Task - Illustrative Mathematics
... we can use any side of the triangle as a base and the key will be finding the height of the triangle with that base. In practice this means finding the line passing through the third vertex meeting the base in a right angle. Two different solutions are presented. The first finds the appropriate alti ...
... we can use any side of the triangle as a base and the key will be finding the height of the triangle with that base. In practice this means finding the line passing through the third vertex meeting the base in a right angle. Two different solutions are presented. The first finds the appropriate alti ...
Sperner`s Lemma and its application
... In dimension one, the question is the same as how many times a function will come cross the x-coordinated if we know f (a) < 0 and f (b) > 0. Then if counted by the order of the zero points, we know it will always be odd!!!! (Need to caution with the tangent points) For higher dimension, we have to ...
... In dimension one, the question is the same as how many times a function will come cross the x-coordinated if we know f (a) < 0 and f (b) > 0. Then if counted by the order of the zero points, we know it will always be odd!!!! (Need to caution with the tangent points) For higher dimension, we have to ...
Formulas Involving Polygons
... Finding Sum of Angles • To find the number of degrees in a polygon, draw all the diagonals possible from one vertex. Then count the number of triangles formed and multiply that by 180 (the number of degrees in one triangle). • Example: When 2 diagonals are drawn in the figure below, 3 triangles are ...
... Finding Sum of Angles • To find the number of degrees in a polygon, draw all the diagonals possible from one vertex. Then count the number of triangles formed and multiply that by 180 (the number of degrees in one triangle). • Example: When 2 diagonals are drawn in the figure below, 3 triangles are ...
Hyperbolic geometry quiz solutions
... (i) This is asking you to prove the Gauss-Bonnet Theorem, given in the lecture notes. Here’s the proof given in lectures. Let ∆ be a hyperbolic triangle with internal angles α, β and γ. We first study the case when at least one of the vertices of ∆ belongs to ∂H, and hence the angle at this vertex i ...
... (i) This is asking you to prove the Gauss-Bonnet Theorem, given in the lecture notes. Here’s the proof given in lectures. Let ∆ be a hyperbolic triangle with internal angles α, β and γ. We first study the case when at least one of the vertices of ∆ belongs to ∂H, and hence the angle at this vertex i ...
The Geometry of Numbers and Minkowski`s Theorem
... A basic fact from linear algebra says: the volume of the fundamental domain is equal to the determinant of the matrix whose columns are the vectors ...
... A basic fact from linear algebra says: the volume of the fundamental domain is equal to the determinant of the matrix whose columns are the vectors ...
Geometry
... a. If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. b. Where the medians of a triangle intersect. c. A perpendicular segment from a vertex of a triangle to the line containing the opposite side. d. When a point is the same distance from two or more objec ...
... a. If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. b. Where the medians of a triangle intersect. c. A perpendicular segment from a vertex of a triangle to the line containing the opposite side. d. When a point is the same distance from two or more objec ...
we will compute formulas for sums of consecutive numbers, or
... for n > 1. Alternatively, by the method applied for question (1), we complete the triangle to a parallelogram and thus double the number of side 2 triangles, but we have also added in those (n − 2) side 2 triangles which intersect the diagonal of the parallelogram and are thus neither included in th ...
... for n > 1. Alternatively, by the method applied for question (1), we complete the triangle to a parallelogram and thus double the number of side 2 triangles, but we have also added in those (n − 2) side 2 triangles which intersect the diagonal of the parallelogram and are thus neither included in th ...
File - Is It Math Time Yet?
... 1. To tessellate (or tile) means to cover a flat surface with a repeated pattern, with no gaps or overlaps. Tessellate comes from the Latin tessera, which refers to a small block of stone, tile, or glass used in making a mosaic. 2. Tessellations often consist of a simple figure which is repeatedly t ...
... 1. To tessellate (or tile) means to cover a flat surface with a repeated pattern, with no gaps or overlaps. Tessellate comes from the Latin tessera, which refers to a small block of stone, tile, or glass used in making a mosaic. 2. Tessellations often consist of a simple figure which is repeatedly t ...