Figurate Numbers. by George Jelliss June 2008 with additions November 2008

Transcription

1 Figurate Numbers by George Jelliss June 2008 with additions November 2008

2 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 n n n + 1

3 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 n n n + 2

4 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 n.(n+1)/2

5 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 = n 2 counters. A nonzero square n 2 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 n 2 is also the sum of two successive triangular numbers, that is: n 2 = 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 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 n^2

6 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: n 2 = [(n 1).n + n.(n+1)]/2. While a metasquare is the geometric mean of two successive squares, that is: n.(n+1) = [(n 2 ).(n+1) 2 ] 1/2 where u 1/2 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 = , 28 = , 36 = , 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) = h 2 k 2 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 n 2 1 = (n 1).(n+1).

7 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 n 2 + (n+1) 2 = 2.n.(n + 1) + 1. That is, one more than twice a metasquare. Figure 6. Diamonds Octagons By an octagon we mean an eight-sided arrangement with n+1 counters in each side, whether horizontal vertical or diagonal. The sequence runs: 1, 12, 37, 76, 129, 196, 277, 372, 481, 604, 741, 892, 1057,... and is generated by the formula 7.(n^2) + 4.n + 1. Greek Crosses and other Polysquares The smallest nontrivial diamond, 5, is also a Greek Cross, that is a shape formed of five equal squares, for which a general form is of course 5.n 2. We can also form other shapes with multiple squares. Shapes formed from squares all of the same size matched edge to edge are termed polyominoes. A single square gives just one shape. Two squares form a domino shape 1 by 2. Three squares form either a rectangle 1 by 3 or an L-shape. Four squares can be combined in five different shapes. Five squares can be combined in twelve shapes. Figure 7. Diagonal crosses. Equal to 5 times diamond plus 4.n = 10.n n

8 Kinds of Pentagonal Number After 3-sided and 4-sided numbers it would seem natural to move to 5-sided numbers, but they do not lend themselves to close-packed arrangements like the triangular and square numbers. The mathematician Leonhard Euler in 1783 studied numbers of the form n.(3n 1)/2 which he called "pentagonal" numbers, but "pentafigural" would be more systematic. Hexagons and Stars The number of counters in a close-packed hexagon with n+1 along each side is n = 6. n + 1 where n denotes the triangular number with n along each side, that is n = n.(n+1)/2, hence n = 3.n.(n+1) + 1 = 3.n n + 1. Thus n is one more than three times a metasquare. A hexagonal array can also be visualised as a cube viewed from above a corner. This shows that it is the difference of two successive cubes: n = (n+1) 3 n 3. To form a six-pointed star we add a further 6 triangles of the same size, so the number of stars is n = 12. n + 1 = 6.n.(n+1) + 1 = 6.n n + 1. Figure 8. Hexagons and Stars n.(n+1) n.(n+1) + 1 A second type of hexagonal number can be defined. I call it a diagonal hexagon by analogy with the diamond which is a diagonal square. The simplest example, apart from 1, is the 13-cell star which is also a diagonal hexagon. The larger diagonal hexagons can be derived from the larger stars by filling in the gaps between the points with a triangular number of counters (shown grey in the illustration). The sequence runs 1, 13, 43, 91, 157, 241, 343, 463, 601, 757, 931,... and a general formula is 9.n^2 + 3.n + 1 or (3.n).(3.n + 1) + 1. [Incidentally 343 = 7^3 ]. This means that 91 is another example of a three-pattern number like 37, being triangular and also hexagonal in both ways. To convert it from the lateral to the diagonal form only the six corner counters have to be moved, to the middles of the sides.

9 Multiply Patterned Numbers This study of Figurate Numbers was initially provoked by the puzzle of finding what numbers can be represented in two different ways, in particular as square, triangle, hexagon or star. The following are all the cases less than The number 0 can be considered a square or triangle (by putting n=0 in the formulas). The number 1 can be considered as of all four shapes (by putting n=1 in the square and triangle formulas, and n=0 in the hexagon and star formulas). The following are the six nontrivial cases. Interestingly they show all six possible pairings of square, triangle, hexagon and star. 36 = square & triangle 37 = hexagon & star 91 = triangle & hexagon 121 = square & star 169 = square & hexagon 253 = triangle & star If we expand the study to include metasquares and diamonds we get 6 = triangle & metasquare 13 = star & diamond 25 = square & diamond 61 = hexagon & diamond 181 = star & diamond 210 = triangle & metasquare 841 = square & diamond If we also look at dominoes (double-squares) we get: 2 = metasquare & domino 72 = metasquare & domino If we include diagonal hexagons and octagons we find triple-shaped numbers: 37 = hexagon, star and octagon 91 = triangle, hexagon and diagonal hexagon (13 could be included, but the star is the same as the diagonal hexagon) as well as other double-pattern numbers: 196 = square and octagon 741 = triangle and octagon Larger Double-Patterned Numbers I first considered whether there are other cases showing hexagon and star. We require 3.n.(n+1) + 1 = 6.m.(m+1) + 1. That is n.(n+1) = 2.m.(m+1); which implies n.(n+1)/2 = m.(m+1) a case of a number that is both triangle and metasquare. This relation between m and n can be put in the form: 2.m m n.(n+1) = 0, which can be solved for m by the quadratic equation formula, giving m = {[1 + 2.n.(n+1)] 1/2 1}/2 (where u 1/2 denotes the square root of u). The expression under the square root is the formula for a diamond shape. For this to be a whole number we require the expression under the square root to be a square; so we have another double pattern number a diamond and square. A diamond is the sum of two consecutive squares so we need to find numbers such that n 2 + (n+1) 2 = m 2. This is a special type of pythagorean triplet (numbers x, y, z that express the lengths of the sides of a right-angled triangle so that z 2 = x 2 + y 2 ). Here the two sides of the right angle (n and n+1) differ by only 1. As the numbers increase the triangle gets closer and closer to a half-square shape. So the ratio 2z/(x+y) is an approximation to root 2. By a well known result, all pythagorean triplets x 2 + y 2 = z 2 can be generated from numbers b and c in the form x = b 2 c 2, y = 2.b.c, z = b 2 + c 2.

