Irregular Penrose Tilings

Consider tilings such as this irregular pattern. The short line segments are all the same length and in one of 5 directions. That is to say that each segment is a translation of one of the sides of some particular regular pentagon. There are two rhombuses in the image up to congruence. If we choose a side of some rhombus it will have a parallel segment on the same rhombus which will abut another rhombus with another parallel segment, etc. Continuing this way across the pattern we are led deterministically on a path that cannot cross another such path based on parallel segments. These paths across the plane divide the plane into equivalence regions and we may thus assign integers to such point in the plane where the integer increases by one each time we cross a path in the same direction. The same may be said for each of the five directions and we thus have may assign 5 integers to each point in the plane. This assignment is obviously continuous in some elementary sense and we have found a map from the plane to a 5 dimensional space.

Another way to see this is to note that the 5 sides of a regular pentagon, as vectors, are incommensurate over the rationals. They are independent as vectors in the vector space over the field of rationals.

Consider a 5 dimensional space packed with 5D cubes in the conventional way. Thru the origin consider the two orthogonal unit vectors (σ, σ, σ, σ, σ) and () where σ = 1/√5