This cheat sheet covers tessellations, symmetry, and the rigid motions used to describe repeating geometric patterns. Students need these ideas to recognize how shapes fill a plane without gaps or overlaps. The reference also helps connect visual patterns to angle measures, transformations, and precise geometry vocabulary.
It is designed for quick review during classwork, homework, or test preparation.
The most important tessellation rule is that the angles meeting at each vertex must add to . For a regular -gon, the interior angle is , so students can test whether copies of one regular polygon will tessellate. Symmetry describes how a figure can be moved and still match itself, using reflections, rotations, translations, and glide reflections.
Rotational symmetry is often measured by the angle when a figure matches itself times in one full turn.
Key Facts
- A tessellation covers the plane with repeated shapes that have no gaps and no overlaps.
- Angles around any point in a tessellation must add to .
- The interior angle of a regular -gon is .
- A regular polygon tessellates by itself when its interior angle divides evenly into .
- The only regular polygons that tessellate by themselves are equilateral triangles, squares, and regular hexagons.
- A reflection flips a figure over a line so corresponding points are the same perpendicular distance from the line of reflection.
- A rotation turns a figure around a fixed center by an angle such as , , or .
- If a figure has rotational symmetry of order , the smallest angle of rotation is .
Vocabulary
- Tessellation
- A pattern of shapes that covers a plane completely with no gaps and no overlaps.
- Regular tessellation
- A tessellation made from only one type of regular polygon, with the same arrangement at every vertex.
- Semi-regular tessellation
- A tessellation made from two or more regular polygons arranged in the same order at every vertex.
- Line symmetry
- A figure has line symmetry when a line divides it into two mirror-image halves.
- Rotational symmetry
- A figure has rotational symmetry when it can be turned less than around a center and still match itself.
- Glide reflection
- A glide reflection is a transformation made by translating a figure and then reflecting it across a line.
Common Mistakes to Avoid
- Adding only some of the angles at a vertex is wrong because every angle touching that point must be included to check whether the total is .
- Assuming every regular polygon tessellates is wrong because the regular polygon's interior angle must divide evenly into .
- Confusing a translation with a reflection is wrong because a translation slides a figure without flipping it, while a reflection reverses its orientation.
- Counting rotational symmetry after a full turn only is wrong because every figure matches itself after a full turn, but true rotational symmetry must occur before .
- Leaving gaps or overlaps in a repeated pattern is wrong because a tessellation must cover the plane completely with no uncovered space and no stacked shapes.
Practice Questions
- 1 A regular hexagon has interior angle . How many regular hexagons meet at one vertex in a regular tessellation?
- 2 Use to find the interior angle of a regular octagon with .
- 3 A design has rotational symmetry of order . What is the smallest angle of rotation that maps the design onto itself?
- 4 Explain why regular pentagons do not make a regular tessellation, even though they can be repeated in a pattern.
Understanding Tessellations & Symmetry Reference
A semi-regular tessellation uses two or more kinds of regular polygons, arranged in the same order at every vertex. The order matters. A pattern with a triangle, square, triangle, square around one vertex has a different structure from one with two triangles followed by two squares.
Students can describe this order by walking around a vertex and naming each polygon. This makes it easier to tell whether a design is truly repeating or only looks balanced in one small area.
Some combinations fit at a vertex but cannot continue across the whole plane in a consistent way. A valid tessellation needs the local arrangement to keep working as the pattern spreads outward.
Rigid motions preserve size, shape, angle measure, and distance. They change position or direction only. A translation slides every point the same distance in the same direction.
This is why matching tiles in a row often come from repeated translations. A rotation keeps one point fixed while every other point moves around it. A reflection reverses orientation.
For example, a letter shape may face the opposite way after a reflection, even though its lengths stay unchanged. Tracking one corner at a time helps students perform transformations accurately. It is useful to mark the center of rotation or the mirror line before drawing the image.
A glide reflection is a two-step motion. The figure is reflected across a line, then shifted along that same line. Footprints in a trail can show this idea because each print is a flipped version of the other and is moved forward.
Many border patterns use glide reflections even when no single reflection line appears obvious. Symmetry in a whole pattern can differ from symmetry in one tile. A tile may have no lines of symmetry, yet repeated copies can create reflection lines across the full design.
Looking at the entire repeated arrangement prevents this common mistake. Rotational symmetry should be checked by imagining a turn around the correct center, not merely by noticing that parts look similar.
These ideas appear in floor tiles, brick walls, fabric prints, wallpaper, mosaics, logos, and computer graphics. Designers use repeated transformations to make patterns that are organized and easy to extend. In class, careful vocabulary matters because slide, flip, turn, and glide describe different motions.
Students should check for gaps or overlaps near every type of vertex, especially where different polygons meet. When drawing on grid paper, use equal spacing for translations and count turns in a consistent direction for rotations.
For reflections, corresponding vertices must lie directly across the mirror line at equal distances. A neat labeled sketch often reveals an error faster than mental guessing.