• polyhedral
  • One stronger form of the circle packing theorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by a polyhedron with a midsphere. (wikipedia.org)
  • face
  • The midsphere is so-called because, for polyhedra that have a midsphere, an inscribed sphere (which is tangent to every face of a polyhedron) and a circumscribed sphere (which touches every vertex), the midsphere is in the middle, between the other two spheres. (wikipedia.org)
  • If O is the midsphere of a polyhedron P, then the intersection of O with any face of P is a circle. (wikipedia.org)
  • The face planes of the polar polyhedron pass through the circles on O that are tangent to cones having the vertices of P as their apexes. (wikipedia.org)
  • Any two polyhedra with the same face lattice and the same midsphere can be transformed into each other by a projective transformation of three-dimensional space that leaves the midsphere in the same position. (wikipedia.org)
  • These are paths on the surface of the polyhedron that avoid the vertices and locally look like a shortest path: they follow straight line segments across each face of the polyhedron that they intersect, and when they cross an edge of the polyhedron they make complementary angles on the two incident faces to the edge. (wikipedia.org)
  • For instance, the graphs of three-dimensional stacked polyhedra are exactly the Apollonian networks, the graphs formed from a triangle by repeatedly subdividing a triangular face of the graph into three smaller triangles. (wikipedia.org)
  • For polyhedra, this operation adds a new hexagonal face in place of each original edge. (wikipedia.org)
  • simplicial
  • The most important open problem in the field is the g-conjecture, formulated by Peter McMullen, which asks about possible numbers of faces of different dimensions of a simplicial sphere. (wikipedia.org)
  • The upper bound theorem gives upper bounds for the numbers fi of i-faces of any simplicial d-sphere with f0 = n vertices. (wikipedia.org)
  • In other words, what are the possible sequences of numbers of faces of each dimension for a simplicial d-sphere? (wikipedia.org)
  • There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. (wikipedia.org)
  • It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. (wikipedia.org)
  • One reason for the significance of stacked polytopes is that, among all d-dimensional simplicial polytopes with a given number of vertices, the stacked polytopes have the fewest possible higher-dimensional faces. (wikipedia.org)
  • Analogously, the simplicial polytopes that maximize the number of higher-dimensional faces for their number of vertices are the cyclic polytopes. (wikipedia.org)
  • pyramid
  • As the orientation of the pyramid of vision changes in response to flight data, the displayed faces are correspondingly displaced, eventually moving out of the pyramid of vision. (patents.com)
  • Faces which are currently not visible (outside the pyramid of vision) are clipped from the data flow. (patents.com)
  • In addition, faces which are only partially outside of pyramid of vision are reconstructed to eliminate the outside portion. (patents.com)
  • The outcodes (O.C.) are systematically processed and examined to determine which faces are completely inside the pyramid of vision (Case A--all signs positive), which faces are completely outside (Case C--All signs negative) and which faces must be reconstructed (Case B--both positive and negative signs). (patents.com)
  • This creation is a truncated triangular pyramid . (hmns.org)
  • topologically
  • With six six-sided faces around each vertex, it is topologically equivalent to a quotient space of the hyperbolic order-6 hexagonal tiling, {6,6} and is an abstract type {6,6}6. (wikipedia.org)
  • Notes: Polyhedra with different names that are topologically identical are listed together. (wikipedia.org)
  • spheres
  • This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and icosahedron (because it has interior room for another sphere). (wikipedia.org)
  • correspond
  • The horizon circles of a canonical polyhedron can be transformed, by stereographic projection, into a collection of circles in the Euclidean plane that do not cross each other and are tangent to each other exactly when the vertices they correspond to are adjacent. (wikipedia.org)
  • In mathematical terms, the panels that form the casing of the traditional soccer ball correspond to the various faces of a regular, truncated icosahedron. (google.com)