The symmetric group is the group defined over “any set whose elements are all the bijections from the set to itself, and who’s group operation is the composition of functions”, while the symmetry group is “the group of all transformations under which the object is invariant, endowed with the group operation of composition” (Wikipedia) “Isometry” comes from the Greek word for “equal measure”: Isometros. An n-dimensional space in the real numbers has an isometry from one function in R to another. The set of isometries that map a function onto itself on the set of real numbers is called a symmetry group. The symmetry group of an object is dependent on both the object itself and the space in which it is viewed. There are four types of symmetries in two-dimensions: rotation, reflection, translation, and glide reflection. Rotation consists of spinning either left or right. The axis of reflection, or mirror, is the line which acts as a two-way mirror between the original structure and it’s reflection. A translation moves all the points an equal distance in the same direction while a glide-reflection consists of both a translation and a reflection.
Farmers have been developing favorable traits by selectively breeding crops since the beginning of agriculture. The age old practice of creating more tolerant and pest resistant plants with higher yield has turned into a complicated, and sometimes secretive and modern process of genetic engineering. With modern technology, alterations can be made directly to the DNA of various plants. With only a few companies leading the market in genetically modified crops, and sometimes entire strains becoming eradicated by a single illness, controversy envelops present-day regulations concerning genetically modified foods. Issues being discussed include the safety and treatment of farmers, gene-patenting, food affordability, the environment, and the quality assurance of the crops being produced. How do genetically modified foods affect humans and other wildlife? Do consumers have the right to know when genetically modified crops are present in food products?
Akira Yoshizawa, a teenager in the nineteen-twenties, quit his factory job and proceeded to turn paper-folding into a fine art. Today, origami kits are common and consist of a booklet of instructions and a set of colored paper. The paper is two-sided with a different color on each side. The instructions, consisting of mostly of arrows and lines, are a part of a worldwide standard of visual language for origami that Akira pioneered. He also invented the wet folding technique, which allows for more sculptural interpretations.