Non-Euclidean - people.stfx.ca
... • The first four postulates are much simpler than the fifth, and for many years it was thought that the fifth could be derived from the first four • It was finally proven that the fifth postulate is an axiom and is consistent with the first four, but NOT necessary (took more than 2000 years!) • Sacc ...
... • The first four postulates are much simpler than the fifth, and for many years it was thought that the fifth could be derived from the first four • It was finally proven that the fifth postulate is an axiom and is consistent with the first four, but NOT necessary (took more than 2000 years!) • Sacc ...
The Word Geometry
... and by his ability to solve extremely complicated mathematical operations Some of his teachers were Gauss,Jacobi, Dirichlet, and Steiner Riemannian geometry ...
... and by his ability to solve extremely complicated mathematical operations Some of his teachers were Gauss,Jacobi, Dirichlet, and Steiner Riemannian geometry ...
Non-Euclidean Geometries
... translated Greek works and tried to prove postulate 5 for centuries ...
... translated Greek works and tried to prove postulate 5 for centuries ...
... line going through the given point outside it, or that there are more than one such parallels. It is easy to prove that if there are more than one parallel, there are infinitely many, so we don't get different geometries with 2, 3, 4 etc. parallels. The "no parallel" geometry is called elliptic plan ...
zero and infinity in the non euclidean geometry
... • "It is this similarity between the whole and its parts, even infinitesimal ones, that makes us consider this curve of von Koch as a line truly marvelous among all. If it were gifted with life, it would not be possible to destroy it without annihilating it whole, for it would be continually reborn ...
... • "It is this similarity between the whole and its parts, even infinitesimal ones, that makes us consider this curve of von Koch as a line truly marvelous among all. If it were gifted with life, it would not be possible to destroy it without annihilating it whole, for it would be continually reborn ...
GeoGebra Konferencia Budapest, január 2014
... • Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly those linked with the angles. But: in Euclidean geometry the sum of the angles of the triangle is constant. Not so in hyperbolic geometry. • Many other properties are not preserved such as those related to the di ...
... • Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly those linked with the angles. But: in Euclidean geometry the sum of the angles of the triangle is constant. Not so in hyperbolic geometry. • Many other properties are not preserved such as those related to the di ...
Course Title
... To study axiomatic system, Euclidean geometry as an axiomatic system and to study some important theorem in Euclidean geometry and finally we study an example of non Euclidean geometry. Teaching and Learning Methods Solving problems with discussion. ...
... To study axiomatic system, Euclidean geometry as an axiomatic system and to study some important theorem in Euclidean geometry and finally we study an example of non Euclidean geometry. Teaching and Learning Methods Solving problems with discussion. ...
HOMEWORK ASSIGNMENT #6 SOLUTIONS
... 1. (Exercises 9.1.22 — 9.1.27, page 369) It can be proved that each of the statements in Exercise 22–27 is equivalent to Euclid’s fifth postulate. Rewrite each sentence, using the negation of the conclusion, to give a statement assume to be true in non-Euclidean geometry. 9.1.22: If a straight line ...
... 1. (Exercises 9.1.22 — 9.1.27, page 369) It can be proved that each of the statements in Exercise 22–27 is equivalent to Euclid’s fifth postulate. Rewrite each sentence, using the negation of the conclusion, to give a statement assume to be true in non-Euclidean geometry. 9.1.22: If a straight line ...
Non-Euclidean Geometry, spring term 2017 Homework 4. Due date
... on a line parallel to the base, have the same area. Problem 2. On a hyperbolic plane, triangles on the same base and with the same midline have the same area. Problem 3. Let S be a unit sphere in R3 , and ∆ ⊂ S a triangle whose sides are arcs of great circles. The angles of ∆ are equal to α, β and γ ...
... on a line parallel to the base, have the same area. Problem 2. On a hyperbolic plane, triangles on the same base and with the same midline have the same area. Problem 3. Let S be a unit sphere in R3 , and ∆ ⊂ S a triangle whose sides are arcs of great circles. The angles of ∆ are equal to α, β and γ ...
Geometry Terms Crossword Puzzle
... 1. A ___ is a way to show something is true. 3. Geometric figures are ___ if they have equal corresponding angles and proportional corresponding sides. 5. Two points are ___ if they are on the same line. 6. Two lines are ___ if they are in the same plane and never intersect. 7. The ___ is a point eq ...
... 1. A ___ is a way to show something is true. 3. Geometric figures are ___ if they have equal corresponding angles and proportional corresponding sides. 5. Two points are ___ if they are on the same line. 6. Two lines are ___ if they are in the same plane and never intersect. 7. The ___ is a point eq ...
Non-Euclidean Geometry
... 2. A segment can be extended indefinitely 3. For any two distinct points A and B, a circle can be drawn with center A and radius AB 4. All right angles are congruent ...
... 2. A segment can be extended indefinitely 3. For any two distinct points A and B, a circle can be drawn with center A and radius AB 4. All right angles are congruent ...
Handout Week 1
... 3. A circle may be described with any given point as its center and any distance as its radius. 4. All right angles are equal to one another. 5. Through a given point P not on a line L, there is one and only one line in the plane of P and L which does not meet L. (Playfair’s version 1795) The fifth ...
... 3. A circle may be described with any given point as its center and any distance as its radius. 4. All right angles are equal to one another. 5. Through a given point P not on a line L, there is one and only one line in the plane of P and L which does not meet L. (Playfair’s version 1795) The fifth ...
3379 NonE hw
... interesting stretching of calculated distances that depends on whether the points are close to the center of the disc or close to the edge of the disc. Points can be the SAME Euclidean distance apart and have different Hyperbolic distances depending on their location in the disc. This is a function ...
