Browsing Mathematics, Department of by Author "Grünbaum, Branko"
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3connected 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 2connected. It has been believed that all geometric 3connected configurations ... 
{4,5} cubic lattice sponges
Gillispie, Steven; Grünbaum, Branko (20090106) 
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 (200811)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 (20080605) 
Euler's ratiosum 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, higherdimensional ... 
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 3connected 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 trianglefree configurations of points and lines
Boben, Marko; Grünbaum, Branko; Pisanski, Tomaz; Zitnik, Arjana (2006)In this paper we show that all combinatorial trianglefree 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) trianglefree ... 
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 ...