polyhedra with equilateral pentagons faces. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T22:41:12Z http://mathoverflow.net/feeds/question/56653 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56653/polyhedra-with-equilateral-pentagons-faces polyhedra with equilateral pentagons faces. jolumij 2011-02-25T17:37:27Z 2011-12-11T22:57:03Z <p>In page <a href="http://loki3.com/poly/isohedra.html" rel="nofollow">http://loki3.com/poly/isohedra.html</a> around six polyhedra with equilateral pentagons as faces are shown: a pyritohedron, icositetrahedrons... Is there a complete list of this kind of polyhedra? How to compute the angles of that pentagons?</p> http://mathoverflow.net/questions/56653/polyhedra-with-equilateral-pentagons-faces/56681#56681 Answer by Igor Rivin for polyhedra with equilateral pentagons faces. Igor Rivin 2011-02-25T21:39:10Z 2011-02-25T21:39:10Z <p>The pentagons are regular (apparently), so their angles should be 144 degrees. As for whether this is a complete list, you should check out </p> <p><a href="http://mathworld.wolfram.com/Isohedron.html" rel="nofollow">http://mathworld.wolfram.com/Isohedron.html</a></p> <p>and references therein.</p> http://mathoverflow.net/questions/56653/polyhedra-with-equilateral-pentagons-faces/56695#56695 Answer by Joseph O'Rourke for polyhedra with equilateral pentagons faces. Joseph O'Rourke 2011-02-26T01:19:53Z 2011-02-26T02:02:39Z <p>This is a nice question, which I cannot answer. But let me draw your attention to a problem posed by Richard Kenyon, <a href="http://cs.smith.edu/~orourke/TOPP/P72.html#Problem.72" rel="nofollow">Problem 72 at The Open Problems Project</a>:</p> <blockquote> <p>Let \$M\$ be a closed polyhedral surface homeomorphic to \$\mathbb{S}^2\$ which is entirely composed of equal regular pentagons. If \$M\$ is immersed in 3-space, is it necessarily the boundary of a union of solid dodecahedra that are glued together at common facets? </p> </blockquote> <p>So in some sense, the classification of all polyhedra with <em>congruent, regular</em> pentagon faces is open. Replacing "immersed" with "embedded" still leaves it open.</p> http://mathoverflow.net/questions/56653/polyhedra-with-equilateral-pentagons-faces/83212#83212 Answer by Scott Sherman for polyhedra with equilateral pentagons faces. Scott Sherman 2011-12-11T22:57:03Z 2011-12-11T22:57:03Z <p>The pentagonal isohedra with sides of equal length listed on <a href="http://loki3.com/poly/isohedra.html" rel="nofollow">http://loki3.com/poly/isohedra.html</a> are the regular dodecahedron, non-convex equilateral pyritohedron, equilateral pentagonal icositetrahedron, non-convex equilateral pentagonal icositetrahedron, non-convex equilateral pentagonal hexecontahedron and non-convex equilateral pentagonal hexecontahedron.</p> <p><img src="http://loki3.com/poly/isohedral-images/dodecahedron.png" width="150" alt="regular dodecahedron" /> <img src="http://loki3.com/poly/isohedral-images/12y_4pz0_0954915z0_6605596.png" width="150" alt="non-convex equilateral pyritohedron" /> <img src="http://loki3.com/poly/isohedral-images/24p_8pz0_3456397z0_031877.png" width="150" alt="equilateral pentagonal icositetrahedron" /> <img src="http://loki3.com/poly/isohedral-images/24p_8pz0_05625923z0_42629483.png" width="150" alt="non-convex equilateral pentagonal icositetrahedron" /> <img src="http://loki3.com/poly/isohedral-images/60p_20pz0_3140921z-0_04663939.png" width="150" alt="non-convex equilateral pentagonal hexecontahedron" /> <img src="http://loki3.com/poly/isohedral-images/60p_20pz0_03413652z0_2976625.png" width="150" alt="non-convex equilateral pentagonal hexecontahedron" /></p> <p>(I should note that the page jolumij references, is a page I built to summarize my findings. I had tried to learn about the isohedra from pages such as <a href="http://mathworld.wolfram.com/Isohedron.html" rel="nofollow">http://mathworld.wolfram.com/Isohedron.html</a>, but all sources I could find were very incomplete when it came to describing the pentagonal isohedra. They offer names such as "octahedral pentagonal dodecahedron" without describing how to construct them or mentioning they may represent an infinite family of shapes.)</p> <p>I assume by "this kind of polyhedra," you're referring to isohedra with pentagonal faces. Mathworld offers this definition of an <a href="http://mathworld.wolfram.com/Isohedron.html" rel="nofollow">isohedron</a>:</p> <blockquote>An isohedron is a convex polyhedron with symmetries acting transitively on its faces with respect to the center of gravity.</blockquote> <p>For my list of isohedra, I relax the definition to include non-convex polyhedra. The <a href="http://loki3.com/poly/transforms.html" rel="nofollow">isohedral transforms</a> can also be used to create polyhedra with intersecting faces "with symmetries acting transitively on [their] faces with respect to the center of gravity."</p> <p>As to whether those 6 are the only isohedra with equilateral pentagonal faces, I believe the list is complete, but I haven't rigorously proven it. What I have done is start from the tetrahedral, octahedral and icosahedral symmetry groups and applied the <a href="http://loki3.com/poly/transforms.html#penta" rel="nofollow">isohedral pentagonal transform</a> to them. This transform has two degrees of freedom. Then I explored the space for equilateral pentagons as well as other interesting symmetries or patterns. I haven't seen references to many of the shapes I found (including 5 of the 6 shapes listed here), so I'd be interested if other people know of any other references.</p> <p>As to how to compute the angles of those pentagons, <a href="http://loki3.com/poly/transforms.html#penta" rel="nofollow">http://loki3.com/poly/transforms.html#penta</a> gives a description of the transform and what my notation means. You can use the parameters and transform to derive the angles. For example, the non-convex equilateral pyritohedron is 4p(0.09549150, 0.6605596), which means you apply the isohedral pentagonal transform (p) to a tetrahedron (4) using the parameters 0.09549150 and 0.6605596. In this case, you get two 36 degree angles and three 108 degree angles.</p>