Joe Fu worked with a colleague in Europe to perform the research, which was recently published in the Annals of Mathematics.
"What we have done is find a new pathway to understand details of certain mathematical structures," Fu said.
The research has its origins in a simple exercise in a so-called "geometric probability" known as Buffon's Needle Problem. This much-studied thought experiment originated in 1770s France and examines the probability issues involved in throwing sticks or needles on a set of parallel equidistant lines, such as planks on a floor.
Buffon's Needle Problem predicts the chance that a needle thrown at random will cross one of the cracks between the planks. It has fascinated mathematicians since its initial publication.
While the probability of how many lines the tossed needles might cross can be as simple as a child's game, studying what happens using advanced mathematics such as linear algebra opens up "a whole world of unexplored problems of the same type," Fu said.
He and his co-author, Andreas Bernig of the Goethe University in Germany, used an algebraic approach to geometry to come to their conclusion.
According to Fu, their research represents the first concrete progress on the collateral mathematical meanings of Buffon's Needle Problem in more than 70 years. Their approach was based on revolutionary new methods in the algebra of measurement, developed at the beginning of the 21st century by Israeli mathematician Semyon Alesker.
"The solution of Buffon's Needle Problem is a consequence of the famous principal kinetic formula, which expresses a variety of geometric probabilities involving convex sets in terms of fundamental measurements such as volume, perimeter, mean breadth and so forth," Fu said. "This opens the door to working out the kinematic formulas... completely algebraically. Beyond the door lies a vast mine of beautiful, important and accessible open problems."
The Fu-Bernig paper may appear abstract, but the precision of its results may well open new inroads into certain problems in math and physics.
"Nature doesn't waste any possibilities," Fu said.
"What we have done is find a new pathway to understand details of certain mathematical structures," Fu said.
The research has its origins in a simple exercise in a so-called "geometric probability" known as Buffon's Needle Problem. This much-studied thought experiment originated in 1770s France and examines the probability issues involved in throwing sticks or needles on a set of parallel equidistant lines, such as planks on a floor.
Buffon's Needle Problem predicts the chance that a needle thrown at random will cross one of the cracks between the planks. It has fascinated mathematicians since its initial publication.
While the probability of how many lines the tossed needles might cross can be as simple as a child's game, studying what happens using advanced mathematics such as linear algebra opens up "a whole world of unexplored problems of the same type," Fu said.
He and his co-author, Andreas Bernig of the Goethe University in Germany, used an algebraic approach to geometry to come to their conclusion.
According to Fu, their research represents the first concrete progress on the collateral mathematical meanings of Buffon's Needle Problem in more than 70 years. Their approach was based on revolutionary new methods in the algebra of measurement, developed at the beginning of the 21st century by Israeli mathematician Semyon Alesker.
"The solution of Buffon's Needle Problem is a consequence of the famous principal kinetic formula, which expresses a variety of geometric probabilities involving convex sets in terms of fundamental measurements such as volume, perimeter, mean breadth and so forth," Fu said. "This opens the door to working out the kinematic formulas... completely algebraically. Beyond the door lies a vast mine of beautiful, important and accessible open problems."
The Fu-Bernig paper may appear abstract, but the precision of its results may well open new inroads into certain problems in math and physics.
"Nature doesn't waste any possibilities," Fu said.
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