Square Weighting In Maths (Long Question)

[TheMacDonaldMan]

I was watching a Numberphile video about making squares into circles using Pascals Triangle as a weighting limit. I saw a mini pattern but I can't entirely figure out how it works. I will have to unfortunately, give a long winded example of how to construct such circle, but it is necessary for the understanding of what I'm talking about. If you do computer Programming, you may already know what this will be, For example: 1) Draw a large square 2) mark on the corners A B C D (starting from the top left going clockwise) 3) Mark the midpoints of each side 4) Connect the midpoints with a dashed line (A diamond within the square) 5)For example, from midpoint AB and midpoint AD, find that midpoint and mark it (You should now be able to draw a line from point A to the midpoint of the midpoint ADAB) 6)From point A to a quarter of the way to B, mark a point, and do the same for AD 7) Connect these two quarter points with a dashed line, and mark the midpoint of the new dashed line 8) where the newest dashed line and one other line intersects it, mark the point 9) repeat process 5 to 8 for all other parts/midpoints 10) You should now have marked the points: AB, BC, CD, DA, and the four other corresponding points 11) Connect all the points in a solid shape. Done. Now, in programming language, you have just used a 1,2,1 weighting limit to construct the beginning of a circle, and this process could be repeated many times to create a perfect circle. I get that for 1,2,1, it's simply splitting the edges, and then dividing by 2 for the new B point, and if it's 1,3,3,1 then you divide by 3 to get thirds of the midpoints and fin their intersections. But how does it work if I was to use a weighting of something like 1,4,6,4,1 or 1,5,10,10,5,1? Is it a division process, or is that rubbish? If it is division, how would I do something like 1,4,6,4,1? Any reply to this will be very much appreciated.

[Dizmo]

I thought that a circle had infinite amount of sides? I'm out my depth here :)

[BlackadderV]

I'm afraid life is too short.

[glenbar]

Oh dear, just shows me how ignorant I am, but good luck with your problem.

[jim360]

I'd have to study the problem further, but I believe that the procedure above, if followed, would lead to the orange shape in the picture below:

http://i59.tinypic.com/9uq9hi.png

Obviously it would be a bit more accurate, as it's drawn free-hand on paint, but hopefully this helps people get the idea.

[TheMacDonaldMan]

Exactly that Jim360. If you could find the Numberphile video on YouTube (I believe that it is something like maths in programming and uses a Disney worker to explain it) You'll understand my circle thing. I just basically want to know if there's a pattern to the circle making, using Pascal's Triangle