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.
Olivia's Butterfly
This is a very simple hat and very quick to make. Using a J hook and worsted weight yarn, I used red heart, and it fits a 20 1/2 inch head very nicely. Very easily adjustible by either adding increase rows or taking away. Rnd 1) ch3, 11dc in 3rd ch from hook (11dc) sl st to top of first dc (now and through out),Rnd 2) ch2, (does not count as dc now and through out) 2dc in each st around, sl st to top of first dcRnd 3) ch 2 * 2dc in first st, 1dc in next st around, repeat from * around slip st to joinRnd 4) ch2, *2dc in first st, dc in next 2dc , repeat from * around, joinRnd 5) ch2, * 2dc in first st, dc in next 3dc , repeat from * around, join Rnd 6) ch2, dc in same st and each st around Rnd 7 and 8) Repeat rnd 6 Rnd 9) ch2, dc in same st, dc in next 21 sts, ch10, skip next 9 sts, dc in next st and in each st accross, slip st to join Rnd 10) ch2, dc in same st and in next 21 sts, ch 10, skip 10 chains, dc in next dc and in each remaining dc, slip st to join Rnd 11 and 12) Repeat rnd 10
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
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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[edit] Early history[edit] 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.
StumbleUpon
WigUsing straight needles or one set of circulars, cast on 96 sts. Work in 2 x 2 rib until the piece is as long as the distance between the bottom of your chin, and your eyebrow (about 6.5"), ending with a WS row. NOTE: The extra stitches along the edge of the work will balance the width of the first and last ribs. Later on, when the bangs are added, these stitches will become parts of purl ribs. If more length in the main body of the wig is desired, work more rows at this point. Bangs With RS facing and using backward loop cast on, loosely CO 32 sts. If more length in the bangs is desired, work more rows at this point. Work Decreases Using diagram B below as a guide, rearrange stitches on needles and add stitch markers. NOTE: Decreases will be worked identically at the front and back of the wig. Begin decrease round by working decreases over bangs.
Free Hobbes Crochet Pattern
I've finally decided to write up my Hobbes pattern and instead of selling it I'd like to give it to you for free. The reason he is free is because the creator of Calvin and Hobbes, Bill Watterson did not want to commercialize his work so keeping that in mind Please don't sell this pattern and don't sell the completed work. ~~~~~~~~~~~~~~~~~~~~ I originally made this up for my son because of his love for Hobbes (he's 23 now) I armed myself with Watterson's brilliant drawings as a reference. I made this pattern to look as close as possible to the Hobbes' drawings . I realized that Hobbes could be broken down into basic shapes. He also had the muzzle of a cat so I looked to the famous Amineko cat. Once I got the shapes down I worked on size. Until I started making him I never realized how different Hobbes looks from a regular cat. His head and body are a lot longer thanan normal cat plus he has the shortest fattest legs. That caused me problems since tubular legs and arms don't want to bend. Resources:
