Exploring Some Geometric Themes — Class 8 Mathematics (Ganita Prakash Part 2)

What the book actually covers (2026-27) This chapter is Chapter 4 of Ganita Prakash Part 2, pages 70–102 — the longest chapter in the book. It explores exactly two themes: 4.1 Fractals — self-similar shapes: the Sierpinski Carpet, the Sierpinski Gasket and the Koch Snowflake, plus fractals in nature and in art. 4.2 Visualising Solids — profiles, nets, shortest paths on a cuboid, projections (front, top and side views), and isometric drawing. It contains no congruence criteria, no similarity of triangles, no circle theorems, no symmetry chapter and no transformations. Congruence appears only as a tool, used in passing to show that four small triangles are identical. All of that other material has been moved to the appendix at the end of this page.

Part A — Fractals

1. What makes a shape a fractal?

A fractal is a shape that contains smaller copies of itself, and those copies contain smaller copies again, without end. The same pattern reappears at smaller and smaller scales. This property is called self-similarity.

A shape is not a fractal because it is complicated. It is a fractal because its complication repeats as you zoom in.

In nature. The fern is the chapter's opening example — the whole frond has leaves shaped like smaller ferns, whose sub-leaves repeat the pattern again. A tree has a trunk with limbs, limbs with branches, branches with branchlets. Clouds, coastlines, mountains and lightning all behave the same way: a magnified piece of coastline looks much like a longer stretch.

In art — and some of the oldest examples are Indian. The Kandariya Mahadev Temple at Khajuraho, Madhya Pradesh, completed around 1025 CE, is a tall structure built from smaller copies of the whole structure, which carry still smaller copies. Fractal-like patterns also appear in temples at Madurai, Hampi, Rameswaram and Varanasi. Nigerian Fulani wedding blankets carry diamonds inside diamonds. The Dutch artist M.C. Escher is the modern master — his print 'Smaller and Smaller' repeats a lizard pattern at ever-decreasing scales.

2. The Sierpinski Carpet

Take a square. Break it into 9 smaller squares and remove the central one. Repeat on each of the 8 that remain.

Let Rₙ be the number of remaining squares and Hₙ the number of holes at step n.

Set up the recurrences from the construction.

• Every surviving square is cut into 9 and loses its centre, so it leaves 8 survivors: Rₙ₊₁ = 8Rₙ
• Every surviving square makes one new hole, and old holes remain: Hₙ₊₁ = Hₙ + Rₙ

Unroll them.

R₀ = 1, R₁ = 8, R₂ = 8², so Rₙ = 8ⁿ

Hₙ = 1 + 8 + 8² + … + 8ⁿ⁻¹. Multiplying by 8 and subtracting gives 7Hₙ = 8ⁿ − 1, so

Hₙ = (8ⁿ − 1) ÷ 7

The area. Each step keeps 8 of 9 parts, so Areaₙ = (8/9)ⁿ, taking the starting square as 1. Counting the other way gives the same: 8ⁿ squares each of area (1/9)ⁿ.

3. The Sierpinski Gasket (Triangle)

Take an equilateral triangle. Join the midpoints — this cuts it into 4 identical equilateral triangles — and remove the middle one. Repeat on the 3 that remain.

Why joining the midpoints gives 4 identical triangles

Let ABC have side 2a, with D, E, F the midpoints of BC, CA, AB.

Take the corner triangle AFE. Since F and E are midpoints, AF = AE = a, so it is isosceles, and its apex angle is ∠A = 60°. The base angles are then (180° − 60°) ÷ 2 = 60° each — so AFE is equilateral with side a. The same holds at B and C.

Since AFE is equilateral, FE = a. Likewise DF = a and ED = a, so the middle triangle DEF is equilateral with side a too. All four are congruent by SSS, each a quarter of the original area.

The counts

By the same method as the Carpet, with 3 survivors instead of 8:

Tₙ = 3ⁿ triangles and Hₙ = (3ⁿ − 1) ÷ 2 holes, with Areaₙ = (3/4)ⁿ

Both Sierpinski fractals have zero area. Each step multiplies the area by a fixed fraction below 1 — 8/9 for the Carpet, 3/4 for the Gasket — so repeating forever drives the area to zero. The number of pieces grows without limit while the area they cover vanishes.

4. The Koch Snowflake

Named after the Swedish mathematician Von Koch, who described it in 1904.

Start with an equilateral triangle. On every side: divide it into 3 equal parts, raise an equilateral triangle on the middle part, and remove the middle part. Each side is replaced by a four-piece bump. Repeat.

How many sides? One side becomes: first third, the two sloping sides of the new triangle, last third — that is 4, not 5. The middle third is gone.

Sₙ = 3 × 4ⁿ

How long is each side? Each step divides sides by 3, so Lₙ = (1/3)ⁿ.

