Tessellation Math Example 2

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Example 2

hard
A proposed semi-regular tiling places 22 triangles and 22 squares at every vertex. Verify whether the vertex angles sum to 360°360°, and if not, find an arrangement of triangles and squares that does work.

Solution

  1. 1
    Interior angle of an equilateral triangle: (3−2)×180°3=60°\dfrac{(3-2)\times180°}{3} = 60°.
  2. 2
    Interior angle of a square: (4−2)×180°4=90°\dfrac{(4-2)\times180°}{4} = 90°.
  3. 3
    Proposed arrangement 2T+2S2T + 2S: 2×60°+2×90°=120°+180°=300°≠360°2\times60° + 2\times90° = 120° + 180° = 300° \neq 360°. ✗
  4. 4
    Try 3T+2S3T + 2S (notation 3.3.4.3.43.3.4.3.4): 3×60°+2×90°=180°+180°=360°3\times60° + 2\times90° = 180° + 180° = 360°. ✓ This tiling works.

Answer

The 2T+2S2T+2S arrangement gives 300°300° and fails. The valid arrangement is 33 triangles +2+ 2 squares (3.3.4.3.43.3.4.3.4), giving 360°360°.
In any valid tessellation, the angles around every vertex must sum to exactly 360°360°. Verifying candidate arrangements by computing their angle sums is the essential check before accepting a proposed tiling.

About Tessellation

A tessellation is a pattern that covers an infinite plane with repeated geometric shapes, leaving no gaps and having no overlaps.

Learn more about Tessellation →

More Tessellation Examples

Example 1 medium

Explain why regular hexagons tessellate the plane but regular pentagons do not.

Example 3 easy

Does an equilateral triangle tessellate the plane? Justify your answer using the interior angle.

Example 4 medium

A tiling uses regular octagons and squares. One proposed vertex arrangement has [formula] octagon an

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