On the growth of the wallpaper groups

Rostislav Grigorchuk1, Cosmas Kravaris1

  • 1Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843-3368, USA.

Related Concept Videos

Masonry Curtain Walls01:20

Masonry Curtain Walls

Masonry curtain walls employ brick or stone veneers supported by the building's structure to form an external cladding system that is both aesthetically appealing and functional. These walls are erected through two principal techniques, first by traditional layering of masonry units and second by using prefabricated panels. Traditional construction relies on steel shelf angles attached to the spandrel beam for support, with high-bond mortars ensuring secure attachment of masonry veneer...
1.2K
Plant Cell Wall01:07

Plant Cell Wall

Plant cells have a cell wall, a rigid outer covering that protects the cell and provides shape and support. During cell division, a mixture of enzymes, proteins, and glucose molecules is transported via vesicles to the center of the cell. These vesicles continuously fuse and build a cell plate between the dividing cells. As the cell plate matures, new polysaccharides are added to it to form the cell walls of the daughter cells. The predominant polysaccharide in the cell wall is cellulose, made...
5.1K
Group Design02:01

Group Design

The most basic experimental design involves two groups: the experimental group and the control group. The two groups are designed to be the same except for one difference— experimental manipulation. The experimental group gets the experimental manipulation—that is, the treatment or variable being tested—and the control group does not. Since experimental manipulation is the only difference between the experimental and control groups, we can be sure that any differences between...
9.5K
Role of Microtubules in Cell Wall Deposition01:02

Role of Microtubules in Cell Wall Deposition

Microtubules are small hollow tubes in eukaryotic cells. The cell wall microtubules are polymerized dimers of two globular proteins, α-tubulin and β-tubulin, two globular proteins. With a diameter of about 25 nm, microtubules are the widest components of the cytoskeleton. They help the cell resist compression and provide a track along which vesicles move through the cell or pull replicated chromosomes to opposite ends of a dividing cell. Microtubules go through quick cycles of...
2.5K
Expansion and Contraction in Masonry Walls01:19

Expansion and Contraction in Masonry Walls

Masonry walls are subject to slight expansion and contraction due to variations in temperature and moisture. Thermal movement in masonry is relatively straightforward to measure and plan for. On the other hand, moisture movement poses more of a challenge. New clay masonry units typically absorb water and expand over time under normal environmental conditions. Conversely, new concrete masonry units tend to shrink as they lose the excess moisture acquired during their production process.
To...
1.0K
Wood Panel Products01:18

Wood Panel Products

Wood panel products are essential materials used in construction for applications such as flooring, siding, and roofing, typically available in standard dimensions of 4 feet by 8 feet, with thicknesses varying from one-quarter of an inch to one and one-eighth inches. Among the most common types of wood panels is plywood, which is produced by gluing multiple layers of thin wood veneers under pressure. The grain of the outer veneers runs lengthwise, while the grains of the interior layers run...
119