Geometry

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INTRODUCTION

Lstening to a mathdoc,  I got knowledge about geometry that connected the geometry of  Euclide to that of Gauss and Riemann. They said:

“The properties of space were first described by the mathematician euclid over 2 000 years ago in his legendary text the elements in it he laid down a set of simple logical rules about space in what today we call euclidean geometry. Euclidean geometry is the geometry we see around us every day. If you’re sitting in a room and it’s the usual rectangular room, what you see is lots of straight lines right angles you see parallel lines. If you have a window the two sides of the window are parallel if you extended them they’d stay exactly the same distance apart they would never meet. And the other thing you would see if you look a little closer, is that any triangle you draw, the angles in the triangle always add up to 180 degrees.Now that’s characteristic of euclidean geometry and people used to think that this was just how geometry was that nothing else was possible for Euclid himself and for almost all mathematicians for the next 2000 years. These rules weren’t just true mathematically. There were also true statements about physical reality itself, so they thought that two parallel lines would remain parallel forever. That a triangle in real space would always have angles adding up to 180 degrees.
But weird as though this might sound it’s not actually always true

almost 250 years ago in a small town in northern Germany, a mathematician was born who had the ability and originality to start to unravel Euclid’s geometry and begin to change our ideas about space his name was carl friedrich auss tackled many great problems in his career but from a young age he began to speculate that the rules of euclid may not be as absolute as everyone had assumed.

Specifically Gauss began to see that in curved spaces other types of geometry could exist with different rules to Euclid’s. For example on the surface of a sphere the angles of a triangle can add up to more than 180 degrees.

Many others would refine and develop gauss’s ideas but one of his greatest achievements will be to give us a cunning method of accurately measuring curvature it would become known simply as the remarkable theorem let me explain with this globe you see we can see that it’s three-dimensional because we can stand back and look at it but what if you are an ant stuck on the surface how would it know that that surface is curved so imagine you’re the ant and you start off at the north pole and facing south you move down towards the equator at the equator you still face south and you shuffle sideways along the equator.

Then you reach a certain point and then you start walking backward so you’re still facing the same direction and head back to the north pole

Now look what’s happened here you’ve been pointing south all the way around and yet when you arrive back at your starting point you’re facing in a different direction understanding this gives us a way of calculating the curvature of a surface without ever leaving it

this was an amazing insight but it only applies to curved surfaces which are two-dimensional it would take a brilliant student of gauss’s bernard Riemann to develop these ideas in a way that could be applied to the three-dimensional space that surrounds us it would be a daring outlandish and to non-mathematicians absurd sounding concept age just 26 Riemann encapsulated his strange new ideas about geometry in a lecture that was to become legendary among mathematicians in June 1854 Riemann delivered his lectures to an enraptured audience in them he detailed how he took gauss’s ideas on curved surfaces and generalized them so they applied not only to curved two-dimensional surfaces but the curvature of space in any dimension

okay so I’m sure this all sounds rather complicated what exactly do we mean by curved space in any dimension so let me try and explain here’s the thing gauss talked about curved two-dimensional surfaces well here we have a sheet of paper and it’s two-dimensional so if I curve it we can visualize and see this curvature but only because it’s embedded in three dimensions now what if we curved three dimensions presumably we’d need a fourth dimension

but how do you get to this four dimensional space it’s impossible to step outside of our three-dimensional world wherever you travel in the universe no matter how far you go you’re always stuck in three dimensions the genius of riemann was to show that you didn’t need to stand in a fourth dimension to tell if space was curved you could actually do it from the inside but for riemann this would always remain a purely mathematical idea it would take albert einstein to tie these mathematical ideas together and apply bendy curved non-euclidean geometries to the real space that surrounds us i think the most important point about the whole story of non-euclidean geometry is it shows how mathematics and the real world relate and it starts out with mathematicians pottering around asking could there be a geometry different from euclid’s and if anyone came to the time and said why are you studying that they’d say haven’t got a clue what’s it useful for no idea it’s just interesting but they potted around and they found a surprising answer that different geometries were possible and even at that point nobody had any real applications for this idea and then when the moment is ripe einstein comes along and says that’s what i need that’s real physics and suddenly this piece of esoteric mathematics becomes vital to the scientific enterprise einstein would reveal that we live not in the flat world of euclid but in the strange curved worlds of gauss and Riemann.

In the space of a few short years, Einstein went from wrestling with some of the most difficult and abstract mathematical ideas to dinner dates with charlie Chaplin and it was all thanks to the pinnacle of his life’s work the general theory of relativity”

INDEX

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Euclidean geometry

Euclidean geometry ruled for 2000 years, until 1798.

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Gauss geometry

In two dimensional curved 2 dimensional spaces, other types of geometry could exist. E.g. the angles of a triangle in curved space can adds up to more than 180 degrees. Read more in https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

In 1854, Gauss selected the topic for Bernhard Riemann‘s inaugural lecture “Über die Hypothesen, welche der Geometrie zu Grunde liegen” (About the hypotheses that underlie Geometry).”

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Riemann’s geometry

Riemann generalized 1884, Gauss geometry of curved spaces to be valid in Space in any dimension, e.g.4 dimensions.

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Einstein

Einstein tied Riemanns math to the real world with his general theory of relativity. We live in a curved space of Gauss and Riemann.

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