his section includes detailed information and references expanding on items linked from the Art Practice: Portfolio and Art Practice: Artist's Statement sections.

Nag Hammadi Cartonnage

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Codex cover

The Nag Hammadi Codices, which are dated to around the 2nd half of the 4th century, were discovered in Egypt, buried in a jar in the sand and rock on the talus of the cliffs of Jabal al-Tarif in 1945. In itself, it is a fascinating set of manuscripts, con­taining 12 books of early biblical documents, many otherwise lost to history until their discovery, and many likely to have been heretical and banned texts of the early Christian era.

Within the leather binding of the books were pieces of papyrus, made from scraps and sheets of manuscripts, accounting records, and other ephemera pasted together to make a thick cardboard called "cartonnage". It was this essentially discarded material that when carefully examined revealed important evidence regarding the books provenance. Of great interest is that within the cartonnage was a good part of another codex which contains five important Gnostic texts.

Lacuna, being an unusual word, seems worth defining in this context. According to the American Heratige Dictionary it is defined as:

    la·cu·na Audio pronunciation of "lacuna" (l-ky‘n)
    n. pl. la·cu·nae (-n) or la·cu·nas
  • An empty space or a missing part; a gap: “self-centered in opinion, with curious lacunae of astounding ignorance” (Frank Norris).
  • Anatomy. A cavity, space, or depression, especially in a bone, containing cartilage or bone cells.

The Golden Game

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The Golden Game is a book by Stanislas Klossowski de Rola which collects and describes al­chem­ical engravings of the Seventeenth Century. From the overleaf:

Alchemy, its mysteries, and the quest for the Philosopher‘s Stone have been a strand in the development of European culture from antiquity onwards. By the seventeenth century the complex pictorial language of alchemical symbols had reached a zenith of elaboration, sophistication and beauty, with the spread of printing enabling this fantastic iconography to flourish as never before.

There would be a flash movie here if you had the player installed. Click here for it.

It has a vast collection of these delightful and intriguing engravings, which are truly strange and mysterious. Here is the cover and a few of the engravings I used in my Transparency Study: The Golden Game piece.

Amazon carries the book here

The Magic
Number, Phi

Phi or Φ, pronounced "fee", is approximately 1.618 and is the number associated with the classical Golden Rectangle. It is an irrational number, and like Pi, has an infinite number of non-repeating numerals after the decimal point. Frequently found in natural formations such as sea shell spirals and sunflower seed patterns.

Discovered by the Pythagoreans, followers of the ancient Greek mathematician Pythagoras, in the 5th century BC. Their discovery was so disturbing to their sense of an ordered universe that it was rumored that the first Pythagorean to discover their existence met with an unfortunate boating "accident."

