User:Roman Czyborra/Hurwitz Life

\documentclass[a4paper,amsmath]{amsart} \begin{document}\author{Roman Czyborra} \title{Maxwell Integral Quaternion Quantized Spacetime Evolution} \keywords{Hurwitz quaternions, grand unified theory, digital physics} \begin{abstract} Picking up Peter Jack's arxiv:math-ph/0307038 left-right-balanced quaternionic reformulation of Maxwellian physics as a $$\phi\in\{\phi\in (\mathbb  R^4\rightarrow\mathbb R^4)\mid \{\{\phi,\nabla\},\nabla\}\} + [[\phi,\nabla],\nabla]=[0,0,0,0]\}$$         Roman Czyborra presents          how this leads to a          countable and computable          deterministic evolution if          the spacetime location          fabric is latticed as          $\frac12\mathbb Z^4$. \end{abstract}\maketitle

Recall that the quaternions $\vec p\in\mathbb Q^4$ and $\vec q\in\mathbb Q^4$ are left-right-multiplied as \begin{equation*} \vec p\times \vec q=\begin{bmatrix}p_0\\p_1\\p_2\\p_3\end{bmatrix} \times\begin{bmatrix}q_0\\q_1\\q_2\\q_3\end{bmatrix} :=\begin{bmatrix} +p_0&-p_1&-p_2&-p_3\\ -p_1&+p_0&-{p_3}&+{p_2}\\ -p_2&+{p_3}&+p_0&-{p_1}\\ -p_3&-{p_2}&+{p_1}&+p_0\\ \end{bmatrix}\times\begin{bmatrix}q_0\\q_1\\q_2\\q_3\end{bmatrix} \end{equation*}

which can be split up into a symmetrically commuting product component \begin{equation*} \{\vec p,\vec q\}:=\frac{[\vec p\times \vec q]+[\vec q\times \vec p]}2 =\begin{bmatrix}+p_0&-p_1&-p_2&-p_3\\ -p_1&+p_0&0&0\\ -p_2&0&+p_0&0\\ -p_3&0&0&+p_0\\ \end{bmatrix}\times\begin{bmatrix}q_0\\q_1\\q_2\\q_3\end{bmatrix} \end{equation*}

and an order-sensitive antisymmetric anticommuting product component \begin{equation*} [\vec p,\vec q]:=\frac{[\vec p\times \vec q]-[\vec q\times \vec p]}2 =\begin{bmatrix} 0&0&0&0\\ 0&0&-{p_3}&+{p_2}\\ 0&+{p_3}&0&-{p_1}\\ 0&-{p_2}&+{p_1}&0\\ \end{bmatrix}\times\begin{bmatrix}q_0\\q_1\\q_2\\q_3\end{bmatrix} \end{equation*}

whose G\^{a}teaux nabla operator $\vec\nabla\in\mathbb R^4\to\mathbb R^4]\to[\mathbb R^4\to\mathbb R^4$ introduced as \begin{equation*} \left[ \vec\nabla \times\vec q   \right] \left[\vec v\right] := \lim_{d\to\pm0} \sum_{i\in\{0,1,2,3\}} \left[ d\vec e_i \right]^{-1} \times\left[ \vec q\left[ \vec v+d\vec e_i \right] -\vec q\left[ \vec v     \right] \right] \end{equation*} \begin{equation*} \left[\vec q\times\vec\nabla\right]\left[\vec v\right]:= \lim_{d\to\pm0} \sum_{i\in\{0,1,2,3\}} \left[ \vec q\left[ \vec v+d\vec e_i \right] -\vec q\left[ \vec v   \right] \right]\times\left[d\vec e_i\right]^{-1} \end{equation*}

