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Quark Space

WikiMechanics begins with the premise that we can understand ordinary space by describing sensation. And, we also need some room to build quark models. So let’s start by defining an algebraic vector space made from the radius vectors  \overline{\rho}, for some finite collection of particles  \mathsf{P}^{1},  \;  \mathsf{P}^{2},  \;  \mathsf{P}^{3}  \ldots   \mathsf{P}^{N}. This mathematical construction is generically written as

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

Radius vectors are defined by describing sense perceptions, so ultimately  \mathbb{Q} is defined by sensation too. We say  \mathbb{Q} is three-dimensional because the three components of a radius vector represent three distinct classes of sensation which may change independently of each other. These three components are not the Cartesian coordinates that we usually use in geometry because instead of being associated with lengths, they are defined by counting quarks. Moreover, this space is explicitly constructed to keep track of quarks, so  \mathbb{Q} is called a quark space.

The following basis vectors are used to make general descriptions; the magnetic axis is defined by \hat{m} \equiv (1, 0, 0), the electric axis from \hat{e} \equiv (0, 1, 0) and the polar axis by \hat{z} \equiv (0, 0, 1). Then, any radius vector in  \mathbb{Q} can be expressed in terms of its components as \overline{\rho} = \rho_{m} \hat{m} + \rho_{e} \hat{e} + \rho_{ z} \hat{z}. The norm of a radius vector in quark space is given by

    \begin{equation*}   \left\| \,   \overline{\rho} \,  \right\|  =  \left( \begin{split} &    \; k_{mm} \rho_{m}^{2} +  k_{ee} \rho_{e}^{2} +  k_{zz} \rho_{z}^{2}  \\  &  + 2 k_{em}  \rho_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z}  \\ & \hspace{30px} + 2 k_{ez}\rho_{e}  \rho_{z} \;  \end{split} \right)^{\frac{1}{2}} \end{equation*}

Quark Metric
centripetal component k_{zz}1
electric component k_{ee}-0.0152286648
magnetic component k_{mm}+0.7453740340
electromagnetic component k_{em}-0.9292374609
electroweak component k_{ez}+1.5428187522
magnetoweak component k_{mz}-1.2742065050

This function compresses all three components of a radius vector into a single quantity that depends on the six numbers shown in the accompanying table. These constants are known collectively as the quark metric.

Particles and anti-particles have symmetrically opposed radius vectors, \overline{\rho} \left( \mathsf{P} \right) = - \overline{\rho} ( \mathsf{\overline{P}} ). So their norms are the same as each other

\left\| \, \overline{\rho}  \left( \mathsf{P} \right) \,  \vphantom{\left(  \mathsf{\overline{P}} \right)}  \right\|  = \left\| \, \overline{\rho} \left(  \mathsf{\overline{P}} \right)  \, \right\|

Quark space is a poor approximation to ordinary space, it is coarse and grainy because quark coefficients are always integers. And  \mathbb{Q} is squashed in a funny way because metric components are not Euclidean. But these messy details are handled mathematically, and in the following articles we make idealized quark models using conventional graphics. Next we consider the work that is required to build a quark model.

Quark space presents a structured array of sensation, somewhat like this tampan from Sumatra presents nine dragons.
Nine Dragon Tampan, Paminggir people. Sumatra circa 1900, 41 x 43 cm. Photograph by D Dunlop.