Johannes Kepler (1571 – 1630) was an Austrian Lutheran mathematician, astronomer and a key figure in the 17th century astronomical revolution. He is best known for his laws of planetary motion, based on his works Astronomia nova and Harmonice Mundi; Kepler's laws provided one of the foundations of Isaac Newton's theory of universal gravitation. Before Kepler, planets' paths were computed by combinations of the circular motions of the celestial orbs; after Kepler astronomers shifted their attention from orbs to orbits—paths that could be represented mathematically as an ellipse.
During his career Kepler was a mathematics teacher at a Graz seminary school (later the University of Graz, Austria), an assistant to Tycho Brahe, court mathematician to Emperor Rudolf II, mathematics teacher in Linz, Austria, and adviser to General Wallenstein. He also did fundamental work in the field of optics and helped to legitimize the telescopic discoveries of his contemporary Galileo Galilei.
Kepler lived in an era when there was no clear distinction between astronomy and astrology, while there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of the more prestigious discipline of philosophy). (Full article...)
The knight's tour
is a mathematical chess problem
in which the piece called the knight
is to visit each square on an otherwise empty chess
board exactly once, using only legal moves. It is a special case of the more general Hamiltonian path problem
in graph theory
. (A closely related non-Hamiltonian problem is that of the longest uncrossed knight's path
.) The tour is called closed
if the knight ends on a square from which it may legally move to its starting square (thereby forming an endless cycle
), and open
if not. The tour shown in this animation is open (see also a static image of the completed tour
). On a standard 8 × 8
board there are 26,534,728,821,064 possible closed tours and 39,183,656,341,959,808 open tours (counting separately any tours that are equivalent by
rotation, reflection, or reversing the direction of travel). Although the earliest known solutions to the knight's tour problem date back to the 9th century CE
, the first general procedure for completing the knight's tour was Warnsdorff's rule
, first described in 1823. The knight's tour was one of many chess puzzles
solved by The Turk
, a fake chess-playing machine exhibited as an automaton
from 1770 to 1854, and exposed in the early 1820s as an elaborate hoax. True chess-playing automatons
(i.e., computer programs) appeared in the 1950s, and by 1988 had become sufficiently advanced to win a match against a grandmaster
; in 1997, Deep Blue
famously became the first computer system to defeat a reigning world champion (Garry Kasparov
) in a match under standard tournament time controls. Despite these advances, there is still debate as to whether chess will ever be "solved"
as a computer problem (meaning an algorithm will be developed that can never lose a chess match). According to Zermelo's theorem
, such an algorithm does exist.