The simplest example of a flat three-dimensional shape is ordinary infinite space — what mathematicians call Euclidean space — but there are other flat shapes to consider too. These shapes are harder ...

"You know you can fill space with cubes," Torquato said, "We were looking for another way." In the article "New Family of Tilings of Three-Dimensional Euclidean Space by Tetrahedra and Octahedra," he ...

In contrast, quasiregular mappings can be found from a Euclidean space to several different ... 2-spheres or the connected ...

over time draws out what is known as a three-dimensional Euclidean space—one which has comparable qualities to the universe overall. The shape created had those same qualities regardless of the ...

Traditional data analysis methods often rely on Euclidean (linear) dependencies among features ... This advancement facilitates nonlinear interpolation and denoising within the high-dimensional ...

Researchers propose that the three dimensions of space may have been “frozen in” during the early moments of the universe. (Left) The Helmholtz free energy density (f) reaches its maximum ...

meaning that space is closed in on itself in all three dimensions like a three-dimensional donut. Such a universe would be finite, and according to their results, our entire cosmos might only be ...