Return to Answer

There is a published paper ("On 2-Reptiles in the Plane") proving that there are exactly 6 rep-2-tiles. But this restricts the definition of a rep-tile to a dissection into directly similar parts. If you allow inversely similar (reflected) parts there are more. These include an infinite number of different parallelograms, parameterised by the value of the smaller angle. If you change the question from the number of rep-2-tiles to the number of classes of rep-2-tiles interconvertible by affine transformations there are at least 10, and possibly 12 (I haven't succeeded in demonstrating that two of them tile the plane).

If you relax the definition to allow irreptiles (dissections in which the parts are similar, but of different sizes) there's another infinite class - of right angled triangles. And there are at least another 8 irreptiles corresponding to the polynomials $x+x^2=1$ (the golden bee shown in Aaron Meyerowitz's answer), $x+x^3=1$ (4 tiles) and $x+x^5=1$ (3 tiles).

There are papers which show that the existence of a Perron number is a necessary and sufficient condition for the existence of a self-similar tile. As the coeffecients of the corresponding polynomials correspond to the number of elements in the dissection this allows constraints of the number of irreptiles of order 2 to be deduced. I don't understand the papers, but I think that they make a restriction to directly similar dissections, so I am not confident that the seven tiles shown at http://www.meden.demon.co.uk/Fractals/dimerIRR.html plus the golden bee and the right-angled triangle are an exhaustive set.

There is a published paper ("On 2-Reptiles in the Plane") proving that there are exactly 6 rep-2-tiles. But this restricts the definition of a rep-tile to a dissection into directly similar parts. If you allow inversely similar (reflected) parts there are more. These include an infinite number of different parallelograms, parameterised by the value of the smaller angle. If you change the question from the number of rep-2-tiles to the number of classes of rep-2-tiles interconvertible by affine transformations there are at least 10, and possibly 12 (I haven't succeeded in demonstrating that two of them tile the plane).

If you relax the definition to allow irreptiles (dissections in which the parts are similar, but of different sizes) there's another infinite class - of right angled triangles. And there are at least another 8 irreptiles corresponding to the polynomials $x+x^2=1$ (the golden bee shown in Aaron Meyerowitz's answer), $x+x^3=1$ (4 tiles) and $x+x^5=1$ (3 tiles).

There are papers which show that the existence of a Perron number is a necessary and sufficient condition for the existence of a self-similar tile. As the coeffecients of the corresponding polynomials correspond to the number of elements in the dissection this allows constraints of the number of irreptiles of order 2 to be deduced. I don't understand the papers, but I think that they make a restriction to directly similar dissections, so I am not confident that the seven tiles shown at http://www.meden.demon.co.uk/Fractals/dimerIRR.html plus the golden bee and the right-angled triangle are an exhaustive set. (Wayback Machine)

There is a published paper ("On 2-Reptiles in the Plane") proving that there are exactly 6 rep-2-tiles. But this restricts the definition of a rep-tile to a dissection into directly similar parts. If you allow inversely similar (reflected) parts there are more. These include an infinite number of different parallelograms, parameterised by the value of the smaller angle. If you change the question from the number of rep-2-tiles to the number of classes of rep-2-tiles interconvertible by affine transformations there are at least 10, and possibly 12 (I haven't succeeded in demonstrating that two of them tile the plane).

If you relax the definition to allow irreptiles (dissections in which the parts are similar, but of different sizes) there's another infinite class - of right angled triangles. And there are at least another 8 irreptiles corresponding to the polynomials $x+x^2=1$ (the golden bee shown in Aaron Meyerowitz's answer), $x+x^3=1$ (4 tiles) and $x+x^5=1$ (3 tiles).

