Class 8 › Mathematics › Chapter
Exploring Some Geometric Themes
Self-similar fractals (Sierpinski carpet and gasket, Koch snowflake), and visualising solids through profiles, nets, shortest paths on a cuboid, projections and isometric drawing.
1. Fractals
Fractal: a self-similar shape that shows the same or a similar pattern again and again at smaller and smaller scales.
Sierpinski carpet: split into 9 squares and remove the centre one; repeat on the 8 left.
Carpet
Sierpinski Carpet
Square $\to$ 9 equal squares $\to$ remove the central one. Repeat on every remaining square.
Gasket
Sierpinski Triangle
Equilateral triangle $\to$ join midpoints to make 4 identical equilateral triangles $\to$ remove the central one. Repeat on the 3 left.
Snowflake
Koch Snowflake
Divide each side into 3 equal parts, raise an equilateral triangle on the middle part, remove that middle part. Repeat on every new side.
- In nature: ferns, trees, clouds, coastlines, mountains, lightning.
- In art: temple towers (e.g. Kandariya Mahadev Temple, Khajuraho, c. 1025 CE), Nigerian Fulani wedding blankets, M.C. Escher's 'Smaller and Smaller'.
- Why joining midpoints works: the midpoint-joins of an equilateral triangle make 4 congruent equilateral triangles (the corner triangles are isosceles with a $60^\circ$ angle).
2. Counting in Fractals
| Fractal | Remaining pieces at step $n$ | Holes at step $n$ | Other |
|---|---|---|---|
| Carpet | $R_n = 8^n$ (since $R_{n+1} = 8R_n$) | $H_{n+1} = H_n + R_n$, so $H_n = 1 + 8 + \dots + 8^{n-1} = \dfrac{8^n - 1}{7}$ | Area left $= \left(\dfrac{8}{9}\right)^n$ (start area $1$) |
| Gasket | $R_n = 3^n$ | $H_n = \dfrac{3^n - 1}{2}$ | Area left $= \left(\dfrac{3}{4}\right)^n$ |
| Koch snowflake | Sides: $3 \times 4^n$ | - | Perimeter $= 3\left(\dfrac{4}{3}\right)^n$ for side $1$ |
- Method: find how a step is built from the previous one (a recurrence), then spot the pattern.
- Every remaining piece at step $n$ creates one new hole at step $n+1$; old holes stay.
- The Koch perimeter keeps growing (each side becomes $4$ sides, each $\tfrac13$ as long), while the carpet and gasket areas keep shrinking.
3. Solids, Faces, Edges and Vertices
Face: a flat surface on the boundary. Edge: a line segment where faces meet. Vertex: a point where edges meet. A cube or cuboid has $6$ faces, $12$ edges, $8$ vertices.
- Profile: what we see of a solid from one viewpoint (like the hole left by a cartoon character running through a wall). A solid can have different profiles from different viewpoints, and different solids can share a profile.
- Prism: two congruent polygons as opposite faces joined by parallelogram faces. Named by the polygon (triangular, pentagonal, hexagonal prism).
- Pyramid: a polygonal base and a point outside it joined to every base vertex. Named by the base. A triangular pyramid is a tetrahedron.
| Solid ($n$-sided polygon) | Faces | Edges | Vertices |
|---|---|---|---|
| Prism | $n + 2$ | $3n$ | $2n$ |
| Pyramid | $n + 1$ | $2n$ | $n + 1$ |
- Check (Euler): $F + V - E = 2$ for both ($n + 2 + 2n - 3n = 2$). A prism on a $10$-sided polygon has $12$ faces, $30$ edges, $20$ vertices; a $10$-sided pyramid has $11$ faces, $20$ edges, $11$ vertices.
4. Nets of Solids
Net: the flat shape obtained by 'unfolding' a solid onto a plane; it can be folded back into the solid. Extra glue flaps are not part of the net.
| Solid | Net |
|---|---|
| Cube | 11 different nets (rotations and flips count as the same) |
| Cuboid | $6$ rectangles in a connected fold-up arrangement (opposite faces equal) |
| Regular tetrahedron | $4$ equilateral triangles; only 2 nets |
| Square pyramid | A square with an isosceles triangle on each side |
| Cylinder | A rectangle (width $2\pi r$, the base circumference; height $h$) plus $2$ circles |
| Cone | A sector of a circle (centre at the apex, radius = slant height $l$) plus the base circle |
| Octahedron (two square pyramids joined at the base) | $8$ equilateral triangles; 11 nets |
| Dodecahedron (all faces pentagons) | $43{,}380$ nets |
| Sphere | No net that wraps it perfectly without gaps, wrinkles or overlaps |
- The net of a solid is not unique; check a candidate by visualising or cutting and folding.
