Congruency, Enlargement and Combined Transformations · 5.4.2
Designing a tessellation
Students create their own tessellation by repeatedly applying isometric transformations, translation, reflection and/or rotation, to a chosen basic tile, checking that the resulting pattern covers the plane completely with no gaps or overlaps.
The official learning standard (5.4.2)
“Design tessellation involving isometric transformation.”
What it means
Students create their own tessellation by repeatedly applying isometric transformations, translation, reflection and/or rotation, to a chosen basic tile, checking that the resulting pattern covers the plane completely with no gaps or overlaps.
How it is examined
This is largely a practical, project-based standard. In the SPM examination it may still appear as a Paper 1 or Paper 2 item asking students to identify which transformation(s) generate a given tessellation from a basic tile, or to complete part of a tessellation diagram.
Worked example
A basic tile is a right-angled triangle with vertices A(0, 0), B(2, 0) and C(0, 2). Describe how isometric transformations can be used, starting with a rotation, to tessellate the plane using this triangle.
- Rotate triangle ABC through 180° about the midpoint of BC, M(1, 1), to form a congruent triangle A′B′C′ (rotation is isometric, so size and shape are unchanged).
- Triangle ABC together with its rotated image A′B′C′ fit together exactly along BC to form a 2 × 2 square with vertices (0, 0), (2, 0), (2, 2) and (0, 2).
- Translate this square repeatedly by the vector (2, 0) to tile a complete row of squares along the x-axis.
- Translate the row repeatedly by the vector (0, 2) to tile the entire plane, producing a pattern with no gaps or overlaps.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Which transformations can be used to design a tessellation?
Any isometric transformation, translation, reflection or rotation, since these preserve the size and shape of the tile, so every repeated copy is congruent and fits together exactly with no gaps or overlaps.
Does every shape tessellate the plane?
No. Only equilateral triangles, squares and regular hexagons tessellate on their own as a regular tessellation.
Other shapes may still tessellate if combined with a rotated or reflected copy of themselves, or with other shapes, but not every shape does.
Why must the transformations used be isometric?
Isometric transformations keep the size and shape of the tile unchanged, changing only its position or orientation. This is essential so that every repeated copy is exactly congruent and fits against its neighbours without gaps or overlaps.