Phi to 10,000 places

1 . 6 1 8 0 3 3 9 8 8 7 4 9 8 9 4 8 4 8 2 0 4 5 8 6 8 3 4 3 6 5 6 3 8 1 1 7 7 2 0 3 0 9 1 7 9 8 0 5 7 6 2 8 6 2 1 3 5 4 4 8 6 2 2 7 0 5 2 6 0 4 6 2 8 1 8 9 0 2 4 4 9 7 0 7 2 0 7 2 0 4 1 8 9 3 9 1 1 3 7 4 8 4 7 5 4 0 8 8 0 7 5 3 8 6 8 9 1 7 5 2 1 2 6 6 3 3 8 6 2 2 2 3 5 3 6 9 3 1 7 9 3 1 8 0 0 6 0 7 6 6 7 2 6 3 5 4 4 3 3 3 8 9 0 8 6 5 9 5 9 3 9 5 8 2 9 0 5 6 3 8 3 2 2 6 6 1 3 1 9 9 2 8 2 9 0 2 6 7 8 8 0 6 7 5 2 0 8 7 6 6 8 9 2 5 0 1 7 1 1 6 9 6 2 0 7 0 3 2 2 2 1 0 4 3 2 1 6 2 6 9 5 4 8 6 2 6 2 9 6 3 1 3 6 1 4 4 3 8 1 4 9 7 5 8 7 0 1 2 2 0 3 4 0 8 0 5 8 8 7 9 5 4 4 5 4 7 4 9 2 4 6 1 8 5 6 9 5 3 6 4 8 6 4 4 4 9 2 4 1 0 4 4 3 2 0 7 7 1 3 4 4 9 4 7 0 4 9 5 6 5 8 4 6 7 8 8 5 0 9 8 7 4 3 3 9 4 4 2 2 1 2 5 4 4 8 7 7 0 6 6 4 7 8 0 9 1 5 8 8 4 6 0 7 4 9 9 8 8 7 1 2 4 0 0 7 6 5 2 1 7 0 5 7 5 1 7 9 7 8 8 3 4 1 6 6 2 5 6 2 4 9 4 0 7 5 8 9 0 6 9 7 0 4 0 0 0 2 8 1 2 1 0 4 2 7 6 2 1 7 7 1 1 1 7 7 7 8 0 5 3 1 5 3 1 7 1 4 1 0 1 1 7 0 4 6 6 6 5 9 9 1 4 6 6 9 7 9 8 7 3 1 7 6 1 3 5 6 0 0 6 7 0 8 7 4 8 0 7 1 0 1 3 1 7 9 5 2 3 6 8 9 4 2 7 5 2 1 9 4 8 4 3 5 3 0 5 6 7 8 3 0 0 2 2 8 7 8 5 6 9 9 7 8 2 9 7 7 8 3 4 7 8 4 5 8 7 8 2 2 8 9 1 1 0 9 7 6 2 5 0 0 3 0 2 6 9 6 1 5 6 1 7 0 0 2 5 0 4 6 4 3 3 8 2 4 3 7 7 6 4 8 6 1 0 2 8 3 8 3 1 2 6 8 3 3 0 3 7 2 4 2 9 2 6 7 5 2 6 3 1 1 6 5 3 3 9 2 4 7 3 1 6 7 1 1 1 2 1 1 5 8 8 1 8 6 3 8 5 1 3 3 1 6 2 0 3 8 4 0 0 5 2 2 2 1 6 5 7 9 1 2 8 6 6 7 5 2 9 4 6 5 4 9 0 6 8 1 1 3 1 7 1 5 9 9 3 4 3 2 3 5 9 7 3 4 9 4 9 8 5 0 9 0 4 0 9 4 7 6 2 1 3 2 2 2 9 8 1 0 1 7 2 6 1 0 7 0 5 9 6 1 1 6 4 5 6 2 9 9 0 9 8 1 6 2 9 0 5 5 5 2 0 8 5 2 4 7 9 0 3 5 2 4 0 6 0 2 0 1 7 2 7 9 9 7 4 7 1 7 5 3 4 2 7 7 7 5 9 2 7 7 8 6 2 5 6 1 9 4 3 2 0 8 2 7 5 0 5 1 3 1 2 1 8 1 5 6 2 8 5 5 1 2 2 2 4 8 0 9 3 9 4 7 1 2 3 4 1 4 5 1 7 0 2 2 3 7 3 5 8 0 5 7 7 2 7 8 6 1 6 0 0 8 6 8 8 3 8 2 9 5 2 3 0 4 5 9 2 6 4 7 8 7 8 0 1 7 8 8 9 9 2 1 9 9 0 2 7 0 7 7 6 9 0 3 8 9 5 3 2 1 9 6 8 1 9 8 6 1 5 1 4 3 7 8 0 3 1 4 9 9 7 4 1 1 0 6 9 2 6 0 8 8 6 7 4 2 9 6 2 2 6 7 5 7 5 6 0 5 2 3 1 7 2 7 7 7 5 2 0 3 5 3 6 1 3 9 3 6 2 1 0 7 6 7 3 8 9 3 7 6 4 5 5 6 0 6 0 6 0 5 9 2 1 6 5 8 9 4 6 6 7 5 9 5 5 1 9 0 0 4 0 0 5 5 5 9 0 8 9 5 0 2 2 9 5 3 0 9 4 2 3 1 2 4 8 2 3 5 5 2 1 2 2 1 2 4 1 5 4 4 4 0 0 6 4 7 0 3 4 0 5 6 5 7 3 4 7 9 7 6 6 3 9 7 2 3 9 4 9 4 9 9 4 6 5 8 4 5 7 8 8 7 3 0 3 9 6 2 3 0 9 0 3 7 5 0 3 3 9 9 3 8 5 6 2 1 0 2 4 2 3 6 9 0 2 5 1 3 8 6 8 0 4 1 4 5 7 7 9 9 5 6 9 8 1 2 2 4 4 5 7 4 7 1 7 8 0 3 4 1 7 3 1 2 6 4 5 3 2 2 0 4 1 6 3 9 7 2 3 2 1 3 4 0 4 4 4 4 9 4 8 7 3 0 2 3 1 5 4 1 7 6 7 6 8 9 3 7 5 2 1 0 3 0 6 8 7 3 7 8 8 0 3 4 4 1 7 0 0 9 3 9 5 4 4 0 9 6 2 7 9 5 5 8 9 8 6 7 8 7 2 3 2 0 9 5 1 2 4 2 6 8 9 3 5 5 7 3 0 9 7 0 4 5 0 9 5 9 5 6 8 4 4 0 1 7 5 5 5 1 9 8 8 1 9 2 1 8 0 2 0 6 4 0 5 2 9 0 5 5 1 8 9 3 4 9 4 7 5 9 2 6 0 0 7 3 4 8 5 2 2 8 2 1 0 1 0 8 8 1 9 4 6 4 4 5 4 4 2 2 2 3 1 8 8 9 1 3 1 9 2 9 4 6 8 9 6 2 2 0 0 2 3 0 1 4 4 3 7 7 0 2 6 9 9 2 3 0 0 7 8 0 3 0 8 5 2 6 1 1 8 0 7 5 4 5 1 9 2 8 8 7 7 0 5 0 2 1 0 9 6 8 4 2 4 9 3 6 2 7 1 3 5 9 2 5 1 8 7 6 0 7 7 7 8 8 4 6 6 5 8 3 6 1 5 0 2 3 8 9 1 3 4 9 3 3 3 3 1 2 2 3 1 0 5 3 3 9 2 3 2 1 3 6 2 4 3 1 9 2 6 3 7 2 8 9 1 0 6 7 0 5 0 3 3 9 9 2 8 2 2 6 5 2 6 3 5 5 6 2 0 9 0 2 9 7 9 8 6 4 2 4 7 2 7 5 9 7 7 2 5 6 5 5 0 8 6 1 5 4 8 7 5 4 3 5 7 4 8 2 6 4 7 1 8 1 4 1 4 5 1 2 7 0 0 0 6 0 2 3 8 9 0 1 6 2 0 7 7 7 3 2 2 4 4 9 9 4 3 5 3 0 8 8 9 9 9 0 9 5 0 1 6 8 0 3 2 8 1 1 2 1 9 4 3 