# Body

updated version is published in Mathematical Intelligencer, Vol. 23, No. 2, pp. 17-28, Spring 2001. David W. Henderson Department of Mathematics, Cornell University, Ithaca, NY, USA, dwh2@cornell.edu Daina Taimiða Department of Mathematics, Cornell University, dtaimina@math.cornell.edu For God's sake, please give it up.  Wolfgang Bolyai urging his son János Bolyai to give up work on hyperbolic geometry. In June of 1997, Daina was in a workshop watching the leader of the workshop, David, helping the participants study ideas of hyperbolic geometry using a paper and tape surface in much the same way that one can study ideas of spherical geometry by using the surface of a physical ball. But, Wait! Constructions of Hyperbolic Planes We will describe three different isometric constructions of the hyperbolic plane (or approximations to the hyperbolic plane) as surfaces in 3-space. 1. This is the paper and tape surface that David learned from William Thruston. Figure 1. 2. Figure 2. Figure 3. 3. 1.

Non-Euclidean Geometry In three dimensions, there are three classes of constant curvature geometries. All are based on the first four of Euclid's postulates, but each uses its own version of the parallel postulate. The "flat" geometry of everyday intuition is called Euclidean geometry (or parabolic geometry), and the non-Euclidean geometries are called hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry (or Riemannian geometry). Spherical geometry is a non-Euclidean two-dimensional geometry. Learning to Hyperbolic Crochet - Experimental Algebra and Geometry Lab Before creating your own Hyperbolic Plane, one must first learn basic crochet skills such as:how to make a chain and how to single crochet. For additional assistance on learning how to make a chain and single crochet, click the following links for instructional videos:Starting ChainSingle Crochet. Once you learn those two crochet skills, it is time to move on to the Hyperbolic Crochet. Follow the next steps to create a the Hyperbolic Plane: Step 0. The following links provide more explanation and detail:Hyperbolic Crochet Video by the Institute for FiguringCrocheting the Hyperbolic Plane by Professors Taimina and HendersonCrochet Hyperbolic Corals by the Institute for Figuring Contact Dr.

Non-Euclidean geometry Version for printing In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems: To draw a straight line from any point to any other. To produce a finite straight line continuously in a straight line. Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by this axiom. To each triangle, there exists a similar triangle of arbitrary magnitude. Here is the Saccheri quadrilateral

Patterns Everyone who has crocheted before has practiced hyperbolic crochet! When you crochet ripples, ruffles, curls and twists you are creating hyperbolic coral reef like forms. Here are some links to various free patterns to help get you started, including a knit shell pattern ( in case you prefer to knit:-) Please keep in mind that this will be a large display. Work of all sizes will be needed including large versions of whatever you want to create. I suggest using these patterns as inspiration. I can't wait to see your creations! note: Please follow any copyright requests made by designers. Basic Hyperbolic Plane pattern - beginner Basic Pseudosphere pattern - beginner Basic Crochet Pseudosphere Coral pattern - beginner Pink Anemone Oasis pattern - advanced beginner (just have to know how to follow a simple pattern) Lion Brand - Basic "How to Crochet" Instructions & video Cobblers Cabin -Chrysanthemum - a basic coral form Lion Brand Kelp Patterns: one, two, three, four, five, and six Sea Mouse

Non-Euclidean geometry Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is set aside. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: History Early history While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work Elements was written.

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