shall now under the possibly oversimplifying assumption that an underlying field $\frac12\mathbb Z^4$ exposes no smaller positive distance than $d\to\min\frac12\mathbb Z^+=\frac12$ be redefined as \begin{equation*} [\vec\nabla\times\vec q][\vec v]:= \sum_{i\in\{0,1,2,3\}} [\vec e_i]^{-1}\times\left[ \vec q\left[\vec v+\frac{\vec e_i}2\right]- \vec q\left[\vec v-\frac{\vec e_i}2\right]\right] \end{equation*} \begin{equation*} [\vec q\times\vec\nabla][\vec v]:= \sum_{i\in\{0,1,2,3\}} \left[\vec q\left[\vec v+\frac{\vec e_i}2\right] -\vec q\left[\vec v-\frac{\vec e_i}2\right]\right]\times[\vec e_i]^{-1} \end{equation*}

such that, using the deltas \begin{equation*} \vec q\vec\Delta_0\begin{bmatrix}w\\x\\y\\z\end{bmatrix} :=\vec q\begin{bmatrix}w+\frac12\\x+0\\y+0\\z+0\end{bmatrix} -\vec q\begin{bmatrix}w-\frac12\\x-0\\y-0\\z-0\end{bmatrix} =\begin{bmatrix} q_0[w+\frac12,x,y,z]-q_0[w-\frac12,x,y,z]\\ q_1[w+\frac12,x,y,z]-q_1[w-\frac12,x,y,z]\\ q_2[w+\frac12,x,y,z]-q_2[w-\frac12,x,y,z]\\ q_3[w+\frac12,x,y,z]-q_3[w-\frac12,x,y,z]\end{bmatrix} \end{equation*}

and \begin{equation*} \vec q\vec\Delta_1\begin{bmatrix}w\\x\\y\\z\end{bmatrix} :=\vec q\begin{bmatrix}w+0\\x+\frac12\\y+0\\z+0\end{bmatrix} -\vec q\begin{bmatrix}w-0\\x-\frac12\\y-0\\z-0\end{bmatrix} =\begin{bmatrix} q_0[w,x+\frac12,y,z]-q_0[w,x-\frac12,y,z]\\ q_1[w,x+\frac12,y,z]-q_1[w,x-\frac12,y,z]\\ q_2[w,x+\frac12,y,z]-q_2[w,x-\frac12,y,z]\\ q_3[w,x+\frac12,y,z]-q_3[w,x-\frac12,y,z] \end{bmatrix} \end{equation*}

and \begin{equation*} \vec q\vec\Delta_2 \begin{bmatrix}w\\x\\y\\z\end{bmatrix} :=\vec q   \begin{bmatrix}w+0\\x+0\\y+\frac12\\z+0 \end{bmatrix} -\vec q   \begin{bmatrix}w-0\\x-0\\y-\frac12\\z-0 \end{bmatrix} =\begin{bmatrix} q_0[w,x,y+\frac12,z]-q_0[w,x,y-\frac12,z]\\ q_1[w,x,y+\frac12,z]-q_1[w,x,y-\frac12,z]\\ q_2[w,x,y+\frac12,z]-q_2[w,x,y-\frac12,z]\\ q_3[w,x,y+\frac12,z]-q_3[w,x,y-\frac12,z] \end{bmatrix} \end{equation*}

and \begin{equation*} \vec q\vec\Delta_3 \begin{bmatrix}w\\x\\y\\z\end{bmatrix} :=\vec q   \begin{bmatrix}w+0\\x+0\\y+0\\z+\frac12 \end{bmatrix} -\vec q   \begin{bmatrix}w-0\\x-0\\y-0\\z-\frac12 \end{bmatrix} =\begin{bmatrix} q_0[w,x,y,z+\frac12]-q_0[w,x,y,z-\frac12]\\ q_1[w,x,y,z+\frac12]-q_1[w,x,y,z-\frac12]\\ q_2[w,x,y,z+\frac12]-q_2[w,x,y,z-\frac12]\\ q_3[w,x,y,z+\frac12]-q_3[w,x,y,z-\frac12] \end{bmatrix} \end{equation*}

