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

WikiMechanics began with the premise that we can understand ordinary space by describing sensation. Our first attempt at this discussed compound quarks situated in a quark space. But that construction was coarse and distorted compared to ordinary rooms 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 that have shapes which can be expressed in Cartesian 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 cylinders, rods or plates. And there must be some variation among these forms. But $\mathbb{S}$ must also be well-stirred so that shapes are not aligned or consistently oriented. Similarly, phase angles must exhibit some variation between atoms, they cannot be coherently related to each other. So for example $\mathbb{S}$ cannot be resonating. Under these conditions, more analysis yields a metric called the Euclidean metric as shown in the accompanying table. In a Euclidean space, a position vector like $\bar{r} = ( x, y, z )$ has a norm given by

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

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

And a norm of the separation vector

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

gives the 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 PythagorasXlink.png.
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