Exploring Some Geometric Themes

1. What is a fractal?

A fractal is a shape that is self-similar: it shows the same (or a similar) pattern over and over again, but at smaller and smaller scales. Zoom into a tiny part of a fractal and you see a copy of the whole.

Nature is full of them. A fern has leaves that are smaller copies of the whole fern, and those leaves have still smaller copies, and so on. The same self-similarity appears in trees (trunk → limbs → branches → branchlets), clouds, coastlines, mountains and lightning. Mathematicians build exact fractals using a simple rule applied again and again.

2. The Sierpinski Carpet

Discovered by the Polish mathematician Sierpinski. The rule, starting from a square:

• Step 0: one full square.
• Each step: break every remaining square into 9 equal smaller squares and remove the central one (leaving $8$).
• Repeat on all the remaining squares.

Let $R_n$ = number of squares that remain at step $n$, and $H_n$ = number of holes. Every remaining square becomes $8$ squares next step, so:

So $R_1=8,\ R_2=8^{2}=64,\ R_3=8^{3}=512$. Holes: every remaining square makes a hole next step and old holes stay, so $H_{n+1}=H_n+R_n$. Hence $H_0=0,\ H_1=1,\ H_2=1+8=9,\ H_3=1+8+8^{2}=73$.

3. The Sierpinski Gasket (Triangle)

Same idea, but with an equilateral triangle. Joining the midpoints of its sides divides it into 4 identical equilateral triangles (the three corner ones plus the central one). Remove the central triangle, leaving $3$. Repeat on each remaining triangle.

Let $T_n$ = triangles remaining at step $n$. Each remaining triangle gives $3$ next step:

So $T_1=3,\ T_2=9,\ T_3=27$. The holes follow $H_{n+1}=H_n+T_n$, giving $H_1=1,\ H_2=1+3=4,\ H_3=1+3+9=13$.

4. Area remaining in the Sierpinski fractals

Take the starting square/triangle to have area $1$ sq. unit. At each step we keep a fixed fraction of the area:

• Carpet: $8$ of every $9$ small squares remain, so area is multiplied by $\dfrac{8}{9}$ each step. After $n$ steps the remaining area is $\left(\dfrac{8}{9}\right)^{n}$.
• Gasket: $3$ of every $4$ small triangles remain, so area is multiplied by $\dfrac{3}{4}$ each step, giving $\left(\dfrac{3}{4}\right)^{n}$ after $n$ steps.

As $n$ grows, both shrink towards $0$ — the famous fractals are "almost all holes"!

5. The Koch Snowflake

Named after the Swedish mathematician Von Koch (1904). Start with an equilateral triangle. The rule on every side:

• (i) divide the side into 3 equal parts, and
• (ii) on the middle part build a small equilateral triangle pointing outward, then remove that middle part.

Effectively each straight side is replaced by a "bump" made of 4 segments, each one-third as long. Repeat on every new side.

Number of sides $S_n$: each side becomes $4$ sides next step. Start with $3$ sides:

Perimeter: each step every side is replaced by $4$ sides each $\tfrac13$ as long, so the perimeter is multiplied by $\dfrac{4}{3}$ each step. If the starting triangle has side $1$ (perimeter $3$), the perimeter at step $n$ is $3\times\left(\dfrac{4}{3}\right)^{n}$ — it grows without bound while the snowflake stays inside a small region.

6. Fractals in art

Fractals are not only natural — they appear in human art too. The Kandariya Mahadev Temple at Khajuraho (~1025 C.E.) is a tall structure built from smaller copies of itself, with even smaller copies again — a fractal in stone. Similar patterns appear in temples at Madurai, Hampi, Rameswaram and Varanasi. Nigerian Fulani wedding blankets show diamonds inside diamonds. The Dutch artist M.C. Escher made tilings such as "Smaller and Smaller" where lizards repeat at ever smaller scales.

7. Visualising solids — profiles

When we look at a solid object we really see its profile (its outline) from one viewpoint. The same solid can have very different profiles from different directions — an elephant looks different from the front, side and top.

• A solid with a square profile: a cube (seen straight on).
• A solid with a circular profile: a sphere from any side, or a cylinder/cone from the right direction.
• A cylinder shows a rectangle from the side but a circle from the top.
• A cone shows a triangle from the side but a circle from the top.

