Fractals
Self-similar shapes with fractional dimension.
Key idea
Self-similar shapes with fractional dimension.
In depth
Fractals are shapes with detail at every scale, often self-similar. Their dimension can be non-integer. Examples: Koch snowflake (infinite perimeter, finite area), Sierpinski triangle.
Worked example
Koch curve: each step multiplies length by 4/3; after n steps length = (4/3)ⁿ, which grows without bound.
Formula
Real-world uses
- Modelling coastlines, tree branches and fern leaves
- Compression in computer graphics
Common mistake
Assuming a finite region must have a finite boundary.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Fractals?
Self-similar shapes with fractional dimension. It is a topic at the expert level in Geometry.
What is the formula for Fractals?
The key formula is D = log N / log s.
Where is Fractals used in real life?
Real-world examples: Modelling coastlines, tree branches and fern leaves; Compression in computer graphics.
What should I learn before Fractals?
Make sure you are comfortable with Golden Ratio, the previous topic in Geometry.
What is a common mistake with Fractals?
Assuming a finite region must have a finite boundary.
How can I practise Fractals?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Fractals?
Move on to Non-Euclidean Geometry next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.