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 Octagon Formulas
and
The Square Root of Two, Minus One.
 
Reekie's, Earlsferry House inspired, Theorem: the Octagon Laid to Rest, without Trig.
 

During my teenage 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 could not find a 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.

 

My ah-ha moment came when I subtracted 1 from the square root of 2 and discovered that the remaining number provided the formula for the multiplier of the Width in order to find the length of the Side of a regular octagon. (It was at a  later moment that I concluded that the square root of 2 is the multiplier of what I called  the B dimension to find the length of the Side)


The Number 0.4142

 

With this number so perceived I made up the following formulas.

 

 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 I deduced that:

   

      S = 0.4142 W                                       W = 2.4142 S

     

      S = 1.4142 B                                          B = 0.7071 S

     

      S = 0.3826 D                                        D = 2.6135 S

     

      W = 3.4142 B                                        B = 0.2929 W

 

      W = 4.8284 C                                        C = 0.2071 W

 

      W = 0.3018 P                                         P = 3.3137 W

 

      W = 0.9239 D                                       D = 1.0824 W

      

     W = The sq. rt. of (1.2071 A)                   A = 0.8284 W2                                      

       -----------------------------------------------------------------

                                                    Also

     W = S + 2B

 

     S = W - 2B

                            

     A = W2 - 2B2  

 -----------------------------------------------------------------    

 All of these formulas include 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 three ?'s)
2.6135 is  ?    

Pertaining to my three numbers 0.3826, 0.9239 and 2.6135 I haven't come up with rationale as to their relationship to the Square Root of 2, but I know there is. There must be one person somewhere who will.

 

Bingo. Eureka.

 

June 1st 2009. An interested viewer took it upon himself to figure out these three questions for which I have not come up with rationale as to their relationship to the square root of 2.

 

______________________________________________________________________

 

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've wrestled with my questions regarding the regular octagon as to what provable relationships could there be pertaining to my observations and deductions as to the square root of 2 minus 1.  

 

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 esoteric

 square root of 2, minus 1

  

2012. Divide the 2nd by the 1st and the answer is the sq. rt. of 2 plus 1.

          Multiply the 2nd by the 1st and the answer is 2 times the sq. rt. of 2.

 

Pertaining to Octagon Formulas: without exception; every equation, that I come up with, includes the Square Root of 2.

 

Fascinating fascination.

 

Octagon birdhouses of my making 

 My most recent 8 plex octagonal bird house has pentagonal entryways 

The first octagonal bird house I made