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

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 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 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 |

$k_{zz} \equiv 1$ | $k_{xz} = 0$ |

$k_{xx} = 1$ | $k_{xy} = 0$ |

$k_{yy} = 1$ | $k_{yz} = 0$ |

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

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 Pythagoras.Related WikiMechanics articles.