Tessellations
Shapes that tile the plane with no gaps or overlaps.
About this topic
A tessellation covers a flat surface with repeating shapes leaving no gaps or overlaps. Only three regular polygons tile alone: equilateral triangles, squares and regular hexagons, because their angles divide evenly into 360° at each vertex.
Semi-regular tessellations mix polygons, and irregular shapes can tile too, as in the artworks of M. C. Escher. Tilings appear in floors, honeycombs, quilts and Islamic art, and they link geometry to transformations and symmetry.
Worked example: at a vertex of squares, four right angles make 360°, so squares tile. Three hexagons give 3 × 120° = 360°, so they tile, as in a honeycomb. Regular pentagons have interior angle 108°, and 360 is not divisible by 108, so they leave gaps. A pattern of one octagon and one square can be combined since 135 + 135 + 90 = 360.
Bathroom tiles, brick paving, beehives and the artwork of M. C. Escher use tessellations. Students often believe any shape can tile just by trying; however, regular polygons must have an interior angle that divides 360 evenly. Any triangle or quadrilateral, even an irregular one, can tile by rotating copies. The check is to add the angles meeting at each vertex and see whether the sum equals exactly 360°, with no gaps or overlaps.
Key ideas
- Triangles, squares and hexagons tessellate regularly
- Angles at a vertex sum to 360°
- Escher's art uses transformations
Quick check
Use cases: where Tessellations helps
- Homework: review the key idea behind Tessellations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Tessellations without notes.
- Visual learning: play with the interactive lab to see how Tessellations changes when you adjust its inputs.
- Teaching a friend: explain Tessellations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Tessellations?
Shapes that tile the plane with no gaps or overlaps.
What will I learn in Tessellations?
Triangles, squares and hexagons tessellate regularly. Angles at a vertex sum to 360°. Escher's art uses transformations.
Is Tessellations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Tessellations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Tessellations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.