Sacred Square Cuts Among Even-Sided Polygons


2p 8-Gon
Angle No.1, Sin (45 degrees / 2) = 0.76536686473018 / 2
Angle No.2, Sin (90 degrees / 2) = 1.41421356237309 / 2
Angle No.3, Sin (135 degrees / 2) = 1.84775906502257 / 2
Angle No.4, Sin (180 degrees / 2) = 2 / 2

When the reciprocal of Angle No.1 (1.30656296487638) is multiplied by Angle No.3 (1.84775906502257), then this equals the length of a diagonal: 2.41421356237309. Likewise, when Angle No.1 (0.76536686473018) is multiplied by the reciprocal of Angle No.3 (0.541196100146197), then this yields the length of another diagonal: - 0.414213562373095. And when the first diagonal is divided by the second diagonal, and when the negation of the second diagonal is divided by the first diagonal, then this yields the two roots of a quadratic polynomial: {2.41421356237309, - 0.414213562373095} = x^2 - 2x - 1.


3p 12-Gon
Angle No.1, Sin (30 degrees / 2) = 0.517638090205041 / 2
Angle No.2, Sin (60 degrees / 2) = 1 / 2
Angle No.3, Sin (90 degrees / 2) = 1.41421356237309 / 2
Angle No.4, Sin (120 degrees / 2) = 1.73205080756888 / 2
Angle No.5, Sin (150 degrees / 2) = 1.93185165257814 / 2
Angle No.6, Sin (180 degrees / 2) = 2 / 2

When the reciprocal of Angle No.1 (1.93185165257814) is multiplied by Angle No.3 (1.41421356237309), then this equals the length of a diagonal: 2.73205080756888. Likewise, when Angle No.3 (1.41421356237309) is multiplied by the reciprocal of Angle No.5 (0.517638090205041), then this yields the length of another diagonal: - 0.732050807568877. And when the first diagonal is divided by the second diagonal, and when the negation of the second diagonal is divided by the first diagonal, then this yields the two roots of a quadratic polynomial: {2.73205080756888, - 0.732050807568877} = x^2 - 2x - 2.

When the reciprocal of Angle No.1 (1.93185165257814) is multiplied by Angle No.5 (1.93185165257814), then this equals the length of a diagonal: 3.73205080756888. Likewise, when Angle No.1 (0.517638090205041) is multiplied by the reciprocal of Angle No.5 (0.517638090205041), then this yields the length of another diagonal: + 0.267949192431123. And when the first diagonal is divided by the second diagonal, and when the negation of the second diagonal is divided by the first diagonal, then this yields the two roots of a quadratic polynomial: {3.73205080756888, + 0.267949192431123} = x^2 - 4x + 1.

When the reciprocal of Angle No.3 (0.707106781186548) is multiplied by Angle No.5 (1.93185165257814), then this equals the length of a diagonal: 1.36602540378444. Likewise, when Angle No.1 (0.517638090205041) is multiplied by the reciprocal of Angle No.3 (0.707106781186548), then this yields the length of another diagonal: - 0.366025403784439. And when the first diagonal is divided by the second diagonal, and when the negation of the second diagonal is divided by the first diagonal, then this yields the two roots of a quadratic polynomial: {1.36602540378444, - 0.366025403784439} = 2x^2 + 2x - 1.


5p 20-Gon
Angle No.1, Sin (18 degrees / 2) = 0.312868930080462 / 2
Angle No.2, Sin (36 degrees / 2) = 0.618033988749895 / 2
Angle No.3, Sin (54 degrees / 2) = 0.907980999479093 / 2
Angle No.4, Sin (72 degrees / 2) = 1.17557050458495 / 2
Angle No.5, Sin (90 degrees / 2) = 1.41421356237309 / 2
Angle No.6, Sin (108 degrees / 2) = 1.61803398874989 / 2
Angle No.7, Sin (126 degrees / 2) = 1.78201304837674 / 2
Angle No.8, Sin (144 degrees / 2) = 1.90211303259031 / 2
Angle No.9, Sin (162 degrees / 2) = 1.97537668119028 / 2
Angle No.10, Sin (180 degrees / 2) = 2 / 2

When the reciprocal of Angle No.2 (1.61803398874989) is multiplied by Angle No.6 (1.61803398874989), then this equals the length of a diagonal: 2.61803398874989. Likewise, when Angle No.2 (0.618033988749895) is multiplied by the reciprocal of Angle No.6 (0.618033988749895), then this yields the length of another diagonal: + 0.381966011250105. And when the first diagonal is divided by the second diagonal, and when the negation of the second diagonal is divided by the first diagonal, then this yields the two roots of a quadratic polynomial: {2.61803398874989, + 0.381966011250105} = x^2 - 3x + 1.

When the reciprocal of Angle No.2 (1.61803398874989) is multiplied by Angle No.10 (2), then this equals the length of a diagonal: 3.23606797749979. Likewise, when Angle No.6 (1.61803398874989) is multiplied by the reciprocal of Angle No.10 (0.5), then this yields the length of another diagonal: - 1.23606797749979. And when the first diagonal is divided by the second diagonal, and when the negation of the second diagonal is divided by the first diagonal, then this yields the two roots of a quadratic polynomial: {3.23606797749979, - 1.23606797749979} = x^2 - 2x - 4.

When the reciprocal of Angle No.4 (0.85065080835204) is multiplied by Angle No.8 (1.90211303259031), then this equals the length of a diagonal: 1.61803398874989. Likewise, when Angle No.4 (1.17557050458495) is multiplied by the reciprocal of Angle No.8 (0.525731112119134), then this yields the length of another diagonal: - 0.618033988749895. And when the first diagonal is divided by the second diagonal, and when the negation of the second diagonal is divided by the first diagonal, then this yields the two roots of a quadratic polynomial: {1.61803398874989, - 0.618033988749895} = x^2 - x - 1.

When the reciprocal of Angle No.6 (0.618033988749895) is multiplied by Angle No.10 (2), then this equals the length of a diagonal: 1.23606797749979. Likewise, when Angle No.2 (0.618033988749895) is multiplied by the reciprocal of Angle No.10 (0.5), then this yields the length of another diagonal: - 3.23606797749979. And when the first diagonal is divided by the second diagonal, and when the negation of the second diagonal is divided by the first diagonal, then this yields the two roots of a quadratic polynomial: {1.23606797749979, - 3.23606797749979} = x^2 + 2x - 4.