![]() So, if you thought that 90 degree angles and straight sides of the same length define a square, then think again. And as it turns out, they are all the same length. In a similar way, the positive curvature of the sphere made the 90 angles in the triangle we saw above look bigger than 90 degrees.Īlthough the sides of the five-sided shape look curved to us, they are straight in the sense that they're segments of geodesics: lines of shortest distance on the pseudosphere. The angles here look smaller than 90 degrees, but this is an illusion caused by the negative curvature. (We only show the top part of the pseudosphere here as it's sufficient for the illustration.) The image is taken from the Numberphile video 5-sided square, on which this article is based. This five-sided shape drawn on the pseudosphere has straight sides of equal length and 90 degree angles. One side consist of half a great circle, and the other of one half of the great circle that's at right angles to the original one. On the sphere we can even draw a two-sided shape whose sides have the same length and meet at 90 degree angles. In our example, they add up to 90+90+90=270 degrees. This also illustrates that the angles of a triangle drawn on a shape with positive curvature add up to more than 180 degrees (which is the case in the plane). What we have created here is a straight-sided shape whose sides all have the same length and meet in 90 degree angles - and it has three sides, not four! (In technical language, a line of shortest distance on a surface is called a geodesic) That's because, just like the shortest distance between two points on a plane is along a straight line, so the shortest distance between two points on a sphere is along a great circle. The great circles on a sphere are the analogues of straight lines on the plane. Each line corresponds to a quarter of a great circle, so they all have the same length. The lines you have walked along all form parts of great circles on the sphere, that is, circles that have the same diameter as the sphere itself. If you sat yourself down at any given point on the sphere, you'd see the sphere curving downwards away from your bottom in all directions.Ī spherical triangle with 90 degree angles and sides of equal length. Intuitively that's because it "curves outwards" in all directions. ![]() A sphere has positive Gaussian curvature at every point. The Euclidean plane has zero constant curvature at every point, as you'd probably expect: the plane isn't curvy, so it should have zero curvature. The details are explained in this article, but for our purpose we distinguish three types of surfaces: those that have zero Gaussian curvature at every point, those that have positive Gaussian curvature at every point, and those that have negative Gaussian curvature at every point. You can measure how bent a surface is at a given point using the notion of Gaussian curvature (named after the mathematician Carl Friedrich Gauss). What happens if you change the curvature of the plane? Will the number of sides of the shape change? How curvy is curvy? However, there are different types of surfaces which can be described using the notion of curvature. It is true that on this plane, any closed, straight-sided figure whose sides are all the same length and whose interior angles are all right angles must have four sides - it must be a square. ![]() The two-dimensional, flat surface we usually do our geometry on at school is known as theĮuclidean plane. And it's similarly impossible to draw a pentagon, hexagon, or shape with more straight sides that has all 90 degree angles. It's impossible to draw a triangle with all 90 degree angles. The 90 degrees and the four straight sides seem to go together. An honest square: it has four straight sides of equal length and four 90 degree angles
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