📐 Geometry
⚫Points, Lines and Planes
A dot from a pen tip, a stretched string, a tabletop — the three basic building blocks of geometry.
Press a pen tip onto paper and you make a point. Drag the point along and its trail is a line. Sweep the line sideways and it spreads into a plane. The whole world of geometry is assembled from these three basic blocks. They have no size and no thickness — pure abstraction — yet every figure you have ever seen stands on them.
Points, segments, rays and lines
A point is a mark with position but no size. Set it moving and you get a line, which comes in three flavors:
- A segment: two endpoints, a measurable length, like the edge of a textbook;
- A ray: one endpoint, stretching away forever in the other direction, like the beam of a flashlight;
- A line: no endpoints at all, endless in both directions — it can never be fully measured.
Endpoints are the whole difference: the more endpoints, the more tame the line. A segment is held down at both ends, so its length can be measured; nothing holds a line back, so it never runs out.
Two ironclad facts
Two basic facts anchor almost every geometric argument:
- Through two points passes exactly one line. When two people each hold one end of a picture-hanging string, it snaps taut into a single straight line — that is the fact at work;
- Two distinct lines meet in at most one point. If two lines shared two points, they would have to be one and the same line.
The second fact has an important exception: parallel lines never meet, so their number of intersections is zero. And when a line meets a plane, the story changes again — a line pierces a plane in exactly one point.
Planes and their seams
A sheet of paper, a wall, a still lake — each plays the role of a plane: flat, straight and endlessly extended. Real paper has edges; a mathematical plane does not. Wherever two planes meet, the seam is always a straight line:
- The ceiling and a wall intersect along a perfectly straight crease;
- In a room corner, two walls and the floor intersect in pairs, producing three lines — and those three lines all pass through a single point.
Solid shapes are built out of exactly such faces, lines and points; the lesson on solid shapes tells that story.
Try the math
A line through A and B
Through point A alone you can draw infinitely many lines — pin your pen at A and swing it around. Through B alone, also infinitely many. But through both A and B passes exactly one line. Infinity plus infinity, yet one extra constraint makes the answer unique.
Count the corner
A room corner is enclosed by three planes: two walls and the floor. Meeting in pairs, they produce one seam each — 3 lines in total — and all three lines squeeze through the same point. Count it backwards: one corner is 1 point, 3 lines and 3 planes. Geometry is never invented from nothing; it hides in the very room around you.
Keep walking along points and lines, and they open up into angles.
Check yourself
Quick quiz
1. How many lines pass through two different points?
2. Which kind of line has a measurable length?
3. Two non-parallel planes intersect in what?