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== In mathematics == 8 is a [[composite number]] and the first number which is neither [[Prime number|prime]] nor [[semiprime]]. By [[Catalan conjecture|Mihăilescu's Theorem]], it is the only nonzero [[perfect power]] that is one less than another perfect power. 8 is the first proper [[Leyland number]] of the form {{math|x<sup>y</sup> + y<sup>x</sup>}}, where in its case {{math|x}} and {{math|y}} both equal 2.<ref>{{Cite OEIS |A076980 |Leyland numbers }}</ref> 8 is a [[Fibonacci number]] and the only nontrivial Fibonacci number that is a [[perfect cube]].<ref>Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 88</ref> [[Sphenic number]]s always have exactly eight divisors.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Sphenic Number |url=https://mathworld.wolfram.com/SphenicNumber.html |access-date=2020-08-07 |website=mathworld.wolfram.com |language=en |quote=...then every sphenic number n=pqr has precisely eight positive divisors}}</ref> 8 is the base of the [[octal]] number system.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Octal |url=https://mathworld.wolfram.com/Octal.html |access-date=2020-08-07 |website=mathworld.wolfram.com |language=en}}</ref> === Geometry === A [[polygon]] with eight sides is an [[octagon]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Octagon |url=https://mathworld.wolfram.com/Octagon.html |access-date=2020-08-07 |website=mathworld.wolfram.com |language=en}}</ref> A regular octagon can fill a [[Euclidean tilings by convex regular polygons#Plane-vertex tilings|plane-vertex]] with a regular [[triangle]] and a regular [[icositetragon]], as well as [[tessellation|tessellate]] two-dimensional space alongside squares in the [[truncated square tiling]]. This tiling is one of eight [[Archimedean tiling]]s that are semi-regular, or made of more than one type of regular [[polygon]], and the only tiling that can admit a regular octagon.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Regular Octagon |url=https://mathworld.wolfram.com/RegularOctagon.html|access-date=2022-06-25|website=mathworld.wolfram.com|language=en}}</ref> The [[Ammann–Beenker tiling]] is a nonperiodic tesselation of [[prototile]]s that feature prominent octagonal ''silver'' eightfold symmetry, that is the two-dimensional [[orthographic projection]] of the four-dimensional [[8-8 duoprism]].<ref>{{Cite book |author =Katz, A |chapter=Matching rules and quasiperiodicity: the octagonal tilings |title=Beyond quasicrystals |publisher=Springer |year=1995 |pages=141–189 |isbn=978-3-540-59251-8 |doi=10.1007/978-3-662-03130-8_6 |editor1-first=F. |editor1-last=Axel |editor2-first=D. |editor2-last=Gratias}}</ref> An [[octahedron]] is a [[regular polyhedron]] with eight [[equilateral triangle]]s as [[face (geometry)|faces]]. is the [[dual polyhedron]] to the cube and one of eight [[Deltahedron|convex deltahedra]].<ref>{{Citation|last1=Freudenthal|first1=H|last2=van der Waerden|first2=B. L.|authorlink1=Hans Freudenthal | authorlink2=B. L. van der Waerden|title=Over een bewering van Euclides ("On an Assertion of Euclid")|journal=[[Simon Stevin (journal)|Simon Stevin]]|volume=25|pages=115–128|year=1947|language=Dutch}}</ref><ref>{{Cite web|url=http://www.interocitors.com/polyhedra/Deltahedra/Convex |author=Roger Kaufman |title=The Convex Deltahedra And the Allowance of Coplanar Faces |website=The Kaufman Website |access-date=2022-06-25}}</ref> The [[stella octangula]], or ''eight-pointed star'', is the only [[stellation]] with [[octahedral symmetry]]. It has eight triangular faces alongside eight vertices that forms a cubic [[faceting]], composed of two self-dual [[Regular tetrahedron|tetrahedra]] that makes it the simplest of five [[Uniform polyhedron compound|regular compound]]s. The [[cuboctahedron]], on the other hand, is a [[rectification (geometry)|rectified]] cube or rectified octahedron, and one of only two convex [[Quasiregular polyhedron|quasiregular polyhedra]]. It contains eight equilateral triangular