Density (polytope)

<h2 id="polygons">Polygons</h2>
<p>The density of a polygon is the number of times that the polygonal boundary winds around its center. For convex polygons, and more generally <a href="/facts/Simple_polygon/C5CfrPWf">simple polygons</a> (not self-intersecting), the density is 1, by the <a href="/facts/Jordan_curve_theorem/zKpty4QT">Jordan curve theorem</a>.
</p><p>The density of a polygon can also be called its <a href="/facts/Turning_number/kaquyUL2">turning number</a>; the sum of the <a href="/facts/Turn_angle/jd78Qkbt">turn angles</a> of all the vertices divided by 360°. This will be an integer for all unicursal paths in a plane.
</p><p>The density of a compound polygon is the sum of the densities of the component polygons.
</p>
<h3>Regular star polygons</h3>
<p>For a regular <a href="/facts/Star_polygon/mDfuIe81">star polygon</a> {<i>p</i>/<i>q</i>}, the density is <i>q</i>. It can be visually determined by counting the minimum number of edge crossings of a ray from the center to infinity.
</p>
<h3>Examples</h3>

<h2 id="polyhedra">Polyhedra</h2>
<p>A polyhedron and its <a href="/facts/Dual_polyhedron/rkIHbmAf">dual</a> have the same density.
</p>
<h3>Total curvature</h3>
<p>A polyhedron can be considered a surface with <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> concentrated at the vertices and defined by an <a href="/facts/Angle_defect/FMHKeUVS">angle defect</a>. The density of a polyhedron is equal to the <a href="/facts/Gaussian_curvature/ARdhnb7o">total curvature</a> (summed over all its vertices) divided by 4π.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>For example, a <a href="/facts/Cube/65lUaAqN">cube</a> has 8 vertices, each with 3 <a href="/facts/Square/JHJeMvmL">squares</a>, leaving an angle defect of π/2. 8×π/2=4π. So the density of the cube is 1.
</p>
<h3>Simple polyhedra</h3>
<p>The density of a polyhedron with <a href="/facts/Simple_polygon/C5CfrPWf">simple faces</a> and <a href="/facts/Vertex_figure/3u2ySK0C">vertex figures</a> is half of the <a href="/facts/Euler_characteristic/JCjzWUr2">Euler Characteristic</a>, χ. If its <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> is <i>g</i>, its density is 1−<i>g</i>. 
</p>
χ = <i>V</i>  − <i>E</i> + <i>F</i> = 2<i>D</i> = 2(1−<i>g</i>).

<h3>Regular star polyhedra</h3>
<p><a href="/facts/Arthur_Cayley/NBd7QkeE">Arthur Cayley</a> used <i>density</i> as a way to modify Euler's <a href="/facts/Polyhedron_formula/JCjzWUr2">polyhedron formula</a> (<i>V</i> − <i>E</i> + <i>F</i> = 2) to work for the <a href="/facts/Kepler%25E2%2580%2593Poinsot_solid/mzUUgBYS">regular star polyhedra</a>, where <i>d</i>v is the density of a <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>, <i>d</i>f of a face and <i>D</i> of the polyhedron as a whole:
</p>

d
          
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    {\displaystyle d_{v}v-e+d_{f}f=2D}
  
