Your first curvahedra ball is the start of an amazing world to explore, with a collection of different branch pieces you can make all sorts of things.
Exploring Curvahedra 3-branch Pieces
The simplest piece is the three branch, but this can make some classic models, here a large ball made up of loops of five towers over a smaller piece, the three connector again, making loops of three. In this case two are woven together.
Tech Speak: You can get more classic than the dodecahedron, one of the regular polyhedra studied in Euclid's elements. The smaller ball adds the curvahedra twist weaving two tetrahedra together.
While a wide collection of spheres is interesting the true curvahedra cognoscenti like to go to the saddle. Though this might be better for some sea creature or alien than a horse as it bends further the more your build. The construction is simple though, just connect 7-branches into loops of three or triangles and keep building.
Tech Speak: This makes a model for a 7-fold tiling of triangles in the hyperbolic plane. That object, discovered in the nineteenth century broke centuries of understanding of geometry. The richer understanding of what space could be was essential in Einstein's development of general relativity. You can hold it in your hand.
Curvahedra Fly's Eye
This model was made to honor our launch, where we were sat next to Buckminster Fuller's Fly's Eye dome at Crystal Bridges. While the dome has pieces connecting in three places making loops of five or six, this model reverses that. It is all triangles but with a careful mix of 5-branches (at the corners) and 6-branches(the flatter region).
Tech Speak: The Fly's Eye is a geodesic polyhedron, this model is its dual. The geometry of the angles (each connector has the same angle in each direction) means that the shape generated is a smoothed icosahedron rather than a sphere. Can you work out how the pieces might change to go to a sphere?
Exploring Curvahedra 4-Branch Pieces
The 4-Branch is quite flexible and loves the diversity of options that it can explore. You can see some of the balls, eggs and a sort of bucket here.
Tech Speak: You can find an octahedron, a cuboctahedron and the simples polyhedral torus lurking in there.