2 0 4 8 1 9 6 4 3 8 7 6 7 5 8 6 3 3 1 4 7 9 8 5 7 1 9 1 1 3 9 7 8 1 5 3 9 7 8 0 7 4 7 6 1 5 0 7 7 2 2 1 1 7 5 0 8 2 6 9 4 5 8 6 3 9 3 2 0 4 5 6 5 2 0 9 8 9 6 9 8 5 5 5 6 7 8 1 4 1 0 6 9 6 8 3 7 2 8 8 4 0 5 8 7 4 6 1 0 3 3 7 8 1 0 5 4 4 4 3 9 0 9 4 3 6 8 3 5 8 3 5 8 1 3 8 1 1 3 1 1 6 8 9 9 3 8 5 5 5 7 6 9 7 5 4 8 4 1 4 9 1 4 4 5 3 4 1 5 0 9 1 2 9 5 4 0 7 0 0 5 0 1 9 4 7 7 5 4 8 6 1 6 3 0 7 5 4 2 2 6 4 1 7 2 9 3 9 4 6 8 0 3 6 7 3 1 9 8 0 5 8 6 1 8 3 3 9 1 8 3 2 8 5 9 9 1 3 0 3 9 6 0 7 2 0 1 4 4 5 5 9 5 0 4 4 9 7 7 9 2 1 2 0 7 6 1 2 4 7 8 5 6 4 5 9 1 6 1 6 0 8 3 7 0 5 9 4 9 8 7 8 6 0 0 6 9 7 0 1 8 9 4 0 9 8 8 6 4 0 0 7 6 4 4 3 6 1 7 0 9 3 3 4 1 7 2 7 0 9 1 9 1 4 3 3 6 5 0 1 3 7 1 5 7 6 6 0 1 1 4 8 0 3 8 1 4 3 0 6 2 6 2 3 8 0 5 1 4 3 2 1 1 7 3 4 8 1 5 1 0 0 5 5 9 0 1 3 4 5 6 1 0 1 1 8 0 0 7 9 0 5 0 6 3 8 1 4 2 1 5 2 7 0 9 3 0 8 5 8 8 0 9 2 8 7 5 7 0 3 4 5 0 5 0 7 8 0 8 1 4 5 4 5 8 8 1 9 9 0 6 3 3 6 1 2 9 8 2 7 9 8 1 4 1 1 7 4 5 3 3 9 2 7 3 1 2 0 8 0 9 2 8 9 7 2 7 9 2 2 2 1 3 2 9 8 0 6 4 2 9 4 6 8 7 8 2 4 2 7 4 8 7 4 0 1 7 4 5 0 5 5 4 0 6 7 7 8 7 5 7 0 8 3 2 3 7 3 1 0 9 7 5 9 1 5 1 1 7 7 6 2 9 7 8 4 4 3 2 8 4 7 4 7 9 0 8 1 7 6 5 1 8 0 9 7 7 8 7 2 6 8 4 1 6 1 1 7 6 3 2 5 0 3 8 6 1 2 1 1 2 9 1 4 3 6 8 3 4 3 7 6 7 0 2 3 5 0 3 7 1 1 1 6 3 3 0 7 2 5 8 6 9 8 8 3 2 5 8 7 1 0 3 3 6 3 2 2 2 3 8 1 0 9 8 0 9 0 1 2 1 1 0 1 9 8 9 9 1 7 6 8 4 1 4 9 1 7 5 1 2 3 3 1 3 4 0 1 5 2 7 3 3 8 4 3 8 3 7 2 3 4 5 0 0 9 3 4 7 8 6 0 4 9 7 9 2 9 4 5 9 9 1 5 8 2 2 0 1 2 5 8 1 0 4 5 9 8 2 3 0 9 2 5 5 2 8 7 2 1 2 4 1 3 7 0 4 3 6 1 4 9 1 0 2 0 5 4 7 1 8 5 5 4 9 6 1 1 8 0 8 7 6 4 2 6 5 7 6 5 1 1 0 6 0 5 4 5 8 8 1 4 7 5 6 0 4 4 3 1 7 8 4 7 9 8 5 8 4 5 3 9 7 3 1 2 8 6 3 0 1 6 2 5 4 4 8 7 6 1 1 4 8 5 2 0 2 1 7 0 6 4 4 0 4 1 1 1 6 6 0 7 6 6 9 5 0 5 9 7 7 5 7 8 3 2 5 7 0 3 9 5 1 1 0 8 7 8 2 3 0 8 2 7 1 0 6 4 7 8 9 3 9 0 2 1 1 1 5 6 9 1 0 3 9 2 7 6 8 3 8 4 5 3 8 6 3 3 3 3 2 1 5 6 5 8 2 9 6 5 9 7 7 3 1 0 3 4 3 6 0 3 2 3 2 2 5 4 5 7 4 3 6 3 7 2 0 4 1 2 4 4 0 6 4 0 8 8 8 2 6 7 3 7 5 8 4 3 3 9 5 3 6 7 9 5 9 3 1 2 3 2 2 1 3 4 3 7 3 2 0 9 9 5 7 4 9 8 8 9 4 6 9 9 5 6 5 6 4 7 3 6 0 0 7 2 9 5 9 9 9 8 3 9 1 2 8 8 1 0 3 1 9 7 4 2 6 3 1 2 5 1 7 9 7 1 4 1 4 3 2 0 1 2 3 1 1 2 7 9 5 5 1 8 9 4 7 7 8 1 7 2 6 9 1 4 1 5 8 9 1 1 7 7 9 9 1 9 5 6 4 8 1 2 5 5 8 0 0 1 8 4 5 5 0 6 5 6 3 2 9 5 2 8 5 9 8 5 9 1 0 0 0 9 0 8 6 2 1 8 0 2 9 7 7 5 6 3 7 8 9 2 5 9 9 9 1 6 4 9 9 4 6 4 2 8 1 9 3 0 2 2 2 9 3 5 5 2 3 4 6 6 7 4 7 5 9 3 2 6 9 5 1 6 5 4 2 1 4 0 2 1 0 9 1 3 6 3 0 1 8 1 9 4 7 2 2 7 0 7 8 9 0 1 2 2 0 8 7 2 8 7 3 6 1 7 0 7 3 4 8 6 4 9 9 9 8 1 5 6 2 5 5 4 7 2 8 1 1 3 7 3 4 7 9 8 7 1 6 5 6 9 5 2 7 4 8 9 0 0 8 1 4 4 3 8 4 0 5 3 2 7 4 8 3 7 8 1 3 7 8 2 4 6 6 9 1 7 4 4 4 2 2 9 6 3 4 9 1 4 7 0 8 1 5 7 0 0 7 3 5 2 5 4 5 7 0 7 0 8 9 7 7 2 6 7 5 4 6 9 3 4 3 8 2 2 6 1 9 5 4 6 8 6 1 5 3 3 1 2 0 9 5 3 3 5 7 9 2 3 8 0 1 4 6 0 9 2 7 3 5 1 0 2 1 0 1 1 9 1 9 0 2 1 8 3 6 0 6 7 5 0 9 7 3 0 8 9 5 7 5 2 8 9 5 7 7 4 6 8 1 4 2 2 9 5 4 3 3 9 4 3 8 5 4 9 3 1 5 5 3 3 9 6 3 0 3 8 0 7 2 9 1 6 9 1 7 5 8 4 6 1 0 1 4 6 0 9 9 5 0 5 5 0 6 4 8 0 3 6 7 9 3 0 4 1 4 7 2 3 6 5 7 2 0 3 9 8 6 0 0 7 3 5 5 0 7 6 0 9 0 2 3 1 7 3 1 2 5 0 1 6 1 3 2 0 4 8 4 3 5 8 3 6 4 8 1 7 7 0 4 8 4 8 1 8 1 0 9 9 1 6 0 2 4 4 2 5 2 3 2 7 1 6 7 2 1 9 0 1 8 9 3 3 4 5 9 6 3 7 8 6 0 8 7 8 7 5 2 8 7 0 1 7 3 9 3 5 9 3 0 3 0 1 3 3 5 9 0 1 1 2 3 7 1 0 2 3 9 1 7 1 2 6 5 9 0 4 7 0 2 6 3 4 9 4 0 2 8 3 0 7 6 6 8 7 6 7 4 