Platonic entanglement

We present a construction of highly entangled states defined on the topology of a platonic solid using tensor networks based on ancillary Absolute Maximally Entangled (AME) states. We illustrate the idea using the example of a quantum state based on AME(5,2) over a dodecahedron. We analyze the entropy of such states on many different partitions, and observe that they come on integer numbers and are almost maximal. We also observe that all platonic solids accept the construction of AME states based on Reed-Solomon codes since their number of facets, vertices and edges are always a prime number plus one.

Path planning for the Platonic solids on prescribed grids by edge-rolling

The five Platonic solids—tetrahedron, cube, octahedron, dodecahedron, and icosahedron—have found many applications in mathematics, science, and art. Path planning for the Platonic solids had been suggested, but not validated, except for solving the rolling-cube puzzles for a cubic dice. We developed a path-planning algorithm based on the breadth-first-search algorithm that generates a shortest path for each Platonic solid to reach a desired pose, including position and orientation, from an initial one on prescribed grids by edge-rolling. While it is straightforward to generate triangular and square grids, various methods exist for regular-pentagon tiling. We chose the Penrose tiling because it has five-fold symmetry. We discovered that a tetrahedron could achieve only one orientation for a particular position.

The Periodic System

In ancient Greek times, philosophers recognized just four elements—earth, water, air, and fire—all of which survive in the astrological classification of the 12 signs of the zodiac. At least some of these philosophers believed that these different elements consisted of microscopic components with differing shapes and that this explained the various properties of the elements. These shapes or structures were believed to be in the form of Platonic solids (figure 1.1) made up entirely of the same two-dimensional shape. The Greeks believed that earth consisted of microscopic cubic particles, which explained why it was difficult to move earth. Meanwhile, the liquidity of water was explained by an appeal to the smoother shape possessed by the icosahedron, while fire was said to be painful to the touch because it consisted of the sharp particles in the form of tetrahedra. Air was thought to consist of octahedra since that was the only remaining Platonic solid. A little later, a fifth Platonic solid, the dodecahedron, was discovered, and this led to the proposal that there might be a fifth element or “quintessence,” which also became known as ether. Although the notion that elements are made up of Platonic solids is regarded as incorrect from a modern point of view, it is the origin of the very fruitful notion that macroscopic properties of substances are governed by the structures of the microscopic components of which they are comprised. These “elements” survived well into the Middle Ages and beyond, augmented with a few others discovered by the alchemists, the precursors of modern-day chemists. One of the many goals of the alchemists seems to have been the transmutation of elements. Not surprisingly, perhaps, the particular transmutation that most enticed them was the attempt to change the base metal lead into the noble metal gold, whose unusual color, rarity, and chemical inertness have made it one of the most treasured substances since the dawn of civilization.

TRUNCATED PLATONIC SOLID DAN TRUNCATED ICOSIDODECAHEDRON

The background of this paper is to attract society interest to know the beauty of mathematics that is represented by the diversity of mathematics geometry. Truncated Platonic Solid is one of those geometry which is expected to represent the beauty of mathematics. Truncated Platonic solid is a cut off Platonic Solid (not perfect). Platonic Solid is a geometry that was built by congruent polygon, all of the ribs have an equal lenght, also that same surface angle. Truncated Platonic Solid consist of cut off Tetrahedron, cut off Hexahedron, cutt of Dodecahedron, cut off Octahedron, and cutt off Icosahedron. Truncared Icosidodecahedron is a cut off Icosidodecahedron