# Golden section

The golden section , usually indicated by the letter φ ( phi ) of the Greek alphabet in the honor of Phidias, sculptor and Greek architect of the Parthenon, is the irrational Nombre:

$\ varphi = \ frac \left\{1 + \ sqrt \left\{5\right\}\right\} \left\{2\right\} \ simeq 1,618033988749894848204586834365\dots$.

## Geometrical properties

### Golden section and pentagon

The golden section appears in the proportions of the regular pentagon, like the relationship between the length on the side of the pentagon and that on the side of the Pentagramme registered. This definite purely geometrical report/ratio on a regular polygon was probably the first Greek definition of the golden section.

### Gold rectangle

One calls gold rectangle a rectangle whose relationship between the length and the width is worth the golden section.

The layout of a gold rectangle is done very simply using a compass; it is enough to point the medium on a side of a square, to point one of the two opposed angles, then to fold back the arc of circle on the line passing by the side of the pointed square (to be noted that this construction was a “secrecy” of Compagnonnage to the Moyen-âge).

Here a possible reason of the attraction caused by the right-angled gold : let us consider a rectangle whose sides lengths has and B is in a report/ratio of the golden section:

So of this rectangle, we remove the square on side length B , then the rectangle remaining is again a gold rectangle, since its sides are in a φ report/ratio. Indeed, according to the algebraic properties,

$\ frac \left\{B\right\} \left\{a-b\right\} = \ frac \left\{1\right\} \left\{a/b -1\right\} = \ frac \left\{has\right\} \left\{B\right\} = \ varphi$.

By reiterating this construction, we obtain a succession of increasingly small gold rectangles. This fact is a geometrical interpretation of the development in Fraction continues golden section (see further).

### Gold triangles

The gold triangles are isosceles triangles of which the report/ratio of the east coasts equal to the golden section. There exists about it of two types. Those for which the side ratio/base is worth φ which gives acute triangles called sometimes money triangles and those for which the report/ratio bases/side is worth φ.

In the united figure:

• isosceles triangles BDA and CAB have a common basic angle in A. ABD is thus similar to BCA in a report/ratio of 1/φ.
• Like φ = 1+ 1/φ, cd. = 1 and DBC is isosceles of top D.
• the angle out of B is thus double angle out of C in ABC.
• the sum of the angles of a triangle being worth 180°, one obtains for the angle C the fifth of the flat Angle, either 36° and for the angle B the two fifths of the flat angle, or 72°.

Since it is a question of cutting out a flat angle into 5, he is not surprising to find these gold triangles in the regular Pentagone and the Pentacle.

In an acute gold triangle, one can draw a blunt gold triangle and an acute gold triangle φ time smaller. One finds this same phenomenon in a blunt gold triangle. These facts explain why one finds these two elements in the pavings of Penrose.

### Gold spirals

One can build, starting from a gold rectangle, a Spirale of gold by tracing quadrants in each square. This spiral approaches a Spirale logarithmic curve of center the intersection of the two diagonals of the two rectangles and of polar equation:

$R \left(\ theta\right) = R. \ varphi^ \left\{- \ frac \left\{\ theta\right\} \left\{\ pi/2\right\}\right\}$