Perimeter = sides × length:

Pₙ = 3 × 4ⁿ × (1/3)ⁿ = 3 × (4/3)ⁿ

An infinite perimeter around a finite area. Each step multiplies the perimeter by 4/3, a factor greater than 1, so the perimeter grows without bound — go far enough and a snowflake fitting on this page has a boundary longer than a kilometre. But the shape always stays inside the circle drawn round the original triangle, so the area stays finite (it settles at 8/5 of the starting triangle). This is why a coastline has no well-defined length: measure it with a shorter ruler and you get a longer answer.

Note the contrast with the Sierpinski fractals, which remove material and lose area. Koch adds material and gains perimeter.

Part B — Visualising Solids

5. Profiles

What you see of a solid is its profile from one viewpoint — the outline it would leave if it punched a hole through a wall. The same solid can have wildly different profiles.

No profile identifies a solid uniquely. A circle could be a sphere, a cylinder end-on, a cone from above or a flat disc. This is the observation the whole of Part B is built on.

Two cutting puzzles worth doing

Cut the four corners of a square, each cut running between the midpoints of adjacent edges. What is left is the tilted midpoint square, of half the area. The four corner triangles are congruent right isosceles triangles; set their right angles together at one point and their hypotenuses form a second square congruent to the first. The original square has been divided into two equal squares — the halving construction of Chapter 9 again.

Cut the corners at the thirds.

The difference is the corner angle. At 60° the cut-off piece is equilateral and the cut edge equals 1/3. At 90° the cut-off piece is right-angled, so its hypotenuse is √2 times the legs. (To make the octagon regular the marks would need to sit at 1/(2 + √2) ≈ 0.293 of the side.)

6. Faces, edges, vertices — and nets

Faces are the flat surfaces bounding a solid, edges the line segments where faces meet, vertices the points where edges meet. A cube has 6 faces, 12 edges and 8 vertices.

A prism has two congruent polygons as opposite faces with the other faces parallelograms. A pyramid has a polygonal base and an apex joined to every base vertex. (A triangular pyramid is a tetrahedron.)

Both satisfy Euler's relation V − E + F = 2 — a free check that costs nothing. Notice a pyramid always has as many faces as vertices; a prism never does.

Nets

A net is the shape obtained by unfolding a solid onto a plane. (Gluing flaps are not part of the net.)

Two nets count as the same if one is a rotation or flip of the other.

Testing a figure for being a cube net:

1. Exactly 6 squares.
2. A band of four that wraps the cube, with the two remaining squares attached on opposite sides of the band.
3. No 2 × 2 block — four squares meeting at one point can never fold, since two land on the same face.
4. All six joined edge to edge in one piece.

Finding all 11 is easiest by sorting on the longest straight row: 6 nets with a row of four, 4 with a row of three, 1 with a row of two, and none with a row of five or six.

Net of a cylinder (radius r, height h): two circles of radius r plus a rectangle 2πr × h — one side the height, the other the base circumference. Peel a label off a tin and you have it. This is where curved surface area = 2πrh comes from.

Net of a cone (radius r, slant height l): a sector of radius l plus a circle of radius r. Every point of the base rim is at distance l from the apex, and unrolling preserves distances, so the boundary is an arc centred at the apex. The arc is the base circumference 2πr, so the sector's angle is (r/l) × 360°.

Why a sphere has no net. A cylinder's curved surface unrolls because it is made of straight lines; a cone's does too. A sphere has no straight lines on it anywhere, so flattening a piece must either tear or wrinkle — try pressing an orange peel flat. This is a genuine impossibility, and it is why every flat map of the Earth distorts area, shape, distance or direction.

7. Shortest paths on a cuboid

An ant on a cuboid must reach a laddu, travelling only along the surface. What is the shortest route?

The key fact: unfolding a cuboid does not stretch the surface. So a path on the solid becomes a path of exactly the same length on the net, and vice versa. On a plane, the shortest route is a straight line. Therefore:

Unfold the cuboid, join the two points with a straight line, and fold back.

This also gives a test: draw any proposed path on the net. If it becomes straight, it is shortest; if it stays bent, it is not.

But the unfolding matters — two traps.

The straight line can leave the net. A segment falling outside the unfolded shape corresponds to no path at all on the cuboid, so that unfolding must be discarded.

Different unfoldings give different lengths. Each one is a separate candidate. Every valid unfolding must be checked and the smallest taken.

Worked example — a room 8 m × 6 m × 4 m, corner to opposite corner.

Crawling along three edges takes 8 + 6 + 4 = 18 m, so the best unfolded route saves over 5 m. The pattern: pair the longest dimension against the sum of the other two, which splits the total most evenly between the legs.

