Chapter 14: Constructions and Tilings
A ruler with no markings and a compass that draws circles , that is all you need to build some of the most elegant shapes in geometry. No protractor. No set square. Just two simple tools and a few clever ideas.
In this chapter you will construct (not measure!) perpendicular bisectors, angles of and , bisect any angle into two equal halves, copy an angle exactly, draw a line parallel to a given line, and finish with a regular hexagon. Every construction is exact in principle , accuracy depends only on how steady your hand is.
The second half explores tiling: how copies of one shape can cover a flat surface with no gaps and no overlaps. This is the geometry behind floor tiles, honeycombs, jali screens and the famous patterns at Humayun's Tomb and Fatehpur Sikri.
These ideas are ancient. The Indian Śulba-Sūtras described many of the same constructions over years ago, often using a rope instead of a compass , proving once again that mathematics is older than the instruments we use to do it.
What's inside
- Perpendicular bisector , the foundation of every other construction.
- Constructing 90°, 60° and 45° angles , building special angles exactly.
- Angle bisection and copying an angle , halving and duplicating angles.
- Constructing parallel lines and regular hexagons , turning bisectors into bigger figures.
- Tilings: covering the plane , when do shapes fit together with no gaps?
Key results
| Construction | One-line idea |
|---|---|
| Perpendicular bisector of | Equal arcs from and above and below ; join intersections. |
| angle at point | Use the perpendicular bisector of a segment whose midpoint is . |
| Angle bisector of | Equal arcs from points equidistant from on each arm meet on the bisector. |
| angle | The angle of an equilateral triangle , built with a single radius. |
| Tiling condition at a vertex | The angles meeting at any point sum to . |
A regular polygon tiles the plane only if its interior angle divides : that gives equilateral triangles, squares and regular hexagons , and no other regular polygon.
How to read this chapter
Don't just read , do. Keep a sharpened pencil, a good compass and a ruler with a straight edge. After each construction, ask: "Why does this work?" The justification is always congruent triangles (SSS, SAS) , geometry's grand machinery hiding behind a few arcs.