# Author archives: David Orden

## We can learn a lot from a walk with real numbers

Questions on the digit expansion of mathematical constants, such as , , or have fascinated mathematicians for centuries. However, many of these questions remain elusive to the efforts of researchers. For instance, it is not known for sure if the number 1 appears infinitely often in the decimal expansion of . Among the most remarkable […]

## How to create the illusion of antigravity motions

Visual illusions, a perceptual behavior where what we “see” differs from the physical reality, are used in vision science to understand the nature of human perception. Most of the known visual illusions come from 2D pictures and their motions, but not so many make use of 3D solid objects. An example of the latter is […]

## Finding a shortest route is not easy for a watchman patrolling streets

Imagine you were a watchman, having to patrol some streets. Today you were assigned to straight, well-illuminated, and wide streets, that can be checked with a glance from the intersection. Before starting the route, you want to determine the shortest route allowing to check all your streets. How difficult can this be? Dumitrescu et al […]

## The complexity of drawing good proportional symbol maps

Proportional symbol maps (also known as graduated symbol maps) are used in Cartography to visualize quantitative data associated with specific locations. For a given point on a geographic map, a symbol (usually a disk or a square) is scaled such that its area is proportional to the numerical value associated to the point. This scaled […]

## The history of the crossing number of the complete graph

We have already introduced the first half of a paper by Lowell Beineke and Robin Wilson, which traces the origins of the crossing number of a graph. After focusing on the brick factory problem, proposed by the Hungarian number-theoretist Paul Turán, the second half of the paper is devoted to the crossing number of the […]

## Progress checking Zarankiewicz’s conjecture on the brick factory problem

A previous post presented the fascinating history of the brick factory problem, which wonders about the smallest possible number of rail crossings when connecting kilns and storage yards, which is mathematically modeled by the crossing number of the complete bipartite graph . Proposed by Paul Turán after the Second World War, the first advances on […]

## The fascinating history of the brick factory problem

The crossing number of a graph, defined as the minimum number of edge crossings arising when the graph is drawn in the plane, turns out to be a mathematical concept with a fascinating history. Lowell Beineke and Robin Wilson trace the origins of the crossing number, with particular emphasis in the war-time experiences of the […]

## Every convex polyhedron can be refolded to a different one

Choose your favorite convex polyhedron in the space. Make sure it is convex, since the current post is restricted to that kind of polyhedra An unfolding of your convex polyhedron is a development of its surface to a single polygon in the plane (possibly overla pping). In order to obtain an unfolding, the surface of […]

## Advances on an Erdős problem on convex boundaries

Among his around 1525 papers, Paul Erdős considered On sets of distances of n points as his most important contribution to discrete geometry. There, he stated: On the boundary of every convex body, there exists a point P such that every circle with center P intersects that boundary in at most 2 points. (Note that […]

## Always look on the light side of graphs

Consider a set of distinct points in the plane, no three of them on the same line. Draw straight-line segments joining pairs of those points. This is called a geometric graph and here we are going to focus on the segments of such a graph. Every segment has two sides, defined by the line in […]