What it does
A tessellation (or tiling) covers a floor with shapes that leave no gaps and never overlap. Where the corners of the tiles meet, their angles must add up to exactly 360°, a full turn. This studio lets you test that idea by hand.
- Polygons around a point: tap shapes to place them corner to corner around one point. The page adds up their corners and tells you if they close exactly, leave a gap (shaded) or overlap (dashed red).
- A patch of tiles: the three regular shapes that tile a floor on their own (triangles, squares and hexagons), drawn as a real patch.
- Rangoli pattern pad: draw a few lines and the pad turns them round the centre k times. With mirrors on, each line is also reflected, as in many rangoli and kolam designs. The page tells you the order of rotational symmetry and the number of lines of symmetry.
It fits Class 8 Mathematics, exploring some geometric themes (tilings, angles of polygons and symmetry), and links back to polygons and symmetry in earlier classes. Solids whose faces are these same regular polygons are in the polyhedron viewer; turning and reflecting a whole grid is in matrix transform.
How to use it
- Tap Triangle, Square, Pentagon and the others to add one at the point (up to 8). Take one off removes the last; Clear starts again.
- The quick buttons load six triangles, four squares, three hexagons, three pentagons and the mixes 4.8.8 (a square and two octagons), 3.12.12 and 3.4.6.4. The numbers name the polygons met going round the point.
- In A patch of tiles, pick triangles, squares or hexagons.
- On the pattern pad, set k (2 to 12 copies), tick or untick Mirrors, then draw. Sample pattern draws a small flower to start with; Undo removes your last line.
A worked example
The corner of a regular polygon with n sides is 180° × (n − 2) ÷ n.
- Hexagon: 180 × 4 ÷ 6 = 120°. Three hexagons make 3 × 120 = 360°, so hexagons tile (as in a honeycomb).
- Pentagon: 180 × 3 ÷ 5 = 108°. Three make 324°, leaving a 36° gap; a fourth would make 432°, an overlap of 72°. So regular pentagons cannot tile alone.
- 3.4.6.4: 60 + 90 + 120 + 90 = 360°, so a triangle, two squares and a hexagon fit together.
On the pad with k = 6 and mirrors on, the pattern looks the same after every turn of 60°, so it has rotational symmetry of order 6, and it has 6 lines of symmetry.
Check yourself
Five fresh questions each time: an interior angle, how many tiles meet at a corner, the gap left by shapes that do not tile, the missing polygon in a mix, and the lines of symmetry of a rangoli. Nothing is saved.
FAQ
Why only triangles, squares and hexagons? Their corners (60°, 90°, 120°) divide 360 exactly. The next shape with a corner that divides 360 would need a corner of 180°, which is a straight line, not a polygon.
Do the mixes always tile a whole floor? Fitting round one point is needed, but not always enough: a few mixes (such as 5.5.10) close round one point but cannot be carried on across a whole floor. The ones in the quick buttons all do.
Is this from the NCERT book? It covers the same topic with fibub’s own pictures and examples. fibub is not part of NCERT.
Where the numbers come from
- Interior angles of regular polygons, regular and semi-regular tilings, and rotational and line symmetry are standard geometry. Wording, pictures and examples are fibub's own.
- Checked against the standard facts: only triangles, squares and hexagons tile on their own, and exactly 17 combinations of regular polygons fit round a point (found by searching every combination up to 42 sides).
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.