Patterns in Mathematics

Ask what mathematics actually is, and one honest answer is this: mathematics is, to a very large extent, the search for patterns, and even more importantly, the search for the reason those patterns exist. Patterns are not rare or hidden away in textbooks -- they are everywhere, in the repeating phases of the moon, in the symmetry of floor tiles at home, in the rhythm of the seasons, in the way a thrown ball always traces the same kind of curve, and in the instructions stored inside every living cell. Long before anyone could explain why, people noticed that the planets and stars seemed to move in repeating, predictable paths across the sky. Centuries of studying that one pattern eventually produced the theory of gravitation, and that single explanation now lets scientists aim a rocket at the Moon or at Mars with real confidence, because the same pattern that guided ancient sky-watchers is trusted to guide a spacecraft today. In a similar way, spotting patterns hidden inside genomes has helped doctors diagnose and treat diseases. Finding a pattern is only half the job in mathematics. The far more powerful half is explaining why the pattern must keep happening -- because a good explanation can be reused in places nobody had even thought of yet.

The simplest patterns in the whole of mathematics are built from whole numbers arranged one after another according to a single rule. Such an ordered list is called a number sequence, and an entire branch of mathematics, called number theory, exists just to study patterns like these. Some sequences are almost too simple to notice, like the sequence of all 1's: 1, 1, 1, 1, 1, .... Others are completely familiar, like the counting numbers 1, 2, 3, 4, 5, ..., the odd numbers 1, 3, 5, 7, 9, 11, ..., and the even numbers 2, 4, 6, 8, 10, .... But some sequences take more digging to notice: the triangular numbers 1, 3, 6, 10, 15, 21, 28, ..., the square numbers 1, 4, 9, 16, 25, 36, 49, ..., and the cube numbers 1, 8, 27, 64, 125, 216, .... There is also a sequence built by doubling every time, the powers of 2: 1, 2, 4, 8, 16, 32, 64, ..., and one built by tripling every time, the powers of 3: 1, 3, 9, 27, 81, 243, 729, .... Recognising which rule is producing a given list of numbers, and being able to say confidently what the next few terms will be without needing anyone to tell you, is the very first skill this chapter is built around.

These names are not arbitrary labels -- they describe exactly how the numbers can be built out of dots. Take 1, 3, 6, 10: arrange 1 dot, then 3 dots as a tiny triangle, then 6 dots as a slightly bigger triangle, then 10, and each new picture is a perfect triangle of dots with one more full row added along the bottom. That is why they are called triangular numbers. Do the same with 1, 4, 9, 16: arrange the dots into square grids -- 1 by 1, 2 by 2, 3 by 3, 4 by 4 -- and every single one packs perfectly into a square, which is why they are called square numbers, or simply squares. Push the idea into three dimensions and 1, 8, 27, 64, 125 become solid cubes of tiny unit cubes: 1 by 1 by 1, 2 by 2 by 2, 3 by 3 by 3, and onward -- hence cube numbers. One especially pleasing fact is that the number 36 is both triangular and square at once: 36 dots can be arranged into a perfect triangle (36 is the 8th triangular number) and, completely separately, into a perfect 6 by 6 square (36 is also the 6th square number). The very same number can play two totally different roles depending on which picture is drawn around it.

Not every sequence grows by a fixed jump. Look at 1, 2, 3, 5, 8, 13, 21: the gap between terms keeps changing, yet there is still a perfectly exact rule hiding inside it. Starting from the third term, every number is simply the sum of the two numbers immediately before it: 3 = 1 + 2, 5 = 2 + 3, 8 = 3 + 5, 13 = 5 + 8, and 21 = 8 + 13. This sequence is called the Virahānka numbers, named after Āchārya Virahānka, a scholar who studied the rhythms of Sanskrit poetry over a thousand years ago and described this exact add-the-last-two-together rule while working out the ways long and short syllables could combine -- centuries before a similar sequence became famous in Europe. Because each new term depends on two previous terms rather than just one, this sequence cannot be built by a simple 'add a fixed number' or 'multiply by a fixed number' rule the way the earlier sequences can. It needs its own kind of rule: look back two steps, add together what is found there, and move forward one step.

