On the left drawing the center of similitude is the intersection of the rectangles' diagonals, the ratio 1/φ, and the angle -π/2 the radius is thus multiplied by φ 4≅6,9 at each turn. On Bernoulli's tombstone is engraved "eadem mutata resurgo" which can be translated by "moved, I rise again the same". These drawings are close to logarithmic spirals, also known as equiangle spirals (constant tangent angle) or Bernoulli's spirals they are invariant by similitude. With sequences of embedded golden rectangles or triangles, we get easily nice " golden spirals". Its eccentricity e is equal to the distance focus-directrix d (if AB=2 then e = d = 1/√φ). Its foci F' and F are easy to construct (M being a point on the ellipse, if AB=1 then MF'+MF = F'F² = φ). The pentagram shows several golden sections and several examples of the two types of golden triangles: isosceles triangles with ratio of the sides equal to φ, their angles measure 72°-36°-72° and 36°-108°-36° (remark: cos π /5 = φ /2).Īn ellipse inscribed in a golden rectangle is a golden ellipse (ratio of the axis equal to φ). The vertices of three golden rectangles two by two orthogonal are the vertices of a regular icosahedron more generally two opposite edges of a regular icosahedron define a golden rectangle (thus there are 15).Ī rhombus whose diagonal's ratio is the golden ratio is a golden rhombus (its vertices are the midpoints of the sides of a golden rectangle). Its construction is simple (with AB=2AU, ABCD and ABC'D' are two golden rectangles, and we get the first by adding a square to the second). The ratio of the sides of this two regular pentagons is the golden ratio φ.Ī rectangle whose length/width ratio is the golden ratio is a golden rectangle. MO=NU=PG (N midpoint of and P of )Ī simple knot made with a strip of paper, and then carefully flatted is a " golden knot" just fold over one of the strip's ends and you get a complete pentagram (convex regular pentagon with its five diagonals which are the sides of a regular star pentagon). Starting from three segments with same length Starting from an equilateral triangle and a square Starting from a rectangular triangle OIV with OV=2×OI We construct 0, U and G such as OG/OU=φ, thus OG=φ if we choose OU as unit. Remark: Any non-zero natural integer is written uniquely as sum of non-consecutive numbers of the Fibonacci sequence (Zeckendorf's theorem).Įxample: 33=21+8+3+1 by successive subtractions of numbers from the Fibonacci sequence (the largest possible): 33- 21=12 12- 8=4 4- 3= 1 three constructions of the golden ratio But the golden ratio is also the irrational number the worst approximated by rational numbers because all the integer parts in its continuous fraction are all equal to 1! It is interesting to point out that every sequence defined as the Fibonacci sequence by f(n+1)=f(n)+f(n-1) leads to the golden ratio, no matter what the two initial values f(0) and f(1) are: f(n+1)/f(n) -> φ.
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