THIUM version 1

1

A simple recursive geometrical construct method and some properties and occurrences of the Golden Ratio (Phi) also related to the transcendentals PI and to Euler’s constant (e) [1,2] * Andrei-Lucian Drăgoi (27 January – February 2017) (study research interval: 2007-2012) [3,4] * This article version: 1.1 * Abstract The Golden Ratio[URL2] (aka “golden mean”, “golden section” or “divine section”: standardly denoted with the capital Greek letter Phi  ) is the only positive (irrational) quadratic solution of the (minimal polynomial) equation 1 / x  x  1  x 2  x  1  x 2  x  1  0 so that  





5  1 / 2  1.618034 .

The purpose of this paper is to demonstrate some aspects recently discovered by the author of this paper: (0) a recursive geometrical method for constructing  (inspired from the Taoist religious iconography) which is more simple and elegant than Odom’s method; (1) an occurrence of  in the water molecule geometry; (2) a quasi-Pythagorean property of the  , e,   triad; (3) some other interesting quasi-exact relations between the elements of the  , e,   triad; (4) the tendency of the human systolic/diastolic blood pressure ratio to stabilize to a value close to  with age Keywords: Golden Ratio/section/mean (Phi), Odom’s method, Taoist religious iconography, water molecule geometry, quasi-Pythagorean property, human blood pressure

[1] Online preprint – version 1.0/1.1: DOI: 10.13140/RG.2.2.25312.69125 [2] Discovered in the 2007-2012 interval and registered in 2012 (in this initial variant) in “Plicul cu idei” (“The envelope of ideas”) (OSIM, Romania) with number: 300323/22.08.2012 (generic URL) [3] Romanian pediatrician specialist (with no additional academic title) undertaking independent research [4] Contact email: [email protected]

2 Introduction The Golden Ratio[URL2] (aka “golden mean”, “golden section” or “divine section”: standardly denoted with the capital Greek letter Phi  ) is the only positive (irrational) quadratic solution of the (minimal polynomial) equation 1 / x  x  1  x 2  x  1  x 2  x  1  0 so that  





5  1 / 2  1.618034 ,

with 1 /     1    1  1 /  and  2    1      1 .  is an algebraic number, more specifically an algebraic integer[URL2]. The property   1  1 /  can be expanded recursively to obtain the continued fraction form of  ,

1

  1

so that

1

. Additionally,  is the simplest continued fraction (with the slowest

1

1 ... convergence) of any irrational number [URL]: for this reason,  is one of the worst cases of Lagrange's approximation theorem and it is an extreme case of the Hurwitz inequality for Diophantine approximations. 1

The property   1   can be expanded recursively to obtain the continued square root (nested radical) form of  , so that   1  1  1  ... . also appears as a limit in the Fibonnaci series  F0   0 , F1   1 , F2   F0  F1  1 , F3   F1  F2  2 ,...Fn   Fn2  Fn1  , so that   lim  Fn / Fn1  . n

The property   1         2

2

0

1

can be generalized by iteratively multiplying with  ,

so that  n   n2   n1   n2   n   n1 , a property which is the main feature of the special case of



Fibonnaci series 0 , 1 ,  2   0  1  , 3   1   2  ,... n   n2   n1



Geometrically,  is the diagonal-to-side length ratio in a regular pentagon and regular pentagram,

so that   2sin  3 / 10  2sin 54o  . Angles close to

u  3 / 10  54o  u  arcsin   / 2   54o

often occur in some natural structures (like the water molecule geometry[URL2,URL3]), but also in Taoist iconography, as I shall demonstrate in the next sections of this paper.  generally appears in all geometrical shapes that have pentagonal symmetry, like in the regular polygons which have a general number of sides N S (n)  5n, with n  * (the decagon for example, in which N S  2   10 sides ).

 is also the defining property of the Golden spiral[URL2a,URL2b] is a special case of logarithmic spiral in which the growth factor of its variable radius is  for each quarter turn  / 2  , so that r (u )  

u2/ 

 3   , with the angle u  , u  0,... ,... ,... ,...2 ,...2 2  ,...2 n  ,...  2 2  

and r (u) “hides” the

special case of the Fibonnaci series r( / 2)  , r(2   / 2)  r( )   2 , r(3 / 2)  3 , ...r(n   / 2)   n 

 is also the defining property of the golden angle[URL2], which is a special case of dividing the circumference C  2 r of a generic circle with radius r in two fractional circumferences with lengths a  b so that a  b  C  2 r AND C / a   a  b  / a  a / b   : the golden angle   corresponds to the angle subtended

