Editing Carl Friedrich Gauss (section)

=== 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}}