How Tessellations Cover a Plane

Begin with the question

How Tessellations Cover a Plane starts with one useful question: Why triangles, squares, and hexagons tile easily—and pentagons are trickier.

A tiling works when angles around every shared vertex add to exactly 360°. The point is not to memorize a label. It is to notice a repeatable relationship, then test what changes when one part of that relationship moves. A friendly toolkit for symmetry, tiling, ratios, and the geometry of looking.

Build a working model

The main diagram is a snap-together tile board with gaps highlighted. It gives the eye a before-and-after view rather than asking it to infer the rule from a paragraph alone.

A working model can be simple and still be powerful: name the parts, hold one choice steady, change another, and compare the result. That is how this topic becomes something you can inspect instead of only admire.

• ✳Name the part that repeats.
• ✳Choose one change to test.
• ✳Compare the new result with the first one.

Try a small experiment

Try triangle, square, and regular pentagon tiles around one point. Open Make a pattern when it fits the question, and keep a note of the setting, rule, or observation that changes the picture most clearly.

If this guide does not need a generator, recreate the diagram on paper: trace the underlying structure first, then add the decorative detail. The structure should still read before the decoration arrives.

• ✳Start with: Snap-together tile board with gaps highlighted.
• ✳Make two variations, changing only one choice each time.
• ✳Keep the version that teaches you the most, not only the prettiest one.

What to notice next

Many irregular shapes tile too; regular polygons are only the simplest starting point. Return to How Tessellations Cover a Plane in a photograph, a built object, or another artwork. Ask where the model fits and where it does not.

That habit—observe, model, test, and revise—connects this lesson to every other guide in the Shape Language collection.