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SQUARE SPIRAL

SQUARE NUMBERS

Square numbers are the result of multiplying any integer by itself. Square numbers can be represented in two dimensions as stacks of square blocks.

ARRANGEMENTS OF SQUARE NUMBERS

Coloring the blocks in the first column reveals a pattern unique to square numbers. Rearranging the blocks in the second column establishes a linear order by way of the associative property.

NUMBER LINE

A square spiral is a segment of a number line which occupies two dimensional space, a spiral vector which can be fit neatly into a square grid. Units along the number line can be any integer, positive or negative.

SQUARE SPIRAL

Alternatively a square spiral can be represented as a series of concentric square rings. Each ring begins with an odd square number and ends with the next square number minus one. Each successive ring contains 8 more cells then the previous ring.


ROOT 2

THE SQUARE ROOT OF 2

The square root of 2 is a number that when multiplied by itself equals 2. It is also the hypotenuse of a right triangle whose legs are 1 unit each.

ADJUSTMENT

Suppose the square root of two could be expressed as the ratio between two numbers A/B. The result would be an equation in which the numerator squared is equal to double the denominator squared.

APPROXIMATING THE SQUARE ROOT OF 2

The square root of 2 is an irrational number, meaning that it cannot be expressed as the ratio of two whole numbers A/B. There is, however, two sequences that when combined together come close to achieving this proportion, always missing the mark by +/- 1.

A COMPOSITE SEQUENCE

Known since ancient times, the series of fractions A/B are a composite of two sequences, the Pell numbers (1,2,5,12…) and the Half-companion Pell numbers (1,3,7,17..). Together they form the closest rational approximations of the square root of 2. Each successive ratio more accurate then the previous, alternating between overestimating and underestimating the square root of 2.


CONCENTRIC POLYGONS

TRIANGULAR NUMBERS

Triangular numbers are equal to the sum of the natural numbers. The next triangular number is generated by adding the next natural number to the base of the triangle.

CONCENTRIC POLYGONS

Inscribed within a set of concentric circles, this series of regular polygons begins with a single chord inscribed at the innermost circle then expands outwards always increasing the number of sides of the next polygon by 1.

CORRELATION

Each successive polygon is inscribed with its vertex facing right. Each node is numbered starting from its vertex and rotating in a counterclockwise direction. Triangular numbered nodes result as each successive ring increases by the amount of the next polygon.

THE FIRST 629 POINTS

Above are the first 629 points inscribed within 34 concentric polygons. The vertex of each polygon line up to form a row of triangular numbered nodes.


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