Fibonacci Numbers


0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811 ...

In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation. The Fibonacci numbers first appeared, under the name matrameru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandah-shastra, the Art of Prosody, 450 or 200 BC). The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics. Fibonacci Numbers




Logarithmic Spiral

The logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral". Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. Jacob Bernoulli wanted such a spiral engraved on his headstone along with the phrase "Eadem mutata resurgo" ("Although changed, I shall arise the same."), but, by error, an Archimedean spiral was placed there instead.




Long-Standing Problem of 'Golden Ratio' and
Other Irrational Numbers Solved with 'Magical Simplicity'
 
Live Science - September 17, 2019




Golden Ratio, Golden Mean, Golden Section, Divine Proportion





The golden ratio is also called the golden section or golden mean. Other names include extreme and mean ratio, medial section, divine proportion, divine section golden proportion, golden cut, and golden number. Many artists and architects have proportioned their works to approximate the golden ratio - especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio - believing this proportion to be aesthetically pleasing.

Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which can be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data. Read more ... 




The Golden Ratio in Space





The Golden Ratio in Architecture




Ancient Pueblo Used Golden Ratio to Build the Sun Temple  
Live Science - February 13, 2017

The Great Pyramids in Giza, the Parthenon in Athens and Chichen Itza in Mexico have something in common. Besides attracting hordes of tourists, all of these architectural wonders appear to use the golden ratio. This mathematical number is often written as 1.618, the first few digits of its infinite decimal form. Expressed another way, two quantities - let's call the larger one "a" and the smaller "b" - are in the golden ratio if "a is to b" as "a + b is to a." The result is a composition with aesthetically pleasing proportions. Now, shapes with the golden ratio, as well as other geometric shapes, have been found in another, unexpected site: the Sun Temple at Mesa Verde National Park in Colorado, built by the ancient Pueblo people who lived in what is now the modern-day Southwest; they had no known written language or written number system.




The Golden Ratio in Art


The Vitruvian Man is a drawing created by Leonardo da Vinci circa 1490. It is accompanied by notes based on the work of the architect Vitruvius. The drawing, which is in pen and ink on paper, depicts a male figure in two superimposed positions with his arms and legs apart and simultaneously inscribed in a circle and square. The drawing and text are sometimes called the Canon of Proportions or, less often, Proportions of Man.




The Golden Ratio in Nature






Fibonacci Numbers or Sequence in Creation - 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811 ...




Vesica Piscis








Seed of Life





Flower of Life





Tree of Life - Qabbalah - Kabbalah





Sacred Geometry






Ouroboros

When things come fun circle





Holographic Universe




SACRED GEOMETRY


MATHEMATICS INDEX


PHYSICAL SCIENCES INDEX


A-Z


CRYSTALINKS HOME PAGE


PSYCHIC READING WITH ELLIE


2012 THE ALCHEMY OF TIME