MATH 498E—Geometry for High School Teachers
... Text: College Geometry Using the Geometer's Sketchpad, 1st Edition, Reynolds and Fenton, Wiley Publishing, 2012 or College Geometry, 1st Edition, with Geometer’s Sketchpad v5 Set by Barbara Reynolds, Nov. 2011. Dates: June 26-July 26, Tuesdays and Thursdays from 9 am to 1:30 pm. Objective. The objec ...
... Text: College Geometry Using the Geometer's Sketchpad, 1st Edition, Reynolds and Fenton, Wiley Publishing, 2012 or College Geometry, 1st Edition, with Geometer’s Sketchpad v5 Set by Barbara Reynolds, Nov. 2011. Dates: June 26-July 26, Tuesdays and Thursdays from 9 am to 1:30 pm. Objective. The objec ...
Strange Geometries
... the fifth, was equivalent to a statement we are all familiar with: that the angles in a triangle add up to 180 degrees. However, this postulate did not seem as obvious as the other four on Euclid’s list, so mathematicians attempted to deduce it from them: to show that a geometry obeying the first fo ...
... the fifth, was equivalent to a statement we are all familiar with: that the angles in a triangle add up to 180 degrees. However, this postulate did not seem as obvious as the other four on Euclid’s list, so mathematicians attempted to deduce it from them: to show that a geometry obeying the first fo ...
Euclidean vs Non-Euclidean Geometry
... from each other even if extended to infinity, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In ellipt ...
... from each other even if extended to infinity, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In ellipt ...
Lesson Plan 1
... Geometry the sum of angles of a triangle is 180. 3) States to the students that in Hyperbolic Geometry the sum of the angles of a triangle is less than 180. Show some examples. 4) States to the students that in Hyperbolic Geometry, all the axioms for neutral geometry hold, but the parallel postulate ...
... Geometry the sum of angles of a triangle is 180. 3) States to the students that in Hyperbolic Geometry the sum of the angles of a triangle is less than 180. Show some examples. 4) States to the students that in Hyperbolic Geometry, all the axioms for neutral geometry hold, but the parallel postulate ...
practice problems
... (e) Show that the summit angles of a Saccheri quadrilateral are equal and acute. (f) Do there exist hyperbolic lines l and m with the property that the distance from a point of m to l is the same for any choice of a point on m? (g) Show that the angle sum of the two interior angles of an omega trian ...
... (e) Show that the summit angles of a Saccheri quadrilateral are equal and acute. (f) Do there exist hyperbolic lines l and m with the property that the distance from a point of m to l is the same for any choice of a point on m? (g) Show that the angle sum of the two interior angles of an omega trian ...
Hyperbolic Spaces
... In hyperbolic geometry, the sum of the angles of a triangle is less than 180°. In hyperbolic geometry, triangles with the same angles have the same areas. There are no similar triangles in hyperbolic geometry. In hyperbolic space, the concept of perpendicular to a line can be illustrated as seen in ...
... In hyperbolic geometry, the sum of the angles of a triangle is less than 180°. In hyperbolic geometry, triangles with the same angles have the same areas. There are no similar triangles in hyperbolic geometry. In hyperbolic space, the concept of perpendicular to a line can be illustrated as seen in ...
Entropy Euclidean Axioms (Postulates) Parallel Postulate Curved
... • Curved space cannot be described by the Euclidean geometry; therefore, it is called non-Euclidean. • In curved space, there is a characteristic length scale, R. – Example: the surface of the Earth – How do we know that the surface of the Earth is curved? ...
... • Curved space cannot be described by the Euclidean geometry; therefore, it is called non-Euclidean. • In curved space, there is a characteristic length scale, R. – Example: the surface of the Earth – How do we know that the surface of the Earth is curved? ...
World Globe
... are equidistant from each other • Proclus (410): If a line intersects one of 2 parallel lines then it intersects the other also • Playfair (1795): Given a line and a point not on a line only one line can be drawn parallel to the given line. ...
... are equidistant from each other • Proclus (410): If a line intersects one of 2 parallel lines then it intersects the other also • Playfair (1795): Given a line and a point not on a line only one line can be drawn parallel to the given line. ...
Non-Euclidean Geometries
... The Most Controversial The Parallel Postulate: In layman’s terms * Given a line and a point not on that line, there is exactly one line through the point that is parallel to the line. ...
