Fractals in art: temple, textile and print built by repeating the whole at a smaller scale

What to assume they know

• Self-similarity: a shape that contains copies of itself — self-similarity, the magnify-a-part test, and the distinction between a repeated motif and a self-similar shape
• The Sierpinski carpet and gasket: what repeated removal leaves behind — the carpet and gasket, as reference constructions the artworks can be compared against
• Tiling, as met earlier in Ganita Prakash — the chapter says the reader has already seen tiling in Escher's prints
• The idea of scaling a shape down without changing its shape
• Reading a photograph as a geometric figure: identifying repeated units and levels in an image

What they should be able to do

• Apply the magnify-a-part test to a photograph or a textile and decide whether the structure is self-similar
• Identify, in the Khajuraho temple, what plays the role of the whole and what plays the role of the copy
• Count the levels of nesting visible in a given artwork and say what limits that count
• Explain why a physical object can only ever be an approximation of a fractal
• Distinguish a tiling from a fractal, given that both repeat a unit
• Describe the Fulani blanket's structure as nested diamonds and say which property makes it self-similar rather than merely patterned
• State, with the chapter's own hedge, the claim about the age of fractal structure in Indian temple architecture
• Design a small self-similar ornament by choosing a shape and a placement rule, and run it two levels

Where it usually goes wrong

• "The temple has a pattern carved on it." The self-similarity is in the massing — the shape of the building itself — not in surface decoration. A pattern carved on a plain box would not be self-similar at all.
• "Any repeated ornament is a fractal." Repetition side by side is a tiling. Repetition inside is self-similarity. The blanket is interesting precisely because it does the second, and the chapter's own sentence about diamonds inside diamonds is what marks the difference.
• "Escher's tilings are fractals." The chapter separates them: tiling is one theme of his work, fractals another. Smaller and Smaller is the fractal one because the same figure recurs at reduced scale, not merely repeated.
• "The print on the page is by Escher." It is stated to be inspired by his work. Getting this wrong is a factual error and an attribution error at once.
• "Real objects are fractals." No physical object can be, because every material has a smallest workable mark. They are finitely many levels of a fractal construction, which is a different and weaker claim — and the honest one.
• "Indian temples are the oldest fractal art." The chapter says perhaps, and an explanation that drops the hedge has strengthened a claim the textbook declined to make.
• "You need the drawing to know what comes next." You need the rule. Given the rule, the next level is determined; given only a drawing of two levels, it is not. This is the section-11 payoff and the reason the topic is mathematics rather than appreciation.

Questions to check understanding

• Given a photograph, decide whether the structure is self-similar and justify the answer by naming the part and the whole
• Distinguish tiling from self-similarity, with one example of each
• Count the levels of nesting visible in a supplied image
• Explain why a carved or woven object cannot have unlimited levels
• Given a shape and a placement rule, draw two levels of the resulting ornament
• Name a monument or craft tradition from the student's own region and say whether its ornament is nested or merely repeated — the competency-based form, and the form the chapter's list of five sites invites
• State a claim about the history of an artwork with the same degree of hedging the source uses

Examples worth working on the board

Values marked not in the book are an added reading of the printed images or reasoning added here; the chapter prints no answers here and this section of the chapter sets no exercises at all.