3 6 3 8 6 5 1 3 2 7 1 0 6 2 8 0 3 2 3 1 7 4 0 6 9 3 1 7 3 3 4 4 8 2 3 4 3 5 6 4 5 3 1 8 5 0 5 8 1 3 5 3 1 0 8 5 4 9 7 3 3 3 5 0 7 5 9 9 6 6 7 7 8 7 1 2 4 4 9 0 5 8 3 6 3 6 7 5 4 1 3 2 8 9 0 8 6 2 4 0 6 3 2 4 5 6 3 9 5 3 5 7 2 1 2 5 2 4 2 6 1 1 7 0 2 7 8 0 2 8 6 5 6 0 4 3 2 3 4 9 4 2 8 3 7 3 0 1 7 2 5 5 7 4 4 0 5 8 3 7 2 7 8 2 6 7 9 9 6 0 3 1 7 3 9 3 6 4 0 1 3 2 8 7 6 2 7 7 0 1 2 4 3 6 7 9 8 3 1 1 4 4 6 4 3 6 9 4 7 6 7 0 5 3 1 2 7 2 4 9 2 4 1 0 4 7 1 6 7 0 0 1 3 8 2 4 7 8 3 1 2 8 6 5 6 5 0 6 4 9 3 4 3 4 1 8 0 3 9 0 0 4 1 0 1 7 8 0 5 3 3 9 5 0 5 8 7 7 2 4 5 8 6 6 5 5 7 5 5 2 2 9 3 9 1 5 8 2 3 9 7 0 8 4 1 7 7 2 9 8 3 3 7 2 8 2 3 1 1 5 2 5 6 9 2 6 0 9 2 9 9 5 9 4 2 2 4 0 0 0 0 5 6 0 6 2 6 6 7 8 6 7 4 3 5 7 9 2 3 9 7 2 4 5 4 0 8 4 8 1 7 6 5 1 9 7 3 4 3 6 2 6 5 2 6 8 9 4 4 8 8 8 5 5 2 7 2 0 2 7 4 7 7 8 7 4 7 3 3 5 9 8 3 5 3 6 7 2 7 7 6 1 4 0 7 5 9 1 7 1 2 0 5 1 3 2 6 9 3 4 4 8 3 7 5 2 9 9 1 6 4 9 9 8 0 9 3 6 0 2 4 6 1 7 8 4 4 2 6 7 5 7 2 7 7 6 7 9 0 0 1 9 1 9 1 9 0 7 0 3 8 0 5 2 2 0 4 6 1 2 3 2 4 8 2 3 9 1 3 2 6 1 0 4 3 2 7 1 9 1 6 8 4 5 1 2 3 0 6 0 2 3 6 2 7 8 9 3 5 4 5 4 3 2 4 6 1 7 6 9 9 7 5 7 5 3 6 8 9 0 4 1 7 6 3 6 5 0 2 5 4 7 8 5 1 3 8 2 4 6 3 1 4 6 5 8 3 3 6 3 8 3 3 7 6 0 2 3 5 7 7 8 9 9 2 6 7 2 9 8 8 6 3 2 1 6 1 8 5 8 3 9 5 9 0 3 6 3 9 9 8 1 8 3 8 4 5 8 2 7 6 4 4 9 1 2 4 5 9 8 0 9 3 7 0 4 3 0 5 5 5 5 9 6 1 3 7 9 7 3 4 3 2 6 1 3 4 8 3 0 4 9 4 9 4 9 6 8 6 8 1 0 8 9 5 3 5 6 9 6 3 4 8 2 8 1 7 8 1 2 8 8 6 2 5 3 6 4 6 0 8 4 2 0 3 3 9 4 6 5 3 8 1 9 4 4 1 9 4 5 7 1 4 2 6 6 6 8 2 3 7 1 8 3 9 4 9 1 8 3 2 3 7 0 9 0 8 5 7 4 8 5 0 2 6 6 5 6 8 0 3 9 8 9 7 4 4 0 6 6 2 1 0 5 3 6 0 3 0 6 4 0 0 2 6 0 8 1 7 1 1 2 6 6 5 9 9 5 4 1 9 9 3 6 8 7 3 1 6 0 9 4 5 7 2 2 8 8 8 1 0 9 2 0 7 7 8 8 2 2 7 7 2 0 3 6 3 6 6 8 4 4 8 1 5 3 2 5 6 1 7 2 8 4 1 1 7 6 9 0 9 7 9 2 6 6 6 6 5 5 2 2 3 8 4 6 8 8 3 1 1 3 7 1 8 5 2 9 9 1 9 2 1 6 3 1 9 0 5 2 0 1 5 6 8 6 3 1 2 2 2 8 2 0 7 1 5 5 9 9 8 7 6 4 6 8 4 2 3 5 5 2 0 5 9 2 8 5 3 7 1 7 5 7 8 0 7 6 5 6 0 5 0 3 6 7 7 3 1 3 0 9 7 5 1 9 1 2 2 3 9 7 3 8 8 7 2 2 4 6 8 2 5 8 0 5 7 1 5 9 7 4 4 5 7 4 0 4 8 4 2 9 8 7 8 0 7 3 5 2 2 1 5 9 8 4 2 6 6 7 6 6 2 5 7 8 0 7 7 0 6 2 0 1 9 4 3 0 4 0 0 5 4 2 5 5 0 1 5 8 3 1 2 5 0 3 0 1 7 5 3 4 0 9 4 1 1 7 1 9 1 0 1 9 2 9 8 9 0 3 8 4 4 7 2 5 0 3 3 2 9 8 8 0 2 4 5 0 1 4 3 6 7 9 6 8 4 4 1 6 9 4 7 9 5 9 5 4 5 3 0 4 5 9 1 0 3 1 3 8 1 1 6 2 1 8 7 0 4 5 6 7 9 9 7 8 6 6 3 6 6 1 7 4 6 0 5 9 5 7 0 0 0 3 4 4 5 9 7 0 1 1 3 5 2 5 1 8 1 3 4 6 0 0 6 5 6 5 5 3 5 2 0 3 4 7 8 8 8 1 1 7 4 1 4 9 9 4 1 2 7 4 8 2 6 4 1 5 2 1 3 5 5 6 7 7 6 3 9 4 0 3 9 0 7 1 0 3 8 7 0 8 8 1 8 2 3 3 8 0 6 8 0 3 3 5 0 0 3 8 0 4 6 8 0 0 1 7 4 8 0 8 2 2 0 5 9 1 0 9 6 8 4 4 2 0 2 6 4 4 6 4 0 2 1 8 7 7 0 5 3 4 0 1 0 0 3 1 8 0 2 8 8 1 6 6 4 4 1 5 3 0 9 1 3 9 3 9 4 8 1 5 6 4 0 3 1 9 2 8 2 2 7 8 5 4 8 2 4 1 4 5 1 0 5 0 3 1 8 8 8 2 5 1 8 9 9 7 0 0 7 4 8 6 2 2 8 7 9 4 2 1 5 5 8 9 5 7 4 2 8 2 0 2 1 6 6 5 7 0 6 2 1 8 8 0 9 0 5 7 8 0 8 8 0 5 0 3 2 4 6 7 6 9 9 1 2 9 7 2 8 7 2 1 0 3 8 7 0 7 3 6 9 7 4 0 6 4 3 5 6 6 7 4 5 8 9 2 0 2 5 8 6 5 6 5 7 3 9 7 8 5 6 0 8 5 9 5 6 6 5 3 4 1 0 7 0 3 5 9 9 7 8 3 2 0 4 4 6 3 3 6 3 4 6 4 8 5 4 8 9 4 9 7 6 6 3 8 8 5 3 5 1 0 4 5 5 2 7 2 9 8 2 4 2 2 9 0 6 9 9 8 4 8 8 5 3 6 9 6 8 2 8 0 4 6 4 5 9 7 4 5 7 6 2 6 5 1 4 3 4 3 5 9 0 5 0 9 3 8 3 2 1 2 4 3 7 4 3 3 3 3 8 7 0 5 1 6 6 5 7 1 4 9 0 0 5 9 0 7 1 0 5 6 7 0 2 4 8 8 7 9 8 5 8 0 4 3 7 1 8 1 5 1 2 6 1 0 0 4 4 0 3 8 1 4 8 8 0 