by

the

smaller

arc



of

 

ba



AND

has

a

value

  360 1  1/    360  2     360 /   180 3  5   3  5 ,   137.508  0.382(2 )  2.4radians . o

o

o

2

o

o

The golden angle   can be used in defining a special case of Fermat's spiral[URL2] (aka parabolic spiral) which often occurs in nature (including phyllotaxis). Fermat's spiral (which is a special case of Archimedean

spiral)

is

 r(u)2  a 2u, with the angle u 

defined

by

the

equation

r(u)  u1/2 , with the angle u 

,

or

and a  * . Fermat’s spiral has the very special property of traverses

3 equal annuli in equal turns: this property explains why Fermat’s spiral appears in the mature-disc phyllotaxis of plants (like the sunflower [Helianthus family] and daisy [Asteraceae family]) in which seeds are generated sequentially and initially have different sizes (also generating variable non-Fermat spirals), BUT finally gain the same approximate size when they all mature AND ALSO tending to approximate the shape of a Fermat’s spiral (the annuli of such a phyllotaxis-disc is composed by all the ~2D transverse sections of all the equal sized seeds AND the angle of succession in a single spiral arrangement approaches   ). The full model of

an “ideal” sunflower head proposed by H. Vogel in 1979 [URL] predicts the exact angle u  n  and radius

r  n  that define the polar coordinates of each n-th floret/seed (indexed with a positive integer n), so that: u  n     n 

2  n, with n  * AND r  c, n   c n , with c  * being a constant scaling factor , which 2

corresponds to a Fermat’s spiral defined by the equation r(c, u)  c u /  , with the angle u  and c  * [1]. The next figures (polar plots) illustrate the progressive construction of Vogel's model of sunflower disk for n=10 / 25 / 50 / 100 / 250 / 500 seeds.

90 120

30

60

120

90 50

60

40

150

30

20

150

30

30 20

r ( c_scale n ) 180

10

0

210

330 240

r ( c_scale n ) 180

10

210

300

330 240 270

u ( n)

u ( n) 90

90

120

60

150

30

40 20

r ( c_scale n ) 180

0

210

330 240

60 60

60 150

300

270

120

0

300

30

40 20

r ( c_scale n ) 180

0

210

330 240

300

270

270

u ( n)

u ( n)

4 90 120

150

90

100

150

120

60

200

60

150

30

30 100

50

r ( c_scale n ) 180

0

210

r ( c_scale n ) 180 210

330 240

0

330 240

300

300

270

270

u ( n)

u ( n)

(Source) In conclusion, Vogel proposed the single fixed (golden) angle   as the “ideal” candidate to produce the optimal design no matter how big the plant grows (this corresponds to   per new seed/leaf/cell). Similarly, once a seed is positioned on a phyllotaxis-disc (aka seed-head), the seed continues out in a straight line pushed out by other ”newborn” seeds, BUT approximately retaining the original angle on that phyllotaxis-disc relatively close to   : this golden angle is the key of the most advantageous packing design in a phyllotaxis-disc. This angle is ALSO very important in the spiral of leafs which is designed by the plant to maximally capture sunlight (for photosynthesis, the chemical process used by plants to convert carbon dioxide and water [using specific frequencies of light] in chemical energy subsequently stored in carbohydrate molecules): in this specific spiral pattern (defined by rotation with   per each new leaf), each new leaf will least obscure the leaves below and be least obscured by any future leaves above it.[URL1,URL2,URL3] [2,3,4] .







5  1 / 2  1.618034 , u  arcsin   / 2   54o and   360o 1  1 /    137.508o appear

in a relatively large palette of natural (including biological) structures/patterns. Shapes with pentagonal (floral) symmetry[URL-Google images] are frequently found in pentamerous petals of some plant flowers: in fact, the great majority of plants and animals has a (quasi-)symmetric anatomical structure, including pentagonal symmetry. The phyllotactic spirals form a distinctive class of patterns in nature, with its special   -based Fermat’s spiral being a tendency for many such phyllotactic spirals.  is also frequently used in arts (architecture, painting, design), mathematics (like in Golden-section search algorithm [discovered by the statistician Jack Kiefer in 1953] or in the algorithm of evenly arranging a given number of nodes on a sphere, which is related to the Thomson problem and has many practical applications) and physics (sunflower-inspired optimized geometries for sunlight capturing mirror layouts in solar power plants based on the   -based Fermat’s spiral). ***

5 Part 1. A recursive geometrical method for constructing  There are very many geometrical methods to construct  [URL1, URL2, URL3, URL4, URL5]. The author of this paper proposes a recursive geometrical method for constructing  (inspired from the Taoist religious iconography) which is more simple and elegant than Odom’s method. Step 1. This method starts from a pair of tangent circles: CR (with center O and radius R) and CR1 (with center O1 and radius 2R).

