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 [[]] is **Euclidean** if almost all of the particles in [[]] 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 [[]] 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 [[]] 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 [[]] has a [[[norm |norm]]] given by

The Euclidean Metric | |
---|---|

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

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

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]].