Patterns in Mathematics · Lesson 5 of 7
Rules in Growing Shapes
“Describe regular polygons and complete graphs, then connect their growth to number sequences.”
• Explain what geometry studies and recognise simple dimensional distinctions. • Describe a shape sequence using a rule for constructing its next figure. • Identify regular polygons from triangle to decagon. • Explain why a polygon has equal counts of sides and corners. • Count the lines of a complete graph and relate them to triangular numbers.
Patterns can be made of shapes
A pattern does not have to be written as numbers. A sequence may show figures changing from one stage to the next. We can ask the same questions as before: what stays the same, what changes, and what rule builds the next figure? Counting selected features of the figures often reveals a number sequence hidden inside the shape sequence.
The study of shapes, their properties, and their spatial relationships, including patterns made from shapes.
Some shapes are one-dimensional, such as an ideal line segment with length but no width. Flat figures such as triangles and squares are two-dimensional: they extend in two directions. Solid figures such as cubes are three-dimensional: they also have depth. Mathematics can study more dimensions, but the figures we investigate here can be drawn or modelled using these familiar ideas.
A shape sequence needs a construction rule. “It gets bigger” usually tells us too little: a figure might get longer, acquire more corners, contain more small pieces, or develop a more detailed boundary. Name the feature that changes so someone else can build the same next stage.
Regular polygons grow by adding a side
A polygon is a closed flat figure whose boundary is made of straight line segments. The segments are its sides; the points where adjoining sides meet are its corners, also called vertices. The regular-polygon sequence begins with a triangle and continues with figures having 4, 5, 6, and more sides.
A polygon with all sides equal in length and all its interior corner angles equal in size. An angle describes the opening where two sides meet.
The regular three-sided figure is an equilateral triangle, and the regular four-sided figure is a square. A four-sided polygon is also called a quadrilateral; it need not be a square unless the required equality of sides and angles is present. The names pentagon, hexagon, heptagon, octagon, nonagon, and decagon describe 5, 6, 7, 8, 9, and 10 sides respectively.
| Sides | Polygon name | Corners |
|---|---|---|
| 3 | Triangle | 3 |
| 4 | Quadrilateral; regular example: square | 4 |
| 5 | Pentagon | 5 |
| 6 | Hexagon | 6 |
| 7 | Heptagon | 7 |
| 8 | Octagon | 8 |
| 9 | Nonagon | 9 |
| 10 | Decagon | 10 |
Why do the side and corner counts agree? Walk once around the boundary. Each side takes you from one corner to the next. After the final side you return to the starting corner. There is one boundary step for each corner, so a polygon has as many sides as vertices. This argument applies to polygons with unequal sides as well as regular polygons.
Problem
How many sides and corners does a regular nonagon have?
- 1.A nonagon is a nine-sided polygon.
- 2.Walking around its boundary passes through one corner for each side.
- 3.It has 9 sides and 9 corners. Being regular additionally tells us that its sides and corner angles are equal within the figure.
Problem
The sequence shows a triangle, square, pentagon, and hexagon. Describe the next figure.
- 1.The side counts are 3, 4, 5, 6. Each new figure has one more side.
- 2.The next count is 7, so the next figure is a heptagon.
- 3.To keep the sequence regular, draw seven equal sides with equal corner angles. A rough freehand sketch can indicate the intended shape, but a carefully regular construction is more demanding.
Equal side lengths alone do not guarantee that a polygon is regular; its corner angles must also be equal. A tilted square remains a square, because rotating a figure changes its position rather than its side lengths or angles.
Every pair joined: complete graphs
Now place a small set of points and join every pair with a straight line. With two points there is one line; with three, there are three lines. With four points, include both the boundary lines and the two crossing diagonals. The resulting figures are complete graphs: “complete” means that every pair of the original points is connected.
A figure formed by joining every pair in a chosen set of vertices with one line. The connecting lines are called edges; each pair is joined only once.
The notation K₂ means a complete graph with 2 vertices, K₃ one with 3 vertices, and so on. In K₄, the two diagonals crossing inside the square are separate edges. Their crossing is not an extra vertex unless the construction specifically declares it one. The vertices are the points selected at the start.
When a new vertex is added, connect it to each existing vertex. Moving from K₂ to K₃ adds 2 new edges; moving to K₄ adds 3; moving to K₅ adds 4. Starting from the single edge of K₂, the totals become 1, 1 + 2 = 3, 1 + 2 + 3 = 6, and 1 + 2 + 3 + 4 = 10. These are triangular numbers.
Problem
A complete graph with 5 vertices has 10 edges. How many edges are in the next graph?
- 1.The next graph has 6 vertices. Keep all 10 existing edges.
- 2.Join the new vertex to each of the 5 old vertices, adding 5 edges.
- 3.The new total is 10 + 5 = 15 edges. The next larger addition explains the triangular-number pattern.
Problem
Explain why K₄ has 6 edges rather than 12.
- 1.Each of the 4 vertices joins to the other 3, which gives 4 × 3 = 12 endpoint-based counts.
- 2.A connection from A to B is the same edge as the connection from B to A, so this method counts every edge twice.
- 3.Halve 12 to obtain 6 distinct edges. A direct count gives four boundary edges and two diagonals, also 6.
Translate a construction into a number pattern
Regular polygons and complete graphs grow differently. A polygon’s boundary gains one side at each stage. A complete graph gains one vertex, and that vertex must be connected to every old one, so the number of new edges keeps increasing. The numerical counts help us compare these growth rules precisely.
| Shape sequence | Feature counted | Counts | Reason |
|---|---|---|---|
| Regular polygons | Sides or corners | 3, 4, 5, 6, … | One more side and corner per stage. |
| Complete graphs K₂, K₃, K₄, … | Edges joining vertex pairs | 1, 3, 6, 10, … | The new vertex joins to 2, then 3, then 4 old vertices. |
Redraw the regular-polygon and complete-graph sequences. Describe each rule in words, then try the next figure. Discuss why an accurate regular polygon is harder to draw than stating its side count, and why a larger complete graph becomes visually crowded even though its rule remains simple.
Quiz
Which field studies patterns in shapes?
A regular polygon must have:
How many corners does a decagon have?
Which is the regular four-sided polygon?
How many new edges are added when K₆ grows to K₇?
Why do complete-graph edge totals form triangular numbers?
Practice Problems
- Sketch regular polygons with 3 through 10 sides. Write the name, side count, and corner count beside each.
- Explain why a polygon has equal counts of sides and corners, even when it is not regular.
- A four-sided figure has four equal sides but unequal corner angles. Explain why it is not regular.
- Draw K₂, K₃, K₄, and K₅. Count every connection once and compare your totals with triangular numbers.
- Extend K₆ to K₇. Find the number of added edges and the total, starting from 15 edges in K₆.
- In K₄, a student counts the crossing of the diagonals as a fifth vertex. Explain why this does not match the stated construction.
- Describe one-dimensional, two-dimensional, and three-dimensional shapes using a segment, square, and cube. Say what kind of extent each adds.
- Compare the rule “add one side” with the rule “add a vertex and connect it to every old vertex.” Why do their number patterns grow differently?
Key Takeaways
• Geometry studies shapes and their relationships, including shape sequences. • A regular polygon has equal sides and equal corner angles. • A polygon has the same number of sides and vertices. • A complete graph joins every pair of chosen vertices once. • Complete-graph edge totals are triangular because each new vertex adds one larger group of connections.