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Euclidean Space

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Under Construction.

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 1k_{xy}  = 0
k_{xx} = 1k_{xz}  = 0
k_{yy} = 1k_{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 ; “]][ Pythagoras][[image /icons/Xlink.png link=”” ]][[/span]].

Ajat basket, Penan people. Borneo 20th century, 15 (cm) diameter by 30 (cm) height. From the Teo Family collection, Kuching. Photograph by D Dunlop.