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Figurate Numbers Figurate Numbers by George Jelliss June 2008 with additions November 2008 Visualisation of Numbers The visual representation of the number of elements in a set by an array of small counters or other standard tally marks is still seen in the symbols on dominoes or playing cards, and in Roman numerals. The word "calculus" originally meant a small pebble used to calculate. Bear with me while we begin with a few elementary observations. Any number, n greater than 1, can be represented by a linear arrangement of n counters. The cases of 1 or 0 counters can be regarded as trivial or degenerate linear arrangements. The counters that make up a number m can alternatively be grouped in pairs instead of ones, and we find there are two cases, m = 2.n or 2.n + 1 (where the dot denotes multiplication). Numbers of these two forms are of course known as even and odd respectively. An even number is the sum of two equal numbers, n+n = 2.n. An odd number is the sum of two successive numbers 2.n + 1 = n + (n+1). The even and odd numbers alternate. Figure 1. Representation of numbers by rows of counters, and of even and odd numbers by various, mainly symmetric, formations. The right-angled (L-shaped) formation of the odd numbers is known as a gnomon. These do not of course exhaust the possibilities. 1 2 3 4 5 6 7 8 9 n 2 4 6 8 10 12 14 2.n 1 3 5 7 9 11 13 15 2.n + 1 Triples, Quadruples and Other Forms Generalising the divison into even and odd numbers, the counters making up a number can of course also be grouped in threes or fours or indeed any nonzero number k. A number m of counters is either an exact multiple of a k or there are some counters, less than k, left over. That is m can be uniquely expressed in the form m = k.n + r where n is called the quotient and r the remainder (and either may be zero). Thus the number k divides the set of all numbers, up to any chosen value, into k classes according as the remainder r is 0, 1, 2, ..., (k–1). That is numbers are of the k forms k.n, k.n + 1, k.n + 2, ..., k.n + (k–1). Numbers that are multiples of k (which we call k-tuples or in specific cases: triples, quadruples, quintuples, sextuples, and so on) can be arranged visually in the form of a k-sided polygonal path. The polygon formed by k.n has n+1 counters along each edge. The polygon can be shown with any angles, but the most popular is regular, with all angles equal (i.e. equiangular) and all sides of equal length (i.e. equilateral) in which case the circular counters can be touching, or at least equally spaced. Numbers of the form k.n + 1 can be visualised by k lines of length n+1 meeting at a common point (in the case of k = 4 we get an equal-armed cross). Numbers of the form k.n + (k–1) can be visualised as k parallel lines each of length n with the (k–1) single counters separating the lines. These patterns do not of course exhaust the possibilities. Figure 2. Representations of Triples 3.n and Triforms 3.n + 1 and 3.n + 2. 0 3 6 9 12 15 18 21 3.n 1 4 7 10 13 16 19 3.n +1 2 5 8 11 14 17 20 3.n + 2 Triangles The term triangular number is applied to a number of counters that can be arranged to form an area bounded by a triangular path and to fill that area in a close-packed fashion. It will be seen, by dividing a triangle into rows (which we may colour light and dark, indicating odd and even) that a triangular number is the sum of all the numbers from 0 (or 1) to n. A formula for the general triangular number is n.(n+1)/2. This can be proved by arranging the numbers 1 to n and n to 1 in two rows and noting that each pair of numbers adds to n+1, and that there are n pairs, so that the sum of the two equal rows is n.(n+1). The fractional expression n.(n+1)/2 is always a whole number since n and n+1 are successive, so one must be even. It is sometimes convenient to denote the nth triangular number as n. Figure 3. The first few nonzero triangular numbers, shown as right-angled or (approximately) equilateral triangles of counters. 1 3 6 10 15 21 28 36 45 n.(n+1)/2 Squares The term square number is applied to numbers that can be shown as an array of n rows and n columns, thus containing n.n = n2 counters. A nonzero square n2 is the sum of all the odd numbers from 1 to (2.n – 1). This can be visualised by cutting up the square into gnomons. A nonzero square n2 is also the sum of two successive triangular numbers, that is: n2 = n = (n– 1).n/2 + n.(n+1)/2 = (n–1) + n, as can also be readily visualised. Figure 4. Illustrating square numbers as a sum of odd numbers, or of two successive triangular numbers. Any square can be regarded as a nesting of quadruples in the form of square paths, around a central 0 or 1, showing squares are of the forms 4.n or 4.n + 1. 1 4 9 16 25 36 49 64 n^2 Figure 5. Squares can also be visualised in rhombic and triangular arrays (of the type sometimes called "pyramids") in which the successive rows are the odd numbers. 1 4 9 16 25 36 49 n^2 Metasquares and other Rectangles Since the sum of the first n numbers is a triangular number, the sum of the first n even numbers is of course twice a triangular number, so it would seem sensible to call such numbers "bitriangular" numbers, in the literature however they are sometimes called "pronic" numbers, but for reasons to be explained below I prefer to call them metasquare numbers. The formula for the nth metasquare is n.(n+1), i.e. the product of two successive numbers. Zero counts as both square and metasquare. A nonzero square is the arithmetic mean of two successive metasquares, that is: n2 = [(n–1).n + n.(n+1)]/2. While a metasquare is the geometric mean of two successive squares, that is: n.(n+1) = [(n2).(n+1)2]1/2 where u1/2 means the square root of u. Written in algebraic form these relations are obvious, but the relationship between squares and metasquares nevertheless seems curiously asymmetric. There is one square between every two sucessive metasquares, and one metasquare between every two successive nonzero squares, hence the name "metasquare". If we colour the rows of a triangular number alternately light and dark we may note that the light counters indicate odd numbers (adding to a square) and the dark counters even numbers (adding to a metasquare). Thus every triangular number is the sum of a square and a metasquare. 3 = 1 + 2, 6 = 4 + 2, 10 = 4 + 6, 15 = 9 + 6, 21 = 9 + 12, 28 = 16 + 12, 36 = 16 + 20, and so on. A number of the form a.b where a and b are greater than 1 is called a composite number and can be represented by a rectangular array. Squares (other than 0 and 1) and metasquares (other than 0 and 2) are special examples of composite numbers. A number greater than 1 that cannot be represented as a rectangle in this way is called a prime number. By this definition 0 and 1 are neither composite nor prime, but all other numbers are either prime or composite. Any number greater than 1 can be expressed uniquely as the product of powers of primes, called its prime factors. The tables that follow list all the numbers less than 1000 together with their prime factorisation in the form (2^a).(3^b).(5^c)... Some simple composite numbers can be represented as a rectangle in only one way. Others can be shown as a rectangle in two or more ways. The number 12 is the first that can be shown as a rectangle in two ways 12 = 2.6 = 3.4. Any multiple of 4 greater than 8 is a multicomposite number since 4.k = 2.(2.k). This implies that we can have no more than three successive simple composite numbers. But such triplets often occur, the first cases are 25, 26, 27 and 33, 34, 35. They consist of numbers of the form 4n + 1, 4n + 2, 4n + 3. The relation (h–k).(h+k) = h2 – k2 enables us to represent a rectangle u.v, in which u and v are both odd or both even, as a difference of two squares [(u+v)/2]2 – [(v–u)/2]2. A particular case of this is n2 – 1 = (n–1).(n+1). Diamonds By a diamond I mean an arrangement of counters on a square lattice in the shape of a square with diagonal sides. From these diagrams the alternate colouring (or division into two pyramids) shows that any diamond is the sum of two successive squares, giving the general form n2 + (n+1)2 = 2.n.(n + 1) + 1.
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