Starting with the Mitsubishi logo (it means three diamonds in Japanese!), what happens when you try to tile it?

The first question that needs to be answered it: does the 'Mitsubishi' tile?

You can interlock three of them in a rotating pattern.

When you do that, each color will fill space separately.

The three sets then combine. In this illustration, the last color, aqua is being dropped into place.

But when the three colors are in place, you cannot tell anymore which is each tile because all the tiles of one color touch to make a continuous network.

The problem can be fixed by slightly modifying the tiles so that they do not touch a tile of the same color anymore. They can now be seen as distinct shapes.

Of course, there are infinitely many ways in which this can be done. In the previous example, the tips were rounded off and consequently, to conserve the space-filling properties, the concavities were also rounded off in the same manner. In this example, the tips and concavities were simply cut off so that now each tile is composed of 4 regular hexagons. Of course hexagons fill space, so this makes sense...

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