Making a Rhombicosidodecahedron: Mathematical Thinking Revisited

A rhombicosidodecahedron (an Archimedean solid with 30 square, 20 triangles, and 12 pentagon faces) was redeemed from 60 pieces by modular origami. This study used a qualitative research case study as it asked about how participants experienced this construction process of rhombicosidodecahedron. Preservice primary mathematics teachers from a mathematics and art course were participants of the study. Additionally, one student; the first student who came out with the totally symmetric and no damaged object was interviewed for the assembly process. Mathematical thinking throughout the process was noted. Student brought her/his previous experiences as much as specific aptıtudes. Student took this project as a creative writing piece so that the process gone through similar phases as intro, progress, and artifact. Deformations and sinking occurred but student investigated the specifics of the real mathematical object did it without a fault. To deal with problems occurred in the phases; students used a creative insight as using paperclips to attach modules and assembly of half spheres. Two main processes; organizational and structural took place in the creative model formation and assembly. Suggestions and future studies are also discussed.

Assembly of silver Trigons into a buckyball-like Ag180nanocage

Buckminsterfullerene (C60) represents a perfect combination of geometry and molecular structural chemistry. It has inspired many creative ideas for building fullerene-like nanopolyhedra. These include other fullerenes, virus capsids, polyhedra based on DNA, and synthetic polynuclear metal clusters and cages. Indeed, the regular organization of large numbers of metal atoms into one highly complex structure remains one of the foremost challenges in supramolecular chemistry. Here we describe the design, synthesis, and characterization of a Ag180nanocage with 180 Ag atoms as 4-valent vertices (V), 360 edges (E), and 182 faces (F)––sixty 3-gons, ninety 4-gons, twelve 5-gons, and twenty 6-gons––in agreement with Euler’s rule V − E + F = 2. If each 3-gon (or silver Trigon) were replaced with a carbon atom linked by edges along the 4-gons, the result would be like C60, topologically a truncated icosahedron, an Archimedean solid with icosahedral (Ih) point-group symmetry. If C60can be described mathematically as a curling up of a 6.6.6 Platonic tiling, the Ag180cage can be described as a curling up of a 3.4.6.4 Archimedean tiling. High-resolution electrospray ionization mass spectrometry reveals that {Ag3}nsubunits coexist with the Ag180species in the assembly system before the final crystallization of Ag180, suggesting that the silver Trigon is the smallest building block in assembly of the final cage. Thus, we assign the underlying growth mechanism of Ag180to the Silver-Trigon Assembly Road (STAR), an assembly path that might be further employed to fabricate larger, elegant silver cages.

Concepts Of The Golden Diamond

The regular diamond whose diagonals are in the golden proportion is commonly known as rhombus aureus or the golden diamond.a Situated in the geometric realm of polytopes, this diamond is the constituent face of a thirty-faceted zonohedron (a polyhedron composed of diamond faces), namely the rhombic triacontahedron, which has a dual relationship with an Archimedean solid and is illustrated in this article. To complete theoretical preliminaries, the triacontahedron is here derived in a new way by a particular group-theoretical approach in order to provide a generalized understanding of regularity with polyhedra. Following this, the golden diamond is observed from an artistic angle since it seems to appear particularly attractive. An explanation of this special aspect is suggested. Further, the golden diamond reveals remarkable geometrical properties, some of which are not so commonly known. These are given closer consideration. And at last, from a constructive point of view, the golden diamond proves to be well applicable to all kinds of design, going from architecture to interior design. The morphological and main part of this article supplies information on some constructive capacities of the golden diamond, in addition to unpublished applications. Two major kinds of examples concerning exterior and interior Architecture are thus presented and richly illustrated. These are: the Great Pyramid on a macro-scale, and personnally constructed art objects (with light within) on a micro-scale.

The Pure Archimedean Polytopes in Six and Seven Dimensions

An Archimedean solid (in three dimensions) may be defined as a polyhedron whose faces are regular polygons of two or more kinds and whose vertices are all surrounded in the same way. For example, the “great rhombicosidodecahedron” is bounded by squares, hexagons and decagons, one of each occurring at each vertex. Thus any Archimedean solid is determined by the faces which meet at one vertex, and therefore by the shape and size of the “vertex figure,” which may be defined as follows. Suppose, for simplicity, that the length of each edge of the solid is unity. The further extremities of all the edges which meet at a particular vertex lie on a sphere of unit radius, and also on the circumscribing sphere of the solid, and therefore on a circle. These points form a polygon, called the “vertex figure,” whose sides correspond to the faces at a vertex and are of length 2 cos π/n for an n-gonal face. Thus the vertex figure of the great rhombicosi-dodecahedron is a scalene triangle of sides .