StudentShare
Contact Us
Sign In / Sign Up for FREE
Search
Go to advanced search...

Reflection Groups in Geometry - Essay Example

Comments (0) Cite this document
Summary
This paper "Reflection Groups in Geometry" focuses on reflection groups generated by a reflection that has been the subject of the significant studies. This paper presents a comprehensive exposition of the concept of reflection groups in Euclidean, spherical, and hyperbolic geometry.  …
Download full paperFile format: .doc, available for editing
GRAB THE BEST PAPER93% of users find it useful
Reflection Groups in Geometry
Read TextPreview

Extract of sample "Reflection Groups in Geometry"

Download file to see previous pages A reflection group is a distinct group produced by multiple reflections of a finite-dimensional (Euclidean) space. Weyl groups of simple Lie algebras and symmetry groups of regular polytypes are examples of finite reflection groups while infinite groups comprise the Weyl groups of infinite-dimensional Kac–Moody algebras and the triangle groups similar to ordinary tessellations of the hyperbolic plane and Euclidean plane. With regard to symmetry, discrete isometry groups of broad Riemannian manifolds that are formed by reflections are grouped into classes leading to hyperbolic reflection groups (corresponding to hyperbolic space), affine (corresponding to Euclidean space) and finite reflection groups (then-sphere). Coxeter groups are reflection groups that are finitely generated. Unlike reflection groups, Coxeter groups are abstract groups that have a certain structure generated by reflections. An investigation of the topology and geometry of reflection groups will help us comprehend the theoretic properties of the group.
The concept of reflection in a Euclidean space and the hypothesis of discrete groups of motions resulting from reflections has its origin in the study of space polyhedral and plane regular polygons that goes back to early mathematics. In the present day, reflection groups are common in many areas of mathematical research, and geometers encounter them as special convex polytopes or discrete groups of isometries of Riemannian spaces with even curvature. On the other hand, an algebraist encounters reflection groups in group theory, particularly in the representation theory, Coxeter groups and invariant theory. Other areas of mathematics where they may be encountered include the theory of arrangements of hyperplanes, a theory of combinations and permutation, a theory of modular forms and quadratic forms, low-dimensional topology, singularity theory, and the theory of hyperbolic real and complex manifolds (Yau 1986).   ...Download file to see next pagesRead More
Cite this document
  • APA
  • MLA
  • CHICAGO
(“Reflection Groups in Geometry Essay Example | Topics and Well Written Essays - 4000 words”, n.d.)
Reflection Groups in Geometry Essay Example | Topics and Well Written Essays - 4000 words. Retrieved from https://studentshare.org/mathematics/1820675-hello-additional-file-will-upload-to-additional-file
(Reflection Groups in Geometry Essay Example | Topics and Well Written Essays - 4000 Words)
Reflection Groups in Geometry Essay Example | Topics and Well Written Essays - 4000 Words. https://studentshare.org/mathematics/1820675-hello-additional-file-will-upload-to-additional-file.
“Reflection Groups in Geometry Essay Example | Topics and Well Written Essays - 4000 Words”, n.d. https://studentshare.org/mathematics/1820675-hello-additional-file-will-upload-to-additional-file.
  • Cited: 0 times
Comments (0)
Click to create a comment or rate a document

CHECK THESE SAMPLES OF Reflection Groups in Geometry

The History of Projective Geometry

...of the of the Concerned 2 May The History of Projective Geometry Introduction Projective geometry is the aspects of mathematical studies that had to do with the study of relationships existing between geometric figures and the mappings or images that are brought into existence as a result of the projection of these geometric figures on some other surface. The focus of the projective geometry is the geometric aspects and properties that are affiliated to changes of perspectives. To put it in simple words, projective geometry delves on geometrical aspects like concurrence and co- linearity and ignores properties like distances and angles. In the everyday life, one often...
3 Pages(750 words)Essay

College Geometry - neutral geometry and Euclidean geometry

...to Euclidean geometry, the parallel postulate when extended makes a three sided diagram that is interior angles sum up to 1800. This is because the subsequent angles are reflections of the points of origin (Eves, 1990). (a) (c) (b) From the diagram above, we can prove that the exterior angle at (a) is equal to the summation of interior angle (b) and (c). The interior angle at pint (a) thus sums up to 180 with the other two angles; (b) and (c). Existence of rectangles in reference to hyperbolic theorem Interior angles in a rectangle add up to 360 degrees. Opposite angles are equal and parallel thus making the figure have four opposite right angles. This is because the base length is a hyperbola and hardly...
3 Pages(750 words)Essay

Comparison and Analysis of Architects: Le Corbousier and Leon Krior

...(1984), Leon Krier is an architect which conceives of cities as a whole, and that his buildings are conceived to be part of the larger whole. He needs to conceptualize the town or the city when he designs his buildings, because simply constructing a disembodied part of the city – a building - is not of an interest to him. This, according to Porphyrios, makes Krier a “heterosexual. All his single buildings have real or implied partners” (Pophyrios, 1984, p. 12). Or, as Pophyrios, his buildings belong to larger groups, and these groups and families are conceived as being a part of the need of the society to which they serve. In this way, Krier is not narcissistic. His buildings are not...
13 Pages(3250 words)Essay

