Patterns in Mathematics

— the art & science of noticing patterns —

📐1.1 What is Mathematics?

Mathematics = the search for patterns + the search for explanations of why those patterns exist.

💡 remember

Patterns exist everywhere — in nature, shopping, cooking, games, weather & technology! Maths is both an art and a science.

🌌Patterns in stars & planets → theory of gravitation → rockets to Moon & Mars!
🧬Patterns in genomes → helped diagnose & cure diseases

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🔢1.2 Patterns in Numbers

Number theory = branch of maths that studies patterns in whole numbers (0,1,2,3,...). The most basic pattern-type: number sequences.

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🎨1.3 Visualising Number Sequences

Pictures make sequences easier to understand! e.g. Triangular numbers form triangles of dots, Squares form square grids, Cubes form 3-D cube stacks.

✏️ fun fact

36 is BOTH a triangular number and a square number — 36 dots arrange perfectly as a triangle and as a 6×6 square!

⬡Hexagonal numbers: 1, 7, 19, 37, ... (dots arranged in hexagon shape)

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🔗1.4 Relations among Number Sequences

Sequences can connect to each other in beautiful ways!

➜

1+3

➜

1+3+5

➜

1+3+5+7

gives 1, 4, 9, 16 ... → the Square Numbers! 🟦

📌 key results

➕Sum of first n odd numbers = n² (Sum of first 10 = 100, first 100 = 10,000)
🔺Counting numbers added up & down (1, 1+2+1, 1+2+3+2+1...) → also gives Square Numbers
📚Sum of counting numbers 1+2+...+n → gives Triangular Numbers
🔷Adding consecutive triangular numbers (1+3, 3+6, 6+10...) → gives Square Numbers
📦Adding powers of 2 (1, 1+2, 1+2+4...) then +1 → gives next Power of 2
🎲Adding hexagonal numbers (1, 1+7, 1+7+19...) → gives Cube Numbers

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🔺1.5 Patterns in Shapes

Geometry = branch of maths that studies patterns in shapes (1D, 2D, 3D or more!). Shape sequences studied:

🔷Regular Polygons: triangle → quadrilateral → pentagon → hexagon → ... → decagon
🕸️Complete Graphs: K2, K3, K4, K5, K6 (dots joined by all possible lines)
🟩Stacked Squares & Stacked Triangles
❄️Koch Snowflake: each line "—" replaced by a speed-bump shape, again & again

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🧩1.6 Shape ↔ Number Sequences

Shape Sequence	Connected Number Sequence
Regular Polygons (sides = corners)	3,4,5,6,7,8,9,10 (Counting nos. from 3)
Complete Graphs (no. of lines)	1,3,6,10,15 (Triangular numbers)
Stacked Squares (little squares)	1,4,9,16,25 (Square numbers)
Stacked Triangles (little triangles)	1,4,9,16,25 (Square numbers)
Koch Snowflake (line segments)	3,12,48,192 (3 × Powers of 4)

⭐ golden rule

Triangular numbers × 6 + 1 → gives Hexagonal numbers (7, 19, 37, 61, 91...)

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I can list all 10 number sequences from Table 1

I can explain why 1,3,6,10,15 are "triangular numbers"

I can prove: sum of odd numbers = square numbers (with a picture)

I know 36 is both triangular AND square

I can name all shape sequences in Table 3

I can connect each shape sequence to its number sequence

I remember: Koch Snowflake → 3 × Powers of 4