Self-similarity: a shape that contains copies of itself

What to assume they know

• Reading a labelled sequence of diagrams as successive states of one process, not as three unrelated pictures
• That a shape can be scaled down without changing its shape
• Congruence and identical copies — that two shapes can be the same shape at different sizes
• Area of a square and of a triangle, at the level of "this piece is one-ninth of that piece"
• Powers with whole-number exponents, enough to read 8² and 8³ as repeated multiplication (Power Play material)
• Comfort with the idea of "keep going forever" as a description of a process, even without any formal notion of a limit

What they should be able to do

• State what self-similarity claims about a shape, in the form "this part, magnified, is the whole"
• Distinguish a self-similar shape from a shape that merely carries a repeated motif, and give a reason for the distinction
• Identify the two ingredients of a fractal construction: a starting shape and one rule that acts on every piece
• Explain why the rule must be applied to its own output, and what would happen if it were applied only once
• Read a step number off a printed figure by counting how many sizes of hole or piece are visible
• Name the natural examples the chapter offers and say, for each, what plays the part of "the whole" and what plays the part of "the copy"
• Explain why a printed picture is always a step in a sequence and never the fractal itself
• Predict the appearance of the next step of a construction given the rule and the current step

Where it usually goes wrong

• "The picture on the page is the fractal." It cannot be. Every printed figure has a smallest visible piece, and the defining property requires there to be no smallest piece. Every picture in §4.1 is a step. Say the step number out loud when you show one.
• "Self-similar just means it has a repeating pattern." Wallpaper repeats and is not self-similar. The test is not "does the motif recur" but "is a part of this shape a shrunk copy of the whole shape". Wallpaper's parts are motifs, not wallpaper.
• "Any shrinking sequence gives a fractal." A square with a smaller square drawn inside it, and a smaller one inside that, is a nest of squares, not a self-similar shape — the whole is not reproduced by its parts. What the carpet does differently is apply the rule to every surviving piece, not just one.
• "The rule is applied to the original shape over and over." It is applied to the output of the previous application. Apply the carpet rule to the original square three times and you get one hole, three times over; apply it to the output each time and you get 1, then 9, then 73 holes. The chapter's recursions on Part II p.71 only make sense on the second reading.
• "Nature's fractals go on forever too." They do not. A fern stops at a few levels, a tree at a handful, a coastline at the size of a grain of sand. The chapter says these are examples of the idea, not instances of the mathematics. This is worth saying because it is the difference between a model and the thing modelled.
• "Removing pieces forever must leave nothing." It does not follow, and the chapter's very next topic shows why: the number of surviving squares grows while the area they occupy shrinks.

Questions to check understanding

• Given a shape and a magnified detail of it, decide whether the shape is self-similar and justify the answer
• Given a starting shape and a rule, draw the next step
• Given a printed step figure, state which step it is and give the reason
• Explain in one or two sentences why no drawing can be a fractal
• Name a self-similar object in nature and identify which part plays the role of the copy
• Distinguish, with an example each, a pattern that repeats from a pattern that is self-similar
• Short competency-based prompts of the "would this construction ever stop" kind, which the chapter's own open questions model

Examples worth working on the board

Values marked not in the book are worked out here on the chapter's stated inputs; the chapter prints no answers anywhere in this chapter.

• The fern (Part II p.70, photograph, right-hand column). A single frond. The chapter's reading of it: the frond carries leaves that are smaller versions of the frond, and each of those carries sub-leaves that are smaller versions again. Two levels are visible in the photograph and the chapter asserts the pattern continues. For the explanation: the useful move is to circle one leaf, blow it up, and land on a picture indistinguishable from the original frond. That single movement is the definition.
• The natural examples the chapter lists (Part II p.70): a tree, where a trunk carries limbs, a limb carries branches, and a branch carries branchlets; and then clouds, coastlines, mountains and lightning, named without diagrams. For the explanation: the tree is the one to show, because the naming of the levels is done for you. Coastline is the one worth a second look — zooming into a bay reveals smaller bays, which is the same claim with no branching involved.
• The carpet rule, stated as three instructions (Part II p.70). Start with a square. Cut it into 9 smaller squares. Delete the central one. Then do the same three things to each of the 8 squares that survive, and keep going. Not in the book: the 9 pieces are a 3 × 3 arrangement, so each has side one-third of the parent's and area one-ninth. Nothing on Part II p.70 states the 3 × 3 layout in words; it is visible in the Step 1 figure.
• The printed step figures (Part II p.70, artwork, three panels plus an ellipsis). Step 0 is a solid filled square. Step 1 is the same square with one square hole at its centre, that hole being one-ninth of the area. Step 2 shows the central hole plus 8 smaller holes, one in each surviving square. Three dots follow Step 2. Not in the book: at Step 2 there are two sizes of hole on the page, one large and eight small; at Step 3 there would be three sizes; so the number of distinct hole sizes visible is the step number. That is the reading rule for section 9.
• The finished-looking picture (Part II p.71, top of page, captioned Sierpinski Carpet). It is drawn with several more levels than Step 2 and looks like the fractal. It is not: it is some later step. Not in the book: count the hole sizes in it and you get a finite number, which is exactly the proof that it is a step.
• The counter-example the explanation needs, which the chapter does not supply. Not in the book: a strip of identical diamonds repeated along a border. Magnify one diamond and you get a diamond, not the strip. So it repeats but is not self-similar. The chapter's Fulani blanket (Part II p.74) is the honest contrast — there the diamonds contain smaller diamonds, and it passes.
• **What the chapter says a fractal *is*** (Part II p.102, SUMMARY, first two bullets). Self-similar objects, found in nature and in art; and the named mathematical ones are reached by applying geometric operations repeatedly to generate a sequence of shapes that approaches the fractal. The second bullet is the chapter's own statement that the fractal is a limit of a sequence, and it is the single most important sentence in §4.1 for this topic.

Figures to have open

• The fern, with a zoom movement on one leaf. This is the chapter's own photograph (Part II p.70); an explanation can use any fern image, but the zoom must land on something that looks like the original, which is a property of the image and needs checking.
• The three carpet step panels, redrawn as a schematic with the 3 × 3 grid lines visible on Step 1 (the printed Step 1 does not show them, and the explanation needs them for section 5). Standard schematic.
• A tree diagram with trunk, limb, branch and branchlet labelled at four levels. Standard schematic; the chapter names the levels in words but prints no tree figure.
• A repeated-motif border to fail the test against. Standard schematic, must be drawn fresh — the chapter supplies no negative example.
• No photograph of a temple, blanket or print is needed here; those belong to Fractals in art: temple, textile and print built by repeating the whole at a smaller scale.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 4, "Exploring Some Geometric Themes", §4.1 "Fractals", Part II pp.70–71. The chapter's unnumbered opening paragraph on Part II p.70 states the two themes of the whole chapter.
• Within §4.1, the printed bold subheading "Sierpinski Carpet" begins on Part II p.70 and the worked recursions continue on Part II p.71.
• Part II p.102 carries the chapter SUMMARY; its first two bullets state what a fractal is and that the named examples arise as limits of shape sequences.
• Forward pointer inside the same chapter: §4.2 begins on Part II p.75, so §4.1 occupies Part II pp.70–75 in total.
• The chapter states that the Koch Snowflake was met in Class 6 Ganita Prakash (Part II p.73), which is the only cross-grade pointer in §4.1.