So a single profile does not fix the solid — many solids can share the same outline.

8. Faces, edges and vertices

For solids whose boundary has flat surfaces:

• Faces — the flat surfaces forming the boundary.
• Edges — the line segments where two faces meet.
• Vertices — the points where edges meet.

9. Prisms and pyramids

Prism: has two congruent polygons as opposite faces, joined by parallelograms. Named after the base polygon (triangular, pentagonal, hexagonal prism…).

Pyramid: has a polygon base and one apex point above it; edges join the apex to each base vertex. Named after its base (triangular, square, pentagonal pyramid…). A triangular pyramid is a tetrahedron.

10. Nets of a solid

A net is the flat shape you get by unfolding a solid onto a plane; folding the net back gives the solid. (Real models often need extra flaps to glue faces, but the net itself is only the unfolded faces.)

• Cube: has 11 different nets (counting rotations/flips as the same).
• Regular tetrahedron (all faces equilateral triangles): only 2 nets.
• Octahedron (two square pyramids joined at the base): 11 nets.
• Dodecahedron (12 pentagon faces): a huge 43,380 nets.
• Cylinder: a rectangle plus two circles. The rectangle's sides are the height and the base circumference.
• Cone: a circle (base) plus a part of a larger circle (the curved surface unrolled).
• Sphere: has no net — paper cannot wrap a ball without wrinkles or gaps.

11. Shortest paths on a cuboid (the ant and the laddu)

On a flat plane the shortest path between two points is a straight line. On the surface of a cuboid the ant can only crawl along faces. The clever trick: unfold the cuboid into a net. A path on the surface becomes a path of the same length on the net, so the shortest surface path is the straight line on the net.

Important catch: a cuboid can be unfolded in several different ways, and the straight line is different in each. We must try all relevant unfoldings and pick the smallest distance. (If the straight line goes outside the net, that unfolding does not give a real path.)

12. Projections and the three views

Drop a perpendicular from a point $P$ onto a plane $M$; the foot $O$ is the projection of $P$. All such feet together form the projection of the object. (Projection length $p\le$ true length $l$; they are equal only when the line is parallel to the plane.)

To capture a solid we use three mutually perpendicular planes and the views on them:

• Front view — projection on the vertical plane.
• Top view — projection on the horizontal plane.
• Side view — projection on the side plane.

One projection is not enough — different solids can share the same projection. Shadows under a faraway light (like the Sun) look just like projections; the projection of a pair of parallel lines stays parallel (so a parallelogram's shadow stays a parallelogram).

13. Isometric projection and drawing

If a cube is balanced on one corner and projected down, all its edges project to equal lengths. This is the isometric projection ("iso" = equal, "metric" = measure). The isometric view of a cube is a regular hexagon.

Tiling the plane with such hexagons gives an isometric grid with edges in three directions ( $/$, $\backslash$, $|$ ) for length, depth and height. Engineers use isometric grids to draw solids while keeping true measurements along all three axes — handy for drawing cube stacks (like Tetris shapes).

14. Worked examples

15. Common mistakes to avoid

• Confusing the carpet rule ($8^{n}$) with the gasket rule ($3^{n}$) — carpet removes $1$ of $9$ squares, gasket removes $1$ of $4$ triangles.
• Thinking the Koch perimeter is finite — it grows by $\tfrac43$ each step, without bound.
• Mixing up prism vs pyramid counts — prism Vertices $=2n$, pyramid Vertices $=n+1$.
• Believing a cube has only one net — it has 11; and a sphere has none.
• Trusting a single unfolding for the ant's shortest path — always compare all unfoldings.
• Saying one projection (view) decides the solid — you need front, top and side views.

16. Quick revision checklist

• Fractal = self-similar, same pattern at smaller scales.
• Carpet $R_n=8^{n}$; Gasket $T_n=3^{n}$; Koch sides $S_n=3\times4^{n}$.
• Cube/cuboid: $6$ faces, $12$ edges, $8$ vertices.
• Prism: $n+2,\ 3n,\ 2n$. Pyramid: $n+1,\ 2n,\ n+1$.
• Cube has $11$ nets; tetrahedron $2$; sphere $0$.
• Shortest cuboid path = straight line on the best net (use Pythagoras).
• Three views: front (vertical), top (horizontal), side (side plane).
• Isometric: all edges equal length; cube projects to a regular hexagon.