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.

  φ6R Extreme and Mean Isosceles Triangle Spiral

This spiral study, suggested by my friend Alexis, was meant to be a sandbox of ideas on different types of spirals. Here are the results of this sandbox study on spirals using Euclid's Extreme and Mean Ratio, the Golden Ratio, Bach's music theory on the 6th root, and drawings by Rafael Araujo. 

Euclid's Elements Book IV Proposition 10

Alexis Doval's drawing of the Isosceles triangles cut into extreme and mean ratios of the Golden Ratio.

If the lines AF and CF were a single straight line, they could be cut into extremes and mean and equal to the Golden Ratio.

If the lines CF and BF were a single straight line, they could be cut into extremes and mean and equal to the Golden Ratio.

After the Isosceles triangles are developed and cut into the Extreme and Mean ratios, the radius of the cone is taken and used to calculate the length of each construction ray in plan view by the 6th root. 

Then, you bring the lengths of the construction rays that define the spiral up to the profile view. Where these lengths intersect, the cone slant line will determine the arcs for the cone surface. Then, use the intersection of the arcs and construction rays on the cone surface to develop the spiral on the cone surface. 

The sixth root of a number is the number that would have to be multiplied by itself 6 times to get the original number.

Golden Ratio Spiral based on Rafael Araujo drawing. 

The vertex of each isosceles triangle is the center point of each arc.

Note:  φ6R Extreme and Mean Isosceles Triangle Spiral is Logarithmic

The Golden Ration Spiral is not a Logarithmic or an Archimedean Spiral.

Archimedean spiral increases in constant distance in plan view but not on the cone's surface.

Logarithmic spiral increases in constant geometric progression in plan view and on the cone's surface.

Fibonacci Icosahedron based on the drawing from Rafael Araujo

Note: The Fibonacci Spiral has nothing to do with the Icosahedron; it just looks neat. 

The Golden Ratio by Rafael Araujo

Golden Shell with tapered torus based on the Golden Ratio

            The golden ratio and Fibonacci numbers in Bach’s music

 Euclid's Elements Book XIII Proposition 16

To completely understand Euclid's Elements Book XIII Proposition 16, you must study Euclid's Elements Book XIII Proposition 9 and Euclid's Elements Book II Proposition 11.  In Proposition 9, Euclid proves that the straight line of the decagon and hexagon inscribed in a circle is cut at extreme and mean, with the hexagon line being the greater segment. When the circle's radius is equal to one, the length of the straight line of the decagon and hexagon is equal to the Golden ratio, 1.61803. However, more importantly, the decagon line length is equal to 0.61803. Which is used in the √5 Triangle to develop the Golden Ratio. 

Euclid's Elements Book XIII Proposition 9

&

Euclid's Elements Book XI

 Proposition 13

If the side of the hexagon and that of the decagon inscribed in the same circle are added together, then the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.

Let ABC be a circle, and of the figures inscribed in the circle, ABC, let BC be the side of a decagon, and CD that of a hexagon, and let them be in a straight line.

Therefore, if the side of the hexagon and that of the decagon inscribed in the same circle are added together, then the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.

Euclid's Elements Book II Proposition 11

 To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

Interactive: Euclid's Extreme and Mean Proportion Constructions

Euclid's Elements Book XIII Proposition 18

based on the Extreme and Mean Ratios

The line from the midpoint of CE to K is the Auron. It's an Orthogon. The root word for Auron is Latin, meaning "gold" and the golden ratio. 

Euclid's Elements Book XIII Proposition 16

Note: Euclid's schematic drawing of the Icosahedron is tough to follow. 

Euclid's Elements Book XIII Proposition 16

Note: ProofWiki has a better schematic drawing of the Icosahedron. However, the vertical lines perpendicular to the plane of the circle  up to points RQUTS,

that are equal to the Hexagon dimension, but they are still tough to visualize. 

Euclid's Elements Book XIII Proposition 16

Euclid starts by developing the line DB in his Proposition 16. The drawing to the right is from Proposition 18. As you can see, the two different techniques both develop the same exact dimension for the radius of the pentagon. Then he draws another pentagon rotated 36°.

In this drawing, the key to drawing out the Icosahedron is the extreme and mean ratio of vertical heights. Making a separate drawing with the decagon and hexagon drawn in a circle with the same radius as the base drawing is easier.
.

Draw the vertical heights for points S, R, Q, U, and T.  Then add decagon heights to the center vertical axis. 

Finally, draw the Icosahedron edge lines ZS, ZR, ZQ, ZU, and ZT to form the equilateral triangles. 

Draw the Icosahedron edge lines for the equilateral triangles on the sides of the Icosahedron SM-SN, RM-RL, QL-QP, UP-UO, and TO-TN . 

An updated drawing of Euclid's Elements Book XIII Proposition 18, with the Golden Ratio drawn in to locate the line GT for the Dodecahedron. This updated version uses the √5 Triangle to develop the Golden Ratio. From point X, a line is drawn to the center of the semicircle at point 3. Then, lay down an arc from center point X using the radius equal to XE to locate the point T. Line XE is half of the dimension of the √5. 

Euclid's Elements Book IV Proposition 10

In this proposition, you can see the relationship between the Hexagon, Decagon, and Pentagon in developing the vertical heights in constructing the Icosahedron. The diagonal in the Extreme and Mean drawing equals one. The circumscribed radius of the sphere of the vertex points of the Icosahedron equals one.