sw00905math--find-c-with-straight-line

36


  r = radius      c = circumference

  Not expecting to finish this entry today, but have gotten into
the substance of it far enough to show the confusion therein.
This search had its origin in March of 1966.  Shortly thereafter
I abandoned it.  Yesterday I began addressing it again.

Circumference = 2 x pi x radius.  Therefore, if the radius is
1/2 inch, the circumference = pi = 3.14159265, which is the
pi value being used here.  My project in 1966 was to see if
the circumference of a circle could be determined without pi.
This led me to considering the 90° arc with the notion that
with the aid of a protractor I could draw interlocking circles
and so establish a straight line that equalled the length of a
circle's circumference divided by "4", or approximately did.  I
may scan my original calculations (which I uncovered several
days back while looking for something else) and show them 
here, even though upon inspecting them then I became uneasy
about my conclusions in 1966.  Could be my conclusions were
correct and I simply no longer understand them.  Moot, moot.
I now have new conclusions.

Let r = one inch.  [ Spotty storms are meandering in. ]           
c = 6.28318530      /4 = 1.57079633 = pi/2      I then found that
4/7 = .57142857      Today I found that 1.57079633 divided
by 11 = .14279967      and that 7 x .14279967 = .99959766
which is quite close to one inch.  However, I then proceeded
to r = 2 (two inches) where c = 12.56637060     and /4 = pi
but pi/11 = .28559933      and 7 x that = 1.99919532  which
comes to a loss of .00040234      And so I then proceeded
to r = 3 where c = 18.84955590     and /4 = 4.712388898
and 4.71238898/11 = .42839900      and 7 x that = 2.99879298
which comes to loss of .00080468 from the r = 1 result, meaning
that there is a loss of .00040234 per inch and consequently a
loss of .00005748 per 1/7 of an inch, and of .00063225 at 11/7.

5:50 PM - The  .00040234 per-inch loss bothered me; so I
ventured on.  1/7 = .14285714      1/56 = .01785714 whereat
I tried x 9 and got .16071429      At this juncture I returned to
my trial-by-error habit because I knew what I was seeking was
far off.  Finally upon dividing .16071429 by 399.45 (oh, yes)
I got .00040234.  So, I needed to know what 56 x 399.45
equalled.  Hmmph!: how does 22369.2 strike you?  Indeed it
was far off.  1/22369.2 = .00004470      Next question: How
important is the .2?  Turns out that at eight out, not seemingly
significant.  1/22369 = .00004470      But still I ventured, as
I wanted to know how 22369 related to 7.  It doesn't.  I then
tried 3, and it didn't either.  This made me wonder if 22369
was either a prime number or a square; so I minused 1 from
it and divided by 8.  Result: 2796.  At that point I stopped.
And now I need to look at my earlier entry regarding terms
in the natural number summation sequence.  Hiatus.

8:47 PM - Fact: The remainder of an odd square - 1 into
which 4 is divided always equals an even integer which is
the product of two adjacent integers, and the sum of those
two adjacent integers always equals the square root of the
odd square.  Example: 17 x 17 = 289    288/4 = 72  and
8 x 9 = 72 and 8 + 9 = 17
-
Given this fact, 22369 is not an odd square/ because 4
divided into 22368 equals 699, which is not an even
integer.  It is still possible 22369 is a prime number since
22368 is evenly divisible by both "3" and "6" but I think
that does not matter.

1/22370 also equals .00004470.  22370/2 = 11185.
1/22372 also equals .00004470.  22372/7 = 3196.
22372/4 = 5593.  3196 x 4 = 12784 and
12784 + 22372 = 35156 and 35156/11 = 3196.
-
.00040234/.00004470 = 9.00089486.
.00004470 x 9 = .00040230 which puts the result off
by .00000004.  But what the heck is 1/22372 of an inch?
The closest unimaginable ruler number is 1/16384.  Then
again, rulers don't bother with sevenths of an inch either.
Whatever, 12784 + 9 = 12793 or 9/22372 more than
4/7 of an inch.  Now what?  Back to 57079633.  ...
Wait!  I should be moving lower.

