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  • How to prove Eulers pentagonal theorem? Some hints will help
    While there is a lot of value to the different bijective proofs known for Euler's pentagonal theorem, perhaps the proof that's easiest to see without having to draw pictures is Euler's original idea
  • Formula for pentagonal numbers - Mathematics Stack Exchange
    Formula for pentagonal numbers Ask Question Asked 12 years, 3 months ago Modified 6 years, 5 months ago
  • Is this a new pentagonal tiling? - Mathematics Stack Exchange
    I discovered this while thinking about the pentagonal tiling of type 15 Is this a new type of tiling? If it is, then I think I have found several other new pentagonal tilings like this one and the pentagonal tiling of type 15 They all have vertices which lie in the field $ \mathbb {Q} (24) $ The internal angles for a pentagon in the image
  • graph theory - Mathematics Stack Exchange
    A polyhedron has all its faces either pentagons or hexagons Show that it must have at least $12$ pentagonal faces I can show that it has exactly $12$ pentagonal faces when exactly $3$ faces meet at
  • Is Cairo pentagonal tiling belong to pentagonal tilings type 8?
    It claims "The Cairo pentagonal tiling has two lower symmetry forms given as monohedral pentagonal tilings types 4 and 8" I am totally confused it Well, I think Cairo pentagonal tiling should be pentagonal tilings types 2 and 4, but not 8 Can someone give me a right answer? Thanks
  • Why are there 12 pentagons and 20 hexagons on a soccer ball?
    The soccer ball shape can be formed by truncating an icosahedron —if each of the 12 vertices is cut off, it will leave a pentagonal face and cause each of the 20 triangular faces to become hexagonal
  • geometry - What are the rules for a Tetartoid pentagon? - Mathematics . . .
    7 The tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry It has twelve identical pentagonal faces It has edges of the form a b b c c The pentagon itself can be completely determined by the sides b c and the angles bb bc cc
  • Why are $10$-sided dice not bipyramids? - Mathematics Stack Exchange
    Commonly used $10$-sided dice are pentagonal trapezohedrons, as opposed to pentagonal bipyramids Given that bipyramids are a more "obvious" shape for a fair die with an even number of faces, it's
  • A New Pentagonal Tiling? Help Me Solve the Mystery
    Thank you for your comment! Indeed, all convex pentagonal tilings have been mapped, and the list is believed to be complete However, for concave pentagons, there are infinitely many possibilities Interestingly, there is a known tiling that uses a shape identical to mine, but the arrangement (or orientation) of the shape in the tiling is different from what I’ve created here
  • Understanding a solution to counting hexagons on a soccer ball
    Each face of a soccer ball is either a pentagon or a hexagon Each pentagonal face is adjacent to five hexagonal faces and each hexagonal face is adjacent to three pentagonal and three hexagonal fa





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