**Authors:** Markos Georgallides

This article explains the correlation between Euclidean Geometry , Complex Numbers and Physics . A Straight line AB is continuous in Points between A and B [ i.e. all points between AB are the elements which fill AB ] , which Points are also , Nothing , Everything , and maybe Anywhere , without any Dimension , and one has to pass the infinite points between A and B . A point C is on line AB only when exists CA+ CB = AB , or the whole AB is equal to the parts CA , CB , and this is an equation , which differentiates geometries . Since points have not any dimension and since only AB has dimension ( the length AB and for ÃC the length AC ) and since on ÃB exist infinite AC → AB , which have infinite Spaces , Anti-Spaces and Sub-Spaces , then 1. Straight line AB is continuous with points as filling ( Infinitively divisible ) . 2. Straight line AB is discontinuous (discrete) with dimensional Units , ds =AB as filling ( that is made up of finite divisible or indivisible parts the Monads ds ) or ds → AB / n , where n = 1 , 2 , → ∞ ) , and for n = ∞ then ds = 0 . 3. Straight line AB is discontinuous (discrete) with dimensional Units ds , or ds = quantum = AB / n [ where n = 1,2,3 → ∞ , = ( a + b.i ) / n , Infinitively divisible and keeping always the conservation of properties at end points A , B ] as filling , and continuous with points as filling ( for n = ∞ then ds = 0 i.e. a point ) . This is the Vector relation of Monads , ds , ( or , as Complex Numbers in their general form , ds = a + b. i ) , which is the Dual Nature of lines AB , ( discrete and continuous ) . So travelling on Points ( ds = 0 ) between AB one never comes to B , on the contrary travelling with ds > 0 one comes in finite time . 4 . Achilles has to pass every point of line AB which is then as passing from the starting point A , ds =0 , where Velocity of Achilles is v(A) = ds/dt = 0 . The same happens for Tortoise at point B where Velocity v(T) = ds/dt = 0 . On the contrary , Achilles passing AB on dimensional Units , ds , then Achilles velocity v(A) = ds/dt(A) is greater than that of Tortoise v(T) = ds / dt(T) . Since in PNS , v = ∞ , T = 0 , meaning infinite velocity and Time not existing , then Arrow AB in [PNS] is constant because AB = ds = Constant = u . 0 = ∞ . 0 Straight line AB is discontinuous (discrete) with dimensional Units ds = AB / n where n = 1 → ∞ and continuous with points [ n = ∞ ] . Continuously on AB happens also with all discrete ds , ( This is the Dual Nature of lines ( Geometry ), discrete and continuous ) . Monads ds = 0 → ∞ are Simultaneously , actual infinite ( because for n = ∞ then ds = [ AB / n = ∞ ] = 0 i.e. a point ) , and potential infinite , ( because for n = 0 then ds = [ AB / n=0 ] = ∞ i.e. the straight line through AB .

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[v1] 18 Sep 2011

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