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.

ARRANGEMENTS OF TRIANGULAR NUMBERS

Triangular, pronic, and square numbers all result from adding successive sets of counting, even or odd numbers. Pronic numbers are double the value of triangular numbers whereas square numbers result of adding two successive triangular numbers together.

TRIANGULAR ARRAYS

The arrangements of blocks in the table above highlight the progression of triangular numbers and its multiples. Each column doubles the amount of blocks from the previous column. Note how x8 triangular numbers forms an odd square number -1.

SQUARE ARRAYS

The arrangements of blocks in the table above highlight the progression of square numbers and its multiples. Each column doubles the amount of blocks from the previous column.

LINEAR SCALE

The square spiral scales to any unit length. The top most spiral counts one unit of length for every grid cell, the next counts one unit of length for every two grid cells, and the last counts one unit of length for every four grid cells. The sequences of numbers generated diagonally across the spiral array scale down proportionally as well.

BALANCE

An unusual balance results from dividing the square spiral diagonally just above the center. The sum of linear units in the bottom portion of the spiral (gray) equal the sum of linear units in the upper section (rainbow).

OBSERVATION

To clarify the observation, separate out the square numbers from the array. Next pair the remaining integers from each side in ascending order and subtract. The final result confirms our original observation and demonstrates how this pattern holds true for each subsequent layer that is added to the array.

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.

PELL AND HALF COMPANION PELL SEQUENCES

Pell and half companion Pell numbers can be generated a using a recursion method. Starting with 1/1 and continuing forward by plugging each successive term back into the fold, adding denominator to numerator to generate the next denominator, then adding the two denominators together to generate the next numerator.

TRIANGULAR SQUARE AND TRIANGULAR PRONIC SEQUENCES

The intersection of triangular and square numbers can be expressed as a ratio between the mth triangular and nth square ordinal numbers, this is true for the intersection of triangular and pronic numbers as well. This proportion can be generated a using a recursion method, starting with a seed and plugging each successive term back into the fold.

INTERLOCKING SEQUENCES

Composite Pell and Half-companion Pell sequences alternate between ratios that overestimate and underestimate the square root of 2. The triangular square progression interlocks additively with the series of ratios that underestimate the square root of 2. The triangular pronic progression interlocks additively with the series of ratios that overestimates.

INTERSECTION

Above are the first 696 triangular numbers inscribed within 492 concentric even-sided polygons. Triangular numbers intersect pronic and square numbers along the x-axis. The subsequent coordinates form a series of line segments whose length from the center is a successive Pell number.

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.

PRONIC AND SQUARE NUMBERS

Pronic numbers are equal to the progressive sum of even numbers whereas square numbers are equal to the progressive sum of odd numbers.

EVEN-SIDED POLYGONS

In this series each successive polygon has an even number of sides. The progression between each ring is 2 nodes and the arrangement is reflected across the x and y axis.

CORRELATION

Again, each successive polygon is inscribed with its vertex facing right. Each ring is numbered counterclockwise bisected on the right side with a square number and ending on the left side with a pronic.

THE FIRST 650 POINTS

Above are the first 650 points inscribed within 25 concentric even-sided polygons. The horizontal axis is split between square numbers on the right and pronics to the left.

TRIANGULAR SQUARE AND TRIANGULAR PRONIC NUMBERS

Numbers which are both triangular and square are called triangular squares, numbers which are both triangular and pronic are called triangular pronics.

TRIANGULAR, SQUARE, AND PRONIC

Inputting the next counting number into the formulae above outputs the next number in either the triangular, square, or pronic sequence. The respective m-valued or n-valued number correlates to the sequential order of the term in the series.

ALTERNATING SEQUENCES

Above is a diagram which maps out the first 4 triangular square and triangular pronic numbers as well as their respective m and n-values. These two series when put in ascending order, alternate back and forth between triangular square and triangular pronic values.

THE FIRST 49 TRIANGULAR NUMBERS

Above are the first 49 triangular numbers inscribed within 35 concentric even-sided polygons. While pronic and square numbers line up in a row, triangular numbers form a galaxy-like spiral.