Step 2a. A line L is drawn intersecting CR1 in a point A and intersecting CR in both points B and C.

Step 2b. The lengths of the resulting segments are: OO1  OB  OC  AO1 / 2  R ,

BC  2OO1  AO1  2 R , AC  AO  OC 

 2R 

2

 R2  R  R

Step 3. It is obvious that AC / BC  BC / AB 



AC / BC  AC / AO1  AO1 / AB  BC / AB  







5 1

5  1 / 2   . Extensively,

6 Step 4. The previous figure can be expanded with an additional circle CR2 (with center O2 and radius 4R), with an additional line L1 drawn intersecting CR2 in a point A1 and intersecting CR1 in both points B1 and C1. Step 4 can be repeated recursively to any n-th step, so that for a circle CR(2,3…n) with center O(2,3…n) and radius (2,22…2n)R:

AnCn / BnCn  BnCn / An Bn 





5  1 / 2   , with

An1Cn1 / AnCn  An1Bn1 / An Bn  Bn1Cn1 / BnCn  2 . Another ways of reiteration is to build circles

CR(2,3…n) with centers O(2,3…n) placed in other positions and/or with radius R/2 (R/23,…R/2n)

The method presented here was inspired from the Taoist iconography: other such links were also discovered by other authors[URL]. This method has also some similarities [URL] with one construct proposed by John Arioni on his Facebook page called “CutTheKnotMath”.

Interestingly, the angle between AO and its corepondent segment (connecting A with the center of the small black inferior disk of Taoist mandala) is 2OAO1  2arctan OO1 / O1 A  2arctan 1 / 2   53.13o , which is an angle relatively close to u  arcsin   / 2   3 / 10  54o and to u 2  arctan





    51.83

from the right triangle defined by the triad of sides L, L  , L   used in the Egyptian pyramids.

o

7 The method proposed here can be also resumed by using two identical tangent circles CR1 (with center O1 and radius R) and CR2 (with center O2 and radius R).

This alternative resumed approach presented in the previous figure is very similar to the one proposed by Elliot McGucken[URL](undated), who probably inspired Tran Quang Hung to create an upside-down variant (undated)[URL], which is strikingly similar to McGucken’s and also a mirror-image rotate with 90 degrees of the my method (found in 2007 and republished in this paper). *** Part 2. The water molecule geometry as a quasi-  construct The water molecule[URL2,URL3] has a geometry that slightly varies with temperature and aggregation state between values in the interval 104o ,110o  , an interval which is relatively centered in 96.7%

2u  6 / 10  108o . At room temperature, the angle HOH is BAC  104.45o  2u (such in the next figure representing the H2O/HOH water molecule as an ABC triangle). HOH  104.45o is slightly smaller 101.4%

than the usual angle uth  109.47o  2u between the bonds of a tetrahedral molecule (with one central atom and other four identical [to each other] peripheral atom).

In the water molecule at room temperature (with a geometry graphed in the previous ABC triangle,



 



with M being the middle point of BC segment), BM / BA  sin BAM  sin BAC / 2 

8





97.7%

BC / BA  2 BM / BA  2sin BAC / 2  1.581   . 100.3%

Anticipating the last part of this paper (and based on the relative closeness 97.3%

 2  e2  1.575   / 2 and

97.1%

 2  e2  1.575   , with  / 2   ), one can find: 100.9% 97.7%       100.6% 103.9%  BC / AB  2sin BAC / 2  1.581    / 2 and 2sin  uth / 2   1.633    / 2 . 103.7% 2 2 100.4% 2 2    e    e  





In conclusion, both HOH / 2  52.23o , 2OAO1  53.13o (from the Taoist mandala) and the angle

u 2  arctan

    51.83

o

(used in the Egyptian pyramids) are all relatively close to each-other and also

close to u  arcsin   / 2   3 / 10  54o . *** Part 3. The quasi-Pythagorean property of the triad o reals  , e,  

A Pythagorean triad/triple[URL2,URL3] (also related to the notorious Fermat’s last theorem) is defined as a triad of positive integers  a, b, c  which a common property that a 2  b2  c 2 . 100.3%

Based on the anticipated relative closeness

97.1%

 2  e2  1.575   / 2 and  / 2   ,  has the

interesting property that it forms a quasi-Pythagorean triad of reals  , e,   with the two well-known 1   dx  3.1416...  AND transcendental numbers[URL2] PI    CircleCircumference / CircleDiameter   1 1  x2  

Euler’s 102.7%

 

number

e  lim 1  1 / x 

 2  e2 .

