BTT Mathematics / Primary Mathematics Learning Hub / Fractals and Self-Similarity
A self-similar construction repeats a rule so that smaller copies resemble the larger structure. Fractal mathematics studies what happens when such rules are repeated many times, and in ideal mathematical models, indefinitely.
This guide works with finite stages. A stage-n figure is a real, countable mathematical object. The infinite fractal is an ideal limit suggested by the repeating rule; it is not something a physical drawing literally completes.
The main questions are structural: How many pieces appear after n stages? How small is each copy? How do total length and total area change? These are repeated-multiplication problems with a geometric meaning.
This guide is Primary Mathematics enrichment. It extends Scale Drawings, Maps and Enlargement/Reduction, Patterns and Early Algebra and Geometric Dissections and Area Proofs.
Formal fractal dimension is optional later-study material. Use the MOE Primary curriculum page and the learner’s school programme for required content.
Self-similarity · Sierpiński triangle · Koch curve · Cantor construction · Sierpiński carpet · Growth and shrinkage · 24 questions · Worked answers
1. Separate number of copies from scale of each copy
A self-similar step often creates N smaller copies, each scaled by a length factor s where 0<s<1. These two numbers control different quantities.
Piece count multiplies by N. A one-dimensional length attached to each copy multiplies by s. Area of each copy multiplies by s².
Worked example A: Three half-size copies
Suppose every filled triangle is replaced by three copies with side length one-half of the parent. After one step there are three copies. After two steps there are 3²=9. After n steps there are 3ⁿ.
Worked example B: Side length after repeated halving
If the original side length is 16 cm, after four half-scale stages each smallest copy has side 16×(1/2)^4=1 cm.
Worked example C: Area scaling
Halving all lengths makes each copy’s area one-quarter of the parent. Three such copies therefore retain 3/4 of the parent’s total area at each stage.
2. Sierpiński triangle: count grows while total area shrinks
Stage 0 is one filled triangle. At each stage, each filled triangle is replaced by three corner triangles with side length one-half of the parent.
Worked example D: Number of small triangles
Stage 0:1
Stage 1:3
Stage 2:9
Stage 3:27
Stage 4:81
Worked example E: Side scale
At Stage n, each smallest triangle has side scale (1/2)^n of the original.
Worked example F: Total filled-area fraction
Each stage keeps three quarters of the previous filled area. Therefore Stage n has filled-area fraction (3/4)^n of Stage 0.
Worked example G: Stage 3 area
If the original triangle area is 64 cm², Stage 3 filled area is 64×(3/4)^3=64×27/64=27 cm².
Worked example H: Removed area after two stages
Filled fraction=(3/4)^2=9/16, so removed fraction=1−9/16=7/16.
3. Koch curve: segment count and total length both change
Stage 0 is one line segment. Each stage replaces every segment by four new segments, each one-third as long as the old segment.
Worked example I: Segment count
Stage n has 4ⁿ segments.
Worked example J: Length of one segment
Each Stage-n segment has length scale (1/3)^n of the original.
Worked example K: Total length
Total length factor at each stage is 4×1/3=4/3. Therefore Stage n total length is original length×(4/3)^n.
Worked example L: Original length 81 cm
At Stage 2, total length=81×(4/3)^2=81×16/9=144 cm. Each small segment has length81/9=9 cm, and there are16 of them.
Count can grow while pieces shrink
There are more segments at every stage, but each segment is shorter. Total length grows because the factor four in count outweighs the factor one-third in each segment length.
4. Cantor construction: count doubles while total retained length falls
Stage 0 is one interval. At each stage, remove the open middle third of every surviving interval, leaving two intervals each scaled by one-third.
Worked example M: Interval count
Stage n contains 2ⁿ surviving intervals.
Worked example N: Length of each interval
Each Stage-n interval has length scale (1/3)^n.
Worked example O: Total retained length
Each stage retains 2/3 of the previous total length. Hence retained length fraction=(2/3)^n.
Worked example P: Start from 81 cm
At Stage 3, each interval is81/27=3 cm. There are8 intervals, so total retained length=24 cm.
5. Sierpiński carpet: eight one-third-scale copies
Start with a square. Divide it into a 3×3 grid and remove the centre square. Repeat the same rule on every remaining filled square.
Worked example Q: Filled-square count
Stage n contains 8ⁿ smallest filled squares.
Worked example R: Side scale
Each smallest square has side scale (1/3)^n.
Worked example S: Filled-area fraction
Each stage keeps8/9 of the previous filled area, so Stage n retains (8/9)^n.
Worked example T: Stage 2 from area81
Filled area=81×(8/9)^2=81×64/81=64 square units. There are64 smallest filled squares, each area one square unit if the original side was9.
6. Repeated scale rules create exponential patterns
If a stage rule multiplies a quantity by the same factor r each time, then after n stages the quantity is initial×r^n.
Piece count may have r>1 while total area or length has r<1. Fractal constructions often combine simultaneous growth and shrinkage.
Compare the four models
Sierpiński triangle: copy count factor3, length scale1/2, area factor3/4.
Koch curve: segment count factor4, segment scale1/3, total-length factor4/3.
Cantor construction: interval count factor2, interval scale1/3, retained-length factor2/3.
Sierpiński carpet: copy count factor8, length scale1/3, area factor8/9.
Finite versus infinite
Every exercise in this guide asks about a finite stage unless it explicitly discusses the limiting trend. Saying a quantity approaches zero is different from saying a finite stage equals zero.
7. Practice: 24 original questions
Questions 1–8: Sierpiński triangle
1. How many smallest filled triangles are in Stage 1?
2. Stage 2?
3. Stage 5?
4. What fraction of the original side length does each Stage-4 smallest triangle have?
5. What fraction of original filled area remains at Stage 2?
6. What fraction remains at Stage 3?
7. Original area64 cm²: find Stage-3 filled area.
8. What fraction of area has been removed by Stage 2?
Questions 9–16: Koch and Cantor constructions
9. How many Koch segments are in Stage 3?
10. What fraction of original length is one Stage-3 segment?
11. What is the Stage-3 total-length factor for the Koch curve?
12. Original Koch length81 cm: find Stage-2 total length.
13. How many Cantor intervals survive at Stage 4?
14. What fraction of original length is each Stage-4 interval?
15. What fraction of total length survives at Stage 4?
16. Start with81 cm. Find total retained Cantor length at Stage 3.
Questions 17–24: Carpet and scaling reasoning
17. How many smallest filled squares are in Sierpiński-carpet Stage 2?
18. Stage 3?
19. What fraction of the original side length does each Stage-3 smallest square have?
20. What fraction of area remains at Stage 2?
21. Original area81: find Stage-2 filled area.
22. Which construction in this guide has total length increasing by factor4/3 per stage?
23. Which two constructions have total retained measure decreasing by factors3/4 and2/3 respectively?
24. Explain why “the infinite fractal has been completed after Stage 10” is mathematically incorrect.
8. Worked answers
Answers 1–8
1. 3.
2. 9.
3. 243. 3^5.
4. 1/16. (1/2)^4.
5. 9/16.
6. 27/64.
7. 27 cm².
8. 7/16.
Answers 9–16
9. 64. 4^3.
10. 1/27.
11. 64/27. (4/3)^3.
12. 144 cm.
13. 16. 2^4.
14. 1/81.
15. 16/81. (2/3)^4.
16. 24 cm.
Answers 17–24
17. 64. 8^2.
18. 512. 8^3.
19. 1/27.
20. 64/81.
21. 64 square units.
22. The Koch curve.
23. Sierpiński triangle:3/4; Cantor construction:2/3.
24. Stage10 is still a finite approximation. The ideal fractal is defined by continuing the rule without a final finite stage.
9. Teaching and transfer
If a learner sees only decorative patterns, record three columns at every stage: number of copies, scale of one copy and total measure. The mathematics becomes a set of linked multiplicative sequences.
When count growth and size shrinkage are mixed
Use separate arrows: copy count may multiply by3 while side length multiplies by1/2. Never combine those factors until the target quantity is named.
When area is scaled like length
Remind the learner that a length scale s gives area scale s². Three half-size triangles therefore contribute total area factor3×(1/2)²=3/4.
When an infinite claim is made from a finite drawing
Name the stage. A computer or paper diagram always shows a finite approximation. The ideal fractal is the mathematical limit of repeating the rule indefinitely.
When total length behaviour feels surprising
For the Koch curve, four times as many segments are created but each is only one-third as long. The product4×1/3=4/3 explains why total length grows.
Connect to scale factors
Every stage is a repeated enlargement/reduction rule, linking directly to Scale Drawings, Maps and Enlargement/Reduction.
Continue through this enrichment collection
For structured fraction approximations, use Continued Fractions, Convergents and Rational Approximation. For non-crossing recursive counts, use Catalan Numbers, Non-Crossing Paths and Balanced Structures. For 3D structural invariants, use Euler Characteristic, Polyhedra, Faces, Edges and Vertices.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. All numerical exercises concern finite construction stages.