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The occurrence of √n in n-sided regular plane figures
Originally called 'Geometry with a Calculator' for reasons that will become clear as you read through this topic.
It can easily be shown, by Pythagoras, that for an equilateral triangle whose sides = 1, the height = √3/2, and for a square whose sides = 1, a diagonal is √2 (or √4/√2).

Is the occurrence of √n a common feature among n-sided regular plane figures?

This topic:
     1. Investigates the regular pentagon and demonstrates its relationship to √5 and the Fibonacci Series.

     2. Investigates the regular heptagon.

     3. Investigates the regular 9 and 11-sided polygons.

     4. Shows examples of polygons where n is a multiple of 3, 4 and 5.

1. The regular pentagon

sin π/5 = 0.5877852
sin 3π/5 = 0.9510565
AC = (sin 3π/5)/(sin π/5) = 1.618034
1/AC = 0.618034
AC - 1 = 0.618034
AC - 1= 1/AC

So, AC² - AC - 1 = 0

 AC = (√5 + 1)/2 or -(√5 - 1)/2

The positive root is the length of the diagonal AC, or the ratio AC/AB.

What does the negative root represent? It is the distance between two adjacent points, A'C', of a regular five pointed star where A'B' = 1, or the ratio A'C'/A'B'.

Note: A'B' is clockwise (positive), and A'C' is anti-clockwise (negative), hence the negative root.

It can be shown that the segments of A'B', as defined by the intersections with C'D' and D'E' are related by the same ratios.

The regular pentagon and the Fibonacci series

When AB = 1,
AC0 =   0AC +   1
AC1 =   1AC +   0
AC2 =   1AC +   1
AC3 =   2AC +   1
AC4 =   3AC +   2
AC5 =   5AC +   3
AC6 =   8AC +   5
AC7 = 13AC +   8
AC8 = 21AC + 13
etc.

If ACn = aAC + b, then ACn+1 = (a + b)AC + a
The values of the terms on the RHS of the equations form two instances of the Fibonacci series.

2. The regular heptagon

sin π/7 = 0.4338837
sin 2π/7 = 0.7818314
sin 4π/7 = 0.9749279
AC = (sin 5π/7)/(sin π/7) = 1.8019377
AD = (sin 4π/7)/(sin π/7) = 2.2469796
AC/AD = 0.8019376
AC - 1 = 0.8019377
AC2- 1 = 2.2469795
AD/AC = 1.2469797
AD - 1 = 1.2469796
1/AD = 0.4450418
AD - AC = 0.4450419

Some relationships:
AC/AD = AC - 1
So, AD = AC/(AC -1) ........................ (1)

AD = AC2 - 1 ............................ (2)

From (1) and (2):
AC/(AC - 1) = AC2 - 1

AC3 - AC2 - 2AC + 1 = 0

Some more relationships:
AD/AC = AD - 1
So, AC = AD/(AD -1) ......................... (3)

1/AD = AD - AC
So, AC = (AD2 - 1)/AD ....................... (4)

From (3) and (4):
AD/(AD -1) = (AD2 - 1)/AD

AD 3 - 2AD2 - AD + 1 = 0

There are two different regular 7-pointed stars
The heptagon can be drawn by joining, in turn, seven dots equally spaced around a circle. The first star can be drawn by joining every second dot, and the second star can be drawn by joining every third dot.

The roots of the equations derived in Section 2. relate to the dimensions of the heptagon and its two companion stars. (When AB = 1)
  • AC3 - AC2 - 2AC + 1 = 0
The roots are: 1.8019377 (=AC)           -1.2469794 (=A’C’)               -0.4450418 (=A"C")
  • AD3 - 2AD2 - AD + 1 = 0
The roots are: 2.2469797 (= AD)           -0.5549579 (= A'D')              -0.8019377 (= A"D")

Note: AC = 1/A’D’, A’C’ = 1/A”D” and A”C” = 1/AD

Super Fibonacci Series
The heptagon has two series associated with it. These are derived in the same way that the Fibonacci series was derived from the dimensions of the pentagon. I call these series 'Super Fibonacci Series'.
When AB = 1
AC0 =   0AC +   0AD  +    1
AC1 =   1AC +   0AD  +    0
AC2 =   0AC +   1AD  +    1
AC3 =   2AC +   1AD  +    0
AC4 =   0AC +   3AD  +    2
AC5 =   5AC +   4AD  +    1
AC6 =   5AC +   9AD  +    5
AC7 = 14AC + 14AD  +    5
AC8 = 19AC + 28AD  +  14
etc.

If ACn =    aAC   +  bAD  +  c

ACn+1 =  (b + c)AC  +  (a + b)AD  +  a
AD0 =     0AC +     0AD  +     1
AD1 =     0AC +     1AD  +     0
AD2 =     1AC +     1AD  +     1
AD3 =     2AC +     3AD  +     1
AD4 =     5AC +     6AD  +     3
AD5 =   11AC +   14AD  +     6
AD6 =   25AC +   31AD  +   14
AD7 =   56AC +   70AD  +   31
AD8 = 126AC + 157AD  +   70
etc.

If ADn =    aAC   +  bAD  +  c

ADn+1 =  (a + b + c)AC  +  (a + b)AD  +  b

3. Other regular polygons

The regular 9-sided polygon
The equations for a regular 9-sided polygon are:
AC3 - 3AC - 1 = 0
AD3 - 3AD2 + 3 = 0
AE3 - 3AE2 + 1 = 0

The regular 11-sided polygon
The equation for AC for a regular 11-sided polygon is:
AC5 - AC4 + 4AC3 - 3AC2 + 3AC - 1 = 0


The increasing values of n seem to follow a pattern:

Odd values of n Power of equation
Prime Non-prime n Prime n Non-prime
5   2  
7   3  
  9    3
11   5  
n   (n - 1)/2  

The instance here that does not follow the pattern is when n = 9. Nine is not only a non-prime number but it is also a square.  The corresponding equations are not only of a lower power, than would be expected if they followed the pattern for primes, but some terms are missing from the equations.


Super Fibonacci Series
More complicated Super Fibonacci Series can be derived from these higher polygons.


Conjecture 1
The  ratios of the dimensions of n-sided regular plane figures involve simple instances of √n only when the equation defining the ratios is a quadratic.


Conjecture 2
The power of the equations, defining the ratios of the dimensions of n-sided regular plane figures, where n is a prime number, is given by (n - 1)/2.

Note: There are no diagonals when n is a prime number less than 5.

4. Dimensions of some more regular polygons where n is non-prime
The figures shown in red have half the number of sides of the black figures. The relationships between these smaller figures and the diagonals of the larger figures are interesting.

n = 4

 n = 6


n = 8 n = 10


Mike Holden - Nov 2005
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