Homework 10 Reading Exercises

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MATH 4050 Homework 10 Due Friday, April 29 Reading • Stillwell, 21.8, 22.9, 24.7*, 24.8*, 24.9 (* these are much too hard, but have some choice tidbits - do the best you can) • Finish Logicomix. • You may find D. Royster's chapters on hyperbolic geometry helpful Exercises Hyperbolic Geometrry 1. A Saccheri quadrilateral is a quadrilateral ABCD where angles C and D are right angles, and sides AD and BC have the same length. Using absolute geometry, prove that angles A and B are equal. 2. Prove the àngle-angle-angle' theorem for hyperbolic geometry: Two triangles whose cor- responding angles are equal are congruent. Hint: line them up at one corner, and if they are not the same triangle the space between them is a quadrilateral with a problem. 3. For each part, draw a Poincarédisk and then draw the indicated diagram. (a) Four hyperbolic lines that don't cross each other (b) A point P , a hyperbolic line `, and three more hyperbolic lines through P that don't cross `. (c) A triangle with angles 90◦ − 5◦ − 5◦. (d) A Saccheri quadrilateral with acute angles of 45◦. (e) A pentagon with five right angles. 4. Find the defect of each hyperbolic polygon in question 3c, d, and e. Hilbert's Problems Choose one of Hilbert's 23 problems and give a short explanation of it (or part of it), and to what extent it has been solved. Some of the more approachable ones are 1, 2, 3, 7, 8, 10, 13, 16, 17 and 18. Russell's Paradox Here is a slight variaton on Russell's paradox. Let S be any set (which contains sets). Let R be the set of x 2 S with x2 = x. That is, R is all sets in S which do not contain themselves. Modern set theory axioms allow R to exist. (a) Show that R is not in S. (b) Explain why there can be no set containing all sets. (Cantor is very excited about this, on pg. 169 of Logicomix)..

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Downloaded from Bookstore.Ams.Org 30-60-90 Triangle, 190, 233 36-72
Index 30-60-90 triangle, 190, 233 intersects interior of a side, 144 36-72-72 triangle, 226 to the base of an isosceles triangle, 145 360 theorem, 96, 97 to the hypotenuse, 144 45-45-90 triangle, 190, 233 to the longest side, 144 60-60-60 triangle, 189 Amtrak model, 29 and (logical conjunction), 385 AA congruence theorem for asymptotic angle, 83 triangles, 353 acute, 88 AA similarity theorem, 216 included between two sides, 104 AAA congruence theorem in hyperbolic inscribed in a semicircle, 257 geometry, 338 inscribed in an arc, 257 AAA construction theorem, 191 obtuse, 88 AAASA congruence, 197, 354 of a polygon, 156 AAS congruence theorem, 119 of a triangle, 103 AASAS congruence, 179 of an asymptotic triangle, 351 ABCD property of rigid motions, 441 on a side of a line, 149 absolute value, 434 opposite a side, 104 acute angle, 88 proper, 84 acute triangle, 105 right, 88 adapted coordinate function, 72 straight, 84 adjacency lemma, 98 zero, 84 adjacent angles, 90, 91 angle addition theorem, 90 adjacent edges of a polygon, 156 angle bisector, 100, 147 adjacent interior angle, 113 angle bisector concurrence theorem, 268 admissible decomposition, 201 angle bisector proportion theorem, 219 algebraic number, 317 angle bisector theorem, 147 all-or-nothing theorem, 333 converse, 149 alternate interior angles, 150 angle construction theorem, 88 alternate interior angles postulate, 323 angle criterion for convexity, 160 alternate interior angles theorem, 150 angle measure, 54, 85 converse, 185, 323 between two lines, 357 altitude concurrence theorem,
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Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used
Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the ﬁrst 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Deﬁnitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent.
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Saccheri and Lambert Quadrilateral in Hyperbolic Geometry
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Tightening Curves on Surfaces Monotonically with Applications † Hsien-Chih Chang∗ Arnaud de Mesmay March 3, 2020 Abstract We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of de Graaf and Schrijver [J. Comb. Theory Ser. B, 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms—the cluster and pipe expansions—to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k-terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and ∆Y -transformations for arbitrary integer k. Previous algorithms only existed in the planar setting when k 4, and all of them rely on extensive case-by-case analysis based on different values of k. Our algorithm≤ makes use of the connection between electrical transformations and homotopy moves, and thus solves the problem in a unified fashion.
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Arxiv:1808.05573V2 [Math.GT] 13 May 2020 A.K.A
THE MAXIMAL INJECTIVITY RADIUS OF HYPERBOLIC SURFACES WITH GEODESIC BOUNDARY JASON DEBLOIS AND KIM ROMANELLI Abstract. We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function. The main results of this paper relate to maximal injectivity radius among hyperbolic surfaces with geodesic boundary. For a point p in the interior of a hyperbolic surface F , by the injectivity radius of F at p we mean the supremum injrad p(F ) of all r > 0 such that there is a locally isometric embedding of an open metric neighborhood { a disk { of radius r into F that takes 1 the disk's center to p. If F is complete and without boundary then injrad p(F ) = 2 sysp(F ) at p, where sysp(F ) is the systole of loops at p, the minimal length of a non-constant geodesic arc in F with both endpoints at p. But if F has boundary then injrad p(F ) is bounded above by the distance from p to the boundary, so it approaches 0 as p approaches @F (and we extend it continuously to @F as 0). On the other hand, sysp(F ) does not approach 0 as p @F .
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Examples of Euclidean Geometry in Real Life
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Euclid's Fifth Postulate
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Homework 27 Answers #1 Hint: Use the Defect Theorem 4.8.2. #2 Hint: Note That the Altitude Splits the Saccheri Quadrilateral
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Geometry Through History Saccheri Quadrilaterals and Consequences Lecture Notes/Solutions from Class Friday, February 12
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