Fractal Geometry

...Fractal Geometry Fractal geometry is rather new mathematical theory which is completely different from the traditional concepts of Euclidean Geometry. Fractal geometry describes elf-similar or scale symmetric objects. It means that if to magnify these objects, their parts will bear an exact resemblance to the whole object. The word "fractal" was created by Benoit Mandelbrot and it means "to break", whereas the form of adjective "fractus" means "fragmentated". (Brandt 24) It is a matter of fact that the word "fractal" has two definite meanings: the first meaning refer to colloquial use and the second is connected with geometry. In colloquial speech fractal...
2 Pages(500 words)Essay

Triangulation in Geometry

...Triangulation in geometry Outline Key definitions 2. Euler's theorem 3. Discovering and generalizations 4. Gauss-Bonnet theorem 5. Surfaces, theirs topology and triangulation 1. Key definitions To make our considerations of extensions of the Euler's theorem and triangulation concepts more pure, we need to preliminary define the key notions from the related topics. First of all, polygon is a closed plane figure with sides (Weisstein 2002). Then, a polyhedron is the union of a finite set of polygons such that: (i) any pair of polygons meets only at their sides or corners; (ii) each side of each polygon meets exactly one other polygon along an edge; (iii) it is possible to travel from the interior of any polygon to the...
12 Pages(3000 words)Research Paper

Geometry Task 3

..., AB=AC Hence triangle ABC is isosceles. AM is perpendicular to BC from construction Given AM is common(Reflexive property by construction) AAS Congruency from 1, 2, and 3 An isosceles triangle is a triangle with at least two congruent sides Conclusion: This proves that, 'If the base angles of a triangle are congruent, then the triangle is isosceles.' WORK CITED Lawrence, S. Leff. (1997). Geometry the Easy Way. APPLYING CONGRUENT TRIANGKS, 7,109-111. Theorems for congruent Triangles. Methods for proving triangles to be congruent. geometry/GP4/Ltriangles.htm>... 07 Feb. 2008 ASSIGNMENT INSTRUCTIONS ment: If the base angles of a triangle are congruent, then the triangle is isosceles. Drawing: ...
2 Pages(500 words)Essay

Geometry

...Quantitative Literacy Steve Beckman WGU QLT1 Task September29, A. Complete the following graphs: 1. Graph the following values on a single number line • Value 1: 1 • Value 2: 0 • Value 3: –6 • Value 4: 3/4 • Value 5: –1.7 2. Graph the following points on a single coordinate plane. Make sure to indicate labels for each quadrant of the coordinate plane. • Point 1: (3, –2) • Point 2: (0, 0) • Point 3: (–1, 7) • Point 4: (3, 5) • Point 5: (–4, –5) (3) Graph the following functions on a separate coordinate planes. Function 1: y = 2x - 1 y Function 2: y = (-3/4)*x + 5 y Function 3: y = x2 - 4... Literacy Steve Beckman...
7 Pages(1750 words)Speech or Presentation

Geometry proj2

...Project 2 Evaluation 32 Geometry 2 (MTHH 036 058) Part A: Other Transformations (24 points Draw XYZ with vertices X(0, 3), Y(2, 0) and Z(4,2). Draw XYZ and its reflection image in the line x = 4. Make sure to label the new triangle at X’Y’Z’. (3 points) 2. Draw XYZ with vertices X(1, 2), Y(0, 5), and Z(-8, 0). Graph XYZ and its image after a 270 rotation about the origin. Name the coordinates of each vertex of the image. (3 points) 3. You are making hand shadows on a wall using a flashlight. You hold your hand 1 foot from the flashlight and 5 feet from the wall. Your hand is parallel to the wall. If the measure from your thumb to ring finger is 4 inches, what will be the distance between...
4 Pages(1000 words)Essay

Geometry

...Geometry Geometry is a branch of mathematics that deals or attempts to explain and describe the shape, size, or relative position of an object in space. The study of geometry dates back to the time of ancient Greek. The Euclidian geometry for example originated in the 6th century and it laid the foundation for modern geometry. Currently, the study has advanced to include other branches of mathematics such as number theory, algebra, and topology. Riemannian geometry for example is among the latest braches of geometry. Each of the above fields of specialization has a different approach to the study of geometry (Mlodinow, 2002). Indeed, a good mathematician will view the different branches of geometry as mere tools designed to enable... ...
2 Pages(500 words)Admission/Application Essay

Geometry

...Step Draw a line AB that we will be divided into 3 (in this case) equal parts. From point A, draw a line segment at an angle to the given line, and about the same length. The exact length is not important. Set the compasses on A, and set its width to a bit less than one third of the length of the new line. Step the compasses along the line, marking off 3 arcs. Label the last one C. With the compasses width set to CB, draw an arc from A just below it. With the compasses width set to AC, draw an arc from B crossing the one drawn in step 4. This intersection is point D. Draw a line from D to B. Using the same compasses width as used to step along AC, step the compasses from D along DB making 3 new arcs across the line Draw lines... Draw a line...
1 Pages(250 words)Speech or Presentation
sponsored ads
We use cookies to create the best experience for you. Keep on browsing if you are OK with that, or find out how to manage cookies.

Let us find you another Essay on topic Reflection Groups in Geometry for FREE!

Contact Us