Projects, puzzles and the wider world of shape

You now know the names, the properties, the patterns and the reasons. This last layer is about using all of it: projects to build with your hands, puzzles that have stumped clever people, olympiad-style problems, shapes in Indian architecture and in nature, the jobs where geometry is used every day, and questions that nobody has answered yet.

Pick the chapters that excite you. None of them needs to be read in order, and several are meant to take an afternoon, not ten minutes.

A regular polyhedron (or Platonic solid) has faces that are all the same regular polygon, with the same number of faces meeting at every vertex. The cube is one: six identical squares, three at each corner. How many others are there?

The astonishing answer, proved by Euclid more than 2,300 years ago, is: exactly five. Not five that people happened to find, but five that are possible, and no more, ever.

In Investigate you learned that exactly 11 of the 35 hexominoes fold into a cube: a long-established result, not a guess. Your project: find all 11 on squared paper, cut each one out and fold it to check. Two nets count as the same if one can be turned or flipped to match the other.

The 11 fall into four families, named by the lengths of their rows:

A shape has rotational symmetry if it looks exactly the same after being turned about its centre by some angle less than a full turn. The number of positions in one full turn where it matches itself is its order.

The Ashoka Chakra has 24 equal spokes, so it matches itself after every turn of 360° ÷ 24 = 15°: order 24. A three-blade ceiling fan has order 3 (every 120°). A playing card like the queen of hearts, printed with the top and bottom halves upside-down copies, has order 2. Rangoli designs often have order 4 or 8.

Make a tangram from a square of side 8 cm (area 64 cm²). Because each piece is made by halving, the areas are simple fractions of the whole:

In Investigate you found that only three regular polygons tile a floor alone: triangles, squares and hexagons. Now mix them. A semi-regular tiling uses two or more kinds of regular polygon, with the same arrangement at every vertex. We describe it by listing the polygons round a vertex: 4.8.8 means a square and two octagons (the common bathroom-floor pattern).

At each vertex the angles must add to exactly 360°. Checking every possibility, Johannes Kepler found in 1619 that there are exactly 8 semi-regular tilings:

This shape is called a truncated icosahedron: take an icosahedron and slice off each of its 12 corners; each cut leaves a pentagon, and each triangle becomes a hexagon. In 1985 chemists discovered a molecule of 60 carbon atoms arranged exactly at the 60 vertices of this shape. They named it buckminsterfullerene (or the buckyball), after the architect Buckminster Fuller, famous for his geodesic domes: huge, light, strong domes made of triangles. Geometry that a child can count on a football turned out to describe a Nobel-prize-winning molecule.

How do you draw a cube on flat paper so that it looks solid? Architects and engineers use two main methods.

Isometric drawing uses a grid of dots arranged in triangles. Vertical edges stay vertical, and the other two directions slope at 30° to the horizontal. Every edge of a cube is drawn the same length, so you can measure from the drawing. Many puzzle books and video games use this style.

Orthographic drawing gives separate, flat views: plan, front elevation and side elevation, lined up with each other. It is less like a picture but more precise, and it is what builders actually use.

Perspective drawing, used by artists since the Renaissance and in Mughal and Rajput miniature paintings in their own ways, makes parallel lines meet at a vanishing point so that far things look smaller, just as the eye sees them.