Hyperbolic Geometry SHIVANI BAISIWALA S O RI BAEK INTRODUCTION What is it? Hyperbolic Geometry is form of non- Euclidean geometry Obtained by considering only the first four postulates of Euclid and replacing the parallel postulate by its negation, the hyperbolic axiom. Postulates of Euclidean Geometry 1. A straight line may be drawn from any given point to any other 2. A straight line may be extended to any finite length 3. A circle may be described with any given point as its center and any distance as its radius 4. All right angles are congruent 5. If a straight line intersects two other straight lines, and so makes the two interior angles on one side of it together less than two right angles, then the other straight lines will meet at a point if extended far enough on the side on which the angles are less than two right angles. REPLACED WITH There exists a line and an external point not lying on this line such that there are at least two lines passing through this point and parallel to this line – aka Hyperbolic Axiom Basic Properties of Hyperbolic Geometry All triangles have angle sum less than 180. 180 minus the angle sum of a triangle is a positive number known as the defect of the triangle. Rectangles do not exist. If two triangles are similar, they are congruent. All convex quadrilaterals have angle sum less than 360. There are an infinite number of parallels for any given line. If two angles are similar, they are also congruent. It is impossible to magnify or shrink a triangle without distortion, so solely similar triangles do not exist. Viewing Hyperbolic Space Hyperbolic space is defined by negative curvature in all directions. In a sense, it can be thought of as the opposite of a sphere. Models Of Hyperbolic Geometry Two major models – Two ways to mathematically “view” hyperbolic space. Beltrami – Klein Model Poincare Model Beltrami – Klein Model Think of a circle Y in the Euclidean plane. If O is the center of circle Y, and OR is a radius, the interior of Y consists of all points X such at OX < OR All these points represent the points of the hyperbolic plane in Klein’s model. Klein Model Continued Remember that a chord is a segment from one end of the circle to another. Now, consider that chord without its endpoints. In Klein’s model, these are the lines of the hyperbolic plane. Parallels in Klein Model The definition of two parallel lines is that they never meet in the plane under discussion. The fact that they may meet outside the circle is irrelevant, because that is not part of the Klein model. Therefore, the two intersecting lines are both parallel to the line at the bottom. The Poincare Model This model is a disk model that also represents the points of the hyperbolic plane by the points interior to a Euclidean circle Y Lines are defined differently. Poincare Model Continued Only open diameters (chords that pass through the center) are considered lines. The other lines are represented by open arcs or circles orthogonal to Y. All arcs are considered parallel as long as they don’t meet. GEOMETRY IN THE HYPERBOLIC PLANE HYPERBOLIC TRIANGLES TRADITIONAL AREA CALCULATIONS In traditional geometry, we calculate areas of triangles by essentially breaking up the area of a rectangle into two. Hence, we have our traditional triangle area formula: However, this cannot work in hyperbolic geometry. Why? Because rectangles do not exist in hyperbolic geometry, so the whole basis of our area formula is removed. CALCULATING AREA AND DEFECT In hyperbolic geometry, the area function of a triangle has the following properties: Invariance under congruence: Congruent triangles have the same area. Additivity: If a polygonT is split into two polygons – T1 and T2 – by a segment joining a vertex to a point of the opposite side, then the area of T is the sum of the areas of T1 and T2. Defect is important This is the difference between the expected angle sum of a given polygon in Euclidean space and its actual angle sum in Hyperbolic space. It is always positive and is given by where n is the number of sides of the polygon. AREA FORMULAS Area formula for hyperbolic geometry for a polygon angles measured in degrees (so your answer will end up in radians) where is a proportionality constant that is typically set at k = 1. SOMETHING INTERESTING Since defect measures how much the angle sum is less than 180, the highest it can be is 180 (if the angle sum of the actual triangle is 0). Therefore, the maximum area of any triangle is . However, no triangle actually achieve this. It is essentially an asymptotic value, since we can approach it as closely as we like by making the shape of our triangle achieve a trebly asymptotic appearance. CACULAT E THE AREA Given a triangle with angle measures A = 14.2, B = 13.4, and C = 43.4, calculate the defect (in degrees) and the area (in radians) of the triangle. CALCULATE Given a triangle with area , and two angle measures of 32 and 40, calculate the third angle measure of the triangle. AREA AND CIRCUMFERENCE The circumference of a circle of radius is given by: The area of a circle with radius is The area of a hyperbolic circle is much larger than the area of a Euclidean circle with the same radius. CALCULATE Calculate the area and the circumference of a circle with radius = 2ln2 (Hint: Recall that ) CALCULATE Given a circle with radius 2, calculate the circumference. TESSELLATIONS A regular tessellation is covering of the plane by regular polygons so that the same number of polygons meet at each vertex. There are infinitely many hyperbolic tessellations.. How do you know if something is a tessellation? Tessellations are written in the notation {n,k}, known as the Schlafli symbol, where n is the number of sides on the polygon and k is the number of polygons that meet at each vertex. If is equal to .5, you have a Euclidean tessellation. If it is less than .5, you have a hyperbolic tessellation. If it is greater than .5, you have an elliptic tessellation. EXAMPLES HYPERBOLIC TESSELLATIONS You can calculate area of the shapes in hyperbolic tessellations using the Schlafli symbol. Keep in mind that tessellations use regular polygons, where all the angles are equal Given a tessellation {n,k}, you know that each angle measure of the polygon used will be . Once you have the angles, you can calculate defect, and then calculate area. CALCULATE Calculate the area of each polygon used in the hyperbolic tessellation given by {3,10}. CALCULATE Given that the area of a decagon used in a hyperbolic tessellation is , find the Schlafli symbol. CREATE YOUR OWN TESSELLATION .