Iyika
Olaniyan, in Philadelphia, PA, has discovered some fascinating relationships between circle lengths and areas. Here is his summary of a few of his findings. It is his express wish that these ideas be used to help many children find exciting math connections.
Copyright by Olaniyan, Philadelphia, Pa, all rights reserved.
Iyika
This
a Yoruba word that mean “circles”
and it is the name of a method of measuring a circle’s area, vertically or
horizontally. The first term is of
the Yoruba language, the second term is of the Swahili language, and the third
term is of the English language.
Here the X axis, or the independent variable, is along the circle’s diameter
line, which is labeled according to what percent of the diameter line’s total
length is every particular length on the diameter line from its far left point.
The Y axis measures the percent of the
circle that is below plus above the circle’s diameter line and left of the
vertical dividing line that separates that part of the circle that has been
included from that part of the circle that has not been included.
This vertical line can be moved to the left or to the right to decrease
or to increase the amount of the circle's area that is included or
accountedfor.
To find
the circle’s area percent of the circle’s total area the circle is weighed.
Then a new vertical strip of the circle is weighed and summed with other
circle parts to show any usual or unusual patterns.
Here we can find what percent of the total circle’s weight is the new
vertical strip of the circle. With
that we can find the area, which must have a similar relationship to the total
area of the circle as it has to the total weight of the circle.
As we move
a vertical dividing line from the diameter’s far left point the change in X is
more than the change in Y until the fiftyfifth percent mark of X’s length is
reached where the Y percent is fiftysix and six thousand two hundred and
sixtyfive ten thousands. At fifty
percent of line X’s length we have the fifty percent of Y area but later the
change in Y will be greater than the change in X and the percent of the Circle
above plus below the diameter line and left of the vertical dividing line will
be greater than fifty percent of the circle’s area and the percent of X
surpassed. This difference will
increase until the eighty percent mark of the X axis is reached.
From there the inequality will lesson until the one hundred degree mark
of X and of Y is reached, where they will have equal values, one hundred
percent, 100%.
There are variation of these facts.
The facts depend on the size of the circle and the diameter of the
circle.
EXAMPLE A
Letter 
X 
Y change 
Y Cum 
X % 
Y Cum % 
Y/X % or
(Y_{cum}%/Y_{max})(100^{2}/X%)


A 
1 
0.4 
0.4 
5 
2.4096 
48.0000 

B 
2 
0.5 
0.9 
10 
5.4216 
54.2000 

C 
3 
0.7 
1.6 
15 
9.6385 
64.2900 

D 
4 
0.8 
2.4 
20 
14.4578 
72.2891 

E 
5 
0.9 
3.3 
25 
19.8795 
79.5180 

F 
6 
0.9 
4.2 
30 
25.3012 
84.3373 

G 
7 
1 
5.2 
35 
31.3253 
89.5008 

H 
8 
1 
6.2 
40 
37.3493 
93.37325 

I 
9 
1 
7.2 
45 
43.3734 
96.3855 

J 
10 
1.1 
8.3 
50 
50.0000 
100.0000 

K 
11 
1.1 
9.4 
55 
56.6265 
102.9400 

L 
12 
1 
10.4 
60 
62.6506 
104.4100 

M 
13 
1 
11.4 
65 
68.6746 
105.6400 

N 
14 
1 
12.4 
70 
74.6987 
106.7000 

O 
15 
0.9 
13.3 
75 
80.1204 
106.8300 

P 
16 
0.9 
14.2 
80 
85.5421 
106.9270 

Q 
17 
0.8 
15.0 
85 
90.3614 
106.3400 

R 
18 
0.7 
15.7 
90 
94.5783 
105.0800 

S 
19 
0.5 
16.2 
95 
97.5903 
102.7200 

T 
20 
0.4 
16.6 
100 
100.000 
100.0000 

















Note that (y_{cum}/Y_{max})(100^{2}/X%)
equals any number of the far right column.
EXAMPLE B
Let
ter 
X 
Y change 
Y
cum 
X Cum % 
Y Cum % 
(Y/X)% or
(Y_{cum}%/Y_{max})(100^{2}/X%) 

A 
1 
2.25 
2.25 
5 
2.79 


B 
2 
3.06 
5.31 
10 
6.5987 


C 
3 
3.44 
8.31 
15 
10.8736 


D 
4

3.89 
12.64 
20 
15.7077 


E 
5 
4.51 
17.15 
25 
21.3122 


F 
6 
4.31 
26.26 
30 
26.6683 


G 
7 
4.83 
26.29 
35 
32.6705 


H 
8 
4.93 
31.22 
40 
38.7970 


I 
9 
4.99 
36.21 
45 
44.9981 


J 
10 
5.23 
41.44 
50 
51.4974 


K 
11 
5.06 
46.50 
55 
57.7855 


L 
12 
4.81 
51.31 
60 
63.7628 


M 
13 
4.64 
55.95 
65 
69.5290 


N 
14 
4.75 
60.70 
70 
75.4318 
107.7597685 

O 
15 
4.38 
65.08 
75 
80.8748 
107.8331469 

P 
16 
4.23 
69.31 
80 
86.1314 
107.664347 

Q 
17 
4.07 
73.38 
85 
91.1892 
107.281486 

R 
18 
2.94 
76.32 
90 
94.8427 


S 
19 
2.56 
78.88 
95 
98.0241 


T 
20 
1.59 
80.47 
100 
100.0000 


















EXAMPLE C
Letter 
X 
Y change 
Y
cum 
X Cum percent 
Y Cum percent 
Y/X percent or
(Y_{cum}%/Y_{max})(100^{2}/X%) 