in their directional contributions \begin{equation*} [1,0,0,0]^{-1}\times[d_0,d_1,d_2,d_3] =[1,0,0,0]\times[d_0,d_1,d_2,d_3] =[d_0,d_1,d_2,d_3] \end{equation*} \begin{equation*} [d_0,d_1,d_2,d_3]\times[1,0,0,0]^{-1} =[d_0,d_1,d_2,d_3]\times[1,0,0,0] =[d_0,d_1,d_2,d_3] \end{equation*} \begin{equation*} \{ [1,0,0,0],[d_0,d_1,d_2,d_3] \} = [d_0,d_1,d_2,d_3] \end{equation*} \begin{equation*} [ [1,0,0,0],[d_0,d_1,d_2,d_3] ] = [0, 0, 0, 0] \end{equation*}

and \begin{equation*} [0,1,0,0]^{-1}\times[d_0,d_1,d_2,d_3] =[0,-1,0,0]\times[d_0,d_1,d_2,d_3] =[d_1,-d_0,d_3,-d_2] \end{equation*} \begin{equation*} [d_0,d_1,d_2,d_3]\times[0,1,0,0]^{-1} =[d_0,d_1,d_2,d_3]\times[0,-1,0,0] =[d_1,-d_0,-d_3,d_2] \end{equation*} \begin{equation*} \{ [0,-1,0,0],[d_0,d_1,d_2,d_3] \} = [d_1,-d_0,0,0] \end{equation*} \begin{equation*} \left[ [0,-1,0,0], [d_0,d_1,d_2,d_3] \right] = [0, 0, d_3, -d_2] \end{equation*}

and \begin{equation*} [0,0,1,0]^{-1}\times[d_0,d_1,d_2,d_3] =[0,0,-1,0]\times[d_0,d_1,d_2,d_3] =[d_2,-d_3,-d_0,d_1] \end{equation*} \begin{equation*} [d_0,d_1,d_2,d_3]\times[0,0,1,0]^{-1} =[d_0,d_1,d_2,d_3]\times[0,0,-1,0] =[d_2,d_3,-d_0,-d_1] \end{equation*} \begin{equation*} \{ [0,0,-1,0], [d_0,d_1,d_2,d_3] \} =[d_2,0,-d_0,0] \end{equation*} \begin{equation*} \left[ [0,0,-1,0],[d_0,d_1,d_2,d_3] \right] =[0,-d_3,0,d_1] \end{equation*}

and \begin{equation*} [0,0,0,1]^{-1}\times[d_0,d_1,d_2,d_3] =[0,0,0,-1]\times[d_0,d_1,d_2,d_3] =[d_3,d_2,-d_1,-d_0] \end{equation*} \begin{equation*} [d_0,d_1,d_2,d_3]\times[0,0,0,1]^{-1} =[d_0,d_1,d_2,d_3]\times[0,0,0,-1] =[d_3,-d_2,d_1,-d_0] \end{equation*} \begin{equation*} \{ [0,0,0,-1],[d_0,d_1,d_2,d_3] \} = [d_3,0,0,-d_0] \end{equation*} \begin{equation*} \left[ [0,0,0,-1], [d_0,d_1,d_2,d_3] \right] = [0,d_2,-d_1,0] \end{equation*}

