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The Allen-Cahn equation on the two-dimensional torus

Submitted by Nils Berglund on

This is a 3D rendering of a solution of the Allen-Cahn equation in a rectangular domain with periodic boundary conditions. Both the z-coordinate and the color hue show the value of the solution, with blue indicating positive values, and red indicating negative ones. By increasing the viscosity (which measures the magnitude of diffusion) over time, one accelerates the coarse-graining dynamics of the system, separating blue and red phases. The luminosity depends on the angle between a normal vector to the surface and a fixed direction, to emulate the effect of a far away light source.

A circular lens, wave height and cumulative energy in 3D

Submitted by Nils Berglund on

This simulation shows the working of a lens made of a circular segment, that is, a disc cut by a straight line. The lens works here like the objective of a camera, by concentrating an incoming planar wave at the focal point of the lens. The index of refraction of the lens is equal to 5/3 = 1.666…, so that according the the lensmaker’s equation, the focal length should be equal to 1.5 times the radius of the circle.

Ceramic tiles and geometry

Submitted by Suzana Nápoles on

Through the observation of tile patterns and the identification of the isometries involved in their composition, we intend to illustrate the natural connections between geometry and art and to provide a stimulating approach to isometries of the plane.

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