Tiling and Tessellation Calculator

Build regular, semi-regular, and custom repeating tilings with live symmetry analysis.

Results are calculated automatically as you enter data.

Pattern controls

Adjust the tiles and watch the plane update instantly.

Tessellation Coloring mode Symmetry focus Tile size 52 px Custom angle 70 deg Motif offset 38%

Outlines Symmetry labels Motif overlay

Base wallpaper group p4m

Formula and explanation

Disclaimer: All tools on this site are intended for educational and informational purposes only. Educational geometry tool; not professional advice. The calculator verifies its predefined repeat cells numerically, but arbitrary user-drawn polygons are not supported. See our Terms of Use.

Assumptions and limitations

Results are educational estimates, not professional engineering, construction, safety, scientific, legal, financial, or academic advice. No guarantee is made that outputs are accurate, complete, suitable, or error-free. Verify units, formulas, assumptions, tolerances, rounding, and limitations independently, and use qualified professional review for technical or safety-sensitive work. Use of this calculator is subject to our Terms of Use.

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Related Calculators

What Is a Tessellation?

A tessellation is a pattern of shapes that covers a flat plane without gaps or overlaps. Floor tiles, honeycomb-like hexagon grids, brick layouts, quilt blocks, and many decorative wall patterns are everyday examples. In geometry, the main question is not only whether the pattern looks attractive, but whether the shapes fit together exactly and can continue indefinitely.

A tessellation usually starts with one tile or a small repeat unit. That unit is copied across the plane by geometric transformations such as translation, rotation, reflection, or glide reflection. When the copies line up cleanly, the pattern can be extended beyond the visible drawing.

The important idea is that a tessellation is not just a picture. It is a rule for repeating shapes. The visible pattern may be finite on a screen or page, but the geometric idea describes how the pattern could continue across the plane.

Why Tessellations Matter

Tessellations connect pure geometry with design, art, architecture, teaching, and visual problem solving. They help students see how angles, polygons, symmetry, and transformations work together.

They are also useful because they make hidden structure visible. A square grid shows translation and reflection symmetry. A honeycomb pattern shows how regular hexagons can meet perfectly around each vertex. Semi-regular patterns show how different polygons can combine as long as their angles still close the space around each meeting point.

For artists and designers, tessellations are a way to build repeating visual systems. For geometry learners, they are a practical way to test whether angle facts and symmetry ideas actually fit together.

Key Terms to Know

Tile: One repeated shape or motif in a tessellation.
Plane: A flat two-dimensional surface that extends indefinitely.
Vertex: A corner point where tiles meet.
Interior angle: The angle inside a polygon at one of its vertices.
Vertex configuration: A notation that records which polygons meet at a typical vertex. For example, 4.8.8 means one square and two octagons meet at that point.
Regular tessellation: A tessellation made from one kind of congruent regular polygon.
Semi-regular tessellation: A tessellation made from more than one kind of regular polygon, arranged in the same order around every vertex.
Translation: Sliding a shape without turning or flipping it.
Rotation: Turning a shape around a point.
Reflection: Flipping a shape across a mirror line.
Glide reflection: A reflection followed by a slide along the direction of the mirror line.
Wallpaper group: A symmetry classification for a repeating pattern in the plane.

How Tessellations Work

The simplest tessellation test is the angle test. Around a point on a flat plane, a full turn is $360^\circ$. For tiles to meet cleanly at a vertex, their interior angles at that point must add to $360^\circ$.

For a regular polygon with $n$ sides, each interior angle is:

$$ \alpha = \frac{(n - 2) \times 180^\circ}{n} $$

Where:

$\alpha$ is one interior angle of the regular polygon.
$n$ is the number of sides.

At a tessellation vertex, the angle condition is:

$$ \alpha_1 + \alpha_2 + \cdots + \alpha_k = 360^\circ $$

Where $\alpha_1, \alpha_2, \ldots, \alpha_k$ are the angles of the tiles that meet at the vertex.

This rule explains why regular triangles, squares, and hexagons tessellate by themselves:

Six equilateral triangles meet because $6 \times 60^\circ = 360^\circ$.
Four squares meet because $4 \times 90^\circ = 360^\circ$.
Three regular hexagons meet because $3 \times 120^\circ = 360^\circ$.

It also explains why a regular pentagon does not tessellate the plane by itself. Each regular pentagon has an interior angle of $108^\circ$, and no whole number of $108^\circ$ angles adds exactly to $360^\circ$.

Symmetry adds another layer. Two tessellations can both fill the plane, but they may have different types of symmetry. A pattern may have mirror lines, rotation centers, glide reflections, or only translations. Wallpaper-group notation summarizes those repeating symmetry features.

