Sacred Square Cuts Among Even-Sided Polygons

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2p 8-Gon
Angle No.1, {Sin(45° ÷ 2)} x 2 = 0.76536686473018
Angle No.2, {Sin(90° ÷ 2)} x 2 = 1.41421356237309
Angle No.3, {Sin(135° ÷ 2)} x 2 = 1.84775906502257
Angle No.4, {Sin(180° ÷ 2)} x 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° ÷ 2)} x 2 = 0.517638090205041
Angle No.2, {Sin(60° ÷ 2)} x 2 = 1
Angle No.3, {Sin(90° ÷ 2)} x 2 = 1.41421356237309
Angle No.4, {Sin(120° ÷ 2)} x 2 = 1.73205080756888
Angle No.5, {Sin(150° ÷ 2)} x 2 = 1.93185165257814
Angle No.6, {Sin(180° ÷ 2)} x 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° ÷ 2)} x 2 = 0.312868930080462
Angle No.2, {Sin(36° ÷ 2)} x 2 = 0.618033988749895
Angle No.3, {Sin(54° ÷ 2)} x 2 = 0.907980999479093
Angle No.4, {Sin(72° ÷ 2)} x 2 = 1.17557050458495
Angle No.5, {Sin(90° ÷ 2)} x 2 = 1.41421356237309
Angle No.6, {Sin(108° ÷ 2)} x 2 = 1.61803398874989
Angle No.7, {Sin(126° ÷ 2)} x 2 = 1.78201304837674
Angle No.8, {Sin(144° ÷ 2)} x 2 = 1.90211303259031
Angle No.9, {Sin(162° ÷ 2)} x 2 = 1.97537668119028
Angle No.10, {Sin(180° ÷ 2)} x 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.