There are papers which show that the existence of a Perron number is a necessary and sufficient condition for the existence of a self-similar tile. As the coeffecients of the corresponding polynomials correspond to the number of elements in the dissection this allows constraints of the number of irreptiles of order 2 to be deduced. I don't understand the papers, but I think that they make a restriction to directly similar dissections, so I am not confident that the seven tiles shown at http://www.meden.demon.co.uk/Fractals/dimerIRR.html pluse the golden bee and the right-angled triangle are an exhaustive set.

There is a published paper ("On 2-Reptiles in the Plane") proving that there are exactly 6 rep-2-tiles. But this restricts the definition of a rep-tile to a dissection into directly similar parts. If you allow inversely similar (reflected) parts there are more. These include an infinite number of different parallelograms, parameterised by the value of the smaller angle. If you change the question from the number of rep-2-tiles to the number of classes of rep-2-tiles interconvertible by affine transformations there are at least 10, and possibly 12 (I haven't succeeded in demonstrating that two of them tile the plane).

If you relax the definition to allow irreptiles (dissections in which the parts are similar, but of different sizes) there's another infinite class - of right angled triangles. And there are at least another 8 irreptiles corresponding to the polynomials $x+x^2=1$ (the golden bee shown in Aaron Meyerowitz's answer), $x+x^3=1$ (4 tiles) and $x+x^5=1$ (3 tiles).

There are papers which show that the existence of a Perron number is a necessary and sufficient condition for the existence of a self-similar tile. As the coeffecients of the corresponding polynomials correspond to the number of elements in the dissection this allows constraints of the number of irreptiles of order 2 to be deduced. I don't understand the papers, but I think that they make a restriction to directly similar dissections, so I am not confident that the seven tiles shown at http://www.meden.demon.co.uk/Fractals/dimerIRR.html plus the golden bee and the right-angled triangle are an exhaustive set.

There is a published paper ("On 2-Reptiles in the Plane") proving that there are exactly 6 rep-2-tiles. But this restricts the definition of a rep-tile to a dissection into directly similar parts. If you allow inversely similar (reflected) parts there are more. These include an infinite number of different parallelograms, parameterised by the value of the smaller angle. If you change the question from the number of rep-2-tiles to the number of classes of rep-2-tiles interconvertible by affine transformations there are at least 10, and possibly 12 (I haven't succeeded in demonstrating that two of them tile the plane).

If you relax the definition to allow irreptiles (dissections in which the parts are similar, but of different sizes) there's another infinite class - of right angled triangles. And there are at least another 8 irreptiles corresponding to the polynomials $x+x^2=1$ (the golden bee shown in Aaron Meyerowitz's answer), $x+x^3=1$ (4 tiles) and $x+x^5=1$ (3 tiles).

There is a published paper ("On 2-Reptiles in the Plane") proving that there are exactly 6 rep-2-tiles. But this restricts the definition of a rep-tile to a dissection into directly similar parts. If you allow inversely similar (reflected) parts there are more. These include an infinite number of different parallelograms, parameterised by the value of the smaller angle. If you change the question from the number of rep-2-tiles to the number of classes of rep-2-tiles interconvertible by affine transformations there are at least 10, and possibly 12 (I haven't succeeded in demonstrating that two of them tile the plane).

If you relax the definition to allow irreptiles (dissections in which the parts are similar, but of different sizes) there's another infinite class - of right angled triangles. And there are at least another 8 irreptiles corresponding to the polynomials $x+x^2=1$ (the golden bee shown in Aaron Meyerowitz's answer), $x+x^3=1$ (4 tiles) and $x+x^5=1$ (3 tiles).

There are papers which show that the existence of a Perron number is a necessary and sufficient condition for the existence of a self-similar tile. As the coeffecients of the corresponding polynomials correspond to the number of elements in the dissection this allows constraints of the number of irreptiles of order 2 to be deduced. I don't understand the papers, but I think that they make a restriction to directly similar dissections, so I am not confident that the seven tiles shown at http://www.meden.demon.co.uk/Fractals/dimerIRR.html pluse the golden bee and the right-angled triangle are an exhaustive set.