Connecting the eight points of intersection of the circle and the sides of the square form an

*octagon.*

Interior view of the Hall of Abencerrajes @ Alhambra

Just about every medieval cathedral has an octagonal baptismal and the Alhambra in Granada Spain has several octagonal ground plans. The roof over the Hall of the Abencerrajes is unique because the roof actually follows the Ad Quadratum ground plan that is used to develop the octagonal ground plan. Squares rotated in the circle.

Ad Quadratum ground plan geometrically developed for hip and valley rafters.

Developing the roof framing plan is pretty straight forward if you use polygon roof framing geometry. The hip rafter is base on a polygon with 4 sides and the valley rafter is based on a polygon with 8 sides.

Using the Rafter Tools for Android app select the Polygon Rafter Angles calculator and enter 4 for the Number of Sides of Polygon to calculate the angles for the hip rafter and use number of sides = 8 for the valley rafter slope angle. These angles can also be found on the Chappell Master Framing Square.

The adjacent radius can be calculated using the law of sines.

Octagon 8 Sides

Ground Plan -- Ad Quadratum

2 - Squares Rotated in Circle

Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C)

Adjacent Radius = ( 144 × sin(45 ) ) ÷ sin(112.5)

110.2128 = ( 144 × sin(45 ) ) ÷ sin(112.5)

Dodecagon - 12 sides

Ground Plan -- Ad Quadratum

3 - Squares Rotated in Circle

Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C)

Adjacent Radius = ( 144 × sin( 45 ) ) ÷ sin( 120)

117.5755 = ( 144 × sin( 45 ) ) ÷ sin( 120)

Hexadecagon - 16 sides

Ground Plan -- Ad Quadratum

4 - Squares Rotated in Circle

Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C)

Adjacent Radius = ( 144 × sin( 45 ) ) ÷ sin( 124.08620)

122.4619 = ( 144 × sin( 45 ) ) ÷ sin( 124.08620)

I'll post more information on the hip rafter and valley rafter head cuts in the next article on the Ad Quadratum ground plan.

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