WikiMechanics begins with the premise that we can understand ordinary space by describing sensation. This is done by objectifying reference sensations as quarks and then considering spaces as collections of quarks. Different kinds of space are defined from different distributions of quark types. And empty space is not defined. So overall, our understanding of a space is based on the particles that are in the space.

Specifically, we assess the shape or*radii*of these particles. Traditionally, a radius is quantified by making a measurement of length. But a full discussion of length measurement requires some ideas that are initially quite vague. So to begin, WikiMechanics evaluates the shape of a particle by counting its quarks, then defining a radius vector $\, \overline{\rho} \,$ from quark inventories. With this quantity an algebraic vector space $\, \mathbb{S} \,$ can be defined from the radius vectors for some collection of particles $\mathbf{P}$

^{1}, $\mathbf{P}$

^{2}, $\mathbf{P}$

^{3}… $\, \mathbf{P}$

^{N}. Such a mathematical construction is generically written as

$\mathbb{S} = \left\{ \overline{\rho}^{ 1 }, \ \overline{\rho}^{2}, \ \overline{\rho}^{3} \ldots \ \overline{\rho}^{\, i} \ldots \ \overline{\rho}^{\, N} \right\}$

$\mathbb{S}$ is characterized using a statistical account of commonalities and variation in the shape of these particles. Averages and standard deviations are given by

$\begin{align} \tilde{\rho}_{\alpha} \equiv \frac{1}{ N} \sum_{i=1}^{N} \rho_{\alpha}^{i} \end{align}$ $\delta \rho_{\alpha} \equiv \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{\alpha}^{i} - \tilde{\rho}_{\alpha} \right)^{2} \; }$

where $\alpha \in \{ m, e, z \}$ notes different components of the radius vector $\overline{\rho} = \rho_{m} \hat{m} + \rho_{e} \hat{e} + \rho_{ z} \hat{z}$. Spaces are also described by correlations between these components using the coefficients

$\chi _{\alpha \beta} \equiv \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{\alpha}^{i} - \tilde{\rho}_{\alpha} \vphantom{\rho_{\beta}^{i}} \right) \left( \rho_{\beta}^{i} - \tilde{\rho}_{\beta} \right) \ }$

where $\beta \in \{ m, e, z \}$. Correlation coefficients may be combined to define some ratios, which are then used to systematically characterize $\mathbb{S}$. For the important special case where $\mathbb{S}$ is the Earth, we have the following.

## The Quark Metric

$\begin{align} k_{\alpha \beta} \equiv \frac{ \chi_{\alpha \beta} ^{ \sf{Earth}} }{ \chi_{zz} ^{ \sf{Earth}} } \end{align}$

and use them to determine the norm of a radius vector. Note that by this definition $k_{zz}$ is exactly one, and $k_{\alpha \beta} = k_{\beta \alpha }$. A set of numbers used to calculate a norm is called a *metric*, and since $k_{\alpha \beta}$ characterizes a quark distribution we call it the **quark metric**. Using this metric to calculate a norm compresses all three components of a radius vector into a single quantity $\, \rho \equiv \left\| \, \overline{\rho} \, \right\|$ that depends on attributes of the Earth. The labels given to numbers in the quark metric are arbitrary, but the names listed in the table above have been chosen for their mnemonic value. They will smoothly fit into our traditional ways of discussing forces and be easy to remember.

## The Euclidean Metric

The Euclidean Metric |

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

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

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

We also consider that $\mathbb{S}$ may be a collection of particles where membership in the set is restricted to certain shapes or other attributes. Then a statistical analysis of shape could yield a different metric. For example, in our laboratories we usually assume that space is filled with room-temperature atoms, not just any composite quark. An extended analysis of this sort of terrestrial space is detailed later but the overall result is easily summarized as the *Euclidean* metric shown in the accompanying table.

Related WikiMechanics articles.

Summary |

Noun | Definition | |

Quark Metric | $\begin{align} k_{\alpha \beta} ^{ \mathbb{Q} } \equiv \frac{ \chi_{\alpha \beta} ^{ \sf{Earth}} }{ \chi_{zz} ^{ \sf{Earth}} } \end{align}$ | 5-15 |

Noun | Definition | |

Centripetal Component of the Quark Metric | $k_{zz} \equiv 1$ | 5-16 |

Noun | Definition | |

Electric Component of Quark Metric | $k_{ee} = -0.0152286648$ | 5-17 |

Noun | Definition | |

Magnetic Component of Quark Metric | $k_{mm} = 0.7453740340$ | 5-18 |

Noun | Definition | |

Electromagnetic Component of Quark Metric | $k_{em} = -0.9292374609$ | 5-19 |

Noun | Definition | |

Magnetoweak Component of Quark Metric | $k_{mz} = -1.2742065050$ | 5-20 |

Noun | Definition | |

Electroweak Component of Quark Metric | $k_{ez} = 1.5428187522$ | 5-21 |