A tessellation is a pattern that covers an infinite plane with repeated geometric shapes, leaving no gaps and having no overlaps.
Like a bathroom floor tile pattern that fits together perfectly and could extend forever in all directions.
Showing a random 20 of 50 problems.
Example 1
medium
How did the artist M.C. Escher use tessellations?
Example 2
challenge
A soccer ball is built from regular pentagons (12) and regular hexagons (20). Using Euler's formula V−E+F=2, verify the count of pentagons.
Example 3
easy
Do all quadrilaterals tessellate the plane?
Example 4
challenge
What is special about Penrose tilings?
Example 5
easy
A tile leaves visible gaps between copies. Is the pattern a tessellation?
Example 6
hard
A vertex configuration 3.7.42 sums to 360° algebraically. Why is it nonetheless not a true vertex of any tiling?
Example 7
medium
Are all quadrilaterals (including non-convex ones) able to tile the plane?
Example 8
medium
A tiling uses regular octagons and squares. One proposed vertex arrangement has 1 octagon and 2 squares. Does this satisfy the 360° vertex condition? What arrangement actually works?Regular octagon: interior angle = 135°; combined with square (90°): 135° + 135° + 90° = 360° ✓
Example 9
easy
How many equilateral triangles meet at each vertex in a triangular tessellation?Interior angle = 60°; six equilateral triangles meet at each vertex: 6 × 60° = 360°
Example 10
medium
A vertex configuration 3.3.3.3.3.3 describes which tiling?
Example 11
medium
At a vertex, can a square (90°), a hexagon (120°), and another shape fit? What angle must the third shape contribute?
Example 12
challenge
Penrose tilings use two prototiles (kite and dart) to cover the plane aperiodically. Why is this impossible with a single regular polygon?
Example 13
medium
Why is the honeycomb (hexagonal tiling) efficient for bees?
Example 14
easy
True or false: every triangle tiles the plane.
Example 15
medium
Solve for n: a regular n-gon tiles the plane alone if and only if n−22n is a positive integer. List the valid n.
Example 16
easy
How many squares meet at each vertex in a square tessellation?Square: interior angle = 90°; four squares meet at each vertex: 4 × 90° = 360°
Example 17
medium
Check the vertex configuration 3.6.3.6: equilateral triangle, hexagon, triangle, hexagon. Does it sum to 360°?
Example 18
hard
Count distinct semiregular (Archimedean) tilings of the plane.
Example 19
medium
What general angle condition must hold at every vertex of any edge-to-edge tessellation?
Example 20
medium
What is a semi-regular (Archimedean) tessellation?