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Here is a simple pentagonal shape:

enter image description here

Using copies of this shape it seems that you can tile the plane, without even needing to flip over the tile.

enter image description here

But can you really?

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    $\begingroup$ I think this tiling was first discovered by Marjorie Rice, and she made a nice artwork version of it which can be seen on her website. This tiling has also been found by others, such as Livio Zucca. As my question implies, there is something wrong with it. $\endgroup$ – Jaap Scherphuis Apr 2 at 8:05
  • $\begingroup$ It's rather fascinating how much this almost works. $\endgroup$ – Rob Watts Apr 2 at 23:06
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    $\begingroup$ @RobWatts: Yes, so it is not too surprising that people didn't notice the problem. I only took a closer look at this tiling after systematic computer search for tilings didn't find this one when I thought it should have. $\endgroup$ – Jaap Scherphuis Apr 2 at 23:18
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Let's annotate the picture to make it easier to identify matching corners:

enter image description here

By vile abuse of notation let A,B,C,D,E denote the angles at the five corners. Inspecting the tiling we can read off:
360 = B+E+B+E = B+E+C+C = A+A+B = E+D+D
It follows
B+E = 180°
C = 90°
A = E+B/2 = C+E/2
D = B+E/2 = C+B/2

Checking incident edges yields:
AB = CD = AE = BC = DE
In other words the pentagon is equilateral. In particular, triangle AED is isosceles and the angle <DAE is B/2. Therefore <DAB=E, hence AD is parallel to BC (because the angles <DAB and <ABC=B sum to 180°) at which point the whole edifice comes crashing down. Indeed, it follows E=B=C=90° and A=D=135° which is incompatible with the pentagon being equilateral.

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  • $\begingroup$ Okay, I see what you mean. Thanks! +1 $\endgroup$ – hexomino Apr 2 at 17:51
  • $\begingroup$ Your second spoiler is rather hard to follow. Two questions - how do you know rot13(gur gevnatyr vf vfbfpryrf), and rot13(jul qbrf gung znxr gur yvar cnenyyry)? I think it would be much easier to understand if you included those intermediate steps. $\endgroup$ – Rob Watts Apr 2 at 22:43
  • $\begingroup$ Done, @RobWatts. Thanks for the suggestions. $\endgroup$ – loopy walt Apr 2 at 23:01
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    $\begingroup$ This is pretty much the same proof as mine, which can be found in this pdf together with two other geometry problems from tilings. $\endgroup$ – Jaap Scherphuis Apr 2 at 23:13
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    $\begingroup$ @justhalf If I remember correctly, on my tile BC and AD were parallel, so side b was a bit longer than the other sides. In this image you can see 6 tiles forming the fundamental unit for the tiling, and here it is repeated without the colours so the overlap is very noticeable. $\endgroup$ – Jaap Scherphuis Apr 3 at 7:35

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