More

x

x



 2.718... ,

101.4%

so

that:

 2  e2   2  100.7%

specifically,

  2  e2  3.163   ,  

105.5%

 2   2  e2  97.3%

  2  e2  1.575    

and

99.1%

  2   2  2.693  e .  

The fact that

100.3%    /2   e  1.575 97.3% makes the hypothetical triangle (with sides defined by the    2

2

 , e,  ) triad to be very close to a 30-60-90 right triangle. Applying the law of cosines in the  , e,  sided

triangle,

one

can

estimate:

99%   2  e2   2  o arcos   89.1  90o ,   2e 

99.8% 103.3%   2   2  e2   e2   2   2  o o o o arcos   59.9  60 arcos and     30.9  30 (see the next figure)  2      2  e  

9

There are also other interesting quasi-exact mathematical relations (discovered by the author) between the

elements

of

 , e, 

the

triad,

such

as:

 4/  21/  2.718241   99.998%  e , 2

2

6  e   3 / 7  100.003%   2   5  e   / 7  3.141623  100.001%     6  e   3 / 7  3.141632  100.001%     etc.   7   / 5  e   1.618018  99.999%    ,  3 7   2 / 6  e  1.61802  99.9992%            7   / 5     2.718255  99.999%  e  2 3   7   /  6     2.718214   99.997%  e *** Part 4. The tendency of human systolic/diastolic blood pressure ratio to stabilize around 

The human systemic blood pressure (BP) has two physiologic extreme values called systolic blood pressure (SBP) and dyastolic blood pressure (DBP). The difference BP  SBP  DBP is called differential (systemic) BP. SBP and DBP vary with age (from birth until death) and the author of this article brings into attention the observation that both ratios ratio1  SBP / DBP and ratio2  with age

DBP DBP tends to  BP SBP  DBP

with age

stabilize around  with age, so that ratio1 /   1 AND ratio2 /   1 Table 4-1. The tendency of both ratios ratio1  SBP / DBP and

DBP DBP tends to stabilize around  with age.  BP SBP  DBP Sources: URL1-BP in children[Table], URL2-BP in adults [Table] ratio2 

Age (years)

1

SBP ( percentile 50 and for average body mass index for age) 85

SBP (percentile 50 and for average body mass index for age) 37

2

88

42

3

91

46

4

93

50

10 5

95

53

6

96

55

7

97

57

8

99

59

9

100

60

10

102

61

11

104

61

12

106

62

13

108

62

14

111

63

15

113

64

16

116

65

17

118

67

25

117

74

30

118

74

35

120

75

40

122

76

45

122

76

50

126

78

55

128

79

60

129

80

65

131

81

69

132

81

11

Acknowledgements I would like to express all my sincere gratitude and appreciation to all my mathematics, physics, chemistry and medicine teachers for their support and fellowship throughout the years, which provided substantial and profound inner motivation for the redaction and completion of this manuscript. *** Competing interests Author has declared that no competing interests exist. *** ENDNOTE ADDITIONAL REFERENCES (in order of citation in this article)

[1] Vogel H. (1979) . “A better way to construct the sunflower head”, Mathematical Biosciences. 44 (44): 179–189. doi:10.1016/0025-5564(79)90080-4 (URL) [2] Ron Knott (1996-2016). “Fibonacci Numbers and Nature - Part 2: Why is the Golden section the "best" arrangement?”, web article extracted from “Dr. Ron Knott's Fibonacci Numbers and the Golden Section” (URL) (as accessed on 26.01.2017) [3] Douady S. and Couder Y.(1996). “Phyllotaxis as a Dynamical Self Organizing Process Part I: The Spiral Modes Resulting from Time-Periodic Iterations”, Journal of Theoretical Biology (1996) 178, 255–274 (URL1, URL2-Douady CV) [4] Douady S. and Couder Y.(1996). “Phyllotaxis as a Dynamical Self Organizing Process Part II: The Spontaneous Formation of a Periodicity and the Coexistence of Spiral and Whorled Patterns”, Journal of Theoretical Biology (1996) 178, 275–294 ((URL1, URL2-Douady CV))