• The Khajuraho claim, with its hedge (Part II p.74). Indian temples are put forward as possibly the earliest place fractal structure was built, and the hedging word the chapter uses is perhaps. The named example is the Kandariya Mahadev Temple, at Khajuraho in Madhya Pradesh, with a completion date given as roughly 1025 C.E. The structure is described as a tall temple form assembled out of reduced versions of the whole form, each of those in turn carrying reduced versions again. That perhaps is printed and must survive into the explanation.
• The photograph (Part II p.74, colour photograph, captioned Kandariya Mahadev Temple, occupying most of the upper half of the page). An added reading of the image: the shot looks steeply upward at the tower, and the self-similar structure is legible as a spire flanked and overlaid by smaller spires of the same profile, with smaller ones again along their flanks.
• The other temple sites the chapter names (Part II p.74): Madurai, Hampi, Rameswaram and Varanasi, listed as places where fractal-like patterns also occur, with no photographs and no further detail. The chapter says among many others.
• The Fulani wedding blanket (Part II p.74, colour photograph, captioned Nigerian Fulani Wedding Blanket; the reading of it is given on Part II p.75). The chapter introduces it as an example from traditional African cultures and then states its structure: diamond-shaped patterns which contain smaller diamond-shaped patterns inside them, and so on. An added reading of the image: the woven field is arranged in vertical bands separated by narrow striped columns, with a red serpentine line running down the left, and the diamond motifs are set in stacked columns within the bands. The nesting the chapter points at is inside the individual diamonds.
• Escher (Part II p.75). Described as the modern master of fractal art and as Dutch. The chapter notes that some of his prints explore tiling, which the reader has met before, and names Smaller and Smaller as the famous fractal example, describing it as the same lizard pattern recurring at ever smaller scales.
• The honesty point a teacher must not miss (Part II p.75). The picture set under that paragraph is not captioned at all. The inspired-by statement is the last sentence of the running text immediately above the image — the chapter's own wording says the print took its inspiration from the Escher work just named, and it is not the print itself. That placement matters practically: anyone told to look for a caption will find none, and reusing the printed image on its own carries no inspired-by wording with it. An added reading of the printed image: interlocking lizard-like forms in green, black and white, arranged in fourfold symmetry about a centre where the forms become very small. An explanation must not caption it as Escher's work, and should not reproduce Escher's own print either.
• The test, applied to each of the three, as the explanation's spine. Not in the book: temple — ring one subsidiary spire, magnify it, land on the whole tower's profile: passes. Blanket — ring one diamond, magnify, land on a diamond containing diamonds: passes. Escher-inspired print — ring one lizard cluster, magnify, land on the same cluster: passes. Then the failure case, which the chapter does not supply: a border of identical repeated diamonds. Magnify one and you get a plain diamond, not the border. Fails.
• The level count, and the reason it is finite. Not in the book: Sierpinski's carpet has a level for every whole number; a carved tower has as many levels as the mason can cut, a blanket as many as the weave will resolve, a print as many as the ink will hold. Three or four is typical. The mathematics has no such floor, and this is exactly the difference between a model and the thing modelled.
• A buildable exercise, since the chapter sets none here. Not in the book: start with an equilateral triangle; on the middle of each side place a copy at half scale; repeat on every copy. Two rounds give 1 + 3 + 9 = 13 triangles. Change "half" to "one-third" and the ornament opens out; change the placement point and it changes character. That is enough to make the point that a rule, not a drawing, is what the artist is choosing.

Figures to have open

• The Kandariya Mahadev Temple photograph, or an equivalent upward view of a Nagara-style temple tower, with the capacity to ring one subsidiary spire and zoom it. The chapter's own photograph is on Part II p.74; an explanation should source its own image rather than reproduce the printed one, but the composition matters — a straight-on elevation will not show the nesting.
• The Fulani wedding blanket, or an equivalent textile with nested diamonds, again with room to ring one diamond. Chapter's own photograph, Part II p.74.
• A drawn negative example: a border of identical diamonds, side by side, no nesting. Must be drawn; the chapter supplies no counter-example anywhere in §4.1.
• A drawn tiling panel for section 8 — a plane covered by one repeated shape. Standard schematic.
• The self-similar-ornament construction of section 11, shown over two rounds. Standard schematic; the chapter sets no such exercise.
• **Do not source or reproduce M.C. Escher's Smaller and Smaller.** It is in copyright, and the textbook itself printed an inspired-by image rather than the work. An explanation should describe the print in words and show its own shrinking-figure pattern.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 4, §4.1 "Fractals", printed subheading "Fractals in Art", Part II pp.74–75. The section ends where §4.2 begins, part-way down Part II p.75.
• The temple claim, the 1025 C.E. date and the list of four further sites are on Part II p.74; the blanket photograph is on the same page and its geometric reading is the first paragraph of Part II p.75.
• The Escher paragraph and the inspired-by image are on Part II p.75.
• §4.1 sets no Figure it Out items in this subsection — the last exercises of §4.1 are on Part II p.73, under the Koch Snowflake.
• Part II p.102, SUMMARY, bullet 1 states that fractals are found in nature and in art, which is the only place the art strand is picked up again.