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Number - The language of the Exact Sciences

 CONTENT: 1) The Fraction Puzzle , Number ONE, Number 2 . . . . .

1) The Fraction Puzzle

ImageRef: FractionAll.jpg

 POSTED: 180411

This Project seeks to find distinctive images for the eight fractions from TWO through NINE

The fraction puzzle comes from the book “More Mathematical Puzzles by Sam Loyd" - a Dover publication (1960) selected and edited by Martin Gardner 1914-2010 (who is probably best known for his monthly article in the Scientific American magazine from 1956 to 1986)

The book was the second of two volumes of puzzles collated from  Sam Loyd’s “Cyclopedia of Puzzles” originally piublished by Franklin Bigelow Corporation 1n 1914.    The puzzle was number 74 in the cyclopedia (as in the accompanying illustration)

Each of the numbers (2-9) is created by the individual numbers

1 to 9 set up as a series of fractions using the numbers in a particular order.

It should be obvious that it's impossible to generate the number 1 by this method (because that would reqiuire the top & bottom of the fraction to be identical - an impossibility!

NUMBER ONE & NUMBER TWO

TemporaryImage - NumberONE.jpg

P O S T E D : 180417

The Old Ones gave great credence to the individual character of the numbers 1 through 9 ; their art of interpreting numbers and letters was called GEMATRIA and is known from Babylonian inscriptions from about 750BC

The early Greeks were unsure as to whether 2 was a number at all. They found it disturbing that '2+2 = 2x2' (or indeed that any number multiplied by 2 is equal to the same number added to itself)        They expected multiplication to do more than mere addition! For example even the next - Number 3 - gives rise to '3+3=6,  3x3=9'

Two is the First Female number - even numbers have always been Female, and Odd numbers always Male -

For the Greeks addition was straight forward (an extension of counting)    But multiplication required a more sophisticated approach and, being a very VISUAL people, they utilised the rectangle to illustrate the principles in terms of areas

For example: a square of side 1 unit defines an area of 1 or: 1x1=1 ( its good to START simple)

Similarly, a rectangle with sides of 3 units and 5 units would define an Area of 3x5 unit squares which equals 15, a solution that can be readily verified by counting the unit squares

The unit square can also illustrate how ALL the numbers (integers) are the progeny of the Unique & Universal Number ONE!

In the sections below that describe each number (2 to 9) in turn, the simple spiraling SEED diagram will be developed for the conception of each sibling number.

—————

Firstly, the creation of number 2 from the seed of 1:

NOTE: the symbol '^' is used to denote 'squared'

The infinite series of natural numbers grows from the Unit Square whose Unit Area (1x1=1) represents ONE, the Mother of ALL numbers.    Within ONE,lies the SEED (indicated by the red diagonal) of length (by Pythagoras) √(1^+1^) = √2 which will become the side that generates the the NEW square whose area (√2x√2 = 2) represents the Number TWO

The new square starts the spiral (not discernable quite yet) of numbers that rotate in ever increasing size but ever decreasing angles about the centre P

ImageRef: NumberTHREE.jpg

NUMBER THREE

POSTED 1804117 (*edited 180528)

Secondly the creation of number 3 from the SEED of 2:

NOTE: the symbol '^' is used to denote 'squared'

To expose the SEED within NumberTWO, the distance 1 is marked off along the side opposite P and the Red Diagonal joins that point with P

The length of the diagonal  (by Pythagoras)  is √(1^+2^) = √3 which will become the side that generates the the NEW square whose area (√3x√3 = 3) and represents the NumberTHREE, conceived directly from the TWO

The new square continues the spiral (still not quite discernable yet) of numbers that rotate in ever increasing size but ever decreasing angles about the centre P

The angle of rotation from TWO to THREE is *ASin[1/√3]= 1/1.732 = 0.5773] = 35.3 degrees

This is Agrippa’s first Magic Square, of order 3 and is dedicated to Saturn using the numbers from 1 to 9. The magic constant (the sum of each column, row and diagonal) is 15 and the total sum of all the numbers is 45. (15 and 45 are triangular numbers) The colours of Saturn’s square are white for the numbers and black for the background. Saturn’s metal is lead    

Heinrich Cornelius Agrippa (1486--1535) was a German Magician, occult writer, theologian, astrologer and alchemist. In his famous sayings, he emphasized in his Law of Resonance: "All things which are similar and therefore connected, are drawn to each other's power."   In his De occulta philosophia libri tres (Three Books Concerning Occult Philosophy), Agrippa joined the seven planets (known at that time) with seven Magic Squares, of orders 3 to 9. In it he expounded on the magical virtues of them, each associated with one of the astrological planets. This book was very influential throughout Europe until the counter- or Catholic-Reformation, and Agrippa's magic squares, sometimes called Kameas, continue to be used within modern ceremonial magic in much the same way as he first prescribed.

The same square, discovered in Chinese is called the LoShu - Story to follow

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