Sequences do not just sit next to each other on a page -- some are secretly built out of others. Start adding up the odd numbers, one at a time: 1 is 1. 1 + 3 is 4. 1 + 3 + 5 is 9. 1 + 3 + 5 + 7 is 16. 1 + 3 + 5 + 7 + 9 is 25. 1 + 3 + 5 + 7 + 9 + 11 is 36. Every single running total lands exactly on a square number. Checking six examples makes this look true, but six examples can never prove it keeps happening forever -- for that, a picture is needed. Picture a square grid of dots and slice it into nested L-shaped bands, starting from one corner: the first band is a single dot, the next band wrapped around it has 3 dots, the next has 5, then 7, then 9, then 11, and together those bands rebuild the entire 6 by 6 square, all 36 dots, with nothing left over and nothing overlapping. Because this same slicing works for a square grid of any size at all, not just size 6, it proves the pattern holds for every square that could ever be drawn, not merely the handful that got checked by hand. That is the real difference between noticing a pattern and truly understanding why it is guaranteed.

There is yet another completely different way to build the square numbers. Instead of only adding upward, count up to a number and then straight back down again: just 1 is 1. 1 + 2 + 1 is 4. 1 + 2 + 3 + 2 + 1 is 9. 1 + 2 + 3 + 4 + 3 + 2 + 1 is 16. 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 is 25. Once again, every running total is a perfect square, arrived at by a completely different route than adding odd numbers. A closely related discovery appears when consecutive triangular numbers are added in pairs: 1 + 3 is 4, 3 + 6 is 9, 6 + 10 is 16, 10 + 15 is 25. Two triangles of dots, one slightly bigger than the other, fit together edge to edge to complete a square exactly -- which is a visual reason why this pairing keeps producing squares. Three completely different-looking patterns -- summing odd numbers, counting up and down, and pairing consecutive triangular numbers -- all arrive at exactly the same sequence of square numbers, which is a strong hint that something deeper connects all three, worth hunting for rather than dismissing as coincidence.

Numbers are not the only place patterns live. Shapes can form sequences too, whether flat, solid, or built from just points and connecting lines. The branch of mathematics that studies patterns in shapes is called geometry. One shape sequence is the regular polygons: an equilateral triangle, then a square, then a regular pentagon, hexagon, heptagon, octagon, nonagon and decagon, each one gaining exactly one more equal side and one more equal corner than the shape before it. A second shape sequence is the complete graphs, built only from dots and every possible connecting line between them: two dots joined by 1 line, three dots with every pair joined by 3 lines forming a triangle, four dots joined by 6 lines, five dots by 10 lines, six dots by 15 lines. A third is stacked squares, growing grids of unit squares -- 1 by 1, then 2 by 2, then 3 by 3 and upward -- and a fourth is stacked triangles, growing triangular arrangements of unit triangles. A fifth, stranger sequence is the Koch snowflake, where every straight edge of a shape is replaced by a small outward bump to produce the next, spikier shape in the sequence, again and again, forever.

Just like number sequences can secretly build each other, shape sequences secretly contain number sequences. Count the sides of the regular polygons -- 3, 4, 5, 6, 7, 8, 9, 10 -- and that is exactly the counting numbers, starting from 3. Count the corners of the very same shapes and the answer is exactly the same list, 3, 4, 5, 6, 7, 8, 9, 10, which makes sense once it is noticed that any closed shape made only of straight sides must have exactly as many corners as it has sides, since each side always joins exactly two corners and the whole loop closes up on itself with nothing left over. Count the connecting lines inside the complete graphs -- 1, 3, 6, 10, 15 -- and that is precisely the triangular numbers, because every one of those lines joins one distinct pair of dots, and counting all the distinct pairs from a set of dots is exactly how triangular numbers are built in the first place. Count the unit squares inside the stacked squares, or the unit triangles inside the stacked triangles, and both give the square numbers, 1, 4, 9, 16, 25, arrived at by two totally different shapes. Even the Koch snowflake hides a number pattern: its total line segments grow as 3, 12, 48, 192, 768, each one exactly 4 times the one before, because every existing segment splits into 4 new ones at every step.