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i got kleetopes working! this means i can now generate several of the catalan and archimedean solids.
the lower row is the triakis tetrahedron, the kleetope of the tetrahedron, and then the triakis octahedron, the kleetop of the octahedron. above them are their duals: the truncated tetrahedron and the truncated cube
the lower row is the tetrakis hexahedron (aka cube) and the triakis icosahedron, a gap for a missing polyhedra, and then the pentakis dodecahedron; and the upper row is their duals, the truncated octahedron, the truncated dodecahedron, and the truncated icosahedron.
the gaps are for the 3rd and 6th polyhedra, since they involve kleetopes+duals of the rhombic dodecahedron and the rhombic triacontahedron, and i haven't figured out how to generate those yet.
except...
so a kleetope is the Geometry Name for taking a polyhedra, adding a point in the center of each face, and dragging all the faces out. so like if you have a cube you add one point to each of the six faces and then pull those points out a little. the thing is, depending on how much you pull you get different kinds of polyhedra, since there's a point when all the new faces 'line up'. so for the tetrahedron, you first get the triakis tetrahedron, and then you get a cube (because each new face lines up perfectly with the new faces on the adjacent old faces to form a square), and then you get a 'stellated tetrahedron', which is just a tetrahedron but bigger, and then you get a caltrop-like shape. and these polyhedra change based on the base shape.
and for the kleetope of the octahedron, the point where all the faces line up turns the shape into a rhombic dodecahedron. and for the kleetope of the icosahedron, the point where all the faces line up turns the shape into a rhombic triacontahedron
it turns out that the rhombic dodecahedron is catalan solid #8, and the rhombic tricontahedron is catalan solid #9, so if i can get 'real' versions of those i can generate the missing #3 and #6 polyhedra and also #8 and #9. but for those polyhedra to be the actual shape i'll need to figure out a way to cut edges and merge planar faces. plus i'll need to do the math to make sure that the faces are actually planar, and not like 0.5% off.
but you know, optimistic.
the lower row is the triakis tetrahedron, the kleetope of the tetrahedron, and then the triakis octahedron, the kleetop of the octahedron. above them are their duals: the truncated tetrahedron and the truncated cube
the lower row is the tetrakis hexahedron (aka cube) and the triakis icosahedron, a gap for a missing polyhedra, and then the pentakis dodecahedron; and the upper row is their duals, the truncated octahedron, the truncated dodecahedron, and the truncated icosahedron.
the gaps are for the 3rd and 6th polyhedra, since they involve kleetopes+duals of the rhombic dodecahedron and the rhombic triacontahedron, and i haven't figured out how to generate those yet.
except...
so a kleetope is the Geometry Name for taking a polyhedra, adding a point in the center of each face, and dragging all the faces out. so like if you have a cube you add one point to each of the six faces and then pull those points out a little. the thing is, depending on how much you pull you get different kinds of polyhedra, since there's a point when all the new faces 'line up'. so for the tetrahedron, you first get the triakis tetrahedron, and then you get a cube (because each new face lines up perfectly with the new faces on the adjacent old faces to form a square), and then you get a 'stellated tetrahedron', which is just a tetrahedron but bigger, and then you get a caltrop-like shape. and these polyhedra change based on the base shape.
and for the kleetope of the octahedron, the point where all the faces line up turns the shape into a rhombic dodecahedron. and for the kleetope of the icosahedron, the point where all the faces line up turns the shape into a rhombic triacontahedron
it turns out that the rhombic dodecahedron is catalan solid #8, and the rhombic tricontahedron is catalan solid #9, so if i can get 'real' versions of those i can generate the missing #3 and #6 polyhedra and also #8 and #9. but for those polyhedra to be the actual shape i'll need to figure out a way to cut edges and merge planar faces. plus i'll need to do the math to make sure that the faces are actually planar, and not like 0.5% off.
but you know, optimistic.
