This article extends an earlier, more general discussion of metrics that considered a particle's shape as described by its radius vector $\, \overline{\rho}$. Initially, the radius was established from a particle's quark content, and a vector space $\mathbb{S}$ was defined from sets of particles and their radii. $\mathbb{S}$ was characterised using statistical averages $\tilde{\rho}$, standard deviations $\delta \rho$ and correlation coefficients noted by $\chi$. These quantities have already been assessed for generic compound quarks.

Now let us consider that $\mathbb{S}$ might be filled with a more restricted set of particles that have special attributes. Specifically we examine collections of atoms where a radius vector can be expressed in Cartesian coordinates. Characterize atomic aggregates using their orbital radius $R$, a wavelength $\lambda$ and $\theta$ the phase angle. Definition: we say that a space is **Euclidean** if almost every particle in $\mathbb{S}$ satisfies the following conditions.

- Atoms and molecules must be fully three dimensional like cylinders or rods so that $R \ne 0$ and $\lambda \ne 0$. Wavelengths and radii are all presumably positive, not nil.
- Particles must exhibit some variation in their shape so that $\delta \rho \ne 0$. The standard deviation for any component of the radius is presumably positive.
- $\mathbb{S}$ must be well-stirred. We require spatial homogeneity at an atomic level so that $\delta \rho _{x} = \delta \rho_{y} = \delta \rho_{z}$.
- The particles in $\mathbb{S}$ cannot all be extremely low temperature, or all resonating like the atoms in a laser. We assume that $\delta \theta \ne 0$ so there must be some variation among atomic phase angles.
- Finally, atoms must interact incoherently so that the variation among phase angles is random. Then we can presume that large sums over odd powers of circular functions of $\theta$ add-up to zero.

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

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

^{3}… $\, \mathbf{P}$

^{N}. Let these particles be described using the rotini model where atomic shapes are like a corkscrew spiral. They approximate a geometric curve called a helicoid so that atomic radii can be mathematically written as

$\rho_{x} = R \cos{\! 2 \theta}$and$\rho_{y} = R \sin{\! 2 \theta }$and$\begin{align} \rho_{z} = \frac{ \lambda \theta}{2 \pi} \end{align}$

The radius vector is expressed in Cartesian coordinates as $\, \overline{\rho} = \rho_{x} \hat{x} + \rho_{y} \hat{y} + \rho_{ z} \hat{z}$ so a space like $\mathbb{S}$ can be generically represented as

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

The average radii of the particles in $\mathbb{S}$ are given by

$\begin{align} \tilde{\rho}_{x} = \frac{R}{ N} \sum_{i=1}^{N} \cos{ 2 \theta^{i} } \end{align}$ and $\begin{align} \tilde{\rho}_{y} = \frac{R}{ N} \sum_{i=1}^{N} \sin{ 2 \theta^{i} } \end{align}$ and $\begin{align} \tilde{\rho}_{z} = \frac{ \lambda \tilde{\theta} }{2 \pi } \end{align}$

As noted above, we presume that the atoms in $\mathbb{S}$ have phase angles that are completely incoherent so that sums over the circular functions add-up to zero. Then

$\begin{align} \tilde{\rho}_{x} \approx 0 \end{align}$ and $\begin{align} \tilde{\rho}_{y} \approx 0 \end{align}$

and the variations in particle radii are

$\delta \rho_{x} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{x}^{i} \right)^{2} \ } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \cos{2 \theta^{i} } \right)^{2} \ }$$\begin{align} \; \approx \frac{R}{2} \end{align}$

$\delta \rho_{y} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{y}^{i} \right)^{2} \ } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \sin{2 \theta^{i} } \right)^{2} \ }$$\begin{align} \; \approx \frac{R}{2} \end{align}$

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

We presume that all the particles in $\mathbb{S}$ are three-dimensional so that $R \ne 0$ and $\lambda \ne 0$. And as noted above, we also assume that phase angles vary. So $\delta \theta \ne 0$ and the standard deviations in particle radii must be positive

$\delta \rho_{x} \ne 0$and$\delta \rho_{y} \ne 0$and$\delta \rho_{z} \ne 0$

We also presume that $\mathbb{S}$ has spatial homogeneity at the atomic-level. Then any variations cannot depend on direction, and so we require that

$\delta \rho _{x} = \delta \rho_{y} = \delta \rho_{z}$

But by definition, correlation coefficients and standard deviations are related as $\; \chi _{\alpha \alpha} = \delta \rho _{\alpha}$ so that

$\chi _{xx} = \chi_{yy} = \chi _{zz} > 0$

The other correlation coefficients are given by

$\chi _{x y} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \rho_{x}^{i} \rho_{y}^{i} \ } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \cos{ 2 \theta^{i} } \right) \left( \sin{ 2 \theta^{i} } \right) \ } \approx 0$

$\chi _{x z} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \rho_{x}^{i} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right) \ } =\sqrt{\frac{R}{N} \sum_{i=1}^{N}\cos{2\theta^{i}} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right) \ } \approx 0$

$\chi _{y z} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \rho_{y}^{i} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right) \ } =\sqrt{\frac{R}{N} \sum_{i=1}^{N}\sin{\,2\theta^{i}} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right) \ } \approx 0$

The Euclidean Metric |

$k_{zz} \equiv 1$ |

$k_{xx} = 1$ |

$k_{yy} = 1$ |

$k_{xy} = 0$ |

$k_{xz} = 0$ |

$k_{yz} = 0$ |

But these coefficients must all be nil because, as noted above, we presume that large sums over odd powers of circular functions of $\theta$ add-up to zero. Then recall that the metric of $\mathbb{S}$ is given by

$\begin{align} k_{\alpha \beta} \equiv \frac{ \chi_{\alpha \beta} ^{ \mathbb{S} } }{ \chi_{zz} ^{ \mathbb{S} } } \end{align}$

where $\alpha$ and $\beta \in \{ x, y, z \}$. So overall

$k_{xx} = k_{yy}= k_{zz} = 1$ and $k_{xy} = k_{xz} = k_{yz} = 0$

and we say that the metric of $\mathbb{S}$ is Euclidean.