we get \begin{equation*} \{\{\vec p,\vec\nabla\},\vec\nabla\} \begin{bmatrix}w\\x\\y\\z \end{bmatrix} =\begin{bmatrix} \sum\left\{\begin{matrix} p_0[w+1,x,y,z]+4p_0[w,x,y,z]+p_0[w-1,x,y,z],\\ -p_0[w,x,y,z-1]-p_0[w,x,y-1,z]-p_0[w,x-1,y,z],\\ -p_0[w,x,y,z+1]-p_0[w,x,y+1,z]-p_0[w,x+1,y,z],\\ 2p_1[w+\frac12,x+\frac12,y,z]-2p_1[w+\frac12,x-\frac12,y,z],\\ 2p_1[w-\frac12,x-\frac12,y,z]-2p_1[w-\frac12,x+\frac12,y,z],\\ 2p_2[w+\frac12,x,y+\frac12,z]-2p_2[w+\frac12,x,y-\frac12,z],\\ 2p_2[w-\frac12,x,y-\frac12,z]-2p_2[w-\frac12,x,y+\frac12,z],\\ 2p_3[w+\frac12,x,y,z+\frac12]-2p_3[w+\frac12,x,y,z-\frac12],\\ 2p_3[w-\frac12,x,y,z-\frac12]-2p_3[w-\frac12,x,y,z+\frac12],\\ \end{matrix}\right\}\\ \sum\left\{\begin{matrix} 2p_0[w+\frac12,x-\frac12,y,z]-2p_0[w+\frac12,x+\frac12,y,z],\\ 2p_0[w-\frac12,x+\frac12,y,z]-2p_0[w-\frac12,x-\frac12,y,z],\\ p_1[w-1,x,y,z]-p_1[w,x+1,y,z],\\ p_1[w+1,x,y,z]-p_1[w,x-1,y,z],\\ p_2[w,x+\frac12,y-\frac12,z]-p_2[w,x+\frac12,y+\frac12,z],\\ p_2[w,x-\frac12,y+\frac12,z]-p_2[w,x-\frac12,y-\frac12,z],\\ p_3[w,x+\frac12,y,z-\frac12]-p_3[w,x+\frac12,y,z+\frac12],\\ p_3[w,x-\frac12,y,z+\frac12]-p_3[w,x-\frac12,y,z-\frac12],\\ \end{matrix}\right\}\\ \sum\left\{\begin{matrix} 2p_0[w+\frac12,x,y-\frac12,z]-2p_0[w+\frac12,x,y+\frac12,z],\\ 2p_0[w-\frac12,x,y+\frac12,z]-2p_0[w-\frac12,x,y-\frac12,z],\\ p_1[w,x+\frac12,y-\frac12,z]-p_1[w,x+\frac12,y+\frac12,z],\\ p_1[w,x-\frac12,y+\frac12,z]-p_1[w,x-\frac12,y-\frac12,z],\\ p_2[w+1,x,y,z]-p_2[w,x,y+1,z],\\ p_2[w-1,x,y,z]-p_2[w,x,y-1,z],\\ p_3[w,x,y+\frac12,z-\frac12]-p_3[w,x,y+\frac12,z+\frac12],\\ p_3[w,x,y-\frac12,z+\frac12]-p_3[w,x,y-\frac12,z-\frac12],\\ \end{matrix}\right\}\\ \sum\left\{\begin{matrix} 2p_0[w+\frac12,x,y,z-\frac12]-2p_0[w+\frac12,x,y,z+\frac12],\\ 2p_0[w-\frac12,x,y,z+\frac12]-2p_0[w-\frac12,x,y,z-\frac12],\\ p_1[w,x+\frac12,y,z-\frac12]-p_1[w,x+\frac12,y,z+\frac12],\\ p_1[w,x-\frac12,y,z+\frac12]-p_1[w,x-\frac12,y,z-\frac12],\\ p_2[w,x,y+\frac12,z-\frac12]-p_2[w,x,y+\frac12,z+\frac12],\\ p_2[w,x,y-\frac12,z+\frac12]-p_2[w,x,y-\frac12,z-\frac12],\\ p_3[w+1,x,y,z]-p_3[w,x,y,z+1],\\ p_3[w-1,x,y,z]-p_3[w,x,y,z-1] \end{matrix}\right\} \end{bmatrix} \end{equation*}

