Metrics

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 And 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 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

Recall that touching the Earth is a reference sensation for WikiMechanics. So we presume that the Earth is implicitly part of every description. And since this celestial body is so big, the law of implies that correlation coefficients will have specific values that do not vary on geological time scales. So we can define five unique constants by

\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.

 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