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Axis, Fisher and Windmill are shape-shifting puzzles, also known as shape mods. Inside they have a regular 3x3 mechanism (or a 2x2 or 4x4 one), but their cutting axes are rotated relative to the faces. When solved, the puzzle looks like a regular cube. After a few moves it changes shape, and it becomes hard to recognize the classic cube in it. If you can solve a 3x3, you can also solve these puzzles with your own method. The only extra challenge is recognizing the pieces and keeping track of center orientation.
Fisher Cube is a classic shape mod invented by Tony Fisher. The 3x3 cuts are rotated by 45° around one axis. The centers of the four side faces take the place of edges, so their orientation matters. This is the easiest one to start with.
Windmill Cube works in a similar way, but the cuts are shifted at an angle other than 45°, so some of them cut through the edges of the cube. When scrambled, the puzzle looks like a windmill. The four side centers have to be placed in the correct orientation, because if one is rotated by 90° or 180°, the puzzle is not solved. On top of that, PLL recognition is harder.
Axis Cube has its core and cuts rearranged so that, when scrambled, it looks completely chaotic and asymmetrical. Both the color and the shape of the pieces matter. Of the three, tracking pieces is usually the hardest here.
Most models are 3x3 versions from QiYi, YJ, MoYu, ShengShou and LanLan. We also have the collectible Bai-Niao-Chao-Feng Fisher from Calvins Puzzle. If you want to start with a smaller puzzle, try the Lee 2x2 models (Axis, Fisher and Windmill). The 4x4 versions from FanXin are much harder, because on top of the shape change you also have the centers and edges known from the 4x4.
The best choice for your first shape mod is the Fisher Cube 3x3, followed by the Windmill and Axis. Simple, affordable models such as the QiYi Axis S, Fisher S and Windmill S are a good place to start. Their Ice Amber versions have transparent pieces that let you see how the mechanism is built. If you want smoother turning for more frequent solving, you can choose YJ models (e.g. YiLeng/New Fisher, KingKong/New Axis, FengHuoLun/Windmill) or the MoYu Meilong Fisher. These are still puzzles for fun and collecting, not cubes designed for setting records.