We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-unfold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron...

regular polytopes, mean value property, finite reflection groups We compute certain polynomial invariants for the finite reflection groups of the types H3, H4 and F4. Using this result, we explicitly determine the solution space of functions satisfying a mean value property related to the exceptional regular polytopes, namely, the icosahedron and dodecahedron in three dimensions and the 24-cell...

the study area is a part of the sanandaj-sirjan metamorphic belt comprising low- to high-graderegional and contact metamorphic rocks that are intruded by various plutonic bodies. garnet crystals areabundant in pelitic schists, amphibole schists, migmatites, hornfelses and aplites. they are mostly almandinerich(xalm>60) pyralspite garnets and show dodecahedron, trapezohedron and mixed (dodecahed...

We provide a new construction of the modes of the Poincaré dodecahedral space S3/I∗. The construction uses the Hopf map, Maxwell’s multipole vectors and orbifolds. In particular, the *235-orbifold serves as a parameter space for the modes of S3/I∗, shedding new light on the geometrical significance of the dimension of each space of k-modes, as well as on the modes themselves.

A robust and advanced N/P co-doped carbon-based dodecahedron catalyst with confined Fe-doped Co 2 P particles was successfully prepared through a multi-doping engineering strategy.

Thirteen simple closed geodesics are found in the lemniscate. Among these are nine “mirrors”—geodesics of reflection symmetry—which generate the full octahedral group and determine a triangulation of the lemniscate as a disdyakis dodecahedron. New visualizations of the lemniscate are presented.

This article examines the universal polytope P (of type {5, 3, 5}) whose facets are dodecahedra, and whose vertex figures are hemiicosahedra. The polytope is proven to be finite, and the structure of its group is identified. This information is used to classifiy the quotients of the polytope. A total of 145 quotients are found, including 69 section regular polytopes with the same facets and ver...

The honeycomb problem on the sphere asks for the partition of the sphere into N equal areas that minimizes the perimeter. This article solves the problem when N = 12. The unique minimizer is a tiling of 12 regular pentagons in the dodecahedral arrangement.