For a rhombus repeat, area can be understood through the parallelogram formula. A rhombus with side length $s$ and included angle $\theta$ has height $s\sin(\theta)$, so its area is:

$$ A = s^2\sin(\theta) $$

In this calculator, rhombus repeat area is shown in square pixels, because the tile size is a screen-space scale rather than a real-world length.

Examples of Tessellations in Practice

Example 1: A Square Tessellation

A square has four sides, so $n = 4$.

$$ \alpha = \frac{(4 - 2) \times 180^\circ}{4} = 90^\circ $$

Four squares meet at each vertex:

$$ 90^\circ + 90^\circ + 90^\circ + 90^\circ = 360^\circ $$

That is why the square tessellation has vertex configuration 4.4.4.4.

Example 2: A Semi-Regular `4.8.8` Tessellation

The notation 4.8.8 means one square and two regular octagons meet around a vertex.

A square contributes $90^\circ$. A regular octagon has:

$$ \alpha = \frac{(8 - 2) \times 180^\circ}{8} = 135^\circ $$

The angles close exactly:

$$ 90^\circ + 135^\circ + 135^\circ = 360^\circ $$

This is why the square-and-octagon arrangement can form a clean semi-regular tessellation.

Example 3: A Custom Rhombus Repeat

Suppose the tile size is $60\text{ px}$ and the rhombus angle is $70^\circ$. The repeat area is computed from the rhombus lattice vectors:

$$ A = 60^2\sin(70^\circ) $$

$$ A \approx 3382.89\text{ px}^2 $$

If the angle moves closer to $0^\circ$ or $180^\circ$, the rhombus becomes very thin and the area gets smaller. The calculator avoids exactly $0^\circ$ and $180^\circ$ because those would collapse the rhombus into a line rather than a usable tile.

How to Interpret the Result

The results describe both the visible drawing and the geometry behind the selected pattern.

Result	What it means	How to read it
Base wallpaper group	The symmetry classification of the built-in geometric construction before optional coloring and overlays	`p4m`, `p6m`, `cmm`, and `pg` are wallpaper-group labels, not measurements
Visible tiles	The number of tiles intersecting the canvas with positive area	A larger tile size usually lowers this count; a smaller tile size usually raises it
Vertex pattern	The sequence of polygon types meeting at a typical vertex	`4.8.8` means one square and two octagons meet there
Local angles	The angle information for the chosen pattern	For a rhombus, it shows the selected angle and its supplement, such as $70^\circ / 110^\circ$
Translation-cell area	The exact area of the displayed pattern's translation cell	It is measured in `px²`, not square centimeters, square inches, or another physical unit

A high visible-tile count does not mean the mathematical tessellation has more tiles overall. A true plane tessellation is unlimited. The visible count is only the number of repeated shapes shown inside the finite drawing area.

A larger repeat area means the pattern’s repeat unit occupies more screen space. It does not mean the pattern is more symmetrical, more accurate, or more mathematically complex.

Color does not change the tile geometry, but it can change the symmetry of the completed visual pattern. Uniform coloring preserves the stated base group; radial coloring is non-periodic, and other decorations may reduce the symmetry.

Common Mistakes and Misconceptions

Mistake 1: Treating the visible tile count as the total number of tiles.
The displayed pattern is only a finite window into a repeating design. The visible tile count depends on canvas size, tile size, and drawing boundaries.

Mistake 2: Confusing pixels with real-world units.
Tile size is measured in pixels, and repeat area is measured in square pixels. These are screen units, not physical print dimensions.

Mistake 3: Assuming color changes the geometry.
Color mode can make symmetry easier to see, but it does not change the tile shape or the angle-sum rule.

Mistake 4: Expecting the angle slider to affect every pattern in the same way.
The angle value is especially important for the custom rhombus. Regular square, triangle, hexagon, and semi-regular patterns use fixed angle relationships.

Mistake 5: Forgetting the $360^\circ$ vertex condition.
A pattern may look repetitive but still fail to tessellate cleanly if the angles around a meeting point leave a gap or overlap.

Mistake 6: Assuming any artistic motif will tile.
The built-in motif is paired by an explicit glide reflection. That verifies this construction, but it does not prove that every freely drawn or deformed motif tessellates.

When to Use Tessellation Concepts

Use tessellation concepts when you want to:

Check whether regular polygons can fill the plane without gaps.
Understand vertex notation such as 4.4.4.4, 6.6.6, 4.8.8, or 3.6.3.6.
Compare translation, rotation, reflection, and glide reflection symmetry.
Explore how mathematical patterns appear in art, tiling, textile design, and architecture.
Teach or study polygon interior angles in a visual way.
Experiment with repeating motifs before drawing or exporting a pattern.

