Rhombic Spirallohedron
Rhombic spirallohedra are composed of rhombic faces (i.e. each edge is the same length) and come in many flavors. The one above is a four-armed spirallohedron, but there are spirallohedra with any number of arms. They were discovered by Russell Towle.
Steps
- Score along the fold lines and cut out four copies of each strip of rhombuses on the pattern.
Join two strips of rhombuses together by gluing the tabs from one strip to the underside of the strip next to it. The three strips are glued to each other in the same order that they appear on the pattern. For example, the tab on the left side of the top rhombus in the middle strip should be glued to the underside of the top rhombus on the strip on the left. The tab on the second rhombus of the middle strip should be glued to the second rhombus of the strip on the left and so on. 
Glue the third strip onto the other two to form one arm of the spirallohedron. In this photo, I finished gluing two red strips together and am ready to start gluing the third strip.
- Repeat to make four arms.
Glue an arm across the top and down the side of another arm. Do this by bringing the top right rhombus of two arms together. Glue the tab at the top of one rhombus to the free side of rhombus on the other arm. In this picture, the yellow and red arms are already glued together and I am showing how I would attach the orange arm. Notice how top rightmost rhombus of the orange arm would attach to the top right rhombus of the yellow arm. 
Glue the other arms onto the model. Here is a view looking down from the top of the completed model.
Notes
- 3-armed and 4-armed spirallohedra fill space.
Links
- I learned about rhombic spirallohedra from an interactive demo on the Wolfram Demonstrations Project site.
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