# Hilbert Book Model Project/Slide S4

hbmp Hilbert Book Model Project S4

We separate all point-like discontinuities from the domain ${\displaystyle {\color {white}\Omega }}$ by encapsulating them in an extra boundary

Symmetry centers ${\displaystyle {\color {yellow}{\mathfrak {S}}_{n}^{x}}}$ represent spherically ordered parameter spaces in regions ${\displaystyle {\color {white}H_{n}^{x}}}$ that float on a background parameter space ${\displaystyle {\color {white}{\mathfrak {R}}}}$

The boundaries ${\displaystyle {\color {white}\partial H_{n}^{x}}}$ separate the regions ${\displaystyle {\color {white}H_{n}^{x}}}$ from the domain ${\displaystyle {\color {white}\Omega }}$

The symmetry centers ${\displaystyle {\color {yellow}{\mathfrak {S}}_{n}^{x}}}$ are encapsulated in regions ${\displaystyle {\color {white}H_{n}^{x}}}$

and the encapsulating boundary ${\displaystyle {\color {white}\partial H_{n}^{x}}}$is not part of the disconnected boundary,

which encapsulates all continuous parts of the quaternionic manifold ${\displaystyle {\color {white}\omega }}$ that exist in the quaternionic model

${\displaystyle {\color {white}\,\int \limits _{\Omega -H}d\omega =\int \limits _{\partial \Omega \cup \partial H}\omega =\int \limits _{\partial \Omega }\omega -\sum _{k=1}^{N}\int \limits _{\partial H_{n}^{x}}\omega }}$

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