Eight-dimensional space

<h2 id="geometry">Geometry</h2>
<h3>8-polytope</h3>
<p class="note">Main article: <a href="/facts/8-polytope/M93v3h3c">8-polytope</a></p>
<p>A <a href="/facts/Polytope/a76nXrL0">polytope</a> in eight dimensions is called an 8-polytope. The most studied are the <a href="/facts/Regular_polytope/RYTnEF6Q">regular polytopes</a>, of which there are only <a href="/facts/List_of_regular_polytopes/Kng25Ymq">three in eight dimensions</a>: the <a href="/facts/8-simplex/QUhcnmk2">8-simplex</a>, <a href="/facts/8-cube/vEpKH3le">8-cube</a>, and <a href="/facts/8-orthoplex/dXFkXtPn">8-orthoplex</a>. A broader family are the <a href="/facts/Uniform_8-polytope/M93v3h3c">uniform 8-polytopes</a>, constructed from fundamental symmetry domains of reflection, each domain defined by a <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a>. Each uniform polytope is defined by a ringed <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a>. The <a href="/facts/8-demicube/sWAomIhw">8-demicube</a> is a unique polytope from the D8 family, and <a href="/facts/4_21_polytope/QFGglCkk">421</a>, <a href="/facts/2_41_polytope/R9z0wNUz">241</a>, and <a href="/facts/1_42_polytope/DTgYDddB">142</a> polytopes from the E8 family.
</p>
Regular and uniform polytopes in eight dimensions(Displayed as orthogonal projections in each <a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a> of symmetry)<table><tbody><tr><th colspan="3">A8</th><th colspan="6">B8</th><th colspan="3">D8</th></tr><tr><td colspan="3"><a href="/facts/8-simplex/QUhcnmk2">8-simplex</a>{3,3,3,3,3,3,3}</td><td colspan="3"><a href="/facts/8-cube/vEpKH3le">8-cube</a>{4,3,3,3,3,3,3}</td><td colspan="3"><a href="/facts/8-orthoplex/dXFkXtPn">8-orthoplex</a>{3,3,3,3,3,3,4}</td><td colspan="3"><a href="/facts/8-demicube/sWAomIhw">8-demicube</a>h{4,3,3,3,3,3,3}</td></tr><tr><th colspan="12"><a href="/facts/E8_(mathematics)/NEaKT71H">E8</a></th></tr><tr><td colspan="4"><a href="/facts/4_21_polytope/QFGglCkk">421</a>{3,3,3,3,32,1}</td><td colspan="4"><a href="/facts/2_41_polytope/R9z0wNUz">241</a>{3,3,34,1}</td><td colspan="4"><a href="/facts/1_42_polytope/DTgYDddB">142</a>{3,34,2}</td></tr></tbody></table>
<h3>7-sphere</h3>
<p>The <a href="/facts/N-sphere/bFu2S7E2">7-sphere</a> or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point, e.g. the origin. It has symbol <i>S</i>7, with formal definition for the 7-sphere with radius <i>r</i> of

S
          
            7
          
        
        =
        
          {
          
            x
            ∈
            
              
                R
              
              
                8
              
            
            :
            ‖
            x
            ‖
            =
            r
          
          }
        
        .
      
    
    {\displaystyle S^{7}=\left\{x\in \mathbb {R} ^{8}:\|x\|=r\right\}.}

</p><p>The volume of the space bounded by this 7-sphere is

V
          
            8
          
        
        
        =
        
          
            
              π
              
                4
              
            
            24
          
        
        
        
          R
          
            8
          
        
      
    
    {\displaystyle V_{8}\,={\frac {\pi ^{4}}{24}}\,R^{8}}

which is 4.05871 × <i>r</i>8, or 0.01585 of the <a href="/facts/8-cube/vEpKH3le">8-cube</a> that contains the 7-sphere.
</p>
<h3>Kissing number problem</h3>
<p class="note">Main article: <a href="/facts/Kissing_number_problem/UE3mtV0d">Kissing number problem</a></p>
<p>The <a href="/facts/Kissing_number_problem/UE3mtV0d">kissing number problem</a> has been solved in eight dimensions, thanks to the existence of the <a href="/facts/4_21_polytope/QFGglCkk">421</a> polytope and its associated <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a>. The kissing number in eight dimensions is <a href="/facts/240_(number)/XXMIa4AK">240</a>.
</p>
<h2 id="octonions">Octonions</h2>
<p class="note">Main article: <a href="/facts/Octonion/nC2sIz3p">Octonion</a></p>
<p>The octonions are a <a href="/facts/Normed_division_algebra/CW1n5sa0">normed division algebra</a> over the real numbers, the largest such algebra. Mathematically they can be specified by 8-tuplets of real numbers, so form an 8-dimensional vector space over the reals, with addition of vectors being the addition in the algebra. A normed algebra is one with a product that satisfies 
</p>

