Triangular faces can be combined to form regular tetrahedra (four triangles, three faces per vertex), octahedra (eight triangles, four faces per vertex), and icosahedra (twenty triangles, five faces per vertex).

This is an attempt at triangular face sections to build regular polyhedra for lampshades or decoration. The primary design goal is to allow simple assembly without the need for fasteners or other materials (e.g. tape, glue, rubber bands, staples, screws, etc.). All pieces slide together, and hold with friction and tension. Secondary considerations are overall aesthetics, hints at other geometries, and scalability and extensibility to other materials.

Faces are laser-cut from sheets of mid-to-heavy card stock. Each sheet (8.5x11 letter; or A4 with no scaling) contains ten trigonal faces. Print two sheets to yield twenty faces. Assembled icosahedron is ~5.5" (~14 cm) in diameter.

Ensure that the slots (six per face) are cut all the way through by the laser; if not, use a knife to clean up the cuts.

To assemble, start with two faces. Line up the slots on both edges, then gently slide together. If the slots won't line up at all, flip the new piece over: all slots on all faces must be oriented the same way: either clockwise or counterclockwise to the face. If one piece is clockwise, and another is counterclockwise, one of them is inside-out: just flip it over.

Take a third face. Slide one edge onto the second face. If the edges are not properly aligned, gently wiggle/pull/push the material until the slots line up.

The fourth piece is attached to the third piece, as if starting to form a ring or bowl. The fifth piece is initially attached to the fourth, continuing the ring/bowl around a single vertex. It is then attached to the first piece, to form a shallow bowl. Note that the vertex (i.e. the corner where all faces meet) has exactly five faces: each vertex in an icosahedron is the intersection of exactly five faces. If you only have four faces meeting at a vertex, add one more. If you have six, take one away. If you have five, join them together, and check that all the edges line up.

It may be necessary to carefully bend the outer tabs a little in order to get the slots to line up: just do it slowly and carefully to avoid tearing or permanently bending the material.

Repeat assembly by adding one face at a time, building up a vertex to five faces. I.e. for the sixth and seventh pieces: choose a point on the bowl, attach the sixth piece to an adjacent edge; attach the seventh piece to the sixth, and then to the existing bowl; notice that the new vertex has five faces. Then move onto another vertex, and add faces until there are five.

The last two faces can be a little tricky to slide into place. Just work slowly to carefully position the face roughly in place, with the tabs appropriately over/under, and gently line up all the slots.

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Octahedral assembly. Join four faces around a single vertex, then attach the fourth to the first. Add a fifth face to an edge of the bowl. Add a sixth face to the fifth, then attach one of its edges to the bowl. Add the seventh and eighth in the same manner as the sixth.