Euclidean Space – EthnoPhysics

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Please notice that //empty// space has not been defined, the foregoing ideas are all based on specific particles and positions.

WikiMechanics began with the [[[premise |premise]]] that we can understand ordinary space by describing sensation. Our first attempt at this discussed [[span style=” display:inline-block ; “]][[[aggregates |compound quarks]]][[/span]] situated in a [[span style=” display:inline-block ; “]][[[quark-space |quark space]]][[/span]]. But that construction was coarse and distorted compared to ordinary classrooms and laboratories. So now let us consider a more restrictive arrangement where all the particles in a space have special attributes. Specifically we examine aggregations of [[[atoms |atoms]]] that have [[span style=” display:inline-block ; “]][[[metrics |shapes]]][[/span]] which can be expressed in [[span style=” display:inline-block ; “]][[[cartesian-coordinates |Cartesian]]][[/span]] coordinates. Definition: we say that a space [[ $\mathbb{S}$ ]] is Euclidean if almost all of the particles in [[ $\mathbb{S}$ ]] satisfy the following requirements. First, they must be fully three-dimensional like [[[shape |cylinders]]], [[[shape |rods]]] or [[span style=” display:inline-block ; “]][[[shape |plates]]].[[/span]] And there must be some variation among these forms. But [[ $\mathbb{S}$ ]] must also be [[[spatial-homogeneity |well-stirred]]] so that shapes are not aligned or consistently oriented. Similarly, [[span style=” display:inline-block ; “]][[[phase-angle |phase angles]]][[/span]] must exhibit some variation between atoms, they cannot be [[[space-time-events |coherently]]] related to each other. So for example [[ $\mathbb{S}$ ]] cannot be resonating. Under these conditions, [[span style=” display:inline-block ; “]][[[euclidean-metric |more analysis]]][[/span]] yields a [[[metrics |metric]]] called the Euclidean metric as shown in the accompanying table. In a Euclidean space, a [[[position |position]]] vector like [[ $\bar{r} = ( x, y, z )$ ]] has a [[[norm |norm]]] given by

The Euclidean Metric
$k_{zz} \equiv 1$	$k_{xy} = 0$
$k_{xx} = 1$	$k_{xz} = 0$
$k_{yy} = 1$	$k_{yz} = 0$

$r \equiv \left\| \; \bar{r} \; \right\| \equiv \sqrt{ \; k_{xx} x^{2} + k_{yy} y^{2} + k_{zz} z^{2} + 2k_{xy} x y + 2 k_{xz} x z + 2k_{yz} y z \; \; } \\ = \sqrt{ \, x^{2} + y^{2} + z^{2} \; }$

And a norm of the [[[position |separation]]] vector

$\Delta \bar{r} = ( \Delta x, \Delta y, \Delta z )$

gives the [[[position |distance]]] between events in a Euclidean space as

$\Delta r \equiv \left\| \, \Delta \bar{r} \vphantom{\sum^{2}} \, \right\| = \sqrt{ \, \Delta x^{2} + \Delta y^{2} + \Delta z^{2} \vphantom{\sum^{2}} \; }$

These relationships express some very old knowledge about geometry that is often attributed to [[span style=” display:inline-block ; “]][https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagoras][[image /icons/Xlink.png link=”https://en.wikipedia.org/wiki/Pythagorean_theorem” ]][[/span]].