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Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt.

enter image description here

I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different directions. According to Wikipedia, 3 out of the 15 currently discovered types of pentagonal tiling belongs to the 4-tile lattice category (type 2, 4 & 6).

https://en.wikipedia.org/wiki/Pentagonal_tiling

I don't think my attempt is an instance of type 4 or 6, since in both type 4 & 6, any side of all pentagons overlaps with only one side of another pentagon, while in my attempt, half of the long sides overlap with two shorter sides.

At the same time, I can't figure out how my attempt is an instance of type 2... Please kindly offer your insight.

enter image description here

enter image description here

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  • $\begingroup$ This is cross-posted at MO. $\endgroup$ – Milo Brandt Aug 24 '15 at 3:36
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As answered on Mathoverflow, where Jacky silently cross-posted this question

There are two questions here:

Q1) Which convex pentagons tile the plane?

Q2) What are all tilings of the plane by copies of a single convex pentagon?

The Wikipedia page you cite concerns Question 1 (though it could make this more explicit); Q1 is contained in Q2, and likely more tractable: once we know that a pentagon tiles the plane, it might still be hard to describe all tilings.

That is the case for your pentagon, which has two parallel sides and is thus contained in Type 1. It is a special case of Type 1 that allows further tilings such as the one you found, but that's a Q2 distinction and doesn't affect Q1.

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A correct generalization, with the angles labelled as in the first diagram, satisfies these equations: B=90, D+E=180, 2C+D=360, a+d=e. The tiling is 2-isohedral so the claim that it is type 1 is false. The only 2-isohedral type with a 4-tile lattice is type 6 which is edge-to-edge which your tiling is not. It is therefore a new discovery (type 16)? Er... no, on second thoughts, the type relates to the type of pentagon that tiles the plane, not the type of tiling. The tile just described is type 1, but it tiles the plane both isohedrally and 2-isohedrally.
The instance shown with D=E=90, A=C=45, a=d, and b=c is a familiar geometric pattern of interlocked St. Andrew's crosses. It is one of 23 periodic tilings by type 1 tiles in Doris Schattschneider, Tiling the Plane with Congruent Pentagons, Mathematics Magazine, Vol. 51, No. 1, Jan. 1978, 29-44.

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Actually, your tiling is not type 2, 4, or 6. It's actually a two-tile lattice. You can tell by looking at the conditions for each tiling, which are found below the picture in the Wikipedia article. The condition for a type one tiling is that there be two consecutive angles whose sum is 180 degrees, or more formally:

$$ B + C = 180^{\circ}$$ $$A + D + E = 360^{\circ} $$

Your tiles meet this with the outside right angles.

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  • $\begingroup$ However as I mentioned, the tiling is composed of pentagons of 4 directions (0, 90, 180, 270 degrees). Could you come up with a 2-tile lattice that can form the pattern by duplicating without rotation? $\endgroup$ – Jacky Aug 24 '15 at 3:41
  • $\begingroup$ [edited only to correct "you're" to "your"] $\endgroup$ – Noam D. Elkies Aug 24 '15 at 18:00

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