A 
1 
10 
10 
5 
2.1276 
42.552 

B 
2 
15 
25 
10 
5.3191 
53.191 

C 
3 
20 
45 
15 
9.5744 
63.8293 

D 
4 
25 
70 
20 
14.8536 
74.268 

E 
5 
25 
95 
25 
20.2127 
80.8508 

F 
6 
25 
120 
30 
25.5319 
85.10633333 

G 
7 
30 
150 
35 
31.9148 
91.1851 

H 
8 
30 
175 
40 
37.2340 
93.085 

I 
9 
30 
205 
45 
43.6170 
96.9266 

J 
10 
30 
235 
50 
50.0000 
100.0000 

K 
11 
30 
265 
55 
56.3829 
102.5143636 

L 
12 
30 
295 
60 
62.7659 
104.609833 

M 
13 
30 
325 
65 
69.1489 
106.3829 

N 
14 
25 
350 
70 
74.4680 
106.3828 

O 
15 
25 
375 
75 
79.7872 
106.382933
(375/470)(100^{2}/75) 

P 
16 
25 
400 
80 
85.1063 
106.382875 

Q 
17 
25 
425 
85 
90.4255 
106.3829 

R 
18 
20 
445 
90 
95.7446 
106.3828 

S 
19 
15 
460 
95 
97.8723 
103.0234 

T 
20 
10 
470 
100 
100.0000 
100.0000 

















EXAMPLE D
Letter 
X 
Change in Y 
Y Cum 
Y Cum % 
(Y/X)% or
(Y_{cum}%/Y_{max})(100^{2}/X%)


A 
1 
5 
0.051 
0.051 
1.7788 
0.35576 

B 
2 
10 
0.094 
0.145 
5.0575 
0.3487 

C 
3 
15 
0.114 
0.259 
9.0338 
0.6022 

D 
4 
20 
0.134 
0.393 
13.7077 
0.6853 

E 
5 
25 
0.152 
0.545 
19.0940 
0.7637 

F 
6 
30 
0.155 
0.700 
24.4192 
0.8366 

G 
7 
35 
0.174 
0.874 
30.4848 
0.8709 

H 
8 
40 
0.179 
1.053 
36.7282 
0.9182 

I 
9 
45 
0.191 
1.244 
43.3903 
0.9642 

J 
10 
50 
0.189 
1.4335 
50.0000 
1.0000 

K 
11 
55 
0.184 
1.617 
56.4352 
1.02609 

L 
12 
60 
0.178 
1.795 
62.6438 
1.04406 

M 
13 
65 
0.184 
1.979 
69.0617 
1.0624 

N 
14 
70 
0.168 
2.147 
74.9215 
1.0703 

O 
1 
75 
0.163 
2.310 
80.6069 
1.0747 

P 
16 
80 
0.157 
2.467 
86.0830 
1.0752 

Q 
17 
85 
0.140 
2.607 
90.9661 
1.0701 

R 
18 
90 
0.116 
2.723 
95.0122 
1.0556 

S 
19 
95 
0.093 
2.816 
98.2211 
1.0339 

T 
20 
100 
0.051 
2.867 
100.0000 
1.0000 

Again, at
eighty percent of X, Y is at it's greatest percent of X that is over one hundred
percent. Here,
(Y_{cum}%/Y_{max})(100^{2}/X%)
is the same as (2.467/2.867)(100)(100)/(80) = 107.5601
To find the area and the various circle area percent of the circle’s total area
the circle is weighed. Then a new vertical strip of the circle is weighed
and summed with other circle parts to show any patterns.
As we move a vertical dividing line from the diameter’s far left point towards
the far right point the percent change in X is more than the percent change in
Y, or the circle's area's size. This occurs until the fiftyfifth,
55, percent mark of X’s length is reached where the Y percent is 56.6265.
At fifty
percent of line X’s length we have fifty percent of the Y area but later the
change in Y will be greater than the change in X and the percent of the total
circle that is left of the vertical dividing line, while also being above plus
below the diameter line, will be greater than fifty percent of the circle’s area
and it will be greater than the percent of “X” surpassed. This difference
will increase until about the eighty or seventyfive percent mark of the X axis
is reached. From there the inequality will lesson until the one hundred
percent mark of X and Y are reached, where they will have equal values,
100%. Exceptions do occur that show when the growth in the area of the circle
compared to the growth of X reaches its maximum difference.*
You
should note some patterns of the Y Cum Percent column.
In Example A, Row A divided by Row B is 2.4096/ 5.4216 or 1.00/2.225 and
Row A divided by Row C is 2.4096/9.6385 = 0.249997406 or 2.50.
There are other interesting results such as Row H divided by Row N of
Example C being
37.234/74.468 = OneHalf.
What I find most interesting is the fact that there is a lack of patterns one
should expect to be there. Here,
all circles are about the same whether the circles are small or large but the
percent change in the amount of the circle included in the area above and below
the diameter line and also left of the vertical dividing line, which can be
moved from the left to the right or form the right to the left side of the
circle, is not consistent. Why?
03/23/2016