Fit the edges
Zero gaps restore the natural spacing between neighboring tile positions.
Tessellation Studio
Adjust repeated polygon spacing and stagger to cover a mat without gaps or overlaps.
Grades 3–5 · Geometry
Your goal
Cover the entire gold mat with repeated square tiles. Adjust their spacing and stagger so no gaps or overlaps remain.
Reduced motion is on. The controls still update the scene.
Square tiles cover a 20 square unit mat. 2.52 square units are still uncovered. Tile edge lengths stay fixed when you change the spacing.
Gap zero means edge fitting spacing. Negative gaps compress the spacing and may create overlaps. Positive gaps spread tiles apart. This model repeats fixed square, equilateral triangle or regular hexagon shapes on a lattice, clips them to the mat and checks polygon areas and intersections. A matching area total alone cannot hide an overlap.
Related resources
Related simulations
Study as flashcards
Learn with Tessellation Studio
Tessellation Studio helps upper elementary learners fit repeated polygons across a rectangular mat. Students adjust horizontal and vertical spacing, slide alternate rows or columns and reverse alternating triangles when needed. Gold regions reveal gaps, and a separate overlap measurement catches tiles that cover each other. Goals use squares, equilateral triangles and regular hexagons across increasingly large mats. Because tile shapes stay fixed, children investigate placement rather than resizing pieces. The area checks show why adding up tile areas is not enough unless the surface also has no overlaps.
What to look for
Alternate triangles
Equilateral triangles need alternating directions to fill each strip.
Stagger hexagons
A column stagger of 2 shifts adjacent columns by half a hexagon height.
For teachers and curious learners
Connect polygon geometry and repeated placement with complete coverage of a surface. Ask students to predict what will happen, change one setting, and explain what they observe. Learn gives a starting point. Practice and Challenge offer new goals with the same scene.
Fixed squares, equilateral triangles and regular hexagons repeat on bounded lattices large enough to cover the mat. Convex polygon clipping computes visible tile area, and pair intersections detect overlaps. Tiles crossing the boundary are clipped. Coverage is accepted within a small floating point tolerance.