‖
        x
        y
        ‖
        ≤
        ‖
        x
        ‖
        ‖
        y
        ‖
      
    
    {\displaystyle \|xy\|\leq \|x\|\|y\|}

<p>for all <i>x</i> and <i>y</i> in the algebra. A normed <a href="/facts/Division_algebra/xp5Bepx3">division algebra</a> additionally must be finite-dimensional, and have the property that every non-zero vector has a unique multiplicative inverse. <a href="/facts/Hurwitz%2527s_theorem_(composition_algebras)/CW1n5sa0">Hurwitz's theorem</a> prohibits such a structure from existing in dimensions other than 1, 2, 4, or 8.
</p>
<h2 id="biquaternions">Biquaternions</h2>
<p>The complexified <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> 
  
    
      
        
          C
        
        ⊗
        
          H
        
      
    
    {\displaystyle \mathbb {C} \otimes \mathbb {H} }
  
, or "<a href="/facts/Biquaternion/YsI2czSu">biquaternions</a>," are an eight-dimensional algebra dating to <a href="/facts/William_Rowan_Hamilton/U86CqUlt">William Rowan Hamilton</a>'s work in the 1850s. This algebra is equivalent (that is, <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a>) to the <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a> 
  
    
      
        C
        
          ℓ
          
            2
          
        
        (
        
          C
        
        )
      
    
    {\displaystyle C\ell _{2}(\mathbb {C} )}
  
 and the <a href="/facts/Pauli_algebra/BdeC2wos">Pauli algebra</a>. It has also been proposed as a practical or pedagogical tool for doing calculations in <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>, and in that context goes by the name <a href="/facts/Algebra_of_physical_space/wdMbXq04">Algebra of physical space</a> (not to be confused with the <a href="/facts/Spacetime_algebra/VD7v4FYt">Spacetime algebra</a>, which is 16-dimensional.)
</p>

<ul><li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H.S.M. Coxeter</a>:
<ul><li>H.S.M. Coxeter, <i>Regular Polytopes</i>, 3rd Edition, Dover New York, 1973</li></ul></li>
<li><i>Kaleidoscopes: Selected Writings of H.S.M. Coxeter</i>, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-01003-6 <a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter</a>
<ul><li>(Paper 22) H.S.M. Coxeter, <i>Regular and Semi Regular Polytopes I</i>, [Math. Zeit. 46 (1940) 380–407, MR 2,10]</li>
<li>(Paper 23) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes II</i>, [Math. Zeit. 188 (1985) 559-591]</li>
<li>(Paper 24) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes III</i>, [Math. Zeit. 200 (1988) 3-45]</li></ul></li>
<li><a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/kiss.html">Table of the Highest Kissing Numbers Presently Known</a> maintained by Gabriele Nebe and <a href="/facts/Neil_Sloane/kLWw29Me">Neil Sloane</a> (lower bounds)</li>
<li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John Horton</a>; Smith, Derek A. (2003), <i>On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry</i>, A. K. Peters, Ltd., <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-56881-134-9. (<a href="http://nugae.wordpress.com/2007/04/25/on-quaternions-and-octonions/">Review</a>).</li>
<li>Duplij, Steven [in Ukrainian]; <a href="/facts/Warren_Siegel/dxEKnuPj">Siegel</a>, Warren; Bagger, Jonathan, eds. (2005), <i>Concise Encyclopedia of Supersymmetry And Noncommutative Structures in Mathematics and Physics</i>, Berlin, New York: <a href="/facts/Springer_Science%252BBusiness_Media/nAesf6nT">Springer</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4020-1338-6 (Second printing)</li></ul>

Eight-dimensional space open-in-new

Eight-dimensional space