Tessellations Using Translations and Reflections

A tessellation is a repeating pattern of one or more shapes that covers a flat surface completely, with no gaps and no overlaps. Think of floor tiles, honeycomb cells, or a checkerboard — each is built from a single shape (or a small group of shapes) repeated over and over according to a rule. Two of the simplest rules for repeating a shape are translation (sliding) and reflection (flipping). Both are covered in more general terms in the lesson on introduction to transformations, and this page focuses on how those two specific moves are used to tile a plane.

A translation slides a shape a fixed distance in a fixed direction without turning it or flipping it. To tessellate using translation, you repeat this same slide over and over, both across and down, so that every copy of the shape lines up edge to edge with the next.

The most familiar example is a grid of squares: pick a square, slide it right by its own width, and keep sliding to fill a row. Then slide the whole row down to fill the plane. Because every copy is an exact, non-rotated repeat of the original, the pattern is easy to check — if the shape tiles once without a gap, sliding it repeats that success forever.

A reflection flips a shape over a line, producing its mirror image. On its own, a reflection creates just one flipped copy, so most reflective tessellations combine a reflection with a translation — this combined move is called a glide reflection: flip the shape, then slide the flipped copy along the mirror line.

Glide reflections are especially useful for shapes that don't tile neatly when only slid, such as many triangles and irregular quadrilaterals. Flipping alternate copies lets their slanted edges match up with their neighbors, closing gaps that a pure translation would leave open. This is different from turning a shape around a point, which is the approach used in tessellations using rotations.

Show that a 3 cm by 3 cm square can tessellate the plane using only translations.

Step 1: Place one square. Its interior angles are all 90 degrees.

Step 2: Translate the square 3 cm to the right, matching edge to edge with no rotation or flip. Repeat across the row.

Step 3: Translate the entire row 3 cm downward and repeat.

Step 4: Check any interior vertex, where four squares meet. Each contributes a 90 degree angle, and 90 × 4 = 360 degrees, exactly filling the space around the point with no gap or overlap. The square tessellates using pure translation.

Show that a scalene right triangle can tessellate the plane using reflections and translations together.

Step 1: Take one right triangle. Reflect it across one of its legs to create a second triangle, forming a rectangle made of the two triangles.

Step 2: Translate this rectangle pair across and down to fill the plane, exactly like the square grid in Example 1.

Step 3: Check a vertex where triangles meet. Around each interior point, the angles from the original triangle and its mirror image together add to 360 degrees, since two triangles' angles (each summing to 180 degrees) combine at each row of vertices. No gaps remain, so the glide-reflected triangle tessellates.

For a single regular polygon to tile the plane on its own, its interior angle must divide evenly into 360 degrees, since that many degrees must fit exactly around every interior vertex. Only three regular polygons pass this test: the equilateral triangle (60 degrees), the square (90 degrees), and the regular hexagon (120 degrees). Other regular polygons, like a regular pentagon at 108 degrees, leave gaps or force overlaps no matter how you translate or reflect them. For the full angle-sum reasoning behind this rule, see the tessellation formula lesson. Irregular shapes can still tessellate, but usually only by combining moves — translating, reflecting, or rotating — so that the angles around each vertex still add to 360 degrees.

When deciding whether translations and/or reflections make a shape tessellate, work through these checks:

• Do the copies of the shape share full edges with their neighbors, with no partial overlaps?
• Do the angles meeting at every interior vertex add up to exactly 360 degrees?
• If a plain translation leaves a gap, does flipping alternate copies (a glide reflection) close that gap?

If you answer yes to all three, the pattern is a true tessellation. Symmetry ideas from line symmetry can help you spot which mirror lines will work before you draw the whole pattern out.

Related lessons

• Geometry Help