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Polygons, Diagonals, and the Bronze Mean

Authors:
  • IES Bachiller Sabuco, Albacete, Spain

Abstract and Figures

. This article furthers the study of the Metallic Means and investigates the question of whether or not there exists a polygon corresponding to the Bronze Mean as the pentagon and the octagon correspond respectively to the Golden and Silver Means.
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Nexus Network Journal 9 (2007) 321-326 NEXUS NETWORK JOURNAL –VOL.9,NO. 2, 2007 321
1590-5896/07/020321-6
© 2007 Kim Williams Books, Turin
Antonia Redondo
Buitrago
I.E.S. Bachiller Sabuco
Departamento de Matemáticas.
Avenida de España 9
02002 Albacete S
PAIN
amcredondo@terra.es
Keywords: Bronze mean, diagonals,
metallic means, polygons
Research
Polygons, Diagonals,
and the Bronze Mean
Abstract. This article furthers the study of the Metallic Means
and investigates the question of whether or not there exists a
polygon corresponding to the Bronze Mean as the pentagon
and the octagon correspond respectively to the Golden and
Silver Means.
The Metallic Means Family (MMF) was introduced in 1998 by Vera W. de Spinadel
[1998; 1999]. Its members are the positive solutions of the quadratic equations
0
2
qpxx , where the parameters p and q are positive integer numbers. The more
relevant of them are the Golden Mean and the Silver Mean, the first members of the
subfamily which is obtained by considering
1 q . The members 3,2,1, p
p
V
of this
subfamily share properties which are the generalization of the Golden Mean properties. For
instance, they all may be obtained by the limit of consecutive terms of certain “
generalized
secondary Fibonacci sequences
” (GSFS) and they are the only numbers which yield
geometric sequences:


,,,,1,
1
,
1
,
1
,
32
23
ppp
p
pp
VVV
V
VV
with additive properties
 

212
,1
k
p
k
p
k
ppp
pp
VVVVV
  
,3,2,1,
11
21
k
p
k
p
k
p
k
p
VVV
.
However, the generalization of geometrical aspects presents some differences. If we
consider the rectangles of ratio
p
V
, they all have the property that the corresponding
gnomon is the union of
p squares. Figs. 1, 2 and 3 show the three first metallic rectangles.
Fig. 1
DOI 10.1007/s000 40 -007-0046-x
322 ANTONIA REDONDO BUITRAGO
Polygons, Diagonals, and the Bronze Mean
Fig. 2
Fig. 3
Also, the Silver Mean,
2
V
V
Ag
, and the Bronze Mean,
3
V
V
Br
, allow us to
construct spirals which generalize to that of the Golden Mean [Redondo Buitrago 2006].
On the other hand, the Golden Mean is linked to pentagonal symmetry, and the Silver
Mean to octagonal symmetry. Actually, in the regular pentagon, the ratio of the lengths of
the first diagonal to that of the side is
I
V
1
, and the Silver Mean,
Ag
V
V
2
, is the ratio
of the lengths of the second diagonal to that of the side in the regular octagon.
2
51
l
d
21
l
d
Fig. 4
So, it is natural to expect that there exists some regular polygon linked with the Bronze
Mean
Br
V
V
3
. However, in the classical literature we have not been able to find any
reference to this fact. Next, we are going to prove that it is not possible to construct some
diagonal in some regular polygon, with a ratio equal to the Bronze Mean. We will only
need very elementary geometrical arguments.
NEXUS NETWORK JOURNAL –VOL.9,NO. 2, 2007 323
Let there be a regular polygon with n sides. When we draw and number its diagonals d
1
,
d
2
,..., d
n1
, including the sides of the polygon as d
1
and d
n1
, as in Figs. 5 and 6, we observe
that the length of
d
i
is equal to d
n1
.
Fig. 5 Fig. 6
So, we are going to consider only the lengths of diagonals d
1
, d
2
,...,
0
n
d
, where n
0
stands for the integer part of

2:2n . Obviously, if n is an even number, as in fig. 6,
then the largest diagonal coincides with the diameter of the circumscribed circumference.
The ratios of the lengths of the diagonals are given by the law of cosines (fig. 7).
Fig. 7
1
2
2
,,3,2,1
2
cos22cos22
»
¼
º
«
¬
ª
n
k
n
k
d
kk
S
D
324 ANTONIA REDONDO BUITRAGO
Polygons, Diagonals, and the Bronze Mean
In particular, taking into account that the side of the polygon is
d
1
, we have
n
n
k
n
n
k
l
d
k
S
S
S
S
sin
sin
2
cos1
2
cos1
.
So, in order to research the possibility of existence of diagonals in some regular polygon
with
n t 5 sides, satisfying
1
2
2
,,3,2,30277563.3
2
133
»
¼
º
«
¬
ª
n
k
l
d
k
we will study the following periodic function family

5,
sin
sin
t n
n
n
x
xf
n
S
Fig. 8
First, we observe that all functions have period nT 2 . Moreover, in [0, 2n] the maxim
value of the function is
 
1
sin2
nnf
S
(fig. 9).
Fig. 9
NEXUS NETWORK JOURNAL –VOL.9,NO. 2, 2007 325
Notice that when the number of sides, n, is fixed by computing the values of the
corresponding function at the integer values of the interval [0,
n], we obtain the ratio of the
successive lengths of the side and diagonals of the corresponding polygon. Nevertheless,
only we need consider the first half of them. That is, the interval [0,
n/2].
If we take a look at the function graph above, we observe that the maximum for
n10 is
less than 3. Therefore, we must search out polygons with
nt10. But, by checking n 10,
11, 12, 13 y 14, we deduce that it is not possible to get

1
2
2
,,3,2,30277563.3
2
133
»
¼
º
«
¬
ª
n
kkf
n
n 10 11 12 13 14
3.236 (k=5) 3.229 (k=4) 2.732 (k=3) 2.771 (k=3) 2.802 (k=3)

kf
n
3.513 (k=5) 3.334 (k=4)
3.732 (k=5)
3.863 (k=6)
3.439 (k=4)
3.907 (k=5)
4.148 (k=6)
3.513 (k=4)
4.049 (k=5)
4.381 (k=6)
4.494 (k=7)
On the other hand, when k is fixed, the function

kfng
nk
is an increasing
function, and fig. 10 shows that we need to study only the values 2, 3 and 4, that is the
second, third and fourth diagonals.
Fig. 10
326 ANTONIA REDONDO BUITRAGO
Polygons, Diagonals, and the Bronze Mean
Hence, we find that
  
2
133
2
sin
2
sin
lim222...95.1
171615
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
fo
n
n
fff
n
S
S
  
2
133
3
sin
3
sin
lim333...82.2
171615
¸
¹
·
¨
©
§
S
¸
¹
·
¨
©
§
S
fo
n
n
fff
n
  
4
sin
4
sin
lim444...57.3
2
133
171615
¸
¹
·
¨
©
§
S
¸
¹
·
¨
©
§
S
fo
n
n
fff
n
.
Consequently, we can conclude: There exists no regular polygon in