Non-Euclidean Geometries
... *Started to wonder could a system of plane geometry be created with more than one line parallel to a given line *Gauss never published his work *Bolyai constructed the foundations of hyperbolic geometry *Lobachevsky first to publish his results of a hyperbolic geometry ...
... *Started to wonder could a system of plane geometry be created with more than one line parallel to a given line *Gauss never published his work *Bolyai constructed the foundations of hyperbolic geometry *Lobachevsky first to publish his results of a hyperbolic geometry ...
Properties of Graphs of Quadratic Functions
... This form gives us the least amount of information about the properties of the parabola but it does give the vertical stretch and the y-intercept. It is the form that is given during regression analysis (on the TI-83) so it is still important. (*Note: to change from the standard form to the general ...
... This form gives us the least amount of information about the properties of the parabola but it does give the vertical stretch and the y-intercept. It is the form that is given during regression analysis (on the TI-83) so it is still important. (*Note: to change from the standard form to the general ...
Unit 7 Powerpoints - Mona Shores Blogs
... A polygon is concave if a line that contains a side of the polygon contains a point in the interior of the polygon. Take any two points in the interior of the polygon. If you can draw a line between the two points that does leave the interior of the polygon, then it is concave. Concave polygons have ...
... A polygon is concave if a line that contains a side of the polygon contains a point in the interior of the polygon. Take any two points in the interior of the polygon. If you can draw a line between the two points that does leave the interior of the polygon, then it is concave. Concave polygons have ...
Study Guide - page under construction
... kite - a quadrilateral with exactly two pairs of congruent consecutive sides trapezoid - a quadrilateral with exactly one pair of parallel sides • base - a parallel side in a trapezoid • base angles - two consecutive angles whose common side is the base of a trapezoid • leg - a nonparallel side in a ...
... kite - a quadrilateral with exactly two pairs of congruent consecutive sides trapezoid - a quadrilateral with exactly one pair of parallel sides • base - a parallel side in a trapezoid • base angles - two consecutive angles whose common side is the base of a trapezoid • leg - a nonparallel side in a ...
Betti Numbers and Parallel Deformations
... • Connected d-regular graph embedded in real Euclidean n-space • Every pair of edges at a vertex determines a planar cycle of edges • These are the geodesics • 1-skeleton of any simple polytope (since any pair of edges at a vertex determines a 2-face) – simplex, cube in any dimension – dodecahedro ...
... • Connected d-regular graph embedded in real Euclidean n-space • Every pair of edges at a vertex determines a planar cycle of edges • These are the geodesics • 1-skeleton of any simple polytope (since any pair of edges at a vertex determines a 2-face) – simplex, cube in any dimension – dodecahedro ...
Kajol Parwani December 13, 2012 Math Investigation #6: Regular
... interlocked and covered the plane without any overlapping or gaps. He had thought that it was beautiful to have tessellation in his art, so after he created his lion block print, he produced tessella ...
... interlocked and covered the plane without any overlapping or gaps. He had thought that it was beautiful to have tessellation in his art, so after he created his lion block print, he produced tessella ...
P6 - CEMC
... This activity works best if students work in small groups with some direction from the teacher. Here are some suggestions. 1. For part a), divide students into six small groups, and have each group do the measurements for one triangle. Then collect the data for the whole class to verify that every t ...
... This activity works best if students work in small groups with some direction from the teacher. Here are some suggestions. 1. For part a), divide students into six small groups, and have each group do the measurements for one triangle. Then collect the data for the whole class to verify that every t ...
Help on Assignment 6
... which contains P and which is parallel to ` (that it is, it does not intersect `). In Hyperbolic geometry there is more than one such line, and in Spherical geometry lines are never parallel. • In Euclidean geometry the sum of angle measures in a single triangle will be exactly 180 degrees, while in ...
... which contains P and which is parallel to ` (that it is, it does not intersect `). In Hyperbolic geometry there is more than one such line, and in Spherical geometry lines are never parallel. • In Euclidean geometry the sum of angle measures in a single triangle will be exactly 180 degrees, while in ...
Lesson Plan Format
... Example 3: Hobby Application Find the length of the model to the nearest tenth of a centimeter. ...
... Example 3: Hobby Application Find the length of the model to the nearest tenth of a centimeter. ...
F E I J G H L K
... 3. Be able to prove that the sum of the angles of any triangle is 180 degrees. Also explain one way to do it using inductive logic and using paper. 4. a. Write down a formula for finding the sum of the measures of the interior angles for any polygon. Explain what the numbers and letters in your form ...
... 3. Be able to prove that the sum of the angles of any triangle is 180 degrees. Also explain one way to do it using inductive logic and using paper. 4. a. Write down a formula for finding the sum of the measures of the interior angles for any polygon. Explain what the numbers and letters in your form ...
Polygons
... less than 180° o Every line segment between two vertices does not go on the exterior of the polygon. (It remains inside or on the boundaries of the polygon) ...
... less than 180° o Every line segment between two vertices does not go on the exterior of the polygon. (It remains inside or on the boundaries of the polygon) ...
List of regular polytopes and compounds
This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an n-polytope equivalently describes a tessellation of a (n-1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example the cube has Schläfli symbol {4,3}, and with its octahedral symmetry, [4,3] or File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, is represented by Coxeter diagram File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png.The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one-lower-dimensional Euclidean space.Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.A more general definition of regular polytopes which do not have simple Schläfli symbols includes regular skew polytopes and regular skew apeirotopes with nonplanar facets or vertex figures.