as a shifted self-application of \begin{equation*} \{\vec q,\vec\nabla\} \begin{bmatrix}w\\x\\y\\z \end{bmatrix} =\begin{bmatrix} \sum\left\{\begin{matrix} q_0[w+\frac12,x,y,z]-q_0[w-\frac12,x,y,z],\\ q_1[w,x+\frac12,y,z]-q_1[w,x-\frac12,y,z],\\ q_2[w,x,y+\frac12,z]-q_2[w,x,y-\frac12,z],\\ q_3[w,x,y,z+\frac12]-q_3[w,x,y,z-\frac12] \end{matrix}\right\} \\ \sum\left\{\begin{matrix} q_0[w,x-\frac12,y,z]-q_0[w,x+\frac12,y,z],\\ q_1[w+\frac12,x,y,z]-q_1[w-\frac12,x,y,z] \end{matrix}\right\} \\ \sum\left\{\begin{matrix} q_0[w,x,y-\frac12,z]-q_0[w,x,y+\frac12,z],\\ q_2[w+\frac12,x,y,z]-q_2[w-\frac12,x,y,z] \end{matrix}\right\} \\ \sum\left\{\begin{matrix} q_0[w,x,y,z-\frac12]-q_0[w,x,y,z+\frac12],\\ q_3[w+\frac12,x,y,z]-q_3[w-\frac12,x,y,z] \end{matrix}\right\} \end{bmatrix} \end{equation*}

and likewise the antisymmetric \begin{equation*} [[\vec p,\vec\nabla],\vec\nabla] \begin{bmatrix}w\\x\\y\\z\end{bmatrix}    =    \begin{bmatrix}0\\      \sum\left\{\begin{matrix}      4p_1[w,x,y,z],\\      -p_1[w,x,y+1,z]-p_1[w,x,y,z+1],\\            -p_1[w,x,y-1,z]-p_1[w,x,y,z-1],\\      p_2[w,x+\frac12,y+\frac12,z]-p_2[w,x-\frac12,y+\frac12,z],\\      p_2[w,x-\frac12,y-\frac12,z]-p_2[w,x+\frac12,y-\frac12,z],\\      p_3[w,x+\frac12,y,z+\frac12]-p_3[w,x+\frac12,y,z-\frac12],\\      p_3[w,x-\frac12,y,z-\frac12]-p_3[w,x-\frac12,y,z+\frac12]      \end{matrix}\right\}      \\      \sum\left\{\begin{matrix}      p_1[w,x+\frac12,y+\frac12,z]-p_1[w,x+\frac12,y-\frac12,z],\\      p_1[w,x-\frac12,y-\frac12,z]-p_1[w,x-\frac12,y+\frac12,z],\\      4p_2[w,x,y,z],\\      -p_2[w,x+1,y,z]-p_2[w,x,y,z+1],      -p_2[w,x-1,y,z]-p_2[w,x,y,z-1],\\      p_3[w,x,y+\frac12,z-\frac12]-p_3[w,x,y+\frac12,z-\frac12],\\      p_3[w,x,y-\frac12,z-\frac12]-p_3[w,x,y-\frac12,z-\frac12]      \end{matrix}\right\}\\      \sum\left\{\begin{matrix}      p_1[w,x+\frac12,y,z+\frac12]-p_1[w,x+\frac12,y,z-\frac12],\\      p_1[w,x-\frac12,y,z-\frac12]-p_1[w,x-\frac12,y,z+\frac12],\\      p_2[w,x,y+\frac12,z+\frac12]-p_2[w,x,y+\frac12,z-\frac12],\\      p_2[w,x,y-\frac12,z-\frac12]-p_2[w,x,y-\frac12,z+\frac12],\\      4p_3[w,x,y,z],\\      -p_3[w,x+1,y,z]-p_3[w,x,y+1,z],\\      -p_3[w,x,y-1,z]-p_3[w,x-1,y,z],\\      \end{matrix}\right\}    \end{bmatrix} \end{equation*}

