Ganita Prakash · Chapter 1 — Patterns in Mathematics

Look around your home. The tiles on the floor repeat. The days of the week come back in the same order. The moon grows from a thin curve to a full circle and back again, month after month. Each of these is a pattern: something that repeats or grows by a rule we can describe.

A large part of mathematics is exactly this: finding patterns, and then explaining why they happen. The finding is fun, like solving a puzzle. The explaining is just as important, because once we know why a pattern works, we can trust it and use it in new places.

Why explanations matter. People watched the sun, moon and planets for thousands of years and noticed regular patterns in how they move. Explaining those patterns led to the idea of gravitation. That same understanding is what lets scientists today send satellites into orbit and rockets to the Moon and Mars. In the same way, spotting patterns in the genes of living things has helped doctors find and treat diseases.

Mathematics is both an art and a science. It is a science because every pattern must be checked and explained carefully. It is an art because discovering a pattern needs imagination, and a good explanation can be as beautiful as a painting.

Where do you meet patterns every day?

• Shopping: prices of 1, 2, 3 packets of biscuits at ₹10 each go 10, 20, 30, …
• Cooking: doubling a recipe doubles every ingredient.
• Games: in a cricket over, balls are counted 1 to 6 and then the count starts again.
• Weather: hot summers and rainy monsoons return every year.
• Technology: a phone stores everything as patterns of 0s and 1s.

How we will work in this chapter. First we look at number patterns, which belong to a branch of mathematics called number theory. Then we look at shape patterns, which belong to geometry. At each step, try to say the rule in your own words and then ask, ‘Why does this happen? Will it happen forever?’ Asking ‘why’ is the habit of a mathematician.

A number sequence is a list of numbers that follow a rule, written in order. The simplest patterns in mathematics use the whole numbers 0, 1, 2, 3, 4, … Here are ten famous sequences. Read each one and try to find the rule before you look at the last column.

Finding the next terms. Once you know the rule, the next terms follow. Triangular numbers after 28: add 8 to get 36, add 9 to get 45, add 10 to get 55. Virahānka numbers after 21: 13 + 21 = 34, 21 + 34 = 55, 34 + 55 = 89. Powers of 3 after 729: 2187, 6561, 19683.

Two kinds of rules. Some rules tell you how to get the next term from the one before, like ‘add 2’ or ‘multiply by 3’. Other rules tell you the term at any position directly: the 7th square is 7 × 7 = 49 without listing the first six. Both kinds are useful. The position rule is faster when you need a far-away term, such as the 20th square, 400.

The Virahānka numbers are named after an Indian scholar who found them while counting the rhythms that can be made from short and long syllables in poetry. The same numbers turn up in the spirals of sunflower seeds and pine cones.

Writing the rule clearly. A good rule is one that someone else can follow without asking you a question. ‘The numbers get bigger’ is not a rule; ‘start at 1 and add 2 each time’ is.

Many sequences become easy to understand when we draw them with dots. A picture shows why a number has its name.

Triangular numbers. Put 1 dot in the first row, 2 dots in the second, 3 in the third and so on. 1 dot, then 1 + 2 = 3 dots, then 1 + 2 + 3 = 6 dots, then 10, then 15. Each picture is a triangle, which is why 1, 3, 6, 10, 15, … are called triangular numbers.

Square numbers. Arrange dots in a square grid: 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16. The dots fill a square, so these are square numbers.

Cubes. Stack small blocks into a solid cube: 1 block, then a 2 × 2 × 2 cube of 8 blocks, then 3 × 3 × 3 = 27 blocks, then 64 and 125. That is why 1, 8, 27, 64, 125, … are called cubes.

Odd and even numbers. Even numbers can be drawn as two equal rows of dots: 2, 4, 6, … Odd numbers always have one dot left over: 1, 3, 5, … This picture explains why odd + odd is even: the two left-over dots pair up.

One number, two pictures. The number 36 is a triangular number (rows 1 + 2 + … + 8 = 36) and also a square number (6 × 6). So the same 36 dots can be arranged as a triangle or as a square. The way we draw a number depends on what we want to show.

Hexagonal numbers. Start with 1 dot. Surround it with a ring of 6 dots to make a hexagon of 7. Add the next ring of 12 to make 19, then a ring of 18 to make 37. The sequence is 1, 7, 19, 37, … The rings grow by 6 each time (6, 12, 18, 24), so the next number is 37 + 24 = 61.

Powers of 2. Take one square. Put two such pieces side by side, then double the whole shape again and again: 1, 2, 4, 8, 16 pieces. Each picture is two copies of the one before.

Sequences can be connected to each other in surprising ways. Watch what happens when we add odd numbers, starting from 1:

Every total is a square number! Adding the first 2 odd numbers gives 2 × 2, the first 3 give 3 × 3, the first 6 give 6 × 6. It looks as if adding the first n odd numbers gives n × n.

Will this go on forever? A picture explains why. Draw a 6 × 6 square of dots. Start at one corner with a single dot. Around it, an L-shaped strip of 3 dots makes a 2 × 2 square. The next L-shape has 5 dots and makes a 3 × 3 square. The next has 7, then 9, then 11. Each L-shape is an odd number of dots, and each one turns a square into the next bigger square.

Why is each L-shape odd? To grow a square of side 3 into a square of side 4, you add one row of 4 and one column of 4, but the corner dot is shared, so you add 4 + 4 − 1 = 7 dots. In general, growing side n − 1 to side n adds n + n − 1, which is always odd. Because this picture can be drawn for a square of any size, the pattern really does continue forever.

Using the pattern. The sum of the first 10 odd numbers, 1 + 3 + … + 19, is 10 × 10 = 100. The sum of the first 100 odd numbers is 100 × 100 = 10,000. No long adding is needed.

