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Reekie's Theorem and Octagon Formulas
 
 The Octagon Laid to Rest, without Trig.
 
All based on 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 as yet 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 had not as yet come up with rationale.

 

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 ]

 

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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 or rather full octagon.

 

 octagonal birdhouses of my making in our garden