faces, whose first [[stellation]] is the [[compound of cube and octahedron|cube-octahedron compound]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Cuboctahedron |url=https://mathworld.wolfram.com/Cuboctahedron.html |access-date=2022-06-25 |website=mathworld.wolfram.com |language=en }}</ref><ref>{{Cite book |last=Coxeter |first=H.S.M. |author-link=Harold Scott MacDonald Coxeter |year=1973 |orig-year=1948 |title=Regular Polytopes |publisher=Dover |place=New York |edition=3rd |pages=18–19 |title-link=Regular Polytopes (book) }}</ref> === Vector spaces=== The [[octonion]]s are a [[Hypercomplex numbers|hypercomplex]] [[normed division algebra]] that are an extension of the [[complex number]]s. They are a [[Double covering group|double cover]] of [[special orthogonal group]] SO(8). The [[special unitary group]] SO(3) has an eight-dimensional [[adjoint representation]] whose colors are ascribed [[Gauge symmetry (mathematics)|gauge symmetries]] that represent the [[Vector (mathematics)|vectors]] of the eight [[gluon]]s in the [[Standard Model]]. [[Clifford algebra]]s display a periodicity of 8.<ref>{{Cite book|last=Lounesto|first=Pertti|url=https://books.google.com/books?id=DTecU6UpkSgC&q=Clifford+algebras+also+display+a+periodicity+of+8.&pg=PA216|title=Clifford Algebras and Spinors|date=2001-05-03|publisher=Cambridge University Press |isbn=978-0-521-00551-7|pages=216|language=en|quote=...Clifford algebras, contains or continues with two kinds of periodicities of 8...}}</ref> === Group theory === The [[Group of Lie type|lie group]] [[E8 (mathematics)|'''E<sub>8</sub>''']] is one of 5 exceptional lie groups.<ref>{{Cite journal |last1=Wilson |first1=Robert A. |author-link=Robert Arnott Wilson |title=Octonions and the Leech lattice |mr=2542837 |year=2009 |journal=Journal of Algebra |volume=322 |issue=6 |pages=2186–2190|doi=10.1016/j.jalgebra.2009.03.021 |doi-access=free }}</ref><ref>{{Cite book |last1=Conway |first1=John H. |author1-link=John Horton Conway |last2=Sloane |first2=N. J. A. |author2-link=Neil Sloane |chapter-url=https://link.springer.com/chapter/10.1007/978-1-4757-2016-7_8 |title=Sphere Packings, Lattices and Groups |chapter=Algebraic Constructions for Lattices |publisher=Springer |location=New York, NY |year=1988 |isbn=978-1-4757-2016-7 |eissn=2196-9701 |doi=10.1007/978-1-4757-2016-7 }}</ref> The order of the smallest [[non-abelian group]] whose subgroups are all normal is 8.{{Citation needed|date=October 2024}} === List of basic calculations === {|class="wikitable" style="text-align: center; background: white" |- ! style="width:105px;"|[[Multiplication]] !1 !2 !3 !4 !5 !6 !7 !8 !9 !10 !11 !12 !13 !14 !15 |- |'''8 × ''x''''' |'''8''' |[[16 (number)|16]] |[[24 (number)|24]] |[[32 (number)|32]] |[[40 (number)|40]] |[[48 (number)|48]] |[[56 (number)|56]] |[[64 (number)|64]] |[[72 (number)|72]] |[[80 (number)|80]] |[[88 (number)|88]] |[[96 (number)|96]] |[[104 (number)|104]] |[[112 (number)|112]] |[[120 (number)|120]] |} {|class="wikitable" style="text-align: center; background: white" |- ! style="width:105px;"|[[Division (mathematics)|Division]] !1 !2 !3 !4 !5 !6 !7 !8 !9 !10 ! style="width:5px;"| !11 !12 !13 !14 !15 |- |'''8 ÷ ''x''''' |'''8''' |4 |2.{{overline|6}} |2 |1.6 |1.{{overline|3}} |1.{{overline|142857}} |1 |0.{{overline|8}} |0.8 ! |0.{{overline|72}} |0.{{overline|6}} |0.{{overline|615384}} |0.{{overline|571428}} |0.5{{overline|3}} |- |'''''x'' ÷ 8''' |0.125 |0.25 |0.375 |0.5 |0.625 |0.75 |0.875 |1 |1.125 |1.25 ! |1.375 |1.5 |1.625 |1.75 |1.875 |} {|class="wikitable" style="text-align: center; background: white" |- ! style="width:105px;"|[[Exponentiation]] !1 !2 !3 !4 !5 !6 !7 !8 !9 !10 ! style="width:5px;"| !11 !12 !13 |- |'''8{{sup|''x''}}''' |'''8''' |64 |512 |4096 |32768 |262144 |2097152 |16777216 |134217728 |1073741824 ! |8589934592 |68719476736 |549755813888 |- |'''''x''{{sup|8}}''' |1 |256 |6561 |65536 |390625 |1679616 |5764801 |16777216 |43046721 |100000000 ! |214358881 |429981696 |815730721 |}
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