10 The first such triplet (3,4,5) with b = 2, c = 1, leads to the triangle and metasquare 6 = (3.4)/2 = 2.3, the diamond and square 25 = = 5 2 = (2+3) 2, and the hexagon and star 37 = 3.(3.4) + 1 = 6.(2.3) + 1. The related triangle and square is 36 = 6 2 = (8.9)/2. The next such triplet is (20, 21, 29) with b = 5, c = 2, leads to the triangle and metasquare 210 = (21.20)/2 = 14.15, the diamond and square 841 = = 29 2 = (14+15) 2, and the hexagon and star 1261 = 3.(21.20) + 1 = 6.(14.15) + 1. The related triangle and square is 1225 = 35 2 = (49.50)/2 The third case is (119, 120, 169) with b = 12, c = 5, giving 7140 = = ( )/2; = = ; = 3.( ) + 1 = 6.(84.85) + 1; = = ( )/2. The fourth case is (696, 697, 985) with b = 29, c = 12, giving = ( )/2 = ; = = ; = 3.( ) + 1 = 6.( ) + 1; = = ( )/2. The values for b and c are two successive terms of the sequence 1, 2, 5, 12, 29, 70,... which has the recurrence relation b(n+2) = 2.b(n+1) + b(n) with b(0) = 1, b(1) = 2. Numbers that are both triangle and square are of the form (b.d) 2 where b is a term of b(n) and d is a term of d(n): 1, 3, 7, 17, 41,... which follows the same recurrence relation as b(n) but has d(0) = 1, d(1) = 3. The ratios b/d are convergents to root 2. The next case, after 91, of a number that is a hexagon of both types appears to be the much larger (19.181) which has 77 cells along a side, and requires 210 cells at each corner to be moved, though I've not fully double-checked this yet. The hexagons, lying on top of each other form a pattern analogous to the Star of David. Key to the Tables. In the following tables, listing all numbers from 0 to 999, we indicate whether a number is prime, and if not prime we express it in terms of its prime factors. The following symbols indicate numbers of a particular shape: Square n = n 2, Triangle n = n.(n+1)/2, Hexagon n = 3.n.(n+1) + 1, Star n = 6.n.(n+1) + 1, Diamond n = 2.n.(n+1) + 1, Metasquare n = n.(n+1). Domino n = 2.n 2. Octagon O. Diagonal Hexagon.

11 Tables of numbers: 0 to n: : : O3 2 prime: : Cube prime: : 7 53 prime : : 2 29 prime prime 5 prime: : : : prime : : 9 7 prime: : 2 Cube prime : prime : prime : prime: : 3 O1 37 prime: 3 2 O prime: ^6: 8 89 prime : : = 4 2 : 4 41 prime: : : prime : 6 67 prime prime prime: : : : prime prime : prime 23 prime prime: : Tables of numbers: 100 to : : Cube prime prime prime prime : O prime prime : prime: prime : prime : prime : : prime prime prime prime : prime prime prime : : : : : : : 14 O prime prime prime prime

12 Tables of numbers: 200 to : prime : prime prime O : prime : prime prime prime prime : prime prime prime : : : : Cube prime : prime prime : prime: : prime Tables of numbers: 300 to : : : prime : prime : : prime: prime prime prime prime : prime: prime: prime : prime : prime : Cube prime O7 397 prime: prime : prime

13 Tables of numbers: 400 to : : prime prime : prime : 15 O prime prime: prime : : prime : : prime prime prime : : : prime prime prime prime : : prime: : prime prime Tables of numbers: 500 to : : : prime 503 prime : : : : : prime prime : : 16 Cube prime prime : prime: prime prime : : prime prime prime: prime prime

14 Tables of numbers: 600 to : : : : 601 prime : prime prime O : : prime: prime prime prime : prime: : prime: prime : prime 617 prime prime prime prime : prime Tables of numbers: 700 to : prime prime : prime : : : : prime prime prime : : prime: prime prime : 38 O prime : prime prime : : prime prime

15 Tables of numbers: 800 to : prime prime prime prime prime prime : prime 809 prime prime prime : : prime prime prime : : : : O : : : prime : prime Tables of numbers: 900 to : : prime : prime prime : : prime prime prime : prime: : prime prime prime : : prime: : prime prime : 997 prime

16 References. W. W. Rouse Ball, revised by H. S. M. Coxeter, Mathematical Recreations and Essays, 11th edition 1939 (reprint 1956), pages has some material on Pythagorean triplets and figurate numbers. I should admit that my interest in this subject has in part been stimulated by the websites of Vernon Jenkins (The Other Bible Code, and Richard McGough (The Bible Wheel, which make use of Figurate Numbers, particularly triangles, hexagons and stars, in connection with Gematria (conversion of biblical words to numerical form by assigning numbers to the letters of the Hebrew and Greek alphabets). Their mathematics is impeccable but their use of it is questionable (to put it mildly). This study by me of Figurate Numbers is a continuing process. A first version was published in June 2008 and further results added on octagons and diagonal hexagons in November It is available to download as a PDF from the Publications page of my Mayhematics website: