Fractals: self-similarity generates a GP

What to assume they know

• The common ratio and arⁿ⁻¹ (Common ratio, and why the nth term is ar^(n−1))
• Explicit and recursive rules, and the role of the seed (A recursive rule builds each term from the ones before it)
• Powers with whole-number exponents, and that 3⁰ = 1
• Multiplying fractions, and comparing fractions below 1 (What "rational" means, and why the denominator cannot be zero)
• The midpoint of a side of a triangle, and that joining midpoints cuts a triangle into four congruent pieces
• Area of a triangle and of a square as a fraction of a whole (A triangle is half a parallelogram)

What they should be able to do

• Carry out the Sierpiński triangle construction and draw the next stage from the current one
• Count the black pieces at each stage and identify the sequence as a GP by its ratio
• Explain why the count must triple, from the construction rather than from the data
• Express the piece count at stage n as a power of 3, matching exponent to stage number
• Compute the black area at each stage and express it as a power of 3/4
• Say what happens to the count and to the area as the stages continue, and why the two go opposite ways
• Distinguish an explicit rule indexed by stage number from a step rule indexed from 1, and convert between them
• Apply the same analysis to the Sierpiński square carpet and report both its sequences

Where it usually goes wrong

• "More pieces means more area." At Stage 5 the triangle has 243 black pieces and less than a quarter of the area it started with. Counting pieces and measuring area are different questions about the same picture, and here they answer in opposite directions.
• "The area reaches zero." Every term of (3/4)ⁿ is positive, so the area is never 0; it gets as close to 0 as you like. The chapter's own wording on p. 191 is that it approaches 0, and an explanation should not upgrade that to arrival.
• "Removing the middle removes a quarter of the length." It removes a quarter of the area. The four sub-triangles are congruent, so each is one quarter of the whole, while each side of a sub-triangle is half the original side. Keeping area and length apart is the hinge of section 6.
• "You must count the pieces at each stage to know the pattern." You do not, and that is the argument. Because every piece is treated identically, the ratio follows from the instruction. Counting only confirms it.
• "Stage 1 is the first term, so the rule is 3ⁿ⁻¹." With the chapter's stage numbering, Stage 0 exists and holds one piece, so the count is 3ⁿ with the exponent equal to the stage. The chapter's own step rules use a different index, which is exactly why section 9 exists.
• "Fractals are just decorative." The boxed passage lists cauliflower, broccoli, branching trees, snowflakes and coastlines, and the chapter's point is that a simple repeated rule can produce that complexity.
• "The carpet works the same way as the triangle." The ratios are 8 and 8/9 rather than 3 and 3/4. The method transfers; the numbers do not.

Questions to check understanding

• Draw the next stage of a given fractal construction
• Count the pieces at each of the first few stages and identify the ratio
• Write the piece count at stage n as a power, and say what the exponent counts
• Compute the remaining area at a stated stage as a fraction and as a decimal
• Give both the explicit and the step rule for a fractal's count and for its area, and state which index base each uses
• Say what happens to the count and to the area as the stages continue, with a reason
• Transfer the whole analysis to a new construction — the square carpet is the chapter's own version of this question

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data; this book prints no answer key.

• The construction, as printed (§8.6.1, p. 188). Stage 0 is a paper equilateral triangle. Join the midpoints of its three sides, which produces four smaller equilateral triangles, and remove the central one — that leaves Stage 1 with a triangular hole. Repeat on each of the three black triangles to reach Stage 2, and again for Stage 3. The page says the process can be continued without end, names the result, and states that fractals are treated properly in a later class.
• Fig. 8.7 (p. 188). Four small figures in a row, labelled Stage 0 to Stage 3, drawn in solid black on white: a filled triangle, then the same with a central hole, then with three more holes, then with nine more. Read off the printed page: the holes are drawn as white gaps, so the black region is what remains rather than what is added.
• The Think and Reflect that drives the section (p. 189), four parts: count the black triangles in Stages 0 to 3; predict Stages 4 and 5; find a rule for stage n; and then, taking Stage 0's area as 1 square unit, find the black area at Stages 1, 2 and 3, predict Stages 4 and 5, find a rule, and say what happens to that area as the stages continue. The whole topic is the answer to this box.
• The counts, as printed (p. 189): 1, 3, 9 and 27 at Stages 0 to 3, then 81 and 243 at Stages 4 and 5. The page states the reason — each black triangle is replaced by three smaller ones — and identifies the sequence as a GP with ratio 3, noting that every term is a power of 3 and that the exponent matches the stage number, so stage n holds 3ⁿ pieces.
• The areas, as printed (p. 189). Stage 0 is 1 square unit. Stage 1 keeps three of four equal parts, so 3/4. Stage 2 keeps three quarters of that, printed as 3/4 × 3/4 = (3/4)². The page then asks the reader to explain why stage n has area (3/4)ⁿ, and states that this too is a GP, with each term three quarters of the last. Verified: Stage 3 is 27/64 = 0.421875, Stage 4 is 81/256 = 0.31640625, and Stage 5 is 243/1024, about 0.237.
• Table 1 (p. 190), printed with a shaded label column. Row 1, Stage (n): 0, 1, 2, 3, 4, 5, an ellipsis column, then n. Row 2, the piece counts, written as both value and power: 1 = 3⁰, 3 = 3¹, 9 = 3², 27 = 3³, 81 = 3⁴, 243 = 3⁵, then 3ⁿ. Row 3, the shaded area: 1, 3/4, (3/4)², (3/4)³, (3/4)⁴, (3/4)⁵, then (3/4)ⁿ. The table is the cleanest artefact in the chapter for showing exponent-matches-stage.
• The four rules, as printed (p. 190). Explicit: tₙ = 3ⁿ for the count and sₙ = (3/4)ⁿ for the area. Step rules: t₁ = 1 with tₙ = 3tₙ₋₁ for n ≥ 2, and s₁ = 1 with sₙ = (3/4)sₙ₋₁ for n ≥ 2.
• The index clash inside those four rules, confirmed on p. 190 and the material of section 9. The explicit rules are indexed by stage, so they give 3⁰ = 1 and (3/4)⁰ = 1 at stage 0. The step rules are indexed from 1 and seeded at 1, so their t₁ and s₁ are the stage-0 values. Both descriptions are right about the sequence and they disagree about what "term 1" names: substituting n = 1 into tₙ = 3ⁿ gives 3, while the step rule's t₁ is 1. Verified by substitution. An explanation must pick one indexing, say which, and stay in it — and this is worth showing openly, because a student who tries to check one rule against the other will otherwise conclude that one of them is wrong.
• The claim about the two directions (p. 190): as the pieces multiply the total black area shrinks. §8.6.2 sharpens this on p. 191, saying the area gets closer and closer to 0 as the stages go on.
• The biographical note (p. 188, a green box with a photograph): Wacław Sierpiński, 1882–1969, a Polish mathematician, with the gasket named as one of the earliest well-known fractals.
• Fig. 8.8 and the fractal box (p. 190). Three photographs — four heads of a cauliflower-like vegetable in one frame, lined up smallest to largest across it, a bare branching tree, and a snowflake — beside a boxed passage stating that fractals repeat at different scales, that zooming into a part shows something like the whole, and listing tree branching, cauliflower and broccoli, snowflakes and coastlines as natural examples. The four heads are worth a beat of their own: the same form at four sizes is what self-similarity looks like in a photograph, so that frame is doing the topic's work rather than decorating it.
• The square carpet (Exercise Set 8.3 item 7 and Fig. 8.12, p. 194). Stage 0 is a paper square. Trisect each side, join the trisection points of opposite sides to make nine smaller squares, remove the centre one and keep the other 8, then repeat on all eight. Fig. 8.12 shows Stages 0 to 3 as solid red squares with white holes. The question asks for the count of red squares in Stages 0 to 3, a prediction for Stages 4 and 5, a rule with both explicit and step forms, and the same four things for the area starting from 1 square unit. Verified: counts 1, 8, 64, 512, then 4096 and 32768, with 8ⁿ at stage n; areas 8/9, (8/9)², (8/9)³, then (8/9)⁴ ≈ 0.624 and (8/9)⁵ ≈ 0.555, with (8/9)ⁿ at stage n. The area here also falls toward 0, but far more slowly than the triangle's — a useful comparison, since 8/9 is much closer to 1 than 3/4 is.

Figures to have open

• Fig. 8.7 redrawn as a schematic: Stages 0 to 3 with the black region solid and the removed triangles left white. The construction must be legible at Stage 1 — midpoints marked, the removed piece indicated — because sections 1 and 6 both read off it. The chapter's own figure is on p. 188.
• A single-piece replacement diagram: one black triangle, an arrow, three smaller black triangles and one white hole. This is the figure that carries section 4, and the chapter does not draw it. Standard schematic.
• Table 1 reproduced as a clean three-row table with the powers shown, from p. 190. This is the chapter's own table and the exponent argument depends on its layout.
• A two-axis comparison for section 8: stage number across, with the piece count and the area drawn as two separate traces on their own scales. Related to the chapter's Figs. 8.10 A and 8.10 B (p. 192), which are treated in A GP plots as a curve, and what that curve tells you.
• Fig. 8.12 redrawn for section 10, Stages 0 to 3 of the square carpet, from p. 194.
• The natural-fractal photographs of Fig. 8.8 (p. 190) are optional. If used, source equivalent images rather than reproducing the printed ones; a branching tree, a cauliflower or broccoli head, and a snowflake are what the passage names.

Where this sits in the book

• Chapter 8, §8.6.1 Fun with Fractals, pp. 188–190. The construction and Fig. 8.7 are on p. 188; the counting and area arguments are on p. 189; Table 1, the four rules and the fractal box are on p. 190.
• The green biographical box on Wacław Sierpiński, p. 188, which is where the word gasket appears.
• The Think and Reflect on p. 189, whose four parts this topic answers in order.
• The boxed passage on fractals and Fig. 8.8, p. 190.
• Exercise Set 8.3 item 7 with Fig. 8.12, p. 194 — the square carpet, which is the transfer task.
• The summary's closing bullet, p. 196, which names both the triangle and the square carpet as sources of geometric progressions.
• Deliberate cross-references outside this topic: the plots of these two sequences are Figs. 8.10 A and 8.10 B on p. 192, treated in A GP plots as a curve, and what that curve tells you; the chapter notes that fractals return in a later class.