Golden Flèche Torse Spiral

Flèche > arrow 

Torse  > torso

Clochers tors

Twisted bell towers 

After some research, it only seems natural that Professor Alexis Doval's exercise in Euclidean geometry on the "Logarithmic Spiral Construction Using Ratio Φ and Isosceles Triangles" could change the way carpenters draw spirals for twisted spires on a pentagonal ground plan.  The research suggests that the Saint Mary's College of California Integral Liberal Arts Program might have also influenced other people in critical thinking. The Integral Program is a complete four-year Great Books course of study covering all mathematics and science. 

Alexis makes and plays guitars, so using the music theory of the 12th octave for the 12th fret on the guitar would be a natural progression to the 6th root.  

The Logarithmic Spiral Construction Using Ratio Φ and Isosceles Triangles is based on Euclid's Elements Book IV Proposition 10.  When I first went through the exercise, it seemed like a natural progression from the Isosceles triangles to the 6th root spiral. After comparing the 6th root spiral to the drawings I used to build the Flèche Torse, with devers: canted spiral hip rafters, the 6th root spiral is pretty close to the spiral construction I used. However, where the 6th root spiral really shines is when it produces twist blocks in the profile view that are almost perfect squares for each twist block. Usually, the twist blocks have different heights. The other aspect of using the 6th roof spiral construction takes the trial and error out of developing the Logarithmic spiral on the roof surface. Developing the spiral edges in the plan view was also a surprise. The spiral edges are parallel to the center spiral. That's not the case for the spiral edges I have in my Flèche Torse drawing.

After studying the 6th root spiral that produces the Golden Spiral on a Flèche Torse, I see no reason not to use this technique on all future spiral drawings for the Flèche Torse.

6th Root Spiral development

Isosceles Triangles:36°:72°:72°

Spiral developed in plan view using the 6th root

Spiral edges developed in plan view

Comparison of spirals in plan view

Comparison of spirals on the roof surface

Galileo used observation and experimentation to interrogate and challenge received wisdom and traditional ideas. For him, it wasn't enough that people in authority had been saying that something was true for centuries; he wanted to test these ideas and compare them to the evidence. 

The evidence is in: the 6th root spiral technique is now the correct course in developing a logarithmic spiral for a Flèche Torse on a pentagonal ground plan with devers : canted spiral hip rafters.