11:44 PM - Just discovered "13" divides into 57079633
evenly, getting 4390741.  I should be in bed.  Tomorrow
perhaps.

May 27, 2008
7:26 AM - Other findings included 12771/22374 = .57079646,
but after truncating 2506218.849 to 2506219 and dividing that
integer by 4390741, .57079636 arrived/ because (I surmise)
that while 13 divides evenly into 57079633 it does not divide
evenly into 4390741.  13 divides evenly into 4390737.  At
nine out, .570796364 arrives.  My calculator does not show
ten zeros beyond the decimal point, and since I'm done with
hunting, somewhere between .00000003 and .00000004 sits
the error.  1/4390741 = .00000023 or .000000228.  2506219
+ 4390741 = 6896960; 1.57079633/6896960 = .00000023;
.00000023 x 4390741 = .99999998 of an inch.  At .000000228
it becomes 1.001088948. 1.001088948 - .99999998 equals
.001088968.  Under, over, the yard north of my bedroom win-
dow is happily banded with white clover, but the dandelions
which banded it last week are now nowhere to be seen.

1:48 PM - Among other activities, went out to take some photos
and did find several d-lions.

At r = 2 inches  c = 12.56637060  and c/4 = pi = 3.14159265
3.14159265/6896960 = .00000046      and 4390741 x
.00000046 = 1.99999995      loss per inch of .00000003 (2.5)
At r = 3 inches  c = 18.8495590      and c/4 = 4.71238975
4.71238975/6896960 = .00000068      and 4390741 x
.00000068 = 2.99999993      loss per inch of .00000002 (2.33)
-
This is much closer, but had to go into outer space to get it.
-
Recap:  4/7 = .57142857      2506219/4390741 = .57079636
              target      .57079633          


Sounds like the robin is practicing a new tune.  It has a falsetto
finishing note.


May 29, 2008  2:12 PM
-
Went out to 9 on my calculator.  3.141592654/2 = 1.570796327
and after some t-and-e chose to use .570796330 because two
integers, both of which happened to be divisible by 15, equalled
it.  So 250621905/439074135 became 16708127/29271609.  Now,
if I use the smaller integers, I first need to divide 1.570796330 by
15 which equals .104719755; and then divide this by 45979736
(16708127 + 29271609).  Either way, the result is .000000002.
250621905 + 439074135 = 689696040.  1.570796330 divided by
689696040.  [ Much of today's entry is calculator-related junk.  My
attempt to go out nine has failed.  Am unable to verify calculations. ]
-
See this Wikipedia page on Pi.

May 30, 2008  4:06 PM
-
About yesterday's numbers:  Dividing 45979736 by 15 was wrong.
The numbers used relate to a circle with a radius of one inch, making
2 x pi the circumference and division by 4 approximately equal to
1.57079633.  After the divisions by 15, the essential integers are
16708127, 29271609, and 45979736.  (However close these are
to 4/7, 7/7, and 11/7, the 1.57079633 straight line represented by
45979736/29271609 can only be imagined.)  Found a way to get
the answers I needed from my calculator.  I eliminated the decimal
point in 1.57079633 when I divided it by 45979736.  That gave me
3.416279576 which truncating to 3.41627958 proved best.  This is
a deceptive number in that it looks a bit like pi, but it has nothing to
do with pi.  When this number is multiplied by 29271609, my c-toy
displays 100,000,000.1 or 1.000000001 after the decimal point is
returned to its proper location.  All these calculations and the need
to verify them entailed more effort than it probably was worth.  If
3.14159265 is allowed to equal pi, 6.2831853 (2 x pi) equals the
circle's c.  If 1.57079633 is allowed to equal c/4, 6.28318532 =
the circle's c. 



#
Brian A. J. Salchert