<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
<p>For example, the <a href="/facts/Great_icosahedron/jaAk3a5Q">great icosahedron</a>, {3, 5/2}, has 20 triangular faces (<i>d</i>f = 1), 30 edges and 12 <a href="/facts/Pentagram/cN0ke8tp">pentagrammic</a> vertex figures (<i>d</i>v = 2), giving
</p>
2·12 − 30 + 1·20 = 14 = 2<i>D</i>.
<p>This implies a density of 7. The unmodified Euler's polyhedron formula fails for the <a href="/facts/Small_stellated_dodecahedron/lX1zlo6C">small stellated dodecahedron</a> {5/2, 5} and its dual <a href="/facts/Great_dodecahedron/LWt3Hgg4">great dodecahedron</a> {5, 5/2}, for which <i>V</i> − <i>E</i> + <i>F</i> = −6.
</p><p>The regular star polyhedra exist in two dual pairs, with each figure having the same density as its dual: one pair (small stellated dodecahedron—great dodecahedron) has a density of 3, while the other (<a href="/facts/Great_stellated_dodecahedron/7efrgSub">great stellated dodecahedron</a>–great icosahedron) has a density of 7.
</p>
<table><tbody><tr><td></td><td></td></tr><tr><td colspan="2">The nonconvex <a href="/facts/Great_icosahedron/jaAk3a5Q">great icosahedron</a>, {3,5/2} has a density of 7 as demonstrated in this transparent and cross-sectional view on the right.</td></tr></tbody></table>
<h3>General star polyhedra</h3>
<p><a href="/facts/Edmund_Hess/HJdajw7h">Edmund Hess</a> generalized the formula for star polyhedra with different kinds of face, some of which may fold backwards over others. The resulting value for density corresponds to the number of times the associated spherical polyhedron covers the sphere.
</p><p>
  
    
      
        
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    {\displaystyle \sum _{i}d_{vi}v_{i}-e+\sum _{i}d_{fi}f_{i}=2D}

</p><p>This allowed Coxeter et al. to determine the densities of the majority of the <a href="/facts/Uniform_polyhedra/PHnprqRC">uniform polyhedra</a>, which have one vertex type, and multiple face types.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

<h3>Nonorientable polyhedra</h3>
<p>For <a href="/facts/Hemipolyhedra/SLbcVFjT">hemipolyhedra</a>, some of whose faces pass through the center, the density cannot be defined.  <a href="/facts/Orientability/1SEQM2Pt">Non-orientable</a> polyhedra also do not have well-defined densities.
</p>
<h2 id="regular-4-polytopes">Regular 4-polytopes</h2>

<p>There are 10 regular star <a href="/facts/4-polytope/mY8vBpec">4-polytopes</a> (called the <a href="/facts/Regular_4-polytope/lFf8IH1C">Schläfli–Hess 4-polytopes</a>), which have densities between 4, 6, 20, 66, 76, and 191. They come in dual pairs, with the exception of the self-dual density-6 and density-66 figures.
</p>
<h2 id="notes">Notes</h2>

<ul><li>Coxeter, H. S. M.; <i><a href="/facts/Regular_Polytopes_(book)/jomuRNha">Regular Polytopes</a></i>, (3rd edition, 1973), Dover edition, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-61480-8</li>
<li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter, H. S. M.</a>; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", <i>Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences</i>, 246 (916): 401–450, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1098%2Frsta.1954.0003">10.1098/rsta.1954.0003</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0080-4614">0080-4614</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/91532">91532</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0062446">0062446</a></li>
<li>Wenninger, Magnus J. (1979), "An introduction to the notion of polyhedral density", <i>Spherical models</i>, CUP Archive, pp. <a href="https://books.google.com/books?id=Olc5AAAAIAAJ&pg=PA132">132–134</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-22279-2</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/PolygonDensity.html">"Polygon density"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Coxeter, H. S. M; The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (206–214, Density of regular honeycombs in hyperbolic space) <a href="/wiki/LCCN_(identifier)" target="_blank">/wiki/LCCN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Geometry and the Imagination in Minneapolis 17. The angle defect of a polyhedron; 20. Curvature of surfaces; 21. Gaussian curvature; 27.3.1 Curvature for Polyhedra pp. 32-51 <a href="https://arxiv.org/abs/1804.03055" target="_blank">https://arxiv.org/abs/1804.03055</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999). (Page 258) <a href="https://books.google.com/books?id=OJowej1QWpoC" target="_blank">https://books.google.com/books?id=OJowej1QWpoC</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Coxeter, 1954 (Section 6, Density and Table 7, Uniform polyhedra) <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Density (polytope) open-in-new

Density (polytope)