The Mathematics of Tiling: Monohedral Convex Pentagonal Tiling

This is a part of the series of posts introducing an active research field called Tessellation – The Mathematics of Tiling, discussing some basic, intuitive tiling patterns. Subscribe now (link) to get notified about more posts in coming days!


We finished the discussion in the previous post with a simple puzzle: “What about monohedral convex pentagonal tiling?”

Turns out it is not too easy to find such tiling! After many years of search, in 1918, Karl Reinhardt, a German manthematician found the first five ‘types’ of this kind of tiling. Each type is a set of pentagons which are convex and forms a perfect tiling on a plane, and follows a general rule. Reinhardt is also credited with solving an interesting problem called Biggest Little Polygon – read more about it from extra reading links.

One of the tiling patterns discovered by Reinhardt

The next type of tiling was found in 1968 – half a century later!

Another interesting development was in late 1970s, when Marjorie Rice, a lady with only high-school education surprised the world by finding 4 new types of tiling! She took her inspiration from an article by science/mathematics communicator Martin Gardner’s column on ongoing research. She sent her results to Gardner, who first couldn’t understand most of it because she had developed her own ‘language’ to express her ideas. On detailed examination, Gardner, along with Doris Schattschneider, identified the four new types of tiling she discovered.

That’s a beauty of mathematics: more than a degree or a doctral, you can still make history if you have a vibrant imagination!

Ignore those Greek and Latin (p2222, pgg22* etc.) there, we will talk about it in future! (from Wikimedia Commons)

The search ended in 2015 with the discovery of 15th type of monohedral convex pentagonal tiling and a claim that the list is exhaustive. But the claim was based on a brute-force search (i.e., trying to confirm a result by checking all possible cases).

A rigerous (using x‘s and y‘s) proof has not yet been found – that means, there is still scope for research in this field. Since it is a highly visual field, it is fairly easy to understand!

In the coming days, we will be looking into different types of convex pentagonal tilings, along with some teaching/downloadable materials.

Extra Reading

  1. Karl Reinhardt (Wikipedia)
    1. Biggest Little Polygon (Wikipedia)
  2. Marjorie Rice (Wikipedia)
    1. Martin Gardner (Wikipedia)
    2. Doris Schattschneider (Wikipedia)
    3. Popular Mathematics (Wikipedia)
    4. List of Amateur Mathematicians (Wikipedia)
    5. Intriguing Tesselations (Marjorie Rice’s Webpage: Archives)
  3. Exhaustive Search of Convex Pentagons which Tile a Plane (ArXiv)
    1. Brute-force Search (Wikipedia)
    2. Problem-solving strategies (Wikipedia)
  4. Pentagonal Tiling (Wolfram | Alpha)
    1. Mathworld (Wolfram)
    2. Pentagonal Tiling (Mathpuzzle.com)
    3. 5 and Penrose TIling – NumberPhile (YouTube)
  5. Interactive Demonstration of Pentagonal Tiling (Wolfram)

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