Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)×(center) is an integer vector. This series of papers explain such...

Descartes (formerly called IRIS) is a software system designed to support visual exploration of spatially referenced data, e.g. demographic, economical, or cultural information about geographical objects or locations such as countries, districts, or cities. Descartes o€ ers two integrated services: automated presentation of data on maps, and facilities to interactively manipulate these maps. Au...

At the beginning of the seventeenth century, two great philosophers, Francis Bacon (1561–1626) in England and René Descartes (1596–1650) in France, proclaimed the birth of modern science. Each of them described his vision of the future. Their visions were very different. Bacon said, “All depends on keeping the eye steadily fixed on the facts of nature.” Descartes said, “I think, therefore I am....

To be able to believe that a dog with a broken paw is not really in pain when it whimpers is a quite extraordinary achievement even for a philosopher. Yet according to the standard interpretation, this is just what Descartes did believe. He held, we are informed, the 'monstrous' thesis that 'animals are without feeling or awareness of any kind'.1 The standard view has been reiterated in a recen...