**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 .

**Comments:** 9 pages

**Download:** **PDF**

[v1] 18 Sep 2011

**Unique-IP document downloads:** 105 times

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful. *