3 ms·

Dodecahedron :-)

• wolfgke 7y ago
  Your snarky remark carries much more mathematical depth than it appears on the surface. There exists a 3-manifold that is homologous to a 3-sphere but not homeomomorphic to it; the so-called Poincaré homology sphere: > https://en.wikipedia.org/w/index.php?title=Homology_sphere&oldid=911674219#Poincar%C3%A9_homology_sphere https://en.wikipedia.org/w/index.php?title=Homology_sphere&o... Among the homology 3-spheres (besides the 3-sphere itself), the Poincaré homology sphere is the only one with a finite fundamental group. Now for the plot twists: 1. A possible construction of the Poincaré homology sphere starts with a dodecahedron. 2. The fundamental group of the Poincaré homology sphere is the binary icosahedral group (https://en.wikipedia.org/w/index.php?title=Binary_icosahedral_group&oldid=940730315 https://en.wikipedia.org/w/index.php?title=Binary_icosahedra...), which is an extension of the symmetry group of the dodecahedron/icosahedron. 3. The huge plot twist: What does this all have to do with the question "What Is the Geometry of the Universe?"? Let me quote the Wikipedia article: "In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft. As of 2016, the publication of data analysis from the Planck spacecraft suggests that there is no observable non-trivial topology to the universe."