Dimensions 1
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Transcribe video | Upload subtitles OR Translate this English subtitles to Transcript belowDimension 2 My name is Hipparchus I lived in the second century before the birth of Christ, and I don't think I'd be bragging if I told you that I am the father of the sciences of Geography and Astronomy.You know, I wrote more than 14 books but unfortunately they have almost all been lost in the mists of time. I was responsible for the first catalogue of the stars, founded the field of mathematics called trigonometry and even invented the astrolabe. Fortunately, my brilliant successor Ptolemy, three centuries after my time inspired by my work, took up where I left off, and nowadays historians sometimes can't determine what was my contribution and what was his. Ptolemy's manuscript the "Almagest" was the first scientific treatise on astronomy and his book "Geography" contains the first map of the known world. Geography and geometry both deal with the study of the Earth Geography is concerned with making visual representations of the Earth whilst geometry is concerned with measuring it. The shape of the Earth is roughly spherical. Let's forget for the moment that it's slightly flattened at the poles and pretend that it really is a perfect sphere. You probably know too that all the points of the sphere are at the same distance from its centre. The arrow that you can see now, starting at the center of the sphere and ending at a point moving on the surface, has a constant length. Let's choose an axis for our sphere: a line through the center. When we cut the sphere along a plane that contains this axis we carve out a great circle which divides the sphere into two hemispheres. If we chop the sphere up using some sort of guillotine that slices down this axis we trace out the meridians. These are half circles going from the north pole to the south pole of the Earth. And now if we slice the sphere up along a plane at right angles to the axis we get a bunch of circles called parallels. So, now our sphere is covered by two networks of curves the meridians and the parallels One of these parallels should be very familiar it's the equator, half-way between the two poles. For historical reasons one of these meridians was chosen to be the principal meridian, it is the one passing through the Greenwich Observatory in England, To specify the position of some point on the surface of the Earth we can start at the point where the Greenwich meridian meets the Equator, and walk round the equator a distance measured by an angle called the longitude - coloured red then you go up along a meridian some way measured by an angle called the latitude - colored green, finally arriving at our desired destination. Any point on the Earth is precisely described by just these two numbers: its latitude and its longitude. Since we need two numbers to specify a location on the surface of the Earth we say that the sphere is two dimensional. and mathematicians often call it S2. Finally, if we let our little plane leave the Earth and fly off into space then to locate it we need to give three numbers latitude, longitude and . . . the altitude above the Earth. Since we now need three numbers, to say where we are in outer space we say that space is 3 dimensional. Look at the paintings on the wall, there's a portrait of Ptolemy -- the father of map making. How do we draw the Earth? One method is to project it on a plane. Let's choose a city, Dakar for example, We draw a straight line from the north pole through Dakar Our line hits the table at a some point that we call its projection onto the table. Any point on the Earth's surface can be projected onto the table in this way. The nearer our town is to the north pole the further away its projection on the table is. In fact it can even end up off the table. For this reason we say that the north pole doesn't have a projection or, more correctly, that its projection is at infinity. The whole Earth, with the exception of the north pole, can be represented on the plane of the table. This map of the world is called the stereographic projection. Of course, our stereographic projection doesn't preserve sizes. South America appears tiny compared to North America. To get a better idea of what this projection does, we'll roll the Earth along just like a giant ball, and will always project from the highest point. The projections of the continents waltz around in the plane, taking turns at becoming bigger and smaller. But if we take a closer look, we see that shapes don't change even if lengths do. For this reason we say that the stereographic projection is conformal. What happens to the meridians and the parallels under the projection? When we project from the north pole, the meridians become radii emanating from the south pole and the parallels, concentric circles. And as the Earth turns, you see that both the meridians and parallels. always project to either circles or straight lines. The stereographic projection transforms circles drawn on the sphere into circles drawn in the plane, except for those circles passing through the pole from which we project, whose projections are in fact straight lines in the plane. Now here's our rolling Earth from below. From this point of view we see the meridians and parallels form two bundles of circles. All of the meridians converge at two points, the north and the south pole. Do you recognize this one here? Yes, it's the Greenwich Meridian, the end of the first stage of our journey towards the fourth dimension.
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