... interesting stretching of calculated distances that depends on whether the points are close to the center of the disc or close to the edge of the disc. Points can be the SAME Euclidean distance apart and have different Hyperbolic distances depending on their location in the disc. This is a function ...
2.7.3 Elliptic Parallel Postulate
... From these properties of a sphere, we see that in order to formulate a consistent axiomatic system, several of the axioms from a neutral geometry need to be dropped or modified, whether using either Hilbert's or Birkhoff's axioms. The incidence axiom that "any two points determine a unique line," ne ...
... From these properties of a sphere, we see that in order to formulate a consistent axiomatic system, several of the axioms from a neutral geometry need to be dropped or modified, whether using either Hilbert's or Birkhoff's axioms. The incidence axiom that "any two points determine a unique line," ne ...
Exam 2
... d. Conclude that the Elliptic Parallel Postulate is not consistent with the axioms and theorems of neutral geometry. e. Guarantees that linear pairs are supplementary. 6. What must you prove to conclude… (Note: there may be more than one correct answer, but you only need to write one.) a. A quadrila ...
... d. Conclude that the Elliptic Parallel Postulate is not consistent with the axioms and theorems of neutral geometry. e. Guarantees that linear pairs are supplementary. 6. What must you prove to conclude… (Note: there may be more than one correct answer, but you only need to write one.) a. A quadrila ...
attached test
... 3. Is the parallel postulate in Euclidean geometry also true in Spherical geometry? Why or why not? 4. Name one unique figure in spherical geometry that is not found in Euclidean geometry. Accurately describe the figure. 5. Name at least two real life applications of spherical geometry. 6. Describe ...
... 3. Is the parallel postulate in Euclidean geometry also true in Spherical geometry? Why or why not? 4. Name one unique figure in spherical geometry that is not found in Euclidean geometry. Accurately describe the figure. 5. Name at least two real life applications of spherical geometry. 6. Describe ...
Lab 3
... So far, we know that in hyperbolic geometries, triangles' angles add up to less than 180 degrees, and in elliptic geometries, triangles' angles add up to more than 180 degrees. In both cases, the difference between that sum and 180 degrees gives the area of the triangle, up to a constant. For the fo ...
... So far, we know that in hyperbolic geometries, triangles' angles add up to less than 180 degrees, and in elliptic geometries, triangles' angles add up to more than 180 degrees. In both cases, the difference between that sum and 180 degrees gives the area of the triangle, up to a constant. For the fo ...
MAT360 Lecture 10
... subsets of l such that no point of either subsets is between two points of the other. Then there exists a unique point O on l such that one of the subsets is equal to a ray of l with vertex O and the other subset is equal to the complement. ...
... subsets of l such that no point of either subsets is between two points of the other. Then there exists a unique point O on l such that one of the subsets is equal to a ray of l with vertex O and the other subset is equal to the complement. ...
Section 22.1
... Given the figure below. Calculate the defects of the four interior polygons. Test the validity of the additive property of defects by calculating the defect of quadrilateral ABCD. B ...
... Given the figure below. Calculate the defects of the four interior polygons. Test the validity of the additive property of defects by calculating the defect of quadrilateral ABCD. B ...
3379 Homework 3
... Given a unit sphere and line AB on the sphere. If point C is between A and B, where is point C. Sketch the situation and discuss the problems with the notion of between. ...
... Given a unit sphere and line AB on the sphere. If point C is between A and B, where is point C. Sketch the situation and discuss the problems with the notion of between. ...
equiangular polygon
... A polygon is equiangular if all of its interior angles are congruent. Common examples of equiangular polygons are rectangles and regular polygons such as equilateral triangles and squares. Let T be a triangle in Euclidean geometry, hyperbolic geometry, or spherical geometry. Then the following are e ...
... A polygon is equiangular if all of its interior angles are congruent. Common examples of equiangular polygons are rectangles and regular polygons such as equilateral triangles and squares. Let T be a triangle in Euclidean geometry, hyperbolic geometry, or spherical geometry. Then the following are e ...
EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY
... a sphere rather than on a plane. Take an orange and a board marker pen. On the surface of the orange draw a great circle – ie a circle which passes through the ‘poles’ of the orange. This could be said to be a straight line. Now draw a great circle around the ‘equator’ of the orange. The triangles t ...
... a sphere rather than on a plane. Take an orange and a board marker pen. On the surface of the orange draw a great circle – ie a circle which passes through the ‘poles’ of the orange. This could be said to be a straight line. Now draw a great circle around the ‘equator’ of the orange. The triangles t ...
Sample Final
... (1) (10 pts) Show that two hyperbolic lines cannot have more than one common perpendicular. (2) (10 pts) Draw a cevian line for a triangle ABC (in hyperbolic geometry). Prove that the angle defect (π radians minus the sum of the angles in the triangle) is equal to the sum of the defects of the two s ...
... (1) (10 pts) Show that two hyperbolic lines cannot have more than one common perpendicular. (2) (10 pts) Draw a cevian line for a triangle ABC (in hyperbolic geometry). Prove that the angle defect (π radians minus the sum of the angles in the triangle) is equal to the sum of the defects of the two s ...
Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry. In hyperbolic geometry the parallel postulate of Euclidean geometry is replaced with:For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.(compare this with Playfair's axiom the modern version of Euclid's parallel postulate)Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature.A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space.When geometers first realised they worked with something else than the standard Euclidean geometry they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry. It was for putting it in the now rarely used sequence elliptic geometry (spherical geometry) , parabolic geometry (Euclidean geometry), and hyperbolic geometry.In Russia it is commonly called Lobachevskian geometry after one of its discoverers, the Russian geometer Nikolai Lobachevsky.This page is mainly about the 2 dimensional or plane hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry.Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases.