Exploring Some Geometric Themes
This chapter takes a stroll through two unrelated-looking corners of geometry and finds that both repay a playful imagination. First we meet fractals, shapes built so that every small part is a miniature of the whole, the same pattern echoing at finer and finer scales. We will grow the Sierpinski Carpet, the Sierpinski Triangle, the Koch Snowflake, and a one-dimensional cousin, the Cantor Set, hunting for the rules behind how many pieces survive at each stage. We will spot fractals carved into a thousand-year-old temple and woven into wedding cloth in West Africa. Then we turn to visualising solids: how one object throws wildly different outlines from different directions, how flat nets fold into cubes and pyramids, how a wandering beetle finds the shortest path across a box, and how designers pin a 3-D solid onto flat paper with projections and isometric grids.
Learning Outcomes
By the end of this chapter, you will be able to:
- recognise self-similar shapes and explain what makes a shape a fractal;
- build the Sierpinski Carpet, Sierpinski Triangle, Koch Snowflake and Cantor Set stage by stage, and find formulas for the count of pieces, holes, sides, perimeter and area at the nth stage;
- describe the profile (outline) of a solid from various viewpoints, and invent solids matching given profiles;
- identify the faces, edges and vertices of prisms and pyramids, and count them for an n-sided figure;
- fold and unfold nets of cubes, cuboids, tetrahedrons, cylinders, cones and octahedrons;
- use a net to find the shortest path between two points on the surface of a cuboid;
- draw the front, top and side views of a solid and read a solid from its three views;
- understand and draw isometric projections on an isometric grid.