Any suggestions for making the vase shown in the pic below? Thinking of making it from mahogany approximately 12″ tall. Hollow so it can hold flowers etc.
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I would turn a 4 inches diameter cylinder and drill a hole on the lathe in the center. Then I would draw lines along the edges and carve the flats with a spokeshave.
I would 3D Print it, like the original!
As a woodworking problem, this is very interesting. The basic shape of each side is a rhombus.
Top and bottom edges are at 90 degrees to the surfaces.
So far, so easy.
Your snag is that the sides are not flat planes - they are curved, and the four corners are not coplanar. This can be seen easily at the 5 minute point in this video: https://www.youtube.com/watch?v=5xT2XDGYPSs
I am sure it is possible to cut a consistent 60 degree angle on a curved piece, but I'm struggling to work it out.
@Gulfstar probably has it right with subtractive techniques being the most practicable way. I'd make a pair of heaxgons, attach them to a cylinder, join with string and carve by hand. Well, I say I'd do that, but I'm nowhere near good enough at carving to come close to an acceptable result, so take that as an 'internet' I would do rather than a 'real' I would do!
It is possible to achieve something very similar with flat surfaces, which would be easier as each piece has the same shape and angles. These images show variations on the theme with triangles instead of a curved surface. As a triangle only has three vertices, all three are always in the same plane so you can make anything out of triangles if you have enough patience. https://www.polyhedra.net/en/model.php?name-en=twisted-hexagonal-prisms
Fusion 360 should be able to calculate the exact shapes and angles you will need very quickly...
Multi-axis spindle turning is ideal for this project. If you are a member of the American Association of Woodturners, there are numerous articles and a few videos on the site. Barbara Dill has a book and several YouTube videos if you are not an AAW member. (I have no personal connection to Ms. Dill).
Multi-Axis Spindle Turning: A Systematic Exploration https://a.co/d/fEHZKmG