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basic_tools:pythagorean_theorem [2018/03/27 16:27] jakobadmin [Why is it interesting?] |
basic_tools:pythagorean_theorem [2018/05/06 14:01] ida [Abstract] |
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<tabbox Abstract> | <tabbox Abstract> | ||
- | <note tip> | + | <blockquote> |
- | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
- | </note> | + | |
+ | Pythagoras’s Theorem, | ||
+ | $$(ds)^2 = (dx)^2 + (dy)^2 $$ | ||
+ | |||
+ | Riemann realized some fundamental things: | ||
+ | |||
+ | - Pythagoras’s Theorem not only told of the properties of right-angled triangles, but the function $(ds)$ (the positive square root of $(ds)^2$), was the ‘distance’ between two points; | ||
+ | - (ii) more than that, it was the shortest ‘distance’ between two points; | ||
+ | - (iii) most amazing of all, Pythagoras’s $(ds)^2$ was but one possibility - more generally, it could be written as: | ||
+ | $$(ds)2 = g_{xx}(dx)(dx) + +g_{xy}(dx)(dy) + g_{yx}(dy)(dx) + g_{yy}(dy)(dy) ,$$ | ||
+ | where the $g_{ij}$ are coefficients to be determined. In Pythagoras’s Theorem, it just so happens that we have the values $g_{xx}=1$, $g_{xy}=0$, $g_{yx}=0$, and $g_{yy}=1$. | ||
+ | |||
+ | In other cases (other spaces),the $g_{ij}$ need not have these particular values, and need not be constants - they could even be functions of x and y. This discussion has, so far, been in a space of only two dimensions, but Riemann generalized his ‘distance function’ still further to n dimensions. | ||
+ | |||
+ | <cite>The Lazy Universe by Coopersmith</cite> | ||
+ | </blockquote> | ||
<tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
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The Pythagorean theorem enables us to draw accurate maps. This is achieved by covering the area that should be mapped with a virtual network of triangles. These triangles, via the Pythagorean theorem, allow us to measure distances and angles between stretches of land. This method is known as triangulation. | The Pythagorean theorem enables us to draw accurate maps. This is achieved by covering the area that should be mapped with a virtual network of triangles. These triangles, via the Pythagorean theorem, allow us to measure distances and angles between stretches of land. This method is known as triangulation. | ||
+ | |||
+ | It is still used today, for example, to deduce the locations from given satellite data. | ||
</tabbox> | </tabbox> | ||