... The Most Controversial The Parallel Postulate: In layman’s terms * Given a line and a point not on that line, there is exactly one line through the point that is parallel to the line. ...
Hyperbolic Triangles
... Area of Hyperbolic Triangle In hyperbolic geometry, a hyperbolic quadrilateral has angle sum less than 2π, therefore cannot have four right angles. Instead, we use triangles as basic figures. The Gauss-Bonnet Formula If the hyperbolic triangle ABC has angles α, β,γ, then its area is Areahyp ...
... Area of Hyperbolic Triangle In hyperbolic geometry, a hyperbolic quadrilateral has angle sum less than 2π, therefore cannot have four right angles. Instead, we use triangles as basic figures. The Gauss-Bonnet Formula If the hyperbolic triangle ABC has angles α, β,γ, then its area is Areahyp ...
Final exam key
... (A) Given any triangle ∆ABC and any segment DE, there exists a triangle ∆DEF (having DE as one of its sides) that is similar to ∆ABC. (B) If two lines cut by a transversal l have a pair of congruent alternate interior angles with respect to l, then the two lines are parallel. (C) If two parallel lin ...
... (A) Given any triangle ∆ABC and any segment DE, there exists a triangle ∆DEF (having DE as one of its sides) that is similar to ∆ABC. (B) If two lines cut by a transversal l have a pair of congruent alternate interior angles with respect to l, then the two lines are parallel. (C) If two parallel lin ...
Hypershot: Fun with Hyperbolic Geometry
... 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the t ...
... 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the t ...
Euclidean/non-Euclidean Geometry
... On a sphere, there are no straight lines. As soon as you start to draw a straight line, it curves on the sphere. In curved space, the shortest distance between any two points (called a geodesic) is not unique. In curved space, (for spherical (riemannian) or hyperbolic geometry)the concept of perpend ...
... On a sphere, there are no straight lines. As soon as you start to draw a straight line, it curves on the sphere. In curved space, the shortest distance between any two points (called a geodesic) is not unique. In curved space, (for spherical (riemannian) or hyperbolic geometry)the concept of perpend ...
Chapter 7: Hyperbolic Geometry
... 1. A straight line may be drawn from a point to any other point. 2. A finite straight line may be produced to any length. 3. A circle may be described with any center and any radius. 4. All right angles are equal. 5. If a straight line meet two other straight lines so that as to make the interior an ...
... 1. A straight line may be drawn from a point to any other point. 2. A finite straight line may be produced to any length. 3. A circle may be described with any center and any radius. 4. All right angles are equal. 5. If a straight line meet two other straight lines so that as to make the interior an ...
Notes on the hyperbolic plane.
... Transformations: Tell whether each of the following is possible on a hyperbolic plane. If it is possible, describe what it’s like. If it’s not possible explain why not. Is it possible to reflect across a hyperbolic line? ...
... Transformations: Tell whether each of the following is possible on a hyperbolic plane. If it is possible, describe what it’s like. If it’s not possible explain why not. Is it possible to reflect across a hyperbolic line? ...
Study Guide - U.I.U.C. Math
... Poincare and Klein models, including distance formulae Parallel Axiom in hyperbolic geometry Angle measure in Poincare model; perpendicularity in Klein model Limiting parallels Ideal points and ideal triangles Angle defect Saccheri and Lambert quadrilaterals Area in hyperbolic geomet ...
... Poincare and Klein models, including distance formulae Parallel Axiom in hyperbolic geometry Angle measure in Poincare model; perpendicularity in Klein model Limiting parallels Ideal points and ideal triangles Angle defect Saccheri and Lambert quadrilaterals Area in hyperbolic geomet ...
Worksheet on Hyperbolic Geometry
... The program KaleidoTile can be found at the website _______________________. Since triangles are ‘thin’ in the hyperbolic geometry setting the symmetry to _____ will put you into the tiling. For a binary tree (part of the hyperbolic plane) each parent has ____ daughters. This means the number of des ...
... The program KaleidoTile can be found at the website _______________________. Since triangles are ‘thin’ in the hyperbolic geometry setting the symmetry to _____ will put you into the tiling. For a binary tree (part of the hyperbolic plane) each parent has ____ daughters. This means the number of des ...
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.