PPT-Equiangular Lines in R

d Moshe Rosenfeld University of Washington Shanghai Jiao Tong University July 1 2013 How wide and how even can you spread your chopsticks Equiangular Lines Definition A set of lines through the origin in R. Tropp Mathematics Department The University of Michigan 530 Church St Ann Arbor MI 481091043 USA ABSTRACT A complex equiangular tight frame ETF is a tight frame cons isting of unit vectors in whose absolute inner products are identical One may Tropp Institute for Computational Engineering and Sciences The University of Texas at Austin Inderjit S Dhillon and Robert W Heath Jr The University of Texas at Austin Thomas Strohmer The University of California at Davis brPage 2br Equiangular Tigh Mixon and Janet C Tremain Department of Mathematics and Statistics Air Force Institute of Technology WrightPatterson AFB OH Program in Applied and Computational Mathematics Princeton University Princeton NJ Department of Mathematics University of Mi SLIDING BEVEL . TRY-SQUARE. Drawing lines around a piece of wood. Drawing lines at 90⁰ to an edge. Testing timber for square. Hardwood stock . Brass strip. Rivets. MORTISE GAUGE. Adjustable thumb screw. Classifying Triangles by Angles. ACUTE. OBTUSE. RIGHT. EQUIANGULAR. ACUTE TRIANGLE. All interior angles are acute (or have a measure less than 90°). Interior Angle. Example of Acute Triangle. Phineas’s. Parallel and Perpendicular Lines. Perpendicular lines. are two lines that intersect to form a 90º. . angle. . Parallel and Perpendicular Lines. Parallel lines. are two lines that, if extended indefinitely, would never cross or touch. Robert Minchin. Spectral Lines. What is a Spectral Line?. Types of Spectral Line. Molecular Transitions. Linewidths. Reference Frames. When a Coconut falls…. Bang!. h. Energy released = . mgh. When an electron falls…. Polygons. Polygon . Definition. : A polygon is a closed figure in a plane, formed by connecting line segments endpoint to endpoint. Each segment intersects exactly two others. . Side. vertex. Polygons!. A point is an exact location in space.. You are here.. A true point has no length, no width, and no height.. In fact, you cannot see a true point.. A point is named by a letter.. P. Point P. Lines are 1-dimensional objects that have only length. Lines continue forever in both directions.. Triangles in Art. Triangles can be used to draw many different body parts such as the nose. Many different cultures use triangles in there art a lot. The natives also used triangles in their cave paintings, mainly for bodies.. Contour lines are lines drawn on a map that connect points of equal elevation. If you walk along a contour line you neither gain or lose elevation.  . Picture walking along a beach exactly where the water. Triangles. Isosceles Triangles. leg. leg. base. Vertex angle. Base angle. Base angle. 2 sides are congruent. 2 theorems. If 2 sides are congruent, then base angles are congruent. If 2 . base angles . Students will be able to. Determine whether two lines are parallel. Write flow proofs. Define and apply the converse of the theorems from the previous section. Objectives. You can use certain angle pairs to determine if two lines are parallel. Geometry. Legs are congruent. Isosceles Triangle. Vertex Angle. Base Angles. Legs. Theorem 4-3 (Isosceles Triangle . Thm. ):. If two sides of a triangle are congruent, then the angles opposite those sides are congruent..