benjamin-rood · March 7, 2017 13:05
diff --git a/abm_env_def b/abm_env_def
 #Definition of the ABM Environment

 Consider the set of all real numbers within $[\,-1,  \,+1\,]$.$\ $ Let $\,F\ $ be the cartesian product of such a set with itself,  *s.t.* individual elements in $\,F\,$ are two-dimensional points, or a cartesian *pair* within a subset of $R^2$.
 $$\therefore \ F\ = \  \{\ (x,y) \in\, \mathbb{R}^{2} \, \lvert\  (\,-1 \leq x \leq 1 \,)\ \wedge\,(\,-1 \leq y \leq 1 \,)\  \}$$

 Let $\ E\ $ be the symbolic representation of our modelling environment, such that $E$ encompasses the following: 

 * A vector space $\ V\ $ over field $\ F$.
 * Variable extensions on the field $F$ which act as "environmental" describers *(to define the properties of the medium the agents exist in)* and modifiers *(to be used to model changes to the world state)*.[^env-mod]

 $ $
 *Note: For the purposes of the initial research goals, it is assumed that all agents exist in a perfect, static environment, with no variations. Improvement or regression of agent abilities or characteristics are entirely independent of environmental factors.* 

 The range of $F$ within $E$ will be sufficient to act as a looping "petrie dish" to observe the behaviour of the two populations of agents, with the origin $\,(0,0)\,$ as the absolute centre of the projection of the environment.

 Rather than agents 'bouncing' or 'avoiding' the boundaries of $E$, values $(x,y) \in  E\,$ **wrap** when exceeding the limits of the surface area, *s.t.* the field behaves  spherical surface.

 $\ \forall \ a \in [-1, +1]\,$ which defines the range of $\,F$,$\,$ we have three cases:

 Case $\ 1 < a \ :$
 : $ a \longmapsto (a-2)$

 <br>
 Case $\ -1 < a < 1 \ :$ 
 : $a = a$

 <br>
 Case $\ a < -1 :$
 : $a \longmapsto (a+2)$

 
 *For example, if an agent moves from position $\,(-0.9, 0.2)\,$ down by 0.3 it would map from  $\,(-1.2, 0.2)\,$ to $\,(0.8, 0.2)$; similarly, if an agent moves from position $\,(1.0, 0.95)$ to $(1.13, 1.025)\,$ the resulting position would be $\,(-0.87,-0.975)$*



 ![planar drawing of E](http://i.imgur.com/WPBefKn.png)

 <CENTER><small>*Planar projection drawing of $E$ with various Colour Polymorphic Prey agents*</small></CENTER><br>


 ![enter image description here](http://i.imgur.com/ONlX5W7.jpg)
 <CENTER><small>*Simplistic partitioning of $E$ into $S$ sectors to reduce search space*</small></CENTER><br>



 [^env-mod]: <small> $ $For example, shifts in the body of water/medium of $E$ like cyclic waves coming in and out according to tidal patterns,  which would dynamically affect agent movement as the body of water shifted; or temperature variations in the water which could affect 'food' abundance.
	#Definition of the ABM Environment

	Consider the set of all real numbers within $[\,-1, \,+1\,]$.$\ $ Let $\,F\ $ be the cartesian product of such a set with itself, s.t. individual elements in $\,F\,$ are two-dimensional points, or a cartesian pair within a subset of $R^2$.
	$$\therefore \ F\ = \ \{\ (x,y) \in\, \mathbb{R}^{2} \, \lvert\ (\,-1 \leq x \leq 1 \,)\ \wedge\,(\,-1 \leq y \leq 1 \,)\ \}$$

	Let $\ E\ $ be the symbolic representation of our modelling environment, such that $E$ encompasses the following:

	* A vector space $\ V\ $ over field $\ F$.
	* Variable extensions on the field $F$ which act as "environmental" describers (to define the properties of the medium the agents exist in) and modifiers (to be used to model changes to the world state).[^env-mod]

	$ $
	Note: For the purposes of the initial research goals, it is assumed that all agents exist in a perfect, static environment, with no variations. Improvement or regression of agent abilities or characteristics are entirely independent of environmental factors.

	The range of $F$ within $E$ will be sufficient to act as a looping "petrie dish" to observe the behaviour of the two populations of agents, with the origin $\,(0,0)\,$ as the absolute centre of the projection of the environment.

	Rather than agents 'bouncing' or 'avoiding' the boundaries of $E$, values $(x,y) \in E\,$ wrap when exceeding the limits of the surface area, s.t. the field behaves spherical surface.

	$\ \forall \ a \in [-1, +1]\,$ which defines the range of $\,F$,$\,$ we have three cases:

	Case $\ 1 < a \ :$
	: $ a \longmapsto (a-2)$

	<br>
	Case $\ -1 < a < 1 \ :$
	: $a = a$

	<br>
	Case $\ a < -1 :$
	: $a \longmapsto (a+2)$


	For example, if an agent moves from position $\,(-0.9, 0.2)\,$ down by 0.3 it would map from $\,(-1.2, 0.2)\,$ to $\,(0.8, 0.2)$; similarly, if an agent moves from position $\,(1.0, 0.95)$ to $(1.13, 1.025)\,$ the resulting position would be $\,(-0.87,-0.975)$



	![planar drawing of E](http://i.imgur.com/WPBefKn.png)

	<CENTER><small>Planar projection drawing of $E$ with various Colour Polymorphic Prey agents</small></CENTER><br>


	![enter image description here](http://i.imgur.com/ONlX5W7.jpg)
	<CENTER><small>Simplistic partitioning of $E$ into $S$ sectors to reduce search space</small></CENTER><br>



	[^env-mod]: <small> $ $For example, shifts in the body of water/medium of $E$ like cyclic waves coming in and out according to tidal patterns, which would dynamically affect agent movement as the body of water shifted; or temperature variations in the water which could affect 'food' abundance.