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...aracteristic#Polyhedra|polyhedron formula]] are arguably the field's first theorems. The term ''topology'' was introduced by [[Johann Benedict Listing]] in the ...

==== 15 and 290 theorems ==== The [[15 and 290 theorems]] describes integer-quadratic matrices that describe all [[positive integer ...

{{Short description|Notation for a polyhedron's vertex figure}} ...1)</ref><ref>[http://people.hws.edu/mitchell/tilings/part3.html 3. General Theorems: Regular and Semi-Regular Tilings] Kevin Mitchell, 1995</ref> '''vertex sym ...

...|link={{filepath:Szilassi_polyhedron_3D_model.svg}}|Interactive [[Szilassi polyhedron]] model with each of 7 faces adjacent to every other. In [{{filepath:Szilas [[Category:Theorems in graph theory]] ...

...}}) of a [[Simplex#Simplices with an "orthogonal corner"|general theorem]] about [[simplex|''n''-simplices]] with a [[right angle|right-angle]] corner, prov [[Category:Theorems about polyhedron]] ...

[[Descartes' theorem on total angular defect]] of a [[polyhedron]] is the piecewise-linear analog: ...which is [[homeomorphic]] to the sphere is 4{{pi}}. More generally, if the polyhedron has [[Euler characteristic]] {{math|''χ'' {{=}} 2 − 2''g''}} (where {{mvar| ...

...Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa.<ref>{{cite conference |last1=Danzer |first1=L. |last2=Grünb ...ny |first1=Imre |last2=Karasev |first2=Roman |date=2012-07-20 |title=Notes About the Carathéodory Number |journal=Discrete & Computational Geometry |languag ...

...d sets formed analogously from the vertices, edges and faces of a [[convex polyhedron]], or more generally of an embedded planar graph: in both cases, the order [[Category:Statements about planar graphs]] ...

...nts who debate the proof of the [[Euler characteristic]] defined for the [[polyhedron]]. A central theme is that [[definition]]s are not carved in stone, but oft <li>Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated con ...

...d them ''after'' Euler.<ref>{{cite book |year=2008 |title=Euler's Gem: The polyhedron formula and the birth of topology |first=David S. |last=Richeson|author-lin ...teristic]], classically the number of vertices minus edges plus faces of a polyhedron. ...

...<ref>{{cite book |last1=Richeson |first1=David S. |title=Euler's gem : the polyhedron formula and the birth of topology |date=23 July 2019 |location=Princeton |i ...Richeson|first=David S.|authorlink= David Richeson |title=Euler's Gem: The Polyhedron Formula and the Birth of Topology|publisher=Princeton University Press|year ...

*Third, ''rotate'' the translated figure about the matched vertex until one pair of [[corresponding sides]] matches. *Fourth, ''reflect'' the rotated figure about this matched side until the figures match. ...

== Notes about the proof == The analogous statement about [[polyhedron|polyhedra]] in three dimensions, known as [[Hilbert's third problem]], is f ...

=== Kuratowski's and Wagner's theorems === ...o cycles of length 3. Therefore, by Theorem 2, it cannot be planar. These theorems provide necessary conditions for planarity that are not sufficient conditio ...

The question about how many vertices/watchmen/guards were needed, was posed to Chvátal by [[Vi ..._with_no_vertex_visible_from_center.svg|thumb|left|upright|An [[orthogonal polyhedron]] with every vertex invisible from its middle <ref>Eryk Lipka, ''[http://ar ...

{{About|Euler characteristic number|Euler characteristic class|Euler class|Euler nu ...ly defined for [[polyhedron|polyhedra]] and used to prove various theorems about them, including the classification of the [[Platonic solid]]s. It was state ...

{{about|the nature of Hamiltonian paths|the question of the existence of a Hamilton ...ar earlier by [[Thomas Kirkman]], who, in particular, gave an example of a polyhedron without Hamiltonian cycles.<ref>{{citation ...

...net/tech_1952_roster.html | title=Tech_alumni_pp48 }}</ref> and has talked about the influence this school had on his career (for instance at his 2014 NIST | title = Maximum matching and a polyhedron with 0,1-vertices ...

...[[Copernican heliocentrism|Copernican system]]) allowed him to explore new theorems. Another important development that allowed Kepler to establish his celest ...regular polyhedra discovered by [[Louis Poinsot]], as the [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]].<ref>Cromwell, P. R. (1995). Kepler's work on po ...

...concepts, [[theorem]]s or [[mathematical structure]]s into other concepts, theorems or structures in a [[Injective function|one-to-one]] fashion, often (but no ...en. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any inte ...