Now showing items 1-17 of 17

    • 3-connected configurations (n3) with no Hamiltonian circuit 

      Grünbaum, Branko (2006)
      For more than a century there have been examples of (n3) configurations without a Hamiltonian circuit. However, all these examples were only 2-connected. It has been believed that all geometric 3-connected configurations ...
    • {4,5} cubic lattice sponges 

      Gillispie, Steven; Grünbaum, Branko (2009-01-06)
    • The Bilinski dodecahedron, and assorted parallelohedra, zonohedra, monohedra, isozonohedra and otherhedra. 

      Grünbaum, Branko (2010)
      Fifty years ago Stanko Bilinski showed that Fedorov's enumeration of convex polyhedra having congruent rhombi as faces is incomplete, although it had been accepted as valid for the previous 75 years. The dodecahedron he ...
    • Can Every Face of a Polyhedron Have Many Sides? 

      Grünbaum, Branko (2008-11)
      The simple question of the title has many different answers, depending on the kinds of faces we are willing to consider, on the types of polyhedra we admit, and on the symmetries we require. Known results and open problems ...
    • A catalogue of simplicial arrangements in the real projective plane 

      Grünbaum, Branko (2005)
      An arrangement is the complex generated in the real projective plane by a family of straight lines that do not form a pencil. The faces of an arrangement are the connected components of the complement of the set of ...
    • Configurations of points and lines 

      Grünbaum, Branko (2005)
      A vigorous study of geometric configurations started in the 1870's but was essentially abandoned early in the twentieth century. New approaches found during the last two decades prompted a renewed interest in the topic, ...
    • An enduring error 

      Grünbaum, Branko (2008-06-05)
    • Euler's ratio-sum theorem and generalizations 

      Grünbaum, Branko; Klamkin, Murray S. (2005)
      A theorem of Euler concerns sums of ratios in which Cevians of a triangle are divided by a common point. Generalizations of this result in three directions are presented: polygons instead of triangles, higher-dimensional ...
    • Graphs of polyhedra; polyhedra as graphs 

      Grünbaum, Branko (2005)
      Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of ...
    • Isogonal prismatoids 

      Grünbaum, Branko (Springer New York, 1997)
      A prismatoid is a polyhedron with all vertices in two parallel planes. A polyhedron P is isogonal if all its vertices form one transitivity class under isometric symmetries of P. Although these restrictions appear very ...
    • Lecture notes on Modern Elementary Geometry 

      Grünbaum, Branko (1997)
    • Lectures on Arrangements 

      Grünbaum, Branko (1974)
    • Lectures on Lost Mathematics 

      Grünbaum, Branko (2010)
    • Omittable lines 

      Grünbaum, Branko (2005)
      Every finite family of (straight) lines in the projective plane, not forming a pencil, is well know to have at least one "ordinary point" –– that is, a point common to precisely two of the lines. A line of a family is ...
    • Small triangle-free configurations of points and lines 

      Boben, Marko; Grünbaum, Branko; Pisanski, Tomaz; Zitnik, Arjana (2006)
      In this paper we show that all combinatorial triangle-free configurations (v_3) for v (is less than or equal to) 8 are geometrically realizable. We also show that there is a unique smallest astral (18_3) triangle-free ...
    • Unsolved Problems in Intuitive Geometry 

      Klee, Victor (1960)
    • What symmetry groups are present in the Alhambra? 

      Grünbaum, Branko (American Mathematical Society, 2006)
      The question which of the seventeen wallpaper groups are represented in the fabled ornamentation of the Alhambra has been raised and discussed quite often, with widely diverging answers. Some of the arguments from these ...