ACARA v9 CONTENT DESCRIPTION “recognise and use combinations of transformations to create tessellations and other geometric patterns, using dynamic geometric software where appropriate”
Builds on: Coordinates on the Plane (AC9M6SP02). Moving a point on the plane leads to moving whole shapes — sliding, flipping and turning them to build patterns.
The three transformations
A transformation moves a shape to a new position without changing its size or form. There are three basic kinds. A translation slides a shape in a straight line, every point moving the same distance in the same direction. A reflection flips a shape across a mirror line, producing a mirror image. A rotation turns a shape about a fixed point through some angle. In all three the shape itself stays the same size and shape; only its position or orientation changes. These three moves are the building blocks of every geometric pattern in this unit.
Three transformations
A shape can be slid, flipped, or turned. Each move keeps the shape the same size.
A translation slides a shape without turning or flipping it. — these are the three transformations that move a shape without changing its size.
Naming the move
Given a shape and its image after a move, the first skill is to name which transformation was used. If the image is shifted along without any turn or flip, it is a translation. If it is a mirror image, reversed across a line, it is a reflection. If it has been turned, so that it points in a new direction about a centre, it is a rotation. Reading the kind of move from the before-and-after positions trains the eye to see transformations everywhere, and it is the basis for describing how any pattern was made.
Name the transformation
Look at how the shape moved: did it slide, flip, or turn?
Which transformation maps the first shape to the second? Pick A, B or C.
Tessellations
A tessellation is a pattern of one or more shapes that covers a flat surface completely, with no gaps and no overlaps. Floor tiles, brick walls and honeycomb are all tessellations. The shapes fit together edge to edge, repeating across the plane in every direction. A tessellation is built by taking a shape and moving copies of it by transformations, sliding, flipping or turning, so that each new copy locks into the last. The idea that a single shape, repeated by transformation, can fill a whole surface is what makes tessellations both useful and beautiful.
Tessellations tile the plane
A tessellation is a pattern of shapes that covers a surface with no gaps and no overlaps.
These squares fit together with no gaps or overlaps — a tessellation covers the plane by repeating a shape.
Which shapes tessellate
Not every shape tessellates on its own. A square does, fitting neatly edge to edge; so does an equilateral triangle and a regular hexagon, which is why honeycomb is hexagonal. A regular pentagon, however, leaves gaps no matter how it is arranged, and a circle cannot tile without spaces between. Whether a regular shape tessellates depends on whether copies of its corners fit exactly around a point. Knowing which shapes tile alone, and which need to be combined with others, is the key to designing a tessellation that truly covers the plane.
Which shapes tessellate
Some regular shapes tile the plane on their own; others leave gaps.
Can a square tile the plane with no gaps? Decide yes or no.
Building a pattern
A geometric pattern grows by repeating a transformation. Slide a motif the same distance again and again and it marches across the page; flip it alternately and it makes a mirrored band; turn it by equal angles about a point and it forms a rosette. Each copy is identical to the first, just moved once more by the same rule. This repetition is exactly how tessellations and decorative patterns are constructed, and dynamic geometry software makes it easy to apply a transformation over and over and watch the pattern build.
Repeat to build a pattern
Apply the same transformation over and over and a regular pattern grows along the row.
Repeating one transformation builds a geometric pattern — each copy is the same shape, moved again the same way.
Combining transformations
Transformations can be combined, one applied after another, and often a combination equals a single simpler move. Two translations in the same direction make one longer translation; reflecting a shape and then reflecting it back across the same line returns it to the start. Rich patterns and tessellations usually need more than one kind of move, mixing translations with reflections or rotations so the shapes interlock. Understanding how transformations combine lets you both predict the result of a sequence of moves and design the precise combination a pattern requires.
Combining transformations
Two transformations in a row often equal a single one, and combinations build patterns.
Choose the best answer about combining transformations. Pick A, B or C.
Where transformations lead
Transformations and tessellations connect geometry to art, design and the natural world, from Islamic tiling patterns to the cells of a honeycomb. The same three moves, translation, reflection and rotation, reappear in later mathematics as the foundation of symmetry, congruence, and the coordinate transformations used in computer graphics and animation. Learning to recognise, apply and combine transformations is a step into a way of thinking about shape and pattern that runs through mathematics, science and the visual world alike.
Quick self-check
1. A shape slides 3 units right without turning or flipping. This transformation is a...
2. A shape is flipped across a mirror line. This is a...
3. A pattern of shapes that covers the plane with no gaps or overlaps is called a...
4. Which regular shape does NOT tessellate on its own?
5. Sliding a shape right, then sliding it right again, is the same as...