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Navier-Stokes equations

Named after Claude-Louis Navier and George Gabriel Stokes, Navier-Stokes equations are a set of partial differential equations that accurately and reliably describe the motion of liquids and gases, allowing them to be recreated in simulation.

The equations work by determining rates of change, and movement in the substance when external forces are applied, rather than modelling the substance itself. This actually considerably simplifies matters, as the shape of the body of fluid or gas, or cluster of particles does not have a bearing on how it reacts. The density does, the shape does not.

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Hacked: The Mathematics of Leaf Vein Architecture
The math by which the veins of leaves are formed has been cracked. By extension, the reasons for leaf size and shape have been cracked, and equations for creating the types of leaves that would service a given plant have been laid bare. Ideal material for anyone wishing to model a realistic but not physically real, plant.

Creating Polygonal Shapes through a Merger with Topology
By merging the traditionally distinct fields of topology and geometry, through the discovery of a new type of mathematics called persistent homology, math researchers have created a set of equations which simply describe the pattern and placement of complex fractals as part of a polygonal model - such as the unique froth on a wave, within the reach of real-time rendering

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The way fabric moves has always been a pain to recapture virtually. Creating realistic folds and drapes in anything from skirts to sails has always been computationally expensive, and impossible to produce in realtime. Now, thanks to Mah...

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Simulation of mechanical systems

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Mathematics might seem like an abstract discipline, remote from real-world applications but their equations can significantly help understand and simulate the functioning of nature. Professor Alfio Quarteroni of the École Polytechnique Fédé...

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