Which figure repeats in the tessellation shown below?
The answer is the star‑shaped tile that appears in every row and column of the pattern. Below we explore why this figure is the only one that can tile the plane without gaps or overlaps, and how you can recognize it in any tessellation.
Introduction
Tessellations—patterns that cover a surface completely without gaps or overlaps—are a favorite topic in geometry, art, and design. When you look at a tessellated floor, a wallpaper, or a puzzle, you might wonder which shapes are being repeated. In the figure shown, several different polygons are visible, but only one of them truly repeats across the entire plane. Identifying the repeating figure requires a mix of observation, symmetry analysis, and a touch of mathematical reasoning.
Observing the Pattern
1. Count the unique shapes
At a glance, the tessellation contains:
- A hexagon that fits snugly beside a square. Which means - A diamond (rhombus) that seems to connect the two. - A star‑shaped tile that sits at the intersection of the other shapes.
Even though all three shapes appear, not all of them repeat in the same orientation or position.
2. Look for symmetry
Tessellations rely on translational symmetry: shifting the pattern by a certain distance reproduces the same arrangement. When you slide the pattern horizontally or vertically, the star tile aligns perfectly with its neighbors, while the hexagon and square do not maintain the same relative position.
3. Check for gaps or overlaps
If you trace the edges of each shape, you'll notice that the star tile’s edges match exactly with adjacent star tiles, leaving no gaps. The hexagons and squares, however, leave small triangular spaces when placed next to each other. Those spaces are filled by the star tile instead Practical, not theoretical..
Why the Star Tile Repeats
A. The Geometry of the Star
The star shape in this tessellation is actually a regular 12‑pointed star (a dodecagon with alternating long and short edges). Its key properties include:
- Equal angles: Each point of the star subtends the same angle at the center, ensuring uniformity.
- Consistent edge lengths: The alternating long and short edges match perfectly with neighboring stars.
- Rotational symmetry: Rotating the star by 30° (one twelfth of a full turn) maps it onto itself, which aligns with the translational symmetry of the tessellation.
These geometric traits allow the star tile to fit into the lattice of the pattern without leaving voids That's the whole idea..
B. Compatibility with Adjacent Shapes
The hexagon and square serve as supporting shapes, filling the spaces that the star cannot. That said, the star’s edges are designed to meet the corners of both the hexagon and the square. When you overlay a star tile onto a hexagon, the long edges of the star align with the hexagon’s sides; when you place it next to a square, the short edges fit snugly into the square’s corners. This complementary fitting is why the star can be the repeating unit Still holds up..
C. Translational Symmetry Explained
In a tessellation, you can shift the entire pattern by a vector (e.g., moving right by the width of one star and up by the height of one star) and the pattern looks unchanged Nothing fancy..
- Horizontal shift: Moving right one star width aligns the star with its neighbor on the right.
- Vertical shift: Moving up one star height aligns the star with the one above.
- Diagonal shift: Combining both moves still lands on a star tile.
The hexagon and square do not satisfy these conditions simultaneously; they would misalign after a single horizontal or vertical shift.
Step‑by‑Step Identification
If you’re unsure which figure repeats, follow these steps:
- Select a reference point: Pick a corner or edge of a shape that appears clearly.
- Track the shape across the pattern: Move horizontally and vertically, noting when the same shape reappears.
- Check orientation: A repeating shape will maintain the same orientation (rotated copies count if the shape is rotationally symmetric).
- Confirm edge matching: check that the edges of the shape match perfectly with neighboring shapes each time it appears.
- Count repetitions: If a shape appears in every row and column without interruption, it’s the repeating figure.
Applying this to the tessellation, the star meets all criteria, while the hexagon and square do not Easy to understand, harder to ignore. Which is the point..
Common Mistakes
| Mistake | Why it Happens | How to Avoid It |
|---|---|---|
| Assuming the most obvious shape repeats (e.g., the hexagon) | Hexagons are common in tessellations and stand out visually | Verify translational symmetry for each shape |
| Ignoring rotational symmetry | Some shapes look identical only after rotation | Check if the shape’s orientation changes after a shift |
| Overlooking small gaps | Tiny gaps can be hard to spot, especially in complex patterns | Zoom in or trace edges carefully |
FAQ
Q1: Can a tessellation have more than one repeating figure?
A1: Yes, a tessellation can be composed of multiple repeatable units, but typically one figure acts as the fundamental repeating tile. In this pattern, the star is the fundamental tile; the hexagon and square are auxiliary shapes that complete the pattern.
Q2: How does the star tile compare to a regular pentagon in tessellations?
A2: Regular pentagons cannot tile the plane by themselves because interior angles (108°) do not divide 360° evenly. The star tile, however, has angles that fit perfectly around a point (e.g., 30° + 30° + 30° + 30° + 30° + 30° = 180°), allowing seamless tiling.
Q3: What if I rotate the entire tessellation? Does the repeating figure change?
A3: Rotating the entire pattern does not change which figure repeats. The star remains the repeating unit; only its orientation relative to the viewer changes.
Q4: Is the star tile a common shape in historical tessellations?
A4: Yes, star polygons have been used in Islamic geometric art, Mesoamerican mosaics, and modern tiling designs due to their aesthetic appeal and mathematical properties Surprisingly effective..
Conclusion
In the tessellation presented, the star‑shaped tile is the figure that truly repeats across the entire plane. That said, its geometric perfection—equal angles, matching edge lengths, and rotational symmetry—allows it to fit naturally with both hexagons and squares, ensuring a gap‑free, overlap‑free pattern. By observing symmetry, checking edge compatibility, and verifying translational invariance, you can confidently identify the repeating figure in any tessellation. Whether you’re a geometry enthusiast, an artist, or a puzzle solver, mastering these techniques opens the door to a deeper appreciation of the beautiful mathematics that underpins tiled designs It's one of those things that adds up..
Looking closely at the tessellation, the star shape stands out as the fundamental repeating unit. That said, its edges match perfectly with adjacent hexagons and squares, and when you shift the pattern by its own width and height, the star tiles align exactly without gaps or overlaps. The hexagon and square, while present, do not possess the same translational symmetry—shifting by their own dimensions misaligns them with the pattern, revealing they are auxiliary rather than fundamental. This makes the star the true "tile" that builds the entire design Surprisingly effective..
In practice, identifying the repeating figure comes down to checking for translational symmetry: does the shape line up with itself after a shift? For the star, the answer is yes; for the hexagon and square, it's no. This distinction is crucial, especially in complex patterns where multiple shapes coexist. The star's ability to fit easily with other polygons while maintaining its own repetition is what makes it the backbone of this tessellation Easy to understand, harder to ignore..
From an artistic and mathematical standpoint, the star's role is significant. Its angles and edges are crafted so that multiple copies fit together perfectly, a property shared by classic tessellations like those of M.C. Here's the thing — recognizing this pattern not only helps in solving puzzles but also deepens appreciation for the underlying geometry in art and design. Escher or traditional Islamic geometric art. In this case, the star is the repeating figure that brings the entire tessellation to life The details matter here..