Limitations and Things to Keep in Mind

A tessellation calculator is a visual and educational tool. It helps you explore patterns, but it does not replace a formal proof for arbitrary tiling problems.

The supported patterns are predefined. The calculator can show regular squares, regular triangles, regular hexagons, selected semi-regular patterns, a custom rhombus, and a verified glide-reflection motif. It does not accept arbitrary user-drawn polygons or classify every possible wallpaper group.

The drawing is finite even though the mathematical idea is infinite. The canvas shows a bounded sample of a pattern that could continue beyond the screen.

Translation-cell area is computed from the determinant of the two translation vectors and cross-checked against the polygon areas in one cell. It is exact up to floating-point rounding, but remains a screen-space measurement rather than a physical area.

Rounding also matters. The visible tile count is displayed as a whole number, while repeat area is displayed with up to two decimal places. Tile size, angle, and offset may be displayed with up to one decimal place. Small visual differences may therefore be rounded away in the text output.

How to Use This Calculator

Choose a tessellation pattern, such as squares, triangles, hexagons, 4.8.8, 3.6.3.6, a rhombus, or the glide-reflection motif.
Choose a color mode to change how repeated tiles are colored.
Select a symmetry focus, such as full, reflection, rotation, or glide.
Adjust the tile size to scale the pattern on the canvas.
For the custom rhombus, adjust the angle to change the rhombus skew and repeat area.
Adjust the offset if you want to change motif placement or deformation effects.
Turn outlines, symmetry labels, and motif marks on or off as needed.
Review the wallpaper group, visible tile count, vertex pattern, local angles, and repeat area.
Use the preset, randomize, reset, or PNG download option when helpful.

Frequently Asked Questions

What makes a pattern a tessellation?

A pattern is a tessellation when its shapes cover the plane without gaps or overlaps. In regular polygon tessellations, the interior angles that meet at each vertex must add to $360^\circ$.

Why do only triangles, squares, and hexagons tessellate by themselves as regular polygons?

Their interior angles divide $360^\circ$ evenly: six equilateral triangles, four squares, or three regular hexagons can meet around a point. Regular pentagons, heptagons, octagons, and most other regular polygons do not work alone because their angles do not close the vertex exactly.

What does vertex pattern `4.8.8` mean?

It means that one square and two regular octagons meet at a typical vertex. The numbers name the polygons by their side counts, read in order around the vertex.

What is a wallpaper group?

A wallpaper group is a classification of a repeating plane pattern by its symmetry. It describes whether the pattern has features such as translations, rotations, reflections, and glide reflections.

Does the color mode change the tessellation?

It does not change the polygon geometry, but it can reduce the symmetry of the colored pattern. Use uniform coloring when you want the displayed decoration to preserve the base wallpaper group.

Why is repeat area measured in `px²`?

The calculator draws patterns on a screen, so its size values are based on pixels. Repeat area therefore describes screen-space area, not a real-world area such as square centimeters or square inches.

Why does the visible tile count change when tile size changes?

The canvas has a fixed visible region. Smaller tiles allow more repeated shapes to fit into that region, while larger tiles allow fewer shapes to fit.

Sources and References

Books

Branko Grünbaum and G. C. Shephard. Tilings and Patterns. 2nd ed., Dover Publications, 2016. Relevant sections: basic tiling terminology, tilings by regular polygons, patterns, and symmetry classification. ISBN: 9780486469812.
John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss. The Symmetries of Things. A K Peters/CRC Press, 2008. Relevant sections: symmetries of plane repeating patterns, planar patterns, orbifolds, and Archimedean tilings. ISBN: 9781568812205.

Online and Official Sources

OpenStax. “10.4 Polygons, Perimeter, and Circumference.” Contemporary Mathematics, accessed July 4, 2026. https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference
OpenStax. “10.5 Tessellations.” Contemporary Mathematics, accessed July 4, 2026. https://openstax.org/books/contemporary-mathematics/pages/10-5-tessellations
OpenStax. “10.6 Area.” Contemporary Mathematics, accessed July 4, 2026. https://openstax.org/books/contemporary-mathematics/pages/10-6-area
OpenStax. “5.4 Right Triangle Trigonometry.” Precalculus 2e, accessed July 4, 2026. https://openstax.org/books/precalculus-2e/pages/5-4-right-triangle-trigonometry
Eric W. Weisstein. “Wallpaper Groups.” Wolfram MathWorld, accessed July 4, 2026. https://mathworld.wolfram.com/WallpaperGroups.html
Museum Escher in The Palace. “The Regular Division of The Plane at the ‘De Roos’ foundation,” accessed July 4, 2026. https://www.escherinhetpaleis.nl/en/about-escher/escher-today/the-regular-division-of-the-plane-at-the-de-roos-foundation