The chapter's own harder example uses a 30 × 12 × 12 box and shows two unfoldings giving 42 cm (= 6 + 30 + 6, a level line) and 40 cm (legs 24 = 12 + 12 and 32 = 1 + 30 + 1, since 24² + 32² = 1600). It then leaves the problem open, noting that all unfoldings must be listed before the true minimum is known.

Every one of these calculations is the Baudhāyana-Pythagoras theorem from Chapter 9, doing the work once the net is drawn.

8. Projections — front, top and side views

Point O is the projection of point P on plane M if OP is perpendicular to the plane. The projections of all an object's points form its projection.

What projection does and does not preserve

Lengths shrink. If a segment has true length l and projection p, then dropping a perpendicular gives a right triangle with l as hypotenuse and p as a leg, so

l² = p² + (something)² , hence p ≤ l

with p = l exactly when the segment is parallel to the plane, and p = 0 when it is perpendicular.

Parallelism survives. Two parallel segments generate two parallel planes of projecting lines, and two parallel planes cut the projection plane in two parallel lines. So:

Parallel lines always project to parallel lines.

Consequences: a parallelogram always projects to a parallelogram (never a trapezium), possibly squashed to a segment. A square projects to a square, rectangle, general parallelogram or segment. A regular n-gon projects to an n-sided convex polygon that is usually not regular — congruent only when its plane is parallel to the projection plane.

Straightness and parallelism survive; lengths and angles do not.

Why three views

A single projection does not determine the object — the chapter shows different lines and different cuboids sharing one. So three mutually perpendicular planes are used:

Each loses a different dimension, so together they usually pin the solid down.

For a line with axis differences a, b, c, the three views have lengths √(a²+c²), √(a²+b²) and √(b²+c²), so

(front)² + (top)² + (side)² = 2 × (true length)²

Views of standard solids

The pyramid and the cone have identical front and side views in outline; only the top view separates them. And notice it is the lines inside a view — the diagonals — not the outline, that identify the pyramid.

Checks when drawing three views: front and top share a width; front and side share a height; top and side share a depth.

Shadows

A torch shadow looks like a projection but is scaled and slightly distorted, because rays spread from a point. Move the torch back and the shadow approaches the true projection. An infinitely distant torch would give exactly the projection — and one exists: the Sun. When sunlight falls perpendicular to a plane, its shadows are projections. So you can test the parallelogram claim outdoors with a cardboard cutout.

Solids built from cubes

Treat each view as a shadow on a grid: shade a cell whenever at least one cube lies along that line of sight. Front view uses (width, height); top view uses (width, depth); side view uses (depth, height).

Hidden cubes count. In the chapter's stack, only 10 cubes are visible — 4, then 3, then 2, then 1. But every bottom cube shows its whole top face, and the row above sits between them rather than squarely on them. Cubes cannot sit half on one cube and half on another, so each row is one step back as well as one step up. It is a step pyramid with square layers: 16 + 9 + 4 + 1 = 30 cubes. The other twenty are hidden — and they must be there to hold the upper layers up.

To rebuild a solid from three views: place a cube at (x, y, z) whenever (x, y) is in the top view, (x, z) is in the front view and (y, z) is in the side view. This maximal solid works if any solid works, since any valid solid must be contained in it.

9. Isometric projection

Balance a cube on one corner, so its long diagonal is vertical, and project it down.

Why all the edges project equally. Rotating a third of a turn about that diagonal maps the cube to itself and sends each edge direction to the next. So the three directions are interchangeable and must be tilted equally to the vertical — hence all twelve edges project to the same length (√6/3 ≈ 0.816 of the true length). Isometric means equal measure in Greek.

The outline is a regular hexagon — six equal sides from six equal edge-projections, and six equal angles from the three-fold symmetry. Tile the plane with hexagons and you get the isometric grid.

Drawing on it

The grid has three directions, and each stands for one axis:

• vertical (|) → height
• one diagonal (↗) → depth
• the other diagonal (↘) → length

Fix this correspondence at the start and never change it — switching part-way is the commonest reason a drawing will not close. Draw edge by edge, counting units along one axis at a time, and decide each edge's direction as well as its axis. Draw faintly first, darken visible edges last.

Because parallel edges project to parallel lines and unit distances along all three axes project equally, lengths can be measured off isometric paper directly along all three directions — which is why engineers use it.

Gluing four cubes

Figure 4.8 shows the 5 ways of arranging four squares in a plane. Gluing four cubes allows more, because you can leave the plane.

First, a subtlety: in 2D, S and Z are different pieces and so are L and J — that is why Tetris has seven. In 3D you can turn a shape over, so those pairs merge and the flat arrangements number 5.

Three more leave the plane: the tripod (an L of three cubes with the fourth on top of the corner) and the two screws, a left-handed and a right-handed version that are mirror images and cannot be rotated into each other.

Total: 8 arrangements counting mirror images separately, or 7 treating them as the same. With real cubes you cannot turn a left screw into a right one, so 8 is the count that matters at the table.

