The Golden Ratio: Phi, 1.618 » Geometry

Phi (Φ) was described by Johannes Kepler as one of the “two great treasures of geometry.” (The other is the Theorem of Pythagoras.) Phi appears in many basic geometric constructions. 3 lines...

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Fibonacci numbers and Phi are related to spiral growth in nature. If you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times the...

https://www.goldennumber.net/spirals/

Phi-based geometric shapes produce interesting mandala designs. A mandala is a geometric design, often symbolic of the universe and used in Eastern religions as an aid to meditation. Phi-based ge...

https://www.goldennumber.net/phi-mandalas/

Tiling in 5-fold symmetry was thought impossible! Areas can be filled completely and symmetrically with tiles of 3, 4 and 6 sides, but it was long believed that it was impossible to fill an area ...

https://www.goldennumber.net/penrose-tiling/

Squaring the Circle comes within four decimal places using the Golden Ratio. Even before the foundations of the Great Pyramids were laid men have tried to “square the circle.” That is, in a f...

https://www.goldennumber.net/squaring-the-circle/

A new fundamental geometric shape with a relationship to Phi. Here’s a challenge to “all the real mathematicians in the back row,” as my college professor often said: Picture the classic so...

https://www.goldennumber.net/the-dor/

Phi is most often calculated using by taking the square root of 5 plus 1 and divided the sum by 2: √5 + 1 2 This mathematical expression can be expressed geometrically as shown below: Three cir...

https://www.goldennumber.net/phi-formula-geometry/

Quasi-crystals represent a newly discovered state of matter. Most crystals in nature, such as those in sugar, salt or diamonds, are symmetrical and all have the same orientation throughout the en...

https://www.goldennumber.net/quasi-crystals/

The Golden Section is an Orthogon called the Auron. The golden section can be constructed from a square with a compass and ruler: This is the most commonly known of twelve orthogons which can be ...

https://www.goldennumber.net/orthogons/

Creating a Triangle based on Phi (or Pythagoras meets Fibonacci): Pythagoras discovered that a right triangle with sides of length a and b and a hypotenuse of length c has the following relation...

https://www.goldennumber.net/triangles/