EINSTEEN PROBLEM

• A set of tile-types (or prototiles) is considered to be aperiodic if copies of these tiles can only form patterns without repetition
• In 1961, mathematician Hao Wang conjectured that aperiodic tilings were impossible
• But his student, Robert Berger, disputed this, finding a set 104 tiles, which when arranged together will never form a repeating pattern
• In the 1970s, Nobel prize-winning physicist Roger Penrose found a set of only two tiles that could be arranged together in a non-repeating pattern ad infinitum
• This is now known as Penrose tiling and has been used in artwork across the world

• Since Penrose’s discovery, mathematicians have been looking for the “holy grail” of aperiodic tiling  a single shape or monotile which can fill a space up to infinity without ever repeating the pattern it creates
• While shapes that can be perfectly fitted on a plane are commonly known  just think of rectangular bathroom tiles or hexagonal tiles which pave footpaths  finding a single shape which can be both perfectly fitting and never repeat the pattern had till now only been theorised about
• Mathematicians call this the einstein problem in geometry
• This problem has stumped mathematicians for decades and many felt that there was simply no answer to this problem

• However, the latest discovery, a 13-sided shape which has been named “the hat” by its proponents, has presented a deceptively simple solution
• The hat comprises eight copies of a 60°–90°–120°–90° kite, glued edge-to-edge, and can be generalised to an infinite family of tiles with the same aperiodic property
• The shape also retains its aperiodic qualities when varying the lengths of the sides, meaning that the solution is actually a continuum of similar shapes
• The shape was first discovered by David Smith, an amateur mathematician from England
• Smith then worked closely with two computer scientists and another mathematician to develop two proofs showing that “the hat” is indeed an aperiodic monotile