925-04054-9 | issn = 0002-9904 | doi-access = free }}</ref> but he strongly forbade Taurinus to make any use of it.{{sfn|Klein|1979|pp=57–60}} Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well.<ref>{{Cite book |last=Bonola |first=Roberto |url=https://archive.org/details/noneuclideangeom00bono/page/64 |title=Non-Euclidean Geometry: A Critical and Historical Study of its Development |publisher=The Open Court Publishing Company |year=1912 |pages=64–67 |language=en}}</ref><ref name=":0" /><ref>{{Cite book |last=Klein |first=Felix |url=https://archive.org/details/elementarymathem0000klei/page/176 |title=Elementary Mathematics from an Advanced Standpoint: Geometry |publisher=Dover Publications |year=1939 |pages=176–177 |language=en}}</ref>

The first publications on non-Euclidean geometry in the history of mathematics were authored by [[Nikolai Lobachevsky]] in 1829 and [[Janos Bolyai]] in 1832.{{sfn|Bühler|1981|pp=100–102}} In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion.{{sfn|Klein|1979|pp=57–60}}<ref>{{Cite journal | last1 = Jenkovszky | first1 = László | last2 = Lake | first2 = Matthew J. | last3 = Soloviev | first3 = Vladimir | date = 12 March 2023 | title = János Bolyai, Carl Friedrich Gauss, Nikolai Lobachevsky and the New Geometry: Foreword | journal = Symmetry| volume = 15 | issue = 3 | pages = 707 | doi = 10.3390/sym15030707 | arxiv = 2303.17011 | bibcode=2023Symm...15..707J | issn = 2073-8994 | doi-access = free}}</ref> Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai<ref>{{Cite web|url=https://archive.org/details/briefwechselzwi00gausgoog/page/n146/mode/2up|title=Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai|first=Carl Friedrich Gauss|last=Farkas Bólyai|date=22 April 1899|publisher=B. G. Teubner|via=Internet Archive}}</ref> claiming that these were congruent to his own thoughts of some decades.{{sfn|Klein|1979|pp=57–60}}<ref>{{cite book | last = Krantz | first = Steven G. | author-link = Steven G. Krantz | title = An Episodic History of Mathematics: Mathematical Culture through Problem Solving|url=https://books.google.com/books?id=ulmAH-6IzNoC&pg=PA171 | access-date = 9 February 2013 | date = 2010 | publisher = [[The Mathematical Association of America]] | isbn = 978-0-88385-766-3| pages = 171f}}</ref> However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his written remarks are vague and obscure.{{sfn|Bühler|1981|pp=100–102}}

[[Wolfgang Sartorius von Waltershausen|Sartorius]] first mentioned Gauss's work on non-Euclidean geometry in 1856, but only the publication of Gauss's [[Nachlass]] in Volume VIII of the Collected Works (1900) showed Gauss's ideas on the matter, at a time when non-Euclidean geometry was still an object of some controversy.{{sfn|Klein|1979|pp=57–60}}

==== Early topology ====
Gauss was also an early pioneer of [[topology]] or ''Geometria Situs'', as it was called in his lifetime. The first proof of the [[fundamental theorem of algebra]] in 1799 contained an essentially topological argument; fifty years later, he further developed the topological argument in his fourth proof of this theorem.{{sfn|Ostrowski|1920|pp=1–18}}

[[File:Carl Friedrich Gauß, Büste von Heinrich Hesemann, 1855.jpg|thumb|left|upright|Gauss bust by Heinrich Hesemann (1855){{efn|Hesemann also took a death mask from Gauss.{{sfn|Dunnington|2004|p=324}}}}]]
Another encounter with topological notions occurred to him in the course of his astronomical work in 1804, when he determined the limits of the region on the [[celestial sphere]] in which comets and asteroids might appear, and which he termed "Zodiacus". He discovered that if the Earth's and comet's orbits are [[Link (knot theory)|linked]], then by topological reasons the Zodiacus is the entire sphere. In 1848, in the context of the discovery of the asteroid [[7 Iris]], he  published a further qualitative discussion of the Zodiacus.<ref name="Epple 1998 45–52">{{cite journal | last = Epple | first = Moritz | title = Orbits of asteroids, a braid, and the first link invariant | journal =  [[The Mathematical Intelligencer]] | volume = 20 | pages = 45–52 | year = 1998 | issue = 1 | doi = 10.1007/BF03024400 | s2cid = 124104367 | url = https://link.springer.com/article/10.1007/BF03024400}}</ref>

In Gauss's letters of 1820–1830, he thought intensively on topics with close affinity to Geometria Situs, and became gradually conscious of semantic difficulty in this field. Fragments from this period reveal that he tried to classify "tract figures", which are closed plane curves with a finite number of transverse self-intersections, that may also be planar projections of [[Knot theory|knot]]s.<ref>{{Cite book | last = Epple | first = Moritz | author-link = Moritz Epple | title = History of Topology | chapter = Geometric Aspects in the Development of Knot Theory | date = 1999 | editor-last = James | editor-first = I.M. | place = Amsterdam | publisher = Elseviwer | pages = 301–357 | chapter-url = https://www.maths.ed.ac.uk/~v1ranick/papers/epple3.pdf}}</ref> To do so he devised a symbolical scheme, the [[Gauss code]], that in a sense captured the characteristic features of tract figures.<ref>{{Cite book | last1 = Lisitsa | first1 = Alexei | last2 = Potapov | first2 = Igor | last3 = Saleh | first3 = Rafiq | title = Language and Automata Theory and Applications | chapter = Automata on Gauss Words | date = 2009 | editor-last = Dediu | editor-first = Adrian Horia | editor2-last = Ionescu | editor2-first = Armand Mihai | editor3-last = Martín-Vide | editor3-first = Carlos | series = Lecture Notes in Computer Science | volume = 5457 | place = Berlin, Heidelberg | publisher = Springer | pages = 505–517 | doi = 10.1007/978-3-642-00982-2_43 | isbn = 978-3-642-00982-2 | chapter-url = https://cgi.csc.liv.ac.uk/~igor/papers/lata2009.pdf}}</ref>{{sfn|Stäckel|1917|pp=50–51}}

