﻿ Iyika

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.

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 accounted-for.

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 fifty-fifth percent mark of X’s length is reached where the Y percent is fifty-six and six thousand two hundred and sixty-five 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 (Ycum%/Ymax)(1002/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 (ycum/Ymax)(1002/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 (Ycum%/Ymax)(1002/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 (Ycum%/Ymax)(1002/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)(1002/75) 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 X Cum % Change in Y Y Cum Y Cum % (Y/X)% or (Ycum%/Ymax)(1002/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,

(Ycum%/Ymax)(1002/X%) is the same as (2.467/2.867)(100)(100)/(80) = 107.5601

Again, Duare or Ayika is a measurement of a circle’s area, vertically or horizontally.  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 a particular length on the diameter line from its far left point, and the Y axis measures the percent of the circle that is below plus above the circle’s diameter line which also is on the left of the inclusion dividing line which is “a vertical dividing line.”  Here, inclusion represents that part of the circle that has been accounted for with a particular percent.

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 fifty-fifth, 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 seventy-five 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 = One-Half.

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