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--- basic_tools:pythagorean_theorem [2018/03/27 16:10]
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+++ basic_tools:pythagorean_theorem [2021/07/04 20:27] (current)
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 <tabbox Intuitive>
-<note tip>
+The Pythagorean theorem states a relationship between the three sides of a right-angled triangle where $a$ and $b$ represent the sides that form the right angle and $c$ is the longest side.
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
-</note>
 <tabbox Concrete>
+$a^2+b^2=c^2$ **if and only if** the triangle is a right triangle
-<note tip>
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
-</note>
 <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. [...]
+Most remarkable of all, Riemann showed that all the geometric properties of an n-dimensional space can be completely determined by just this ‘distance’, $ds$ (the square root of the above equation. For example, if any of the coefficients, $g_{ij}$ , are not constant but are functions of the coordinates, then the corresponding space is not ‘flat’ (Euclidean) but ‘curved’. This, finally, is what ‘curved’ means: it is a measure of the departure from Euclidean space, and is determined by certain functions (Riemann’s ‘curvature functions’) of the $g_{ij}$ -coefficients. While the ‘distance’ is specific to the given space, its value is invariant as regards which coordinate representation has been adopted.
+<cite>The Lazy Universe by Coopersmith</cite>
+</blockquote>
+<tabbox Why is it interesting?>
-<tabbox Why is it interesting?>
+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.
+It is still used today, for example, to deduce the locations from given satellite data.
 </tabbox>

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