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How can I construct hyperbolic solids (hyperbolic cube, tetrahedron, etc.) in Blender?

Below is a picture of a hyperbolic cube:

enter image description here

Here is some more information on hyperbolic solids.

Just to be clear, I would ultimately like to construct a hyperbolic dodecahedron, but I want to learn how to use a general method.

EDIT: The solution does not have to be mathematically perfect, but it would be nice. I am trying to 3D print a gift for someone.

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    $\begingroup$ How accurate does it need to be? Or does it just need to look good enough for visualization purposes? $\endgroup$ – Fweeb Jun 2 '14 at 17:27
  • $\begingroup$ @Fweeb, good catch. I am trying to make a gift for a math-oriented friend. The mathematical quality does not have to be perfect (though it would be nice). $\endgroup$ – Qu0rk Jun 2 '14 at 18:40
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If you're just in it for the looks, the easiest thing would be to take advantage of proportional editing. Using your cube as an example, do the following steps:

  1. Sub-divide the cube multiple (Edit mode > Select All [A] > Specials [W] > Subdivide... then use the Last Operator panel [F6] to increase the number of subdivisions to taste).

  2. Select just the eight corner vertices on the cube.

  3. Enable proportional editing (O).

  4. Change the proportional fall-off to Sharp (Shift+O 3 times if you're at the default of Smooth. Alternatively, there's a drop-down menu in the 3D View's header where you can choose the fall-off using your mouse).

  5. Scale (S) those vertices. As you do, scroll your mouse wheel to adjust the fall-off influence of the proportional edit area.

  6. To get sharp edges, select all the edges around the model and then crease them with (ShiftE) and drag until the value in the header is at 1.0 or just press 1 on your keyboard.

Finally, you may want to also use the Subdivision Surface modifier and the Edge Split modifier to help smooth things out a bit.

enter image description here

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