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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. $\endgroup$ Commented Apr 2, 2021 at 8:05
  • $\begingroup$ It's rather fascinating how much this almost works. $\endgroup$ Commented Apr 2, 2021 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. $\endgroup$ Commented Apr 2, 2021 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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    $\begingroup$ This is pretty much the same proof as mine, which can be found in this pdf together with two other geometry problems from tilings. $\endgroup$ Commented Apr 2, 2021 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. $\endgroup$ Commented Apr 3, 2021 at 7:35
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    $\begingroup$ @JaapScherphuis Argh, those animated gifs have given me a dizzy spell! $\endgroup$ Commented Apr 3, 2021 at 16:16
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    $\begingroup$ How does Livio Zucca's example fail? Using your point-labelling, & his angles (B=103.644, A=129.486, E=76.356, D=140.514, C=90) we can place the points B=(0,1), C=(0,0), D=(1,0), A=(.97178,1.23589), E=(1.77178, .63589). Indeed AE=1 near enough, & DAE=EDA=CBA/2. But BAD=77.664 & ADC=88.692, not 90. $\endgroup$ Commented Apr 2, 2023 at 16:38
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    $\begingroup$ @RosieF It satisfies most contraints but not 360 = A+A+B = E+D+D. In fact 2A+B=362.616 and 2D+E=357.384 so at some vertices in the tiling the angles don't add up to 360. $\endgroup$ Commented Apr 3, 2023 at 12:18

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