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Exact trigonometric constants. Exact algebraic expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification.

Exact trigonometric constants

All values of the sines, cosines, and tangents of angles at 3° increments are derivable in radicals using identities—the half-angle identity, the double-angle identity, and the angle addition/subtraction identity—and using values for 0°, 30°, 36°, and 45°. Note that 1° = π/180 radians. According to Niven's theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2, and 1. Fermat number[edit] The list in this article is incomplete in at least two senses.

Wythoff construction. Wythoffian constructions from 3 mirrors forming a right triangle.

Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process[edit] ENCYCLOPEDIA OF TRIANGLE CENTERS. X(585) = 1st CONGRUENT SHRUNK INSQUARES POINT Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*area*(1/b + 1/c - 1/a) + b + c - a]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2*area*(1/b + 1/c - 1/a) + b + c - a X(585) lies on this line: 8,192 For a discussion, see Floor van Lamoen, "Vierkanten in een driehoek: 5.


Coxeter–Dynkin diagram. Coxeter–Dynkin diagrams for the fundamental finite Coxeter groups Coxeter–Dynkin diagrams for the fundamental affine Coxeter groups Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams.

Coxeter–Dynkin diagram

Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6.