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Posers and Puzzles

Posers and Puzzles

  1. Standard member talzamir
    Art, not a Toil
    03 Nov '11 18:12
    A cube has 12 edges; an octahedron has 12 edges.
    A cube has 8 vertexes; an octahedron has 8 faces.
    A cube has 6 faces; an octahedron has 6 vertexes.
    Each face of a cube has 4 sides; each vertex of an octahedron touches 4 edges.
    Each vertex of a cube touches 3 edges; each face of an octahedron has 3 sides.

    So, cubes and octahedrons have something to do with each other.

    Do tetrahedrons or icosahedrons have a similar pair?
  2. 03 Nov '11 19:01
    Originally posted by talzamir
    A cube has 12 edges; an octahedron has 12 edges.
    A cube has 8 vertexes; an octahedron has 8 faces.
    A cube has 6 faces; an octahedron has 6 vertexes.
    Each face of a cube has 4 sides; each vertex of an octahedron touches 4 edges.
    Each vertex of a cube touches 3 edges; each face of an octahedron has 3 sides.

    So, cubes and octahedrons have something to do with each other.

    Do tetrahedrons or icosahedrons have a similar pair?
    So in the case of the tetrahedron, are we looking for a corresponding polyhedron whose edges are 6, faces are 4, vertexes are 4, sides of each face are 3, and each face has 3 sides?
  3. Standard member talzamir
    Art, not a Toil
    03 Nov '11 20:35
    Indeed we are. The case for a tetrahedron is pretty obvious.. the icosahedron perhaps a bit less so.

    A tetrahedron has 6 edges; a ? has 12 edges.
    A tetrahedron has 4 vertexes; a ? has 8 faces.
    A tetrahedron has 4 faces; a ? has 4 vertexes.
    Each face of a tetraedron has 3 sides; each vertex of a ? touches 3 edges.
    Each vertex of a tetrahedron touches 3 edges; each face of a ? has 3 sides.
  4. Standard member talzamir
    Art, not a Toil
    03 Nov '11 22:54 / 1 edit
    Oops. That's what I get from posting in a hurry. The above should read,

    A tetrahedron has 6 edges; a ? has 6 edges.
    A tetrahedron has 4 vertexes; a ? has 4 faces.
    A tetrahedron has 4 faces; a ? has 4 vertexes.
    Each face of a tetraedron has 3 sides; each vertex of a ? touches 3 edges.
    Each vertex of a tetrahedron touches 3 edges; each face of a ? has 3 sides.
  5. 05 Nov '11 16:31
    Originally posted by talzamir
    A cube has 12 edges; an octahedron has 12 edges.
    A cube has 8 vertexes; an octahedron has 8 faces.
    A cube has 6 faces; an octahedron has 6 vertexes.
    Each face of a cube has 4 sides; each vertex of an octahedron touches 4 edges.
    Each vertex of a cube touches 3 edges; each face of an octahedron has 3 sides.

    So, cubes and octahedrons have something to do with each other.

    Do tetrahedrons or icosahedrons have a similar pair?
    Shades of Escher!

    Although I don't think he used the icosahedron case - but he did use the tetrahedron one to great effect.

    Richard
  6. Standard member talzamir
    Art, not a Toil
    05 Nov '11 17:29
    Indeed he did. An art gallery here was once filled with the works of Escher - a real delight. =)

    As for the puzzle.. a tetrahedron (a four-sided die or D4) flips into another tetrahedron. A cube and an octahedron (d6 and d8) convert into each other. An icosahedron (D12) flips into a dodecahedron (D20). When I saw that I was rather stunned about it, and wondered why it works so. There is a clear reason for it.

    Some gamers also use a ten-side die (D10) that looks like two pyramids of a base and five sides each, with the bases glued together. It has seven vertexes, ten faces, and 15 edges. It converts into a cylinder with a pentagon as base. 15 edges, 7 faces, 10 vertexes. In that, not every face looks the same, but then, the d10 is not one of Plato's perfect five.
  7. Standard member talzamir
    Art, not a Toil
    10 Nov '11 22:01
    The reason.. if you take the center point each face of a polyhedron and assign that as a vertex, and make an edge where two faces are adjacent, you get a face where the vertexes were. That projection converts
    * vertexes into faces and vice versa
    * maintains the number of edges
    * replaces the number of sides of each face by edges from a vertex and vice versa

    So a tetrahedron becomes a tetrahedron, a cube an octahedron, a dodecahedron an icosahedron, and vice versa.