In a fragment from 1833, Gauss defined the [[linking number]] of two space curves by a certain double integral, and in doing so provided for the first time an analytical formulation of a topological phenomenon. On the same note, he lamented the little progress made in Geometria Situs, and remarked that one of its central problems will be "to count the intertwinings of two closed or infinite curves". His notebooks from that period reveal that he was also thinking about other topological objects such as [[Braid theory|braid]]s and [[Tangle (mathematics)|tangle]]s.<ref name="Epple 1998 45–52"/>

Gauss's influence in later years to the emerging field of topology, which he held in high esteem, was through occasional remarks and oral communications to Mobius and Listing.{{sfn|Stäckel|1917|pp=51–55}}

=== Minor mathematical accomplishments ===
Gauss applied the concept of complex numbers to solve well-known problems in a new concise way. For example, in a short note from 1836 on geometric aspects of the ternary forms and their application to crystallography,<ref>Printed in ''Collected Works '' Volume 2, pp. 305–310</ref> he stated the [[Axonometry|fundamental theorem of axonometry]], which tells how to represent a 3D cube on a 2D plane with complete accuracy, via complex numbers.<ref>{{cite journal | last1 = Eastwood | first1 = Michael | last2 = Penrose | first2 = Roger | author-link2 = Michael Eastwood | author-link = Roger Penrose | title = Drawing with Complex Numbers  | journal = The Mathematical Intelligencer | year = 2000 | volume = 22 | issue = 4 | pages = 8–13 | doi = 10.1007/BF03026760 | arxiv=math/0001097| s2cid = 119136586 }}</ref> He described rotations of this sphere as the action of certain [[Mobius transformation|linear fractional transformations]] on the extended complex plane,{{sfn|Schlesinger|1933|p=198}} and gave a proof for the geometric theorem that the [[Altitude (triangle)|altitudes]] of a triangle always meet in a single [[orthocenter]].<ref>Carl Friedrich Gauss: Zusätze.II. In: {{cite book | last = Carnot | first = Lazare | author-link = Lazare Carnot | translator =  H.C. Schumacher | title = Geometrie der Stellung | pages = 363–364 | year = 1810 | publisher =  Hammerich | place = Altona | language=de}} (Text by Schumacher, algorithm by Gauss), republished in [https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B397%5D%2C%22pan%22%3A%7B%22x%22%3A0.529%2C%22y%22%3A0.4%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.657%7D ''Collected Works'' Volume 4, p. 396-398]</ref>

Gauss was concerned with [[John Napier]]'s "[[Pentagramma mirificum]]" – a certain spherical [[pentagram]] – for several decades;<ref>{{cite journal | last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter | title = Frieze patterns | journal = [[Acta Arithmetica]] | volume = 18 | pages = 297–310 | year = 1971 | url = http://matwbn.icm.edu.pl/ksiazki/aa/aa18/aa18132.pdf | doi = 10.4064/aa-18-1-297-310 | doi-access = free}}</ref> he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects.<ref>[https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B489%5D%2C%22pan%22%3A%7B%22x%22%3A0.547%2C%22y%22%3A0.533%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.548%7D ''Pentagramma mirificum''], printed in ''Collected Works'' Volume III, pp. 481–490</ref> In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane.<ref>{{cite journal | last = Schechtman | first = Vadim | author-link = Vadim Schechtman | title = Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...) | journal = Annales de la faculté des sciences de Toulouse Mathématiques | volume = 22 |  pages = 353–375 | year = 2013 | issue = 2  | doi = 10.5802/afst.1375 | url = https://eudml.org/doc/275393| doi-access = free }}</ref>

Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given [[quadrilateral]],<ref>[https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B389%5D%2C%22pan%22%3A%7B%22x%22%3A0.549%2C%22y%22%3A0.675%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.609%7D ''Bestimmung der größten Ellipse, welche die vier Ebenen eines gegebenen Vierecks berührt''], printed in ''Collected Works'' Volume 4, pp. 385–392; [https://zs.thulb.uni-jena.de/rsc/viewer/jportal_derivate_00237703/Monatliche_Correspondenz_130168688_22_1810_0115.tif?logicalDiv=jportal_jpvolume_00203050&q=August%201810 original] in ''Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde'', Volume 22, 1810, pp. 112–121</ref>{{sfn|Stäckel|1917|p=71-72}} and discovered a surprising result about the computation of area of [[pentagon]]s.<ref>Printed in ''Collected Works '' Volume 4, pp. 406–407</ref>{{sfn|Stäckel|1917|p=76}}