How many odd numbers are there? To use the rule, count the odd numbers. 1 + 3 + 5 + … + 25: the last odd number is 25, and 25 is the 13th odd number (because 2 × 13 − 1 = 25), so the sum is 13 × 13 = 169.

Check with arithmetic. 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 = 7 × 7. Seven odd numbers, seven squared. The picture and the sum agree.

Odd numbers and squares were only the start. Here are more links. In each case, check the first few terms with arithmetic, then try to see a picture.

1. Adding up and then down gives squares. 1 = 1; 1 + 2 + 1 = 4; 1 + 2 + 3 + 2 + 1 = 9; 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16. Picture a square of dots tilted on its corner like a diamond: its rows have 1, 2, 3, 4, 3, 2, 1 dots. So 1 + 2 + … + 99 + 100 + 99 + … + 2 + 1 = 100 × 100 = 10,000.

2. Adding counting numbers gives triangular numbers. 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10. Each sum is a row-by-row triangle. Adding the All 1’s sequence up (1, 1 + 1, 1 + 1 + 1, …) gives the counting numbers, and adding it up and down (1, 1 + 1 + 1, 1 + 1 + 1 + 1 + 1, …) gives the odd numbers.

3. Two neighbouring triangular numbers make a square. 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25. Picture: cut a square of dots along a diagonal staircase. One part is a triangle with rows 1 to n, the other a triangle with rows 1 to n − 1.

4. Powers of 2 add to one less than the next power. 1 = 1, 1 + 2 = 3, 1 + 2 + 4 = 7, 1 + 2 + 4 + 8 = 15, then 31. Add 1 to each: 2, 4, 8, 16, 32, the powers of 2 again. Why? 1 + 1 = 2, 2 + 2 = 4, 4 + 4 = 8: the extra 1 keeps doubling its way up.

5. Six triangles and a dot make a hexagon. Multiply triangular numbers by 6 and add 1: 6 × 1 + 1 = 7, 6 × 3 + 1 = 19, 6 × 6 + 1 = 37, 6 × 10 + 1 = 61. These are the hexagonal numbers. Picture: a hexagon of dots is a centre dot plus six equal triangles around it.

6. Adding hexagonal numbers gives cubes. 1 = 1, 1 + 7 = 8, 1 + 7 + 19 = 27, 1 + 7 + 19 + 37 = 64. Picture a cube built from blocks seen from one corner: it looks like a hexagon, and each new layer of the corner is a hexagonal number of blocks.

Numbers are not the only things that form patterns. Shapes do too, and the branch of mathematics that studies shapes is geometry. Shapes can be in one dimension (lines), two dimensions (flat figures) or three dimensions (solids). A shape sequence is a list of shapes in which each one follows from the one before by a rule.

1. Regular polygons. Triangle, quadrilateral (square), pentagon, hexagon, heptagon, octagon, nonagon, decagon. Each shape has one more side than the one before. They are called regular because all their sides have equal length and all their corners (angles) look the same.

2. Complete graphs. Mark some dots and join every dot to every other dot with a straight line. With 2 dots we get K₂, a single line. With 3 dots we get K₃, a triangle. Then come K₄ (a square with both diagonals), K₅ (a five-pointed star inside a pentagon) and K₆.

3. Stacked squares. A 1 × 1 square, then a 2 × 2 block of little squares, then 3 × 3, and so on. Each shape is a bigger square made of unit squares.

4. Stacked triangles. A triangle, then a bigger triangle made of small triangles with rows of 1 and 3, then rows of 1, 3 and 5, and so on. Some small triangles point up and some point down.

5. The Koch snowflake. Begin with a triangle. Replace every straight side ‘—’ by a ‘speed bump’: divide it into three equal parts and raise a small triangle on the middle part, so one segment becomes four. Do this again and again. The edges get more and more crinkly, with tinier and tinier segments, like the edge of a snowflake.

Describing the rule. As with numbers, a shape sequence is understood when you can say the rule in words and draw the next shape. For regular polygons: ‘add one side, keep all sides and corners equal’. For stacked squares: ‘make the side one unit longer’. For the Koch snowflake: ‘put a speed bump on every segment’.

Can you always draw the next shape? For polygons and stacked squares it is easy. For the Koch snowflake, after a few steps the segments become too small to draw by hand, even though the rule is perfectly clear. Mathematics can describe shapes we cannot fully draw.

Shape sequences become even more interesting when we count something in each shape. The counts often turn out to be number sequences we already know.

Why these numbers?

Polygons. Walk around a polygon: after each side you reach a corner, and after each corner a side. Sides and corners take turns, so a closed shape has exactly as many corners as sides.

Complete graphs. When a new dot is added to K₄ to make K₅, it must be joined to all 4 old dots, adding 4 lines: 6 + 4 = 10. Adding a 6th dot adds 5 lines: 10 + 5 = 15. So the counts are 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, …, the triangular numbers.

Stacked squares. A shape with side n has n rows of n little squares, so n × n of them.

Stacked triangles. Count by rows. The shape with 3 rows has 1, 3 and 5 little triangles in its rows: 1 + 3 + 5 = 9. That is adding odd numbers again, which we know gives squares. (The hint: row k has k triangles pointing up and k − 1 pointing down, so 2k − 1 in all.)

Koch snowflake. The first shape, a triangle, has 3 segments. Each step turns every segment into 4, so the count is multiplied by 4: 3, 12, 48, 192, 768. These are 3 × 1, 3 × 4, 3 × 16, 3 × 64, 3 × 256, that is, 3 times the powers of 4.

The big lesson. A shape and a number sequence can describe the same idea. Knowing one helps you understand the other: the stacked-triangle picture is another proof that odd numbers add to squares.