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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 angle π / 3 radians (60 degrees, written 60°) is constructible. The argument below shows that it is impossible to construct a 20° angle. This implies that a 60° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected. Denote the set of rational numbers by Q.
Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, [1]: p. xi and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.
90° Hexahemioctacron (Dual of cubohemioctahedron) — V(4.6. 4 / 3 .6) π − π / 3 120° Octahemioctacron (Dual of octahemioctahedron) — V(3.6. 3 / 2 .6) π − π / 3 120° Small dodecahemidodecacron (Dual of small dodecahemidodecacron) — V(5.10. 5 / 4 .10) π − π / 5 144° Small ...
Fig 1. Construction of the first isogonic center, X(13). When no angle of the triangle exceeds 120°, this point is the Fermat point. In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible [1] or ...
To find the angle of a rotation, once the axis of the rotation is known, select a vector v perpendicular to the axis. Then the angle of the rotation is the angle between v and Rv. A more direct method, however, is to simply calculate the trace: the sum of the diagonal elements of the rotation matrix.
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A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane with four pentagons around each vertex ( or more ) and sphere with three pentagons ; the latter produces ...