There is no greater geometric solid than the simplex. It is the paragon of efficiency, the pinnacle of symmetry, and the prototype of simplicity. If the universe were not constructed of continuous coordinates, then surely it would be tiled by tessellations of simplices.
Indeed, simplices, or simplexes, arise in a wide range of geometrical problems and real-world applications. For instance, metallic alloys are described on a simplex to identify the constituent elements [1]. Zero-sum games in game theory and ecosystems in population dynamics are described on simplexes [2], and the Dantzig simplex algorithm is a central algorithm for optimization in linear programming [3]. Simplexes also are used in nonlinear minimization (amoeba algorithm), in classification problems in machine learning, and they also raise their heads in quantum gravity. These applications reflect the special status of the simplex in the geometry of high dimensions.
… It’s Simplexes all the way down!
The reason for their usefulness is the simplicity of their construction that guarantees a primitive set that is always convex. For instance, in any space of d-dimensions, the simplest geometric figure that can be constructed of flat faces to enclose a d-volume consists of d+1 points that is the d-simplex.
Or …
In any space of d-dimensions, the simplex is the geometric figure whose faces are simplexes, whose faces are simplexes, whose faces are again simplexes, and those faces are once more simplexes … And so on.
In other words, it’s simplexes all the way down.
Simplex Geometry
In this blog, I will restrict the geometry to the regular simplex. The regular simplex is the queen of simplexes: it is the equilateral simplex for which all vertices are equivalent, and all faces are congruent, and all sub-faces are congruent, and so on. The regular simplexes have the highest symmetry properties of any polytope. A polytope is the d-dimensional generalization of a polyhedron. For instance, the regular 2-simplex is the equilateral triangle, and the regular 3-simplex is the equilateral tetrahedron.
The N-simplex is the high-dimensional generalization of the tetrahedron. It is a regular N-dimensional polytope with N+1 vertexes. Starting at the bottom and going up, the simplexes are the point (0-simplex), the unit line (1-simplex), the equilateral triangle (2-simplex), the tetrahedron (3-simplex), the pentachoron (4-simplex), the hexateron (5-simplex) and onward. When drawn on the two-dimensional plane, the simplexes are complete graphs with links connecting every node to every other node. This dual character of equidistance and completeness give simplexes their utility. Each node is equivalent and is linked to each other. There are N•(N-1)/2 links among N vertices, and there are (N-2)•(N-1)/2 triangular faces.
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