The Mighty Simplex
5 days ago
- #simplex
- #geometry
- #optimization
- The simplex is the most efficient and symmetric geometric solid, fundamental in high-dimensional geometry.
- Simplices have wide applications, including in metallic alloys, game theory, population dynamics, linear programming, and quantum gravity.
- A simplex in d-dimensions is constructed from d+1 points, forming the simplest figure to enclose a d-volume.
- Regular simplexes are equilateral with congruent faces and sub-faces, representing the highest symmetry in polytopes.
- Construction of a d-simplex is recursive, building from lower-dimensional simplexes.
- Simplices can be represented in barycentric coordinates, useful in alloy phase diagrams and crystallography.
- Replicator dynamics on simplexes model ecosystems, showing behaviors like fixed points, spirals, and saddles.
- The Dantzig simplex algorithm efficiently solves linear programming problems by walking along polytope vertexes.
- The amoeba algorithm uses a crawling simplex for nonlinear minimization, adapting its shape to find minima.
- Simplices are foundational in various fields, from materials science to machine learning and optimization algorithms.