4 0 7 2 5 2 4 4 0 6 1 6 4 2 9 0 2 2 4 7 8 2 2 7 1 5 2 7 2 4 1 1 2 0 8 5 0 6 5 7 8 8 8 3 8 7 1 2 4 9 3 6 3 5 1 0 6 8 0 6 3 6 5 1 6 6 7 4 3 2 2 2 3 2 7 7 6 7 7 5 5 7 9 7 3 9 9 2 7 0 3 7 6 2 3 1 9 1 4 7 0 4 7 3 2 3 9 5 5 1 2 0 6 0 7 0 5 5 0 3 9 9 2 0 8 8 4 4 2 6 0 3 7 0 8 7 9 0 8 4 3 3 3 4 2 6 1 8 3 8 4 1 3 5 9 7 0 7 8 1 6 4 8 2 9 5 5 3 7 1 4 3 2 1 9 6 1 1 8 9 5 0 3 7 9 7 7 1 4 6 3 0 0 0 7 5 5 5 9 7 5 3 7 9 5 7 0 3 5 5 2 2 7 1 4 4 9 3 1 9 1 3 2 1 7 2 5 5 6 4 4 0 1 2 8 3 0 9 1 8 0 5 0 4 5 0 0 8 9 9 2 1 8 7 0 5 1 2 1 1 8 6 0 6 9 3 3 5 7 3 1 5 3 8 9 5 9 3 5 0 7 9 0 3 0 0 7 3 6 7 2 7 0 2 3 3 1 4 1 6 5 3 2 0 4 2 3 4 0 1 5 5 3 7 4 1 4 4 2 6 8 7 1 5 4 0 5 5 1 1 6 4 7 9 6 1 1 4 3 3 2 3 0 2 4 8 5 4 4 0 4 0 9 4 0 6 9 1 1 4 5 6 1 3 9 8 7 3 0 2 6 0 3 9 5 1 8 2 8 1 6 8 0 3 4 4 8 2 5 2 5 4 3 2 6 7 3 8 5 7 5 9 0 0 5 6 0 4 3 2 0 2 4 5 3 7 2 7 1 9 2 9 1 2 4 8 6 4 5 8 1 3 3 3 4 4 1 6 9 8 5 2 9 9 3 9 1 3 5 7 4 7 8 6 9 8 9 5 7 9 8 6 4 3 9 4 9 8 0 2 3 0 4 7 1 1 6 9 6 7 1 5 7 3 6 2 2 8 3 9 1 2 0 1 8 1 2 7 3 1 2 9 1 6 5 8 9 9 5 2 7 5 9 9 1 9 2 2 0 3 1 8 3 7 2 3 5 6 8 2 7 2 7 9 3 8 5 6 3 7 3 3 1 2 6 5 4 7 9 9 8 5 9 1 2 4 6 3 2 7 5 0 3 0 0 6 0 5 9 2 5 6 7 4 5 4 9 7 9 4 3 5 0 8 8 1 1 9 2 9 5 0 5 6 8 5 4 9 3 2 5 9 3 5 5 3 1 8 7 2 9 1 4 1 8 0 1 1 3 6 4 1 2 1 8 7 4 7 0 7 5 2 6 2 8 1 0 6 8 6 9 8 3 0 1 3 5 7 6 0 5 2 4 7 1 9 4 4 5 5 9 3 2 1 9 5 5 3 5 9 6 1 0 4 5 2 8 3 0 3 1 4 8 8 3 9 1 1 7 6 9 3 0 1 1 9 6 5 8 5 8 3 4 3 1 4 4 2 4 8 9 4 8 9 8 5 6 5 5 8 4 2 5 0 8 3 4 1 0 9 4 2 9 5 0 2 7 7 1 9 7 5 8 3 3 5 2 2 4 4 2 9 1 2 5 7 3 6 4 9 3 8 0 7 5 4 1 7 1 1 3 7 3 9 2 4 3 7 6 0 1 4 3 5 0 6 8 2 9 8 7 8 4 9 3 2 7 1 2 9 9 7 5 1 2 2 8 6 8 8 1 9 6 0 4 9 8 3 5 7 7 5 1 5 8 7 7 1 7 8 0 4 1 0 6 9 7 1 3 1 9 6 6 7 5 3 4 7 7 1 9 4 7 9 2 2 6 3 6 5 1 9 0 1 6 3 3 9 7 7 1 2 8 4 7 3 9 0 7 9 3 3 6 1 1 1 1 9 1 4 0 8 9 9 8 3 0 5 6 0 3 3 6 1 0 6 0 9 8 7 1 7 1 7 8 3 0 5 5 4 3 5 4 0 3 5 6 0 8 9 5 2 9 2 9 0 8 1 8 4 6 4 1 4 3 7 1 3 9 2 9 4 3 7 8 1 3 5 6 0 4 8 2 0 3 8 9 4 7 9 1 2 5 7 4 5 0 7 7 0 7 5 5 7 5 1 0 3 0 0 2 4 2 0 7 2 6 6 2 9 0 0 1 8 0 9 0 4 2 2 9 3 4 2 4 9 4 2 5 9 0 6 0 6 6 6 1 4 1 3 3 2 2 8 7 2 2 6 9 8 0 6 9 0 1 4 5 9 9 4 5 1 1 9 9 5 4 7 8 0 1 6 3 9 9 1 5 1 4 1 2 6 1 2 5 2 5 7 2 8 2 8 0 6 6 4 3 3 1 2 6 1 6 5 7 4 6 9 3 8 8 1 9 5 1 0 6 4 4 2 1 6 7 3 8 7 1 8 0 0 0 1 1 0 0 4 2 1 8 4 8 3 0 2 5 8 0 9 1 6 5 4 3 3 8 3 7 4 9 2 3 6 4 1 1 8 3 8 8 8 5 6 4 6 8 5 1 4 3 1 5 0 0 6 3 7 3 1 9 0 4 2 9 5 1 4 8 1 4 6 9 4 2 4 3 1 4 6 0 8 9 5 2 5 4 7 0 7 2 0 3 7 4 0 5 5 6 6 9 1 3 0 6 9 2 2 0 9 9 0 8 0 4 8 1 9 4 5 2 9 7 5 1 1 0 6 5 0 4 6 4 2 8 1 0 5 4 1 7 7 5 5 2 5 9 0 9 5 1 8 7 1 3 1 8 8 8 3 5 9 1 4 7 6 5 9 9 6 0 4 1 3 1 7 9 6 0 2 0 9 4 1 5 3 0 8 5 8 5 5 3 3 2 3 8 7 7 2 5 3 8 0 2 3 2 7 2 7 6 3 2 9 7 7 3 7 2 1 4 3 1 2 7 9 6 8 2 1 6 7 1 6 2 3 4 4 2 1 1 8 3 2 0 1 8 0 2 8 8 1 4 1 2 7 4 7 4 4 3 1 6 8 8 4 7 2 1 8 4 5 9 3 9 2 7 8 1 4 3 5 4 7 4 0 9 9 9 9 9 0 7 2 2 3 3 2 0 3 0 5 9 2 6 2 9 7 6 6 1 1 2 3 8 3 2 7 9 8 3 3 1 6 9 8 8 2 5 3 9 3 1 2 6 2 0 0 6 5 0 3 7 0 2 8 8 4 4 7 8 2 8 6 6 6 9 4 0 4 4 7 3 0 7 9 4 7 1 0 4 7 6 1 2 5 5 8 6 5 8 3 7 5 2 9 8 6 2 3 6 2 5 0 9 9 9 8 2 3 2 3 3 5 9 7 1 5 5 0 7 2 3 3 8 3 8 3 3 2 4 4 0 8 1 5 2 5 7 7 8 1 9 3 3 6 4 2 6 2 6 3 0 4 3 3 0 2 6 5 8 9 5 8 1 7 0 8 0 0 4 5 1 2 7 8 8 7 3 1 1 5 9 3 5 5 8 7 7 4 7 2 1 7 2 5 6 4 9 4 7 0 0 0 5 1 6 3 6 6 7 2 5 7 7 1 5 3 9 2 0 9 8 4 0 9 5 0 3 2 7 4 5 1 1 2 1 5 3 6 8 7 3 0 0 9 1 2 1 9 9 6 2 9 5 2 2 7 6 5 9 