Lines in the wider world

Lines are not only for geometry homework. Artists bend the rules of parallel lines to make flat pictures look deep. Engineers keep railway rails exactly parallel across thousands of kilometres. Map makers draw lines on a round Earth. Computer games fire millions of rays every second to draw each frame. And some puzzles about straight lines have kept mathematicians busy for centuries.

This layer takes the ideas from the first four depths into the wider world: three dimensions, maps, art, sport, careers, puzzles and a few questions that nobody has fully answered.

Stand in the middle of a long, straight railway platform and look along the track. The rails are parallel: 1,676 mm apart on India's broad gauge, all the way. Yet they appear to close in and meet at a single point on the horizon. The sleepers look shorter and shorter, and closer together.

This is perspective. Things further away make a smaller image in your eye, so a fixed gap looks smaller the further off it is. Any set of parallel lines running away from you seems to meet at one point, called the vanishing point, which lies on the horizon line at the height of your eyes.

Mathematicians took the artist's trick seriously. In the 1600s Girard Desargues, and later Jean-Victor Poncelet in 1822, developed projective geometry. It adds one extra "point at infinity" for every direction, where all lines of that direction meet, and a "line at infinity" made of all such points: the horizon of the whole plane.

In projective geometry there are no parallel lines: every two different lines meet in exactly one point, either an ordinary one or one at infinity. It sounds like cheating, but it makes many theorems simpler, and it is the geometry that cameras, computer graphics and virtual-reality headsets use every day.

On a page, two lines either meet or are parallel. In a room there is a third possibility. Look at the edge where the front wall meets the floor, running left to right. Now look at the edge where the left wall meets the ceiling, running from front to back. They never meet. But they are not parallel either: they point in different directions. Such lines are called skew lines.

Skew lines can only exist in three dimensions, because two lines that lie in one plane must either meet or be parallel. The edges of every box, cupboard and building give plenty of examples.

Maps are covered in lines. Lines of latitude run east–west and tell you how far north or south you are; lines of longitude (meridians) run north–south and tell you how far east or west.

Lines of latitude never meet, and stay the same distance apart along the Earth's surface, so they behave like parallel lines: that is why they are also called parallels. Lines of longitude are different: they are far apart at the equator, get closer together as you go north or south, and all meet at the North and South Poles. Every line of longitude crosses every line of latitude at a right angle.

Sports grounds are some of the most precisely drawn line diagrams in everyday life. Rules decide which lines are parallel, which are perpendicular and exactly how far apart they are.

Physics borrows the word ray directly from geometry. Light travels in straight lines, so a thin beam from a torch or the Sun is drawn as a ray: an end point at the source and an arrow showing the direction. That is why shadows have sharp straight edges, why you cannot see round corners, and why a pinhole camera works: rays from the top of a tree pass through the tiny hole and land at the bottom of the screen, making an upside-down image.

Look at a circuit diagram in a science book: every wire is drawn as a straight segment, running either horizontally or vertically and turning at right angles. The real wires may be tangled, but drawing them as parallel and perpendicular segments makes the diagram far easier to read. Overhead power lines are strung parallel to each other between pylons, kept a safe distance apart so that they never touch.

Straight lines lead to some of the best puzzles in mathematics. Try each one before reading the hints.

These problems are in the style of mathematics olympiads. Each one needs an idea from this topic (pairs, collinearity, parallel lines) used in a new way. Try each for at least ten minutes before reading the solution.

Artists have played with lines for as long as people have made pictures. Kolam and rangoli start from a grid of dots, and many designs are built from straight segments between the dots, often with lines of symmetry running through the middle. Warli paintings from Maharashtra build people and animals out of triangles, circles and straight lines. Temple walls and jaali screens repeat parallel and perpendicular lines to make patterns that let light through.

In the 1920s the Dutch painter Piet Mondrian made famous paintings from nothing but black horizontal and vertical lines and blocks of colour: a whole art style built on perpendicular lines. Graphic designers today still use a hidden grid of parallel lines to line up text and pictures on every page and screen.

Here is something that sounds impossible: a curve made entirely of straight segments. Draw two segments meeting at a corner, like an L. Mark 10 equally spaced points on each, numbered 1 to 10 from the corner outward on one arm and from the far end inward on the other. Join 1 to 1, 2 to 2, and so on.

No single segment is curved, yet a smooth curve appears where the segments crowd together. The curve (a parabola) is touched by every one of the segments. Mathematicians call such a curve an envelope. The same idea makes the curved look of a suspension bridge's cables and many kolam and rangoli designs drawn from straight strokes.

Some of the most useful straight lines in daily life are the ones that make grids: two families of parallel lines crossing at right angles. Map makers, town planners, sports officials and electricians all rely on them.