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#### The Bilinski dodecahedron, and assorted parallelohedra, zonohedra, monohedra, isozonohedra and otherhedra.

(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 discovered will be used here to document errors by several mathematical luminaries. It also prompted an ...

#### Can Every Face of a Polyhedron Have Many Sides?

(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 about this topic are presented.

#### Graphs of polyhedra; polyhedra as graphs

(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 nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and ...

#### An enduring error

(2008-06-05)

*No abstract or description.*

#### A catalogue of simplicial arrangements in the real projective plane

(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 lines generating the arrangement. An arrangement is simplicial if all faces are triangles. Simplicial ...

#### Euler's ratio-sum theorem and generalizations

(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 polytopes, and several kinds of ratios.

#### Omittable lines

(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 said to be "omittable" if it contains no ordinary point of the family. Despite the large extent of work on ...

#### What symmetry groups are present in the Alhambra?

(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 discussions will be presented in detail. This leads to the more general problem about the validity and ...

#### Lecture notes on Modern Elementary Geometry

(1997)

*No abstract or description.*

#### Lectures on Lost Mathematics

(2010)

*No abstract or description.*