Anthropometric Sizing Essay Example

manifolds; these abstract spaces resemble Euclidean space at small scales but can be furnished with additional structure that allows us to determine length.

The visual aspect of geometry makes it easier to comprehend compared to other branches of mathematics like algebra or number theory. Moreover, geometric language is utilized in diverse fields not limited to Euclidean geometry including fractal geometry and algebraic geometry. Geometry initially emerged as a practical discipline that dealt with surveying, measurements, areas, and volumes. Its notable achievements involve formulas for calculating measurements like the Pythagorean theorem, circumference and area of a circle, area of a triangle, and volume of various shapes including a cylinder, sphere, and pyramid.

Thales is credited with developing a technique for computing distances or heights by comparing geometric figures. The emergence of astronomy led to the development of computational techniques such as trigonometry and spherical trigonometry. Ancient scientists focused on constructing geometric objects that had been previously described, primarily using classical instruments like compasses and straightedges.

While conventional methods for problem-solving have been effective, some problems remain too difficult or impossible to solve using such techniques. As a result, novel designs that employ curves like parabolas and mechanical devices have been developed. In the field of geometry, the Pythagoreans introduced the concept of incommensurable lengths for sides of triangles.

The reintroduction of numbers in geometry happened through the use of coordinates by Descartes. He observed that representing geometric shapes algebraically can simplify their study. Analytic geometry therefore applies algebraic techniques to geometric problems, often by connecting geometric curves with algebraic equations.

In ancient times, geometric figures and shapes were analyzed in terms of their relative position or

spatial relationship. This is demonstrated through examples such as inscribed and circumscribed circles of polygons, lines that intersect and are tangent to conic sections, the Pappus and Menelaus configurations of points and lines. During the Middle Ages, more complex questions emerged regarding this type of analysis. For instance, there arose the kissing number problem, which sought to determine the maximum number of spheres that can touch a given sphere of equal size.

One of the questions in geometry is about the densest packing of spheres that are equal in size in space, known as Kepler conjecture. This falls under the category of 'rigid' shapes along with lines. Present day geometry has three sub-disciplines, namely projective, convex and discrete geometry, which are concerned with these types of questions.

Geometry of place, otherwise known as topology, was introduced by Euler as a new branch of geometry. While it originated from geometry, topology has since become its own discipline. It does not distinguish between objects that can undergo continuous deformation and may still retain some geometry, such as in the case of hyperbolic knots. Differential geometry employs calculus to investigate geometric problems beyond Euclid's domain.

For almost two millennia, geometric inquiries had increased while fundamental comprehension of space remained constant since the time of Euclid. According to Immanuel Kant, there existed only one geometry that was absolute and known to be true a priori through an internal faculty of the mind; synthetic a priori was Euclidean geometry. However, this widely accepted perspective was disrupted by the groundbreaking discovery of non-Euclidean geometry by Gauss (though never published), Bolyai, and Lobachevsky. They demonstrated that Euclidean space was merely one possibility

for geometric development.

Riemann's lecture entitled ?ber die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), delivered prior to his demise, elucidated a comprehensive outlook on the topic of geometry. His innovative concept of space, which is fundamental in Einstein's theory of general relativity, and Riemannian geometry - a field that explores spaces with diverse definitions of length, have become fundamental concepts in contemporary geometry. The lecture was only published after Riemann's death.

According to the theorem of invariance of domain, connected topological manifolds possess an established dimension, rather than being assumed beforehand.

The definition of geometry, according to Felix Klein's Erlangen program, is rigorously established by symmetry through the concept of a transformation group.

Both topology and geometric group theory heavily rely on discrete symmetries, while Lie theory and Riemannian geometry make use of continuous symmetries. Projective geometry also features a significant type of symmetry called the principle of duality. This concept involves swapping points with planes and vice versa, as well as join for meet and lies-in for contains in any theorem to obtain an equally true theorem.