Deep-cut, or origin-cut, twisty puzzles are interesting because a large chunk of the puzzle moves all at once, usually making it difficult to solve. There is no center face that stays put like in the 3x3 Rubik's cube. They are also difficult to make.

Most of the following list came from deep-cut.net/system/deep_cut.htm

- Face-turning tetrahedron and vertex-turning tetrahedron (In a tetrahedron, a face is opposite a vertex.): Jing's Pyraminx - mass produced, shape mod of Skewb. I disagree with deep-cut.net's classification of the Mini Pyraminx as this object, because the cuts ought to all intersect at a single point at the center.
- Edge-turning tetrahedron: Pyramorphix - mass produced, faces subdivided into triangles, shapeshifting, shape mod of Pocket Cube
- Face-turning cube: Rubik's Pocket Cube, a.k.a. 2x2 Rubik's Cube - mass produced, faces subdivided into squares
- Vertex-turning cube: Skewb - mass produced
- Edge-turning cube: 24 Cube, a.k.a. Rua Tekau Ma-wha, a.k.a. Little Chop - jumbling
- Face-turning octahedron: Skewb Diamond - mass produced, faces subdivided into triangles, shape mod of Skewb
- Vertex-turning octahedron: Pyradiamond - faces not subdivided at all, shape mod of Pocket Cube
- Edge-turning octahedron: 24-octahedron
- Face-turning dodecahedron: Pentultimate
- Vertex-turning dodecahedron: Chopasaurus - jumbling
- Edge-turning dodecahedron: Big Chop
- Face-turning icosahedron: unnamed
- Vertex-turning icosahedron: Icosamate - face subdivided into triangles
- Edge-turning icosahedron: unnamed

Other polyhedra are possible.

Deep cuts not aligning with faces, edges, nor vertices are possible. For example, there are many more shape mods of the Pocket Cube and Skewb. One can also combine multiple sets of deep cuts, e.g., both vertex- and face-turning.

Deep cut results in a 2-layer puzzle. Adding additional parallel cuts creates a 4-layer "master" puzzle, though there are options of where to put the parallel cuts. I think the most difficult both to make and solve have the cuts as deep as possible, intersecting with each other as much as possible, so the cuts are at 1/2-epsilon, 1/2, and 1/2 epsilon.

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