1 3 1 6 3 7 0 9 3 9 6 8 6 0 7 2 7 1 3 4 2 6 9 2 6 2 3 1 5 4 7 5 3 3 0 4 3 7 9 9 3 3 1 6 5 8 1 1 0 7 3 6 9 6 4 3 1 4 2 1 7 1 9 7 9 4 3 4 0 5 6 3 9 1 5 5 1 2 1 0 8 1 0 8 1 3 6 2 6 2 6 8 8 8 5 6 9 7 4 8 0 6 8 0 6 0 1 1 6 9 1 8 9 4 1 7 5 0 2 7 2 2 9 8 7 4 1 5 8 6 9 9 1 7 9 1 4 5 3 4 9 9 4 6 2 4 4 4 1 9 4 0 1 2 1 9 7 8 5 8 6 0 1 3 7 3 6 6 0 8 2 8 6 9 0 7 2 2 3 6 5 1 4 7 7 1 3 9 1 2 6 8 7 4 2 0 9 6 6 5 1 3 7 8 7 5 6 2 0 5 9 1 8 5 4 3 2 8 8 8 8 3 4 1 7 4 2 9 2 0 9 0 1 5 6 3 1 3 3 2 8 3 1 9 3 5 7 5 6 2 2 0 8 9 7 1 3 7 6 5 6 3 0 9 7 8 5 0 1 5 6 3 1 5 4 9 8 2 4 5 6 4 4 5 8 6 5 4 2 4 7 9 2 9 3 5 7 2 2 8 2 8 7 5 0 6 0 8 4 8 1 4 5 3 3 5 1 3 5 2 1 8 1 7 2 9 5 8 7 9 3 2 9 9 1 1 7 1 0 0 3 2 4 7 6 2 2 2 0 5 2 1 9 4 6 4 5 1 0 5 3 6 2 4 5 0 5 1 2 9 8 8 4 3 0 8 7 1 3 4 4 4 3 9 5 0 7 2 4 4 2 6 7 3 5 1 4 6 2 8 6 1 7 9 9 1 8 3 2 3 3 6 4 5 9 8 3 6 9 6 3 7 6 3 2 7 2 2 5 7 5 6 9 1 5 9 7 2 3 9 5 4 3 8 3 0 5 2 0 8 6 6 4 7 4 7 4 2 3 8 1 5 1 1 0 7 9 2 7 3 4 9 4 8 3 6 9 5 2 3 9 6 4 7 9 2 6 8 9 9 3 6 9 8 3 2 4 9 1 7 9 9 9 5 0 2 7 8 9 5 0 0 0 6 0 4 5 9 6 6 1 3 1 3 4 6 3 3 6 3 0 2 4 9 4 9 9 5 1 4 8 0 8 0 5 3 2 9 0 1 7 9 0 2 9 7 5 1 8 2 5 1 5 8 7 5 0 4 9 0 0 7 4 3 5 1 8 7 9 8 3 5 1 1 8 3 6 0 3 2 7 2 2 7 7 2 6 0 1 7 1 7 4 0 4 5 3 5 5 7 1 6 5 8 8 5 5 5 7 8 2 9 7 2 9 1 0 6 1 9 5 8 1 9 3 5 1 7 1 0 5 5 4 8 2 5 7 9 3 0 7 0 9 1 0 0 5 7 6 3 5 8 6 9 9 0 1 9 2 9 7 2 1 7 9 9 5 1 6 8 7 3 1 1 7 5 5 6 3 1 4 4 4 8 5 6 4 8 1 0 0 2 2 0 0 1 4 2 5 4 5 4 0 5 5 4 2 9 2 7 3 4 5 8 8 3 7 1 1 6 0 2 0 9 9 4 7 9 4 5 7 2 0 8 2 3 7 8 0 4 3 6 8 7 1 8 9 4 4 8 0 5 6 3 6 8 9 1 8 2 5 8 0 2 4 4 4 9 9 6 3 1 8 7 8 3 4 2 0 2 7 4 9 1 0 1 5 3 3 5 7 9 1 0 7 2 7 3 3 6 2 5 3 2 8 9 0 6 9 3 3 4 7 4 1 2 3 8 0 2 2 2 2 0 1 1 6 2 6 2 7 7 1 1 9 3 0 8 5 4 4 8 5 0 2 9 5 4 1 9 1 3 2 0 0 4 0 0 9 9 9 8 6 5 5 6 6 6 5 1 7 7 5 6 6 4 0 9 5 3 6 5 6 1 9 7 8 9 7 8 1 8 3 8 0 4 5 1 0 3 0 3 5 6 5 1 0 1 3 1 5 8 9 4 5 8 9 0 2 8 7 1 8 6 1 0 8 6 9 0 5 8 9 3 9 4 7 1 3 6 8 0 1 4 8 4 5 7 0 0 1 8 3 6 6 4 9 5 6 4 7 2 0 3 2 9 4 3 3 4 3 7 4 2 9 8 9 4 6 4 2 7 4 1 2 5 5 1 4 3 5 9 0 5 8 4 3 4 8 4 0 9 1 9 5 4 8 7 0 1 5 2 3 6 1 4 0 3 1 7 3 9 1 3 9 0 3 6 1 6 4 4 0 1 9 8 4 5 5 0 5 1 0 4 9 1 2 1 1 6 9 7 9 2 0 0 1 2 0 1 9 9 9 6 0 5 0 6 9 9 4 9 6 6 4 0 3 0 3 5 0 8 6 3 6 9 2 9 0 3 9 4 1 0 0 7 0 1 9 4 5 0 5 3 2 0 1 6 2 3 4 8 7 2 7 6 3 2 3 2 7 3 2 4 4 9 4 3 9 6 3 0 4 8 0 8 9 0 5 5 4 2 5 1 3 7 9 7 2 3 3 1 4 7 5 1 8 5 2 0 7 0 9 1 0 2 5 0 6 3 6 8 5 9 8 1 6 7 9 5 3 0 4 8 1 8 1 0 0 7 3 9 4 2 4 5 3 1 7 0 0 2 3 8 8 0 4 7 5 9 8 3 4 3 2 3 4 5 0 4 1 4 2 5 8 4 3 1 4 0 6 3 6 1 2 7 2 1 0 9 6 0 2 2 8 2 4 2 3 3 7 8 2 2 8 0 9 0 2 7 9 7 6 5 9 6 0 7 7 7 1 0 8 4 9 3 9 1 5 1 7 4 8 8 7 3 1 6 8 7 7 7 1 3 5 2 2 3 9 0 0 9 1 1 7 1 1 7 3 5 0 9 1 8 6 0 0 6 5 4 6 2 0 0 9 9 0 2 4 9 7 5 8 5 2 7 7 9 2 5 4 2 7 8 1 6 5 9 7 0 3 8 3 4 9 5 0 5 8 0 1 0 6 2 6 1 5 5 3 3 3 6 9 1 0 9 3 7 8 4 6 5 9 7 7 1 0 5 2 9 7 5 0 2 2 3 1 7 3 0 7 4 1 2 1 7 7 8 3 4 4 1 8 9 4 1 1 8 4 5 9 6 5 8 6 1 0 2 9 8 0 1 8 7 7 8 7 4 2 7 4 4 5 6 3 8 6 6 9 6 6 1 2 7 7 2 4 5 0 3 8 4 5 8 6 0 5 2 6 4 1 5 1 0 3 0 4 0 8 9 8 2 5 7 7 7 7 5 4 4 7 4 1 1 5 3 3 2 0 7 6 4 0 7 5 8 8 1 6 7 7 5 1 4 9 7 5 5 3 8 0 4 7 1 1 6 2 9 6 6 7 7 7 1 0 0 5 8 7 6 6 4 6 1 5 9 5 4 9 6 7 7 6 9 2 7 0 5 4 9 6 2 3 9 3 9 8 5 7 0 9 2 5 5 0 7 0 2 7 4 0 6 9 9 7 8 1 4 0 8 4 3 1 2 4 9 6 5 3 6 3 0 7 1 8 6 6 5 3 3 7 1 8 0 6 0 5 8 7 4 2 2 4 2 5 9 8 1 6 5 3 0 7 0 5 2 5 7 3 8 3 4 5 4 1 5 7 7 0 5 4 2 9 2 1 6 2 9 9 