I always thought that 5-fold symmetry can only be obtained by sophisticated constructions (as with Penrose tiles, see the image on the top right of http://en.wikipedia.org/wiki/Penrose_tiling ) Now a childishly simple contruction gives 5-fold symmetry with just one tile: Our tile is a triangle: /\ / \ /====\ The two angles at the base shall equal 2*Pi/k for k>=5. (Drawing obviously not to scale). Assemble tiles as follows: /\====/\====/\====/\== / \ / \ / \ / \ etc. ad inf. /====\/====\/====\/====\/ This is "plank A" Now arrange planks into an infinite wedge: / /\====/\ etc ad inf. to right and top / \ / \ / \/ /\====/\====/\====/\== / \ / \ / \ / \ / \/ \/ \/ /\====/\====/\====/\== / \ / \ / \ / \ / \/ \/ \/ /\====/\====/\====/\=== / \ / \ / \ / \ /====\/====\/====\/====\/== This is "wedge A" Arrange k such wedges around the origin to fill the plane. This arrangement has k-fold rotational symmetry but (except for k=6) no translational symmetry. Rotational symmetry can be avoided by taking for one wegde the "wedge B" obtained by reflecting wedge A over its horizontal base (we still use one tile only, reflection of tiles is obtained by rotation them by 180 degrees). Wedge B cannot be transformed to wedge A by a rotation. This is surely known since about Silurian years, but why doesn't it appear anywhere, e.g. in http://tilings.math.uni-bielefeld.de/ ?