Cubiakis Icosohedron: Hinged and Dissected

The distinguishing feature of the Cubiakis Icosahedron is that it can be constructed by dissecting 5 cubes and hinging pieces together. Each cube is dissected such that four of the eight corners of the cube become a point or vertex of a triangular pyramid. Each dissected triangular pyriamid is then hinged with another triangular pyriamid. The circumscribed tetrahedron is not used. The animation as well as the video clip below demonstrates how five dissected and hinged cubes can morph into a Cubiakis Icosahedron.

Five Dissected and Hinged Cubes

Five Cubes Morphing into a Cubiakis Icosahedron (2009)

Sources for 3D Animated Graphics

Colorado School of Mines 3D-3D Coordinate Transforms, Prof. William Hoff, Engineering Division, EGGN 512 Computer Vision

LiveGraphics3D by Martin Kraus. It is a non-commercial Java applet to display and rotate three-dimensional graphics in HTML pages.

CheerPJ enables legacy Java Applets to run in current browsers.

References on Dissecting and Hinging Polyhedra

Peter Cromwell, in his 1997 book Polyhedra, explains that the Chinese developed mathematics around the third century BC. (page 24.) The Chinese used a process of dissecting polyhedra into known shapes so as to compute volume.

Dissecting Polyhedra has a following within the puzzle making community. For example, consider Stewart Coffin's The Puzzling World of Polyhedral Dissections.

Wolfram Research has several demonstrations of how polyhedrons can be dissected into other polyhedrons. For example, see Dissection of a Cube into Five Polyhedra.

Not surprisingly, the prolific mathematician and sculptor George Hart also addresses polyhedral dissections. For example, he considers the dissection of the rhombic triacontahedron.

Rubic's Snake is an example of dissected and polyhedra that pivot.

However, the previous examples address dissections but do not involve hinging.

Any consideration of dissected and hinged polygons or polyhedrons must consider the works of Greg N. Frederickson. Professor Frederickson has written numerous papers and three definitive books on this subject Hinged Dissections: Swinging and Twisting and Dissections Plane & Fancy and Piano-Hinged Dissections: Time to Fold!. Be sure to see his video gallery. The videos includes one by Daniel Wyllie and two videos of beautiful wooden models created by Walt van Ballegooijen.

Daniel Wyllie also has a photo of another dissected and hinged polyhedra on Flickr and several interesting demonstrations on YouTube. Here is one called Icosahedron Inside Out and another called Rhombic Dodecahedron. Check them out.

Robert Webb has a nice video showing a model of a regular dodecahedron morphing into a rhombic dodecahedron using hinged polyhedra. He calls his model Dodecamorph - The Dodecahedron Shape-Shifter.

More recent advances in the analysis of dissecting and hinging polyhedra is found in the computer science and computational geometry literature. For example, a paper by Erik D. Demaine, Martin Demaine, Jeffrey Lindy and Diane Souvaine "Hinged Dissection of Polypolyhdra" presents a "general family of 3D hinged dissections for polypolyhedra, i.e., connected 3D solids formed by joining several rigid copies of the same polyhedron along identical faces."

See also Timothy G. Abbott, Abel, Charlton, E. Demaine, M. Demaine and Kominers Hinged dissections exist-This citation has many references to this literature