8 1 1 4 9 1 7 5 0 8 6 1 1 3 1 1 7 6 5 7 7 3 1 7 2 0 9 5 6 1 5 6 5 6 4 7 8 6 9 5 4 7 4 4 8 9 2 7 1 3 2 0 6 0 8 0 6 3 5 4 5 7 7 9 4 6 2 4 1 4 5 3 1 0 6 6 9 8 3 7 4 2 1 1 3 7 9 8 1 6 8 9 6 3 8 2 3 5 3 3 3 0 4 4 7 7 8 8 3 1 6 9 3 3 9 7 2 8 7 2 8 9 1 8 1 0 3 6 6 4 0 8 3 2 6 9 8 5 6 9 8 8 2 5 4 4 3 8 5 1 6 6 7 5 8 6 2 2 8 9 9 3 0 6 9 6 4 3 4 6 8 4 8 9 7 5 1 4 8 4 0 8 7 9 0 3 9 6 4 7 6 0 4 2 0 3 6 1 0 2 0 6 0 2 1 7 1 7 3 9 4 4 7 0 2 6 3 4 8 7 6 3 3 6 5 4 3 9 3 1 9 5 2 2 9 0 7 7 3 8 3 6 1 6 7 3 8 9 8 1 1 7 8 1 2 4 2 4 8 3 6 5 5 7 8 1 0 5 0 3 4 1 6 9 4 5 1 5 6 3 6 2 6 0 4 3 0 0 3 6 6 5 7 4 3 1 0 8 4 7 6 6 5 4 8 7 7 7 8 0 1 2 8 5 7 7 9 2 3 6 4 5 4 1 8 5 2 2 4 4 7 2 3 6 1 7 1 3 7 4 2 2 9 2 5 5 8 4 1 5 9 3 1 3 5 6 1 2 8 6 6 3 7 1 6 7 0 3 2 8 0 7 2 1 7 1 5 5 3 3 9 2 6 4 6 3 2 5 7 3 0 6 7 3 0 6 3 9 1 0 8 5 4 1 0 8 8 6 8 0 8 5 7 4 2 8 3 8 5 8 8 2 8 0 6 0 2 3 0 3 3 4 1 4 0 8 5 5 0 3 9 0 9 7 3 5 3 8 7 2 6 1 3 4 5 1 1 9 6 2 9 2 6 4 1 5 9 9 5 2 1 2 7 8 9 3 1 1 3 5 4 4 3 1 4 6 0 1 5 2 7 3 0 9 0 2 5 5 3 8 2 7 1 0 4 3 2 5 9 6 6 2 2 6 7 4 3 9 0 3 7 4 5 5 6 3 6 1 2 2 8 6 1 3 9 0 7 8 3 1 9 4 3 3 5 7 0 5 9 0 0 3 8 1 4 8 7 0 0 8 9 8 6 6 1 3 1 5 3 9 8 1 9 5 8 5 7 4 4 2 3 3 0 4 4 1 9 7 0 8 5 6 6 9 6 7 2 2 2 9 3 1 4 2 7 3 0 7 4 1 3 8 4 8 8 2 7 8 8 9 7 5 5 8 8 8 6 0 7 9 9 7 3 8 7 0 4 4 7 0 2 0 3 1 6 6 8 3 4 8 5 6 9 4 1 9 9 0 9 6 5 4 8 0 2 9 8 2 4 9 3 1 9 8 1 7 6 5 7 9 2 6 8 2 9 8 5 5 6 2 9 7 2 3 0 1 0 6 8 2 7 7 7 2 3 5 1 6 2 7 4 0 7 8 3 8 0 7 4 3 1 8 7 7 8 2 7 3 1 8 2 1 1 9 1 9 6 9 5 2 8 0 0 5 1 6 0 8 7 9 1 5 7 2 1 2 8 8 2 6 3 3 7 9 6 8 2 3 1 2 7 2 5 6 2 8 7 0 0 0 1 5 0 0 1 8 2 9 2 9 7 5 7 7 2 9 9 9 3 5 7 9 0 9 4 9 1 9 6 4 0 7 6 3 4 4 2 8 6 1 5 7 5 7 1 3 5 4 4 4 2 7 8 9 8 3 8 3 0 4 0 4 5 4 7 0 2 7 1 0 1 9 4 5 8 0 0 4 2 5 8 2 0 2 1 2 0 2 3 4 4 5 8 0 6 3 0 3 4 5 0 3 3 6 5 8 1 4 7 2 1 8 5 4 9 2 0 3 6 7 9 9 8 9 9 7 2 9 3 5 3 5 3 9 1 9 6 8 1 2 1 3 3 1 9 5 1 6 5 3 7 9 7 4 5 3 9 9 1 1 1 4 9 4 2 4 4 4 5 1 8 3 0 3 3 8 5 8 8 4 1 2 9 0 4 0 1 8 1 7 8 1 8 8 2 1 3 7 6 0 0 6 6 5 9 2 8 4 9 4 1 3 6 7 7 5 4 3 1 7 4 5 1 6 0 5 4 0 9 3 8 7 1 1 0 3 6 8 7 1 5 2 1 1 6 4 0 4 0 5 8 2 1 9 3 4 4 7 1 2 0 4 4 8 2 7 7 5 9 6 0 5 4 1 6 9 4 8 6 4 5 3 9 8 7 8 3 2 6 2 6 9 5 4 8 0 1 3 9 1 5 0 1 9 0 3 8 9 9 5 9 3 1 3 0 6 7 0 3 1 8 6 6 1 6 7 0 6 6 3 7 1 9 6 4 0 2 5 6 9 2 8 6 7 1 3 8 8 7 1 4 6 6 3 1 1 8 9 1 9 2 6 8 5 6 8 2 6 9 1 9 9 5 2 7 6 4 5 7 9 9 7 7 1 8 2 7 8 7 5 9 4 6 0 9 6 1 6 1 7 2 1 8 8 6 8 1 0 9 4 5 4 6 5 1 5 7 8 8 6 9 1 2 2 4 1 0 6 0 9 8 1 4 1 9 7 2 6 8 6 1 9 2 5 5 4 7 8 7 8 9 9 2 6 3 1 5 3 5 9 4 7 2 9 2 2 8 2 5 0 8 0 5 4 2 5 1 6 9 0 6 8 1 4 0 1 0 7 8 1 7 9 6 0 2 1 8 8 5 3 3 0 7 6 2 3 0 5 5 6 3 8 1 6 3 1 6 4 0 1 9 2 2 4 5 4 5 0 3 2 5 7 6 5 6 7 3 9 2 5 9 9 7 6 5 1 7 5 3 0 8 0 1 4 2 7 1 6 0 7 1 4 3 0 8 7 1 8 8 6 2 8 5 9 8 3 6 0 3 7 4 6 5 0 5 7 1 3 4 2 0 4 6 7 0 0 8 3 4 3 2 7 5 4 2 3 0 2 7 7 0 4 7 7 9 3 3 1 1 1 8 3 6 6 6 9 0 3 2 3 2 8 8 5 3 0 6 8 7 3 8 7 9 9 0 7 1 3 5 9 0 0 7 4 0 3 0 4 9 0 7 4 5 9 8 8 9 5 1 3 6 4 7 6 8 7 6 0 8 6 7 8 4 4 3 2 3 8 2 4 8 2 1 8 9 3 0 6 1 7 5 7 0 3 1 9 5 6 3 8 0 3 2 3 0 8 1 9 7 1 9 3 6 3 5 6 7 2 7 4 1 9 6 4 3 8 7 2 6 2 5 8 7 0 6 1 5 4 3 3 0 7 2 9 6 3 7 0 3 8 1 2 7 5 1 5 1 7 0 4 0 6 0 0 5 0 5 7 5 9 4 8 8 2 7 2 3 8 5 6 3 4 5 1 5 6 3 9 0 5 2 6 5 7 7 1 0 4 2 6 4 5 9 4 7 6 0 4 0 5 5 6 9 5 0 9 5 9 8 4 0 8 8 8 9 0 3 7 6 2 0 7 9 9 5 6 6 3 8 8 0 1 7 8 6 1 8 5 5 9 1 5 9 4 4 1 1 1 7 2 5 0 9 2 3 1 3 2 7 9 7 7 1 1 3 8 0 3

