Penultimate Units

Penultimate units are extremely simple, versatile, and rigid. They are perfect for creating sturdy, exact models of most polyhedra. The original creation is by Robert Neale.

Base Unit

Materials: Only paper is needed. Start with a sheet of letter-sized paper. Fold one corner diagonally to meet the opposite edge, then cut off the excess to leave the largest possible square. Divide this square into 4 smaller squares; each unit uses one.

Step 1. Fold the paper into quarters in a wavelike pattern, as shown. Collapse it, and you should end up with a flat 1x4 horizontal strip.

Small square of paper folded into quarters in a wavelike (accordion) pattern, collapsing into a flat 1x4 horizontal strip

Step 2. Fold the entire top right corner down (all flaps) to meet the bottom edge, as shown. Similarly, fold the entire bottom-left corner up.

(For advanced folders, fold ONLY the top flap if you are folding square or hexagon modules.)

Top right corner of the folded strip brought down to the bottom edge, and bottom left corner brought up to the top edge, forming diagonal creases at each end

Step 3. Connect the top and bottom crease endpoints created in Step 2 using a valley fold.

Valley fold connecting the top and bottom crease endpoints from Step 2, completing the finished base unit

Variations

At this point, you have already finished the pentagon module. See Assembly section to create a dodecahedron using 30 of these modules!

However, to create other polygonal faces, different connector variants must be used. Linked below are module extensions for triangular, square, and hexagonal faces.

With a bit of trigonometry knowledge and a calculator, you can actually easily design your own variations to fit your needs. I encourage you to check out the Trigonometry Behind Penultimate Units page to learn the principles behind Penultimate piece design.

Assembly

To put pieces together, simply insert the end of one piece into the pocket of another. Make sure that the two red creases coincide after you insert.

Shown on the right are pentagon units, but all other variations work analogously.

To create a polyhedron, just connect many units together while adhering to the polyhedron's structure. Remember that each Penultimate unit is an edge of the polyhedron. Feel free to explore the interactive visualizer to better understand polyhedron structures in 3D.

Two pentagon Penultimate units connected by inserting the end of one piece into the pocket of the other, with matching red creases aligned

(The visualizer is functional but still under development.)