Toroidal polyhedra are polyhedral structures that are topologically equivalent to a torus, meaning they have a shape resembling a doughnut. In mathematical terms, a toroidal structure has a genus of one, indicating it has one hole.

Amorphous solids are a class of materials that lack a long-range ordered crystalline structure. Unlike crystalline solids, which have a well-defined and repeating arrangement of atoms or molecules, amorphous solids possess a disordered arrangement. This means that their atomic or molecular structure does not exhibit the regular repeating patterns characteristic of crystals. Key features of amorphous solids include: 1. **Disordered Structure**: Amorphous solids do not have a definitive geometric shape or long-range order.

Guo Shoujing (1231–1316) was a prominent Chinese astronomer, engineer, and hydrologist during the Yuan Dynasty. He is best known for his work in advancing astronomical observation and developing more accurate calendars. Guo played a crucial role in the construction of large astronomical instruments, including the armillary sphere, and improved the accuracy of measurements for time and celestial events.

Physics beyond the Standard Model refers to theoretical and experimental research that seeks to address limitations and unanswered questions associated with the Standard Model of particle physics. The Standard Model successfully describes the fundamental particles and their interactions through the electromagnetic, weak, and strong forces, but it has several significant shortcomings: 1. **Gravity**: The Standard Model does not incorporate gravity, which is described by General Relativity. A complete theory of quantum gravity remains elusive.

The 19th century was a significant period for physics in Austria, with notable contributions from several prominent physicists. Here are a few key figures: 1. **Ernst Mach (1838-1916)**: Mach was an influential physicist and philosopher known for his work in mechanics, particularly in relation to the concept of shock waves and supersonic speeds. He is also known for the Mach principle and contributions to the philosophy of science, questioning the foundations and interpretations of physical theories.

Heinz Rutishauser refers to a notable Swiss mathematician known for his contributions to mathematical logic and computational theory, particularly in the areas of recursive functions and general recursive mathematics. He is best recognized for his work on algorithms and computability during the mid-20th century.

In the 18th century, the Polish-Lithuanian Commonwealth was home to several notable physicists and scholars, though the term "physicist" as a distinct professional title was not as common or well-defined as it is today. Instead, scientists often held broader roles in natural philosophy, mathematics, or engineering.

The 18th century was an important period for the development of physics, and Austria contributed several notable physicists during this time. Here are a few key figures: 1. **Georg Simon Ohm (1789-1854)**: Although he made most of his contributions in the 19th century, Ohm was born in Germany and became a prominent physicist known for Ohm's Law, which relates voltage, current, and resistance in electrical circuits.

"Da Ruan" (大阮) typically refers to a traditional Chinese musical instrument known as the "ruan." The ruan is a plucked string instrument that has a round body and is often described as having a sweet, resonant sound. It is commonly used in Chinese classical and folk music.

The 18th century was a significant period for the development of physics in France, marked by the contributions of several key figures. Some prominent French physicists from this era include: 1. **Émilie du Châtelet (1706–1749)**: An influential physicist and mathematician, du Châtelet is best known for her translation and commentary on Isaac Newton's "Principia Mathematica." She contributed to the understanding of kinetic energy and the conservation of energy.

The 18th century was a significant period for the development of physics, particularly in Germany, as it was a time of transition from classical physics to more modern approaches. Some notable German physicists from this period include: 1. **Christoph Friedrichs** (1740–1815) - He contributed to experimental physics and optics, particularly in the field of thermodynamics.

The commutator subgroup of a group \( G \), often denoted as \( [G, G] \) or sometimes as \( G' \), is a specific subgroup of \( G \) that captures the "non-abelian" structure of the group. It is constructed using the commutators of elements from \( G \).