as a shifted self-application of \begin{equation*} [\vec q,\vec\nabla] \begin{bmatrix}w\\x\\y\\z\end{bmatrix} =   \begin{bmatrix}0\\ \sum\left\{\begin{matrix} q_2[w,x,y,z+\frac12]-q_2[w,x,y,z-\frac12],\\ q_3[w,x,y-\frac12,z]-q_3[w,x,y+\frac12,z] \end{matrix}\right\} \\     \sum\left\{\begin{matrix} q_1[w,x,y,z-\frac12]-q_1[w,x,y,z+\frac12],\\ q_3[w,x+\frac12,y,z]-q_3[w,x-\frac12,y,z] \end{matrix}\right\}\\ \sum\left\{\begin{matrix} q_1[w,x,y+\frac12,z]-q_1[w,x,y-\frac12,z],\\ q_2[w,x-\frac12,y,z]-q_2[w,x+\frac12,y,z] \end{matrix}\right\} \end{bmatrix} \end{equation*}

yielding, when demanding the Maxwell balance of matter-like local thermoelectric deceleration and local spatial curvature to even out \begin{equation*} \{\{\vec\phi,\vec\nabla\},\vec\nabla\} +[[\vec\phi,\vec\nabla],\vec\nabla]=\vec0 \end{equation*} a deterministic time evolution \begin{equation*} \vec\phi  \begin{bmatrix}w\\x\\y\\z  \end{bmatrix}  =  \begin{bmatrix}    \sum\left\{\begin{matrix}    \phi_0[w-1,x+1,y,z]+\phi_0[w-1,x,y+1,z]+\phi_0[w-1,x,y,z+1],\\    \phi_0[w-1,x-1,y,z]+\phi_0[w-1,x,y-1,z]+\phi_0[w-1,x,y,z-1],\\    -4\phi_0[w-1,x,y,z]-\phi_0[w-2,x,y,z],\\    2\phi_1[w-\frac12,x-\frac12,y,z]-2\phi_1[w-\frac12,x+\frac12,y,z],\\    2\phi_1[w-\frac32,x+\frac12,y,z]-2\phi_1[w-\frac32,x-\frac12,y,z],\\    2\phi_2[w-\frac12,x,y-\frac12,z]-2\phi_2[w-\frac12,x,y+\frac12,z],\\    2\phi_2[w-\frac32,x,y+\frac12,z]-2\phi_2[w-\frac32,x,y-\frac12,z],\\    2\phi_3[w-\frac12,x,y,z-\frac12]-2\phi_3[w-\frac12,x,y,z+\frac12],\\    2\phi_3[w-\frac32,x,y,z+\frac12]-2\phi_3[w-\frac32,x,y,z-\frac12],\\    \end{matrix}\right\}\\\sum\left\{\begin{matrix}    2\phi_0[w-\frac12,x+\frac12,y,z]-2\phi_0[w-\frac12,x-\frac12,y,z],\\    2\phi_0[w-\frac32,x-\frac12,y,z]-2\phi_0[w-\frac32,x+\frac12,y,z],\\    \phi_1[w-1,x+1,y,z]+\phi_1[w-1,x,y+1,z]+\phi_1[w-1,x,y,z+1],\\    \phi_1[w-1,x-1,y,z]+\phi_1[w-1,x,y-1,z]+\phi_1[w-1,x,y,z-1],\\    -4\phi_1[w-1,x,y,z]-\phi_1[w-2,x,y,z]    \end{matrix}\right\}\\\sum\left\{\begin{matrix}    2\phi_0[w-\frac12,x,y+\frac12,z]-2\phi_0[w-\frac12,x,y-\frac12,z],\\    2\phi_0[w-\frac32,x,y-\frac12,z]-2\phi_0[w-\frac32,x,y+\frac12,z],\\    \phi_2[w-1,x+1,y,z]+\phi_2[w-1,x,y+1,z]+\phi_2[w-1,x,y,z+1],\\    \phi_2[w-1,x-1,y,z]+\phi_2[w-1,x,y-1,z]+\phi_2[w-1,x,y,z-1],\\    -4\phi_2[w-1,x,y,z]-\phi_2[w-2,x,y,z]    \end{matrix}\right\}\\\sum\left\{\begin{matrix}    2\phi_0[w-\frac12,x,y,z+\frac12]-2\phi_0[w-\frac12,x,y,z-\frac12],\\    2\phi_0[w-\frac32,x,y,z-\frac12]-2\phi_0[w-\frac32,x,y,z+\frac12],\\    \phi_3[w-1,x+1,y,z]+\phi_3[w-1,x,y+1,z]+\phi_3[w-1,x,y,z+1],\\    \phi_3[w-1,x-1,y,z]+\phi_3[w-1,x,y-1,z]+\phi_3[w-1,x,y,z-1],\\    -4\phi_3[w-1,x,y,z]-\phi_3[w-2,x,y,z],    \end{matrix}\right\}  \end{bmatrix} \end{equation*}

