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The strictest version of the problem was solved in 2023, after an initial discovery in 2022. The einstein problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral. [3]
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations).
When we recently wrote about the toughest math problems that have been solved, we mentioned one of the greatest achievements in 20th-century math: the solution to Fermat’s Last Theorem. Sir ...
In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying approximations of the equations, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter.
Einstein himself considered the introduction of the cosmological constant in his 1917 paper founding cosmology as a "blunder". [3] The theory of general relativity predicted an expanding or contracting universe, but Einstein wanted a static universe which is an unchanging three-dimensional sphere, like the surface of a three-dimensional ball in four dimensions.
In 2023, Kaplan was part of the team that solved the einstein problem, a major open problem in tiling theory and Euclidean geometry. The problem is to find an "aperiodic monotile", a single geometric shape which can tesselate the plane aperiodically (without translational symmetry) but which cannot do so periodically. The discovery is under ...
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