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For a polyhedron, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If a polyhedron is convex, then the defect of each vertex is always positive. If the sum of the angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is ...
Since the area of a triangle cannot be negative the spherical excess is always positive. It is not necessarily small, because the sum of the angles may attain 5 π (3 π for proper angles). For example, an octant of a sphere is a spherical triangle with three right angles, so that the excess is π /2.
Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves, space curves, polyhedra, ordinary differential equations, partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films), conformal ...
Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2 π /5 radians (72° = 360°/5) can be trisected, but the angle of π /3 radians (60°) cannot be trisected. [8]
The gauche effect is very sensitive to solvent effects, due to the large difference in polarity between the two conformers.For example, 2,3-dinitro-2,3-dimethylbutane, which in the solid state exists only in the gauche conformation, prefers the gauche conformer in benzene solution by a ratio of 79:21, but in carbon tetrachloride, it prefers the anti conformer by a ratio of 58:42. [9]
In the Gauss diagram of a knot, every chord crosses an even number of other chords, or equivalently each pair in the diagram connects a point in an even position of the cyclic order with a point in an odd position, and sometimes this is used as a defining condition of Gauss diagrams. [5] In algebraic geometry, chord diagrams can be used to ...
Negative rake: A tool has a negative rake angle when the face of the cutting tool slopes away from the cutting edge at outer side. Positive rake angles generally: Make the tool more sharp and pointed. This reduces the strength of the tool, as the small included angle in the tip may cause it to chip away. Reduce cutting forces and power ...
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.. A pseudosphere of radius R is a surface in having curvature −1/R 2 at each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature 1/R 2.
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