Type 1 pentagon pairs · Cairo-style Type 4 pinwheel · Colour by orientation · Pan the canvas
A regular pentagon has an interior angle of exactly 108°, and 360° ÷ 108° = 3.33…, which is not a whole number. That means you can never fit an exact integer number of regular pentagons snugly around a single point with no gap and no overlap — so regular pentagons alone can never tile the plane, unlike equilateral triangles, squares and regular hexagons. Convex irregular pentagons are a different story: mathematicians have identified 15 known families ("types") of convex pentagons that can tile the plane monohedrally (using only congruent copies of one shape). Each family is defined by a set of angle and/or side-length constraints. This simulator renders two genuinely gap-free constructions built from real vertex coordinates: a Type 1 tiling, where two pentagons — one the 180° rotation of the other — combine edge-to-edge into a parallelogram that tiles by translation alone, and a Cairo-style "Type 4" pinwheel tiling, where a square grid cell is split into two point-symmetric pentagons and each cell is independently rotated by a multiple of 90°, producing the four-pentagons-around-a-point pinwheel pattern famous from Cairo's street paving.
For nearly 30 years after the 14th type of convex pentagon tile was found in 1985, mathematicians wondered whether the list was complete. Then in 2015, a team at the University of Washington Bothell — Casey Mann, Jennifer McLoud-Mann and David Von Derau — used a computer search to discover a genuinely new, 15th type of convex pentagon that tiles the plane. The discovery made international news as a rare, real breakthrough in a very old and seemingly well-trodden area of geometry, and it remains an open question whether more types remain to be found or whether 15 is the complete list.
This simulator draws two genuinely gap-free convex pentagon tilings, computed as real vertex coordinates rather than pre-drawn images. Each tile is built by splitting a quadrilateral (a parallelogram for Type 1, a square for the Cairo-style Type 4) into two pentagons around its centre point, so the two pentagons always reassemble exactly into the original quadrilateral — and quadrilaterals are already known to tile the plane by translation, which is what guarantees no gaps or overlaps anywhere on the canvas.
Type 1 splits an oblique parallelogram into two pentagons, one the 180° rotation of the other about the parallelogram's centre; the pair then tiles by translation along the parallelogram's two side vectors. Type 4 does the same split on a square, then rotates each grid cell independently by 0°, 90°, 180° or 270°, producing the four-way pinwheel pattern seen in the real Cairo pentagonal tiling.
Pick a Pentagon Type, drag the Pentagon Size slider to rescale, and switch Display between filled colour, outline-only, or colour-by-orientation to see the distinct rotation classes highlighted in different colours. Drag the canvas to pan, click Redraw to refresh, or Reset View to recentre.
The 15th and (so far) most recent type of convex pentagon tile was discovered in 2015 by a University of Washington Bothell team using a computer search, ending a nearly 30-year gap since the 14th type was found in 1985. It is still an open question whether any more types exist.
No. A regular pentagon's interior angle is 108°, and 360° divided by 108° is about 3.33, not a whole number. You can never arrange an exact integer number of regular pentagons around a single point without leaving a gap or forcing an overlap, so regular pentagons cannot tile the plane on their own.
Irregular convex pentagons have more freedom: their five angles and five side lengths do not all have to be equal, so it is possible to design a shape whose angles add up correctly around every vertex of the tiling. Mathematicians have found 15 distinct families of convex pentagon, each defined by its own set of angle or side constraints, that tile the plane using only copies of one shape.
Type 1 pentagons are characterised by having two angles that sum to 180°. That "straight angle" condition lets a pentagon and a 180°-rotated copy of itself meet edge to edge along a straight line, so the pair combines into a parallelogram (or hexagon) that then tiles the plane by simple translation, exactly as this simulator constructs it.
It is built the same way as Type 1 — splitting a quadrilateral into two point-symmetric pentagons — but starting from a square instead of an oblique parallelogram, and rotating each grid cell independently by a multiple of 90°. Because a square maps onto itself under any 90° rotation about its centre, every cell still fits its grid square exactly, so the tiling stays gap-free while producing the pinwheel arrangement, with four pentagons meeting at many vertices, that resembles the real Cairo pentagonal paving pattern.
Yes. Casey Mann, Jennifer McLoud-Mann and David Von Derau at the University of Washington Bothell found it using a computer search of the possible angle and side combinations, and it was widely reported as a genuine mathematical discovery. It followed a nearly 30-year gap after the 14th type was found in 1985, and it remains unknown whether the classification of convex pentagon tiles is now complete.