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Octahedral pyramid
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<h2 id="occurrences-of-the-octahedral-pyramid">Occurrences of the octahedral pyramid</h2> <p>The regular <a href="/facts/16-cell/X0YAfobw">16-cell</a> has <i>octahedral pyramids</i> around every vertex, with the <a href="/facts/Octahedron/G38q92xW">octahedron</a> passing through the center of the 16-cell. Therefore placing two regular octahedral pyramids base to base constructs a 16-cell. The 16-cell tessellates 4-dimensional space as the <a href="/facts/16-cell_honeycomb/lbktV8Nr">16-cell honeycomb</a>. </p><p>Exactly 24 regular octahedral pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a <a href="/facts/24-cell/tfoGnlAD">24-cell</a> with octahedral bounding cells, surrounding a central vertex with 24 edge-length long radii. The 4-dimensional content of a unit-edge-length 24-cell is 2, so the content of the regular octahedral pyramid is 1/12. The 24-cell tessellates 4-dimensional space as the <a href="/facts/24-cell_honeycomb/foFoK5YX">24-cell honeycomb</a>. </p><p>The octahedral pyramid is the <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> for a <a href="/facts/Truncated_5-orthoplex/EMvxIKtY">truncated 5-orthoplex</a>, . </p> <p>The graph of the octahedral pyramid is the only possible minimal counterexample to <a href="/facts/Negami%27s_conjecture/TX1NupDf">Negami's conjecture</a>, that the connected graphs with <a href="/facts/Planar_cover/TX1NupDf">planar covers</a> are themselves projective-planar.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>Example 4-dimensional coordinates, 6 points in first 3 coordinates for cube and 4th dimension for the apex. ( ± 1 , 0 , 0 ; 0 ) ( 0 , ± 1 , 0 ; 0 ) ( 0 , 0 , ± 1 ; 0 ) ( 0 , 0 , 0 ; 1 ) {\displaystyle {\begin{array}{lllr}(\pm 1,&0,&0;&0)\\(0,&\pm 1,&0;&0)\\(0,&0,&\pm 1;&0)\\(0,&0,&0;&\ 1)\end{array}}} </p> <h2 id="other-polytopes">Other polytopes</h2> <h3>Cubic pyramid</h3> <p>The dual to the octahedral pyramid is a <a href="/facts/Cubic_pyramid/xmotrN5d">cubic pyramid</a>, seen as a cubic base and 6 <a href="/facts/Square_pyramid/wDetDohM">square pyramids</a> meeting at an <a href="/facts/Apex_(geometry)/SrMxmHJk">apex</a>. </p> <p>Example 4-dimensional coordinates, 8 points in first 3 coordinates for cube and 4th dimension for the apex. ( ± 1 , ± 1 , ± 1 ; 0 ) ( 0 , 0 , 0 ; 1 ) {\displaystyle {\begin{array}{lllr}(\pm 1,&\pm 1,&\pm 1;&0)\\(0,&0,&0;&1)\end{array}}} </p> <h3>Square-pyramidal pyramid</h3> <p> The square-pyramidal pyramid, ( ) ∨ [( ) ∨ {4}], is a bisected octahedral pyramid. It has a <a href="/facts/Square_pyramid/wDetDohM">square pyramid</a> base, and 4 <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedrons</a> along with another one more square pyramid meeting at the apex. It can also be seen in an edge-centered projection as a <a href="/facts/Square_bipyramid/G38q92xW">square bipyramid</a> with four tetrahedra wrapped around the common edge. If the height of the two apexes are the same, it can be given a higher symmetry name [( ) ∨ ( )] ∨ {4} = { } ∨ {4}, joining an edge to a perpendicular square.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>The <i>square-pyramidal pyramid</i> can be distorted into a <i>rectangular-pyramidal pyramid</i>, { } ∨ [{ } × { }] or a <i>rhombic-pyramidal pyramid</i>, { } ∨ [{ } + { }], or other lower symmetry forms. </p><p>The <i>square-pyramidal pyramid</i> exists as a vertex figure in uniform polytopes of the form , including the <a href="/facts/Bitruncated_5-orthoplex/EMvxIKtY">bitruncated 5-orthoplex</a> and <a href="/facts/Bitruncated_tesseractic_honeycomb/CVRrn5Fy">bitruncated tesseractic honeycomb</a>. </p> <p>Example 4-dimensional coordinates, 2 coordinates for square, and axial points for pyramidal points. ( ± 1 , ± 1 ; 0 ; 0 ) ( 0 , 0 ; 1 ; 0 ) ( 0 , 0 ; 0 ; 1 ) {\displaystyle {\begin{array}{lllr}(\pm 1,&\pm 1;&0;&0)\\(0,&0;&1;&0)\\(0,&0;&0;&\ \ 1)\end{array}}} </p> <h2 id="external-links">External links</h2> <ul><li>Olshevsky, George. <a href="https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Pyramid">"Pyramid"</a>. <i>Glossary for Hyperspace</i>. Archived from <a href="http://members.aol.com/Polycell/glossary.html#Pyramid">the original</a> on 4 February 2007.</li> <li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/../explain/segmentochora.htm">"4D Segmentotopes"</a>. <ul><li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/..//incmats/octpy.htm">"Segmentotope octpy, K-4.3"</a>.</li></ul></li> <li>Richard Klitzing, <a href="http://bendwavy.org/klitzing/pdf/edge-facetings_color.pdf">Axial-Symmetrical Edge Facetings of Uniform Polyhedra</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Klitzing, Richard. "3D convex uniform polyhedra x3o4o - oct". 1/sqrt(2) = 0.707107 <a href="https://bendwavy.org/klitzing/dimensions/polyhedra.htm" target="_blank">https://bendwavy.org/klitzing/dimensions/polyhedra.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, CiteSeerX 10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645 <a href="http://www.fi.muni.cz/~hlineny/papers/plcover20-gc.pdf" target="_blank">http://www.fi.muni.cz/~hlineny/papers/plcover20-gc.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Klitzing, Richard. "Segmentotope squasc, K-4.4". <a href="https://bendwavy.org/klitzing/dimensions/..//incmats/squasc.htm" target="_blank">https://bendwavy.org/klitzing/dimensions/..//incmats/squasc.htm</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>