Octagon
Formulas
Reekie's
Theorem, the Octagon Laid to Rest, without Trig.
The
Square Root of Two Minus One
During my years at Waid Academy because
of my admiration for Earlsferry House with its polygon turret I became interested
in the parameters of the regular octagon; an eight sided
polygon in which all the sides are of equal length and all
the angles are of equal degree. My interested peaked because
I had a difficult time in finding any math book that told me what I wanted to know relating to
the subject. All referred to the use of trigonometric
functions and trigonometric tables in which one had to be
well versed and knowledgeable to use.
Concurrent with this interest the
subject of unity (one) was a fascination of mine. By using 1 as being the length of the sides of a right angled
isosceles triangle, the length of the hypotenuse of such a
triangle is the square root of 2 which is the irrational
number 1.4142.
Then it hit me. By sight alone
and the knowledge of how to arrive at the parameters of the
octagon by the trigonometry method I saw that I had
stumbled on to how to find any and all of the parameters of
octagons without the knowledge and or use of trigonometric
functions or trigonometric tables.
I saw that if I merely subtracted unity or 1 from the
square root of 2 I had the key to the whole subject.
The Number 0.4142
With this number so perceived I
made up the following formulas that are good for every size of
regular octagon.
With the width of an octagon being the dimension between the parallel sides,
|
Let W be the width of the octagon
W
is also the diameter of the inscribed circle
S be
the length of the side
A be the
area
P be the
perimeter
B be the dimension at right angles to the
sides
C
be the dimension at right angles from a side to the point of
intersection at right
angles
from
the adjacent two sides
D
be the diagonal dimension
D is also the
diameter of the circumscribed circle
-P-
|
|
Then by
Reekie's Theorem
S =
0.4142 W
B = 0.2929 W
S
= 1.4142 B
B = 0.7071 S
W = 2.4142 S W
= 0.3018 P
W = 3.4142 B
D = 1.0824 W
A = 0.8284 W2
P = 3.3137 W
W = 4.8284 C S = 0.3826 D
C = 0.2071 W
W = 0.9239 D
W = The square root of 1.2071 A D = 2.6135 S
-----------------------------------------------------------------
W = S + 2B
S = W - 2B
A = W2 - 2B2
-----------------------------------------------------------------
|
 |
|
This is how I arrived
at these numbers. Since 1.4142 is the square root
of
2
|
0.4142 |
is the square root of 2 then - 1 |
|
1.4142 |
is
the square root of 2 |
|
2.4142 |
is the square root of 2 then + 1 |
|
3.4142 |
is
the square root of 2 then + 2 |
|
0.8284 |
is 2 times (the square root of 2
then - 1) |
|
4.8284 |
is (2 times the square root of
2) then + 2 |
|
0.2071 |
is (the
square root of 2 then - 1) then
divided by 2 |
|
1.2071 |
is the square root of 2
then - 1 then divided by 2 then + 1 |
|
0.2929 |
is 1 - ( the square root of 2
then divided
by 2) |
|
0.7071 |
is the square root of 2
then divided by 2 |
|
0.3018 |
is the square root of 2
then + 1 then divided by 8 |
|
1.0824 |
is
the square root of [ (the square root
of 2 then - 1)2
then + 1 ] |
|
3.3137 |
is 8 times (the square root of 2
then -
1) |
|
0.3826 |
is
? |
|
0.9239 |
is
?
(I'm looking for help on these ?) |
|
2.6135 |
is
? |
Pertaining
to my three numbers 0.3826, 0.9239 and 2.6135 I
haven't been able to come up with a rationale as
to their relationship to the Square Root of 2,
but I know there is. There must be one person
somewhere who can.
Bingo. Eureka.
June
1st 2009. An interested viewer took it upon
himself to figure out these three questions for
which I could not come up with answers.
He writes
:
2.6135 is 0.9239 x 2 x the square root of 2
0.3826
is the square root of [ 1 divided by ( 2 x the
square root of 2 ) then + 4 ]
0.9239
is the square root of [ 1 divided by ( negative
2 x the square root of 2 ) then + 4 ]
For an octagon with a
Width of 12 feet,
The Side length is
4.970 feet
The Area is 119.290
square feet
The Perimeter is
39.764 feet
The B dimension is
3.514 feet
The
C dimension is 2.485 feet
The
D dimension is 12.989 feet
The
diameter of the inscribed circle is 12 feet.
The
diameter of the circumscribed circle is 12.989
feet.
Note,
In this example my chosen starting point is the
Width.
However since the
Width, the Side, the Area, the Perimeter, the
B, C, D, dimensions and
the inscribed and circumscribed circles are
all related, by using my formulas, any one or all of these parameters can just as
easily be found by starting from any one of them.
|
For
many years I
wrestled with my question regarding the regular octagon
as to what provable relationships could there be pertaining to
my observations and deductions.
AND NOW AFTER ALL THESE
YEARS, Read
my brother, Noel Reekie's Octagon Proof
2008.
Back to Square One. My
latest discovery and mystery.
1.0824
is my multiplier of the Width to get the Diagonal.
2.6135
is my multiplier of the Side to get the Diagonal.
Divide
the 1st by the 2nd and the answer is
0.4142
again,
that
square root
of 2 minus 1
Looks
like I've gone full circle
