The Holographic Universe and Pi = 4 in Kinematics!

[MarkE]

http://theory.uwinnipeg.ca/physics/circ/node6.html

That theory page does not show rigorously that when "the direction of the centripetal acceleration is inwards along the radius vector"  then circular motion is produced - it just asserts it, like you.

The question how can V_║(t₁) be smaller than the tangential velocity V_T(t₀) if the force acting on it was always perpendicular between t₀ and t₁, still stands unanswered.

http://dev.physicslab.org/Document.aspx?doctype=3&filename=CircularMotion_CentripetalAcceleration.xml

[verpies]

LOL, you can enjoy yourself misstating what I have said if that pleases you. 

It does not please me to misstate you, but drawing conclusions from your statements and synthesizing a larger statement from several of your statements is not a misstatement. It is an acceptable way of conducting conversation.  If I make an error along the way I expect you to point it out.

Gravityblock's method does not reproduce the path of travel along the circumference.  ...

The shortest path between two points is only orthogonal line segments when one of those segments is zero length.  The distance following a Manhattan route between vertices on the circumference is therefore always greater than the straight line distance between the same vertices.

This is the same as stating that at the limit the chord approaches the curve/arc.
BTW: Note that this is a condensation and interpretation of your words and your animation - a valid debating technique, not a misstatement.

And you would be correct if all of the points on the curve/circle had the same time coordinates - like in an abstract geometric circle.
But neither Gravityblock not I are analyzing abstract circles.  We are analyzing real circles made by real physical processes where the points on the circle do not have the same temporal coordinates.
In such real circles, the chord does not approach the curve and this very issue is the subject of this paper.

You are welcome to prove your assertion that at the limit the chord approaches the curve/arcs created by real physical processes.

[MarkE]

It does not please me to misstate you, but drawing conclusions from your statements and synthesizing a larger statement from several of your statements is not a misstatement. It is an acceptable way of conducting conversation.  If I make an error along the way I expect you to point it out.

LOL, no that's called making things up.  If you can't win the point, just build a man of straw to slay.

This is the same as stating that at the limit the chord approaches the curve/arc.
BTW: Note that this is a condensation and interpretation of your words and your animation - a valid debating technique, not a misstatement.

And you would be correct if all of the points on the curve/circle had the same time coordinates - like in an abstract geometric circle.

Plane geometry does not involve time.

But neither Gravityblock not I are analyzing abstract circles.  We are analyzing real circles made by real physical processes where the points on the circle do not have the same temporal coordinates.

Here we go:  A special pleading to "circles" that are not "circles" except when you two want them to be circles.  Yet they fail tests for basic properties of circles.

In such real circles, the chord does not approach the curve and this very issue is the subject of this paper.

Neither you nor Gravityblock have established that your special "circles" are in fact circles.

You are welcome to prove your assertion that at the limit the chord approaches the curve/arcs created by real physical processes.

LOL, now you don't believe first semester calculus.  You are free at any time to as you say use actual facts to argue your specious and silly case.

[verpies]

Plane geometry does not involve time.

But we are not discussing abstract timeless plane geometry.  You are assertions about Pi are correct in abstract time geometry.
We are not discussing abstract geometry, we are discussing real physical problems from the start. 

http://dev.physicslab.org/Document.aspx?doctype=3&filename=CircularMotion_CentripetalAcceleration.xml

This link you posted refers to an article describing a physical circle, created by real forces acting on a real mass.  There is no avoiding the time variables in this one.

That article states:
"Note that in both cases, Δv points to the center of the circle reflecting that the acceleration is also directed towards the center of the circle"
..but it is just an empty assertion. 

That article does not prove that the acceleration vector and force that causes the circle lays on a line that passes through the center of the circle.
That article correctly subtracts two tangent velocity vectors.  On my diagram that is V_T(t₀) - V_T(t₁) but it fails to prove that the result of this subtraction lays on a line that passes through the center of the circle.
What proof did you or that article give that the acceleration/force vector lays on such line?  What proof did you or that article give that the acceleration/force vectors do not lay on the dashed lines depicted on the diagram below that does not pass through the center ?

[gravityblock]

Each one of your jaunts along two edges of the approximating squares travels along one segment towards the perimeter and one away from it.  Dividing into a larger quantity of smaller squares does not change the path length.  It does not make the path a better approximation of the circumference.

MarkE,

The plot of a convergent sequence {a_n} is shown in blue in the illustration below. Visually we can see the sequence is converging to the limit 0 as n increases.  Similarly, we can visually see the exponentially larger quantity of smaller squares in each successive squaring method is converging while the path length does not change.

Gravock