rings, since the face and degeneracy maps are all polynomial). The algebraic n-simplices are used in higher K-theory and in the definition of higher Chow groups. Applications In statistics, simplices are sample spaces of compositional data and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot. In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such mixtures, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using response surface methodology, and then a local maximum can be computed using a nonlinear programming method, such as sequential quadratic programming. In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig. In geometric design and computer graphics, many methods first perform simplicial triangulations of the domain and then fit interpolating polynomials to each simplex. In chemistry, the hydrides of most elements in the p-block can resemble a simplex if one is to connect each atom. Neon does not react with hydrogen and as such is a point, fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a bent fashion resembling a triangle, nitrogen reacts to form a tetrahedron, and carbon forms a structure resembling a Schlegel diagram of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a halogen atom. In some approaches to quantum gravity, such as Regge calculus and causal dynamical triangulations, simplices are used as building blocks of discretizations of spacetime; that is, to build simplicial manifolds. See also 3-sphere Aitchison geometry Causal dynamical triangulation Complete graph Delaunay triangulation Distance geometry Geometric primitive Hill tetrahedron Hypersimplex List of regular polytopes Metcalfe's law Other regular n-polytopes Cross-polytope Hypercube Tesseract Polytope Schläfli orthoscheme Simplex algorithm—a method for solving optimization problems with inequalities. Simplicial complex Simplicial homology Simplicial set Spectrahedron Ternary plot Notes References (See chapter 10 for a simple review of topological properties.) pp. 120–121, §7.2.