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Untitled, folded paper [32 F, 90 E, 60V] This shape corresponds to the usual system of panels used to make a soccer ball, with every face either a pentagon or hexagon. One result of Euler’s Polyhedron Formula is that every polyhedron that has only pentagonal and hexagonal sides must have exactly twelve pentagons. Can you…

Untitled, cardstock and glue. Contributed by Thomas Sibley. [180 F, 270 E, 92 V] The producers of Polyplane are honored to include work of the renowned Father Magnus Wenninger (1919-2017). Fr. Magnus, a Benedictine monk, broke new ground by constructing physical realizations of all 75 uniform polyhedra, including many challenging self-intersecting shapes. He went on…

Boxycomb, laser-cut fiberboard and glue [44 F, 88 E, 46 V] Copies of the rhombic dodecahedron (the Catalan solid dual to the cuboctahedron) fill space without gaps or overlaps, as illustrated by this cluster of five rhombic dodecahedral custom-cut cardboard boxes.

Metahedron, 3D-printed polylactic acid plastic [60 F, 120 E, 62 V] This construction celebrates how some polyhedra can be connected face-to-face to create larger structures, including the frameworks of other polyhedra. In this case, tiny regular dodecahedra are used to create the 62 vertices and 120 edges of this star-shaped polyhedron, first described mathematically in…

Untitled, injection-molded plastic. Contributed by Studio Infinity. [107 F, 210 E, 105 V] Start with an icosidodecahedron, one of the Archimedean solids. (Its faces are shown in blue in this model). Succesively “stretch” or elongate it in five different directions, interpolating parallelograms (some of which are rectangles) along a belt or “zone” each time it…