Impossible figures

Isometric projection throws away depth — near and far edges are drawn the same size. Two points far apart in space can therefore be drawn touching, and the eye has no evidence to object.

The rolling ball appears to descend along every stretch and yet returns to its starting point. Impossible: a ball that only descends must end lower than it began.

The impossible triangle shows three bars each apparently perpendicular to the next, closed into a loop. Follow the loop and after three perpendicular turns you should be pointing along a third axis, not back at the start — no such object exists. You can build a sculpture that looks like it from one viewpoint, by leaving a real gap between two arms, one far behind the other; its top and side views give the trick away by showing the shape is open.

Why the illusion works: every part is locally correct — cover any corner and the rest is perfectly buildable — but your brain reads each apparent junction as a real join and tries to assemble one consistent object. None exists. The figure is locally honest and globally impossible.

That is the price of parallel projection, and it is precisely why engineering drawings carry three orthographic views as well as an isometric one.

10. Common mistakes

1. Counting five pieces per Koch side. The middle third is removed — it is 4, giving 3 × 4ⁿ.
2. Counting only visible cubes. Ask what supports the upper rows. The chapter's stack looks like 10 and is 30.
3. Accepting a 2 × 2 block in a cube net. Four squares at a point can never fold.
4. Trusting one unfolding for a shortest path. Every unfolding gives a different candidate.
5. Believing one view identifies a solid. It never does.
6. Views that disagree on shared dimensions. Check width, height and depth across pairs.
7. Calling the corner-cut octagon regular. Equiangular is not regular.
8. Changing the axis-to-grid correspondence mid-drawing.

11. Chapter summary

• A fractal is self-similar — smaller copies of itself at every scale.
• Sierpinski Carpet: Rₙ = 8ⁿ, Hₙ = (8ⁿ − 1)/7, area (8/9)ⁿ.
• Sierpinski Gasket: 3ⁿ triangles, (3ⁿ − 1)/2 holes, area (3/4)ⁿ. Both areas → 0.
• Koch Snowflake: 3 × 4ⁿ sides, perimeter 3 × (4/3)ⁿ → ∞, with finite area.
• Prism: F = n + 2, E = 3n, V = 2n. Pyramid: F = n + 1, E = 2n, V = n + 1. Both give V − E + F = 2.
• Nets: tetrahedron 2, cube 11, octahedron 11, dodecahedron 43,380, sphere none.
• Shortest path on a cuboid: unfold, straight line, Baudhāyana theorem — try every unfolding.
• Projection preserves straightness and parallelism, never lengths or angles; p ≤ l always.
• Three views are needed because one never determines a solid.
• Isometric projection: a cube on its corner gives a regular hexagon; depth information is lost, which is what impossible figures exploit.

Appendix — beyond the current syllabus

Everything below is real geometry, but none of it is in Chapter 4 of Ganita Prakash Part 2. Do not present it as this chapter's content in a Class 8 exam.

Congruence criteria (Class 9)

SSS, SAS, ASA, AAS and RHS are the standard tests for two triangles being congruent. This chapter uses congruence twice — to show the four midpoint triangles of an equilateral triangle are identical, and that the corner triangles cut from a square are identical — but it does not teach the criteria, which belong to Class 9.

Similarity of triangles (Class 10)

AA, SSS and SAS similarity, the Basic Proportionality Theorem, and the fact that areas of similar triangles are in the ratio of the squares of corresponding sides. Similarity is genuinely relevant to fractals — self-similar shapes are made of similar copies — but the formal treatment is Class 10.

Circle theorems (Class 9–10)

Equal chords subtend equal angles at the centre; the angle at the centre is twice the angle at the circumference; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; the tangent is perpendicular to the radius at the point of contact. None of this appears in Ganita Prakash Class 8.

Symmetry and transformations

Line symmetry, rotational symmetry, and translations, reflections, rotations and enlargements. Ganita Prakash Class 8 does not have a symmetry chapter, though symmetry is used informally — for example the three-fold symmetry of the cube about its long diagonal, which is what makes the isometric projection work.

Fractal dimension

Fractals sit between the usual dimensions. If a shape breaks into N copies of itself each scaled by 1/s, its dimension is log N / log s:

More than a line, less than a surface — which is a precise way of saying what the areas and perimeters were hinting at.

Where this chapter goes next

• Class 9–10 Surface Areas and Volumes — every formula there comes from a net. The cylinder's 2πrh and the cone's πrl are literally the areas of the rectangle and the sector in this chapter.
• Class 9 Coordinate Geometry — projections onto axes are the coordinates themselves.
• Engineering drawing — front, top and side views plus isometric projection are the working language.
• Computer graphics — fractal rules generate mountains, clouds and coastlines; projection matrices turn 3D scenes into 2D screens.