that can easily be simulated on a computer as in: \begin{footnotesize} \begin{verbatim}

module Czyborra.Czyborspace20160901 (q) where import Data.Function.Memoize (memoFix) q=memoFix$ \q(v,w,x,y,z)->if(w>0)then case v of 0->q(0,w-2,x+2,y,z)+q(0,w-2,x,y+2,z)+q(0,w-2,x,y,z+2)+ q(0,w-2,x-2,y,z)+q(0,w-2,x,y-2,z)+q(0,w-2,x,y,z-2)+ (0-4)*q(0,w-2,x,y,z)-q(0,w-4,x,y,z)+ 2*q(1,w-1,x-1,y,z)-2*q(1,w-1,x+1,y,z)+ 2*q(1,w-3,x+1,y,z)-2*q(1,w-3,x-1,y,z)+ 2*q(2,w-1,x,y-1,z)-2*q(2,w-1,x,y+1,z)+ 2*q(2,w-3,x,y+1,z)-2*q(2,w-3,x,y-1,z)+ 2*q(3,w-1,x,y,z-1)-2*q(3,w-1,x,y,z+1)+ 2*q(3,w-3,x,y,z+1)-2*q(3,w-3,x,y,z-1) 1->2*q(0,w-1,x+1,y,z)-2*q(0,w-1,x-1,y,z)+ 2*q(0,w-3,x-1,y,z)-2*q(0,w-3,x+1,y,z)+ q(1,w-2,x+2,y,z)+q(1,w-2,x,y+2,z)+q(1,w-2,x,y,z+2)+ q(1,w-2,x-2,y,z)+q(1,w-2,x,y-2,z)+q(1,w-2,x,y,z-2)+ (0-4)*q(1,w-2,x,y,z)-q(1,w-4,x,y,z) 2->2*q(0,w-1,x,y+1,z)-2*q(0,w-1,x,y-1,z)+ 2*q(0,w-3,x,y-1,z)-2*q(0,w-3,x,y+1,z)+ q(2,w-2,x+2,y,z)+q(2,w-2,x,y+2,z)+q(2,w-2,x,y,z+2)+ q(2,w-2,x-2,y,z)+q(2,w-2,x,y-2,z)+q(2,w-2,x,y,z-2)+ (0-4)*q(2,w-2,x,y,z)-q(2,w-4,x,y,z) 3->2*q(0,w-1,x,y,z+1)-2*q(0,w-1,x,y,z-1)+ 2*q(0,w-3,x,y,z-1)-2*q(0,w-3,x,y,z+1)+ q(3,w-2,x+2,y,z)+q(3,w-2,x,y+2,z)+q(3,w-2,x,y,z+2)+ q(3,w-2,x-2,y,z)+q(3,w-2,x,y-2,z)+q(3,w-2,x,y,z-2)+ (0-4)*q(3,w-2,x,y,z)-q(3,w-4,x,y,z) else if (v,w,x,y,z)==(0,0,0,0,0) then 1 else 0 \end{verbatim} \end{footnotesize} \end{document}