An alternating harmony of φ
1 = φ¹ – φ⁻¹
3 = φ² + φ⁻²
4 = φ³ – φ⁻³
7 = φ⁴ + φ⁻⁴
11 = φ⁵ – φ⁻⁵
18 = φ⁶ + φ⁻⁶
etc.
Interestingly, the Luca numbers show up in this golden ladder.
The Golden Ladder
Discover the Golden Ellipse
A golden ellipse is one where the axes are in golden proportion, meaning the ratio of the major axis (a) to the minor axis (b) is the golden ratio:
φ = (1 + √5)/2.
To visualize this, draw a golden ellipse along with its inscribed and circumscribed circles: the smallest circle fitting inside the ellipse and the largest circle surrounding it.
Interestingly, the area of the ellipse matches the area of the “annulus” formed between these two circles!
Here’s how it works:
Let a be the semi-major axis and b the semi-minor axis, with a = φb.
The area of the annulus is:
π(a² − b²) = πb²(φ² − 1)
The area of the ellipse is:
πab = πφb²
And as φ² − 1 = φ, then πb²(φ² − 1) = πφb².
Isn’t it fascinating how geometry intertwines with the golden ratio?
Circles & Roots
Delve into the realm of Sacred Geometry, where circles unveil the elegance of successive square roots from 1 to 6. Extend your exploration with the enigmatic charm of the square root of Phi.
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φibonacci formula
Nested Radicals
Some remarkable infinite nested square roots of 2
Read more.
Kepler Triangle
A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio.
So, the sides of such a triangle are in the ratio 1 : √ φ : φ [where φ = ( 1 + √5 )/ 2 is the golden ratio.]
Inverse Powers of Phi
The Kepler Triangle, Phi and Pi
A “Kepler triangle” is a right triangle having edge lengths in a geometric progression, in which the common ratio is √ϕ, where ϕ represents the golden ratio.
Well, let’s construct a square with side length √ϕ that inscribes a Kepler triangle, that is, a right triangle with edges 1 : √ϕ : ϕ (or approximately 1 : 1.272 : 1.618), as shown in the picture. Draw then the circumcircle of the Kepler triangle (highlighted in orange in the picture) whose diameter is the hypotenuse of the triangle.
Then, the perimeters of the square (4√ϕ≈5.0884) and the circle (πϕ≈5.083) coincide up to an error less than 0.1%. From this, we can get the approximation coincidence π≈4/√ϕ
Golden Ratio (And Its Inverse) In Yin Yang
The philosophy of the Yin Yang is depicted by the the “taichi symbol” (taijitu). In fact, Yin Yang is a concept of dualism, describing how seemingly opposite or contrary forces may actually be complementary,
Curiously enough, in the taichi symbol are hidden the golden ratio and its reverse. As shown in the picture. Continue reading “Golden Ratio (And Its Inverse) In Yin Yang”