- To draw a net of a cuboid $a \times b \times c$, draw pairs of rectangles $a \times b$, $b \times c$, $a \times c$ with matching edges.
5. Shortest Paths on a Cuboid
Idea
A path on the surface of the cuboid maps to a path of the same length on the net, and every net path maps back. So the shortest surface path $=$ the straight line between the two points on a suitable net.
The way you unfold matters. The straight line must stay inside the net, and different unfoldings give different lengths, so list all valid unfoldings and take the shortest.
Calculation: in a right triangle on the net, use the Baudhayana (Pythagoras) theorem. Example: legs $24$ and $32$ give $d^2 = 24^2 + 32^2 = 1600$, so $d = 40$ cm.
6. Projections and Views
Projection of a point $P$ on a plane $M$: the foot $O$ of the perpendicular from $P$ to $M$. The projection of an object is the set of projections of all its points.
- Length of a projected line: projected length $p \le$ actual length $l$; they are equal only when the line is parallel to the plane (in the right triangle, $p$ is a leg and $l$ the hypotenuse).
- Parallel lines project to parallel lines, so a parallelogram always projects to a parallelogram.
- Not unique: different objects can give the same projection (different cuboids or lines, a cone and other solids).
- Three standard views: front view (on the vertical plane), top view (on the horizontal plane), side view (on the side plane). Together they describe a solid well.
- Shadows: sunlight is a very distant light source, so shadows on a plane perpendicular to the rays look like projections. A nearby torch makes scaled or distorted shadows.
- Hole shape: an object passing perpendicularly through a plane leaves a hole with the shape of its projection.
- Examples: a cube can project to a square, rectangle or hexagon depending on orientation; a cylinder gives a rectangle and a circle; a cone a triangle and a circle.
7. Isometric Projections
Isometric projection: a projection in which the projections of all three edge directions have equal length ('isometric' means 'equal measure'). A cube balanced on one corner projects to a regular hexagon.
- Isometric grid: a plane tiled with hexagons (equilateral triangles); gives three directions of grid lines.
- Axes: vertical lines $=$ height; the two slanted directions $=$ length and depth. Count unit edges along each axis to draw a solid (e.g. $1 \times 1 \times 1$ or $2 \times 2 \times 2$ cube; Tetris shapes made of $4$ cubes).
- Why it works: parallel edges stay parallel, and unit lengths along the three axes project to equal lengths.
- Tip: draw faint lines first, then darken the visible edges.
- Impossible figures (e.g. the impossible triangle) can be drawn on an isometric grid but cannot be built from real cubes.
8. Key Results and Quick Revision
| Topic | Key fact |
|---|---|
| Carpet | $R_n = 8^n$, $H_n = \tfrac{8^n - 1}{7}$, area $(\tfrac89)^n$ |
| Gasket | $R_n = 3^n$, $H_n = \tfrac{3^n - 1}{2}$, area $(\tfrac34)^n$ |
| Koch snowflake | Sides $3 \cdot 4^n$, perimeter $3(\tfrac43)^n$ |
| $n$-gon prism / pyramid | $(n+2,\ 3n,\ 2n)$ / $(n+1,\ 2n,\ n+1)$ for (F, E, V) |
| Cube / tetrahedron / octahedron nets | $11$ / $2$ / $11$ |
| Shortest surface path | Straight line on a suitable net, then Baudhayana theorem |
| Views | Front, top and side projections; isometric grid for 3D drawing |
Common mistakes:
• Counting the flaps as part of a net.
• Treating all arrangements of $6$ squares as cube nets (rotations and flips are the same net, and many arrangements do not fold into a cube).
• Using only one unfolding when finding the shortest path on a cuboid.
• Assuming a projection identifies the solid uniquely.
• Mixing up which grid direction stands for height, length and depth.
• Forgetting that $H_n$ adds up the geometric series, not $R_n$ alone.
• Counting the flaps as part of a net.
• Treating all arrangements of $6$ squares as cube nets (rotations and flips are the same net, and many arrangements do not fold into a cube).
• Using only one unfolding when finding the shortest path on a cuboid.
• Assuming a projection identifies the solid uniquely.
• Mixing up which grid direction stands for height, length and depth.
• Forgetting that $H_n$ adds up the geometric series, not $R_n$ alone.