A golden rectangle.

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Fibonacci Numbers

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The Fibonacci Numbers were first published by Leonardo Pisano, nicknamed Fibonacci, in 1202. They are a sequence of numbers generated by adding the two preceding numbers to get the next number, staring from 0 and 1: 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… add the last two to get the next Fibonacci number. The ratio of any two numbers of the sequence approximate the ratio of the Golden Ratio, 1.618903…

Calculate A Fibonacci Number

Enter a number for the position of the Fibonacci number:

This is the Fibonacci number at that position:

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Leonardo Pisan, AKA Fibonacci.

Brownian Motion

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The unceasing and irregular motion of microscopic particles suspended in a heated fluid, caused by the random bombardment of the fluid's molecules. First observed by Robert Brown in 1827 but was first turned into a practical mathematical model by Albert Einstein in 1905. The mathematics of Brownian motion has led to major discoveries in the world of physics.

There is an excellent link on the history and implications of Brownian motion here.

The Sound of Noise

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Pink Noise (left) compared to White Noise (right) in an FFT spectrogram.

"White noise is a sound that contains every frequency within the range of human hearing (generally from 20 hertz to 20 kHz) in equal amounts. Most people perceive this sound as having more high-frequency content than low, but this is not the case. This perception occurs because each successive octave has twice as many frequencies as the one pre­ceding it. For example, from 100 Hz to 200 Hz, there are one hundred dis­crete frequencies. In the next octave (from 200 Hz to 400 Hz), there are two hundred frequencies.

White noise can be generated on a sound synthesizer. Sound designers can use this sound, with some processing and filtering, to create a multitude of effects such as wind, surf, space whooshes, and rumbles.

Pink noise is a variant of white noise. Pink noise is white noise that has been filtered to reduce the volume at each octave. This is done to compensate for the increase in the number of frequencies per octave. Each octave is reduced by 6 decibels, resulting in a noise sound wave that has equal energy at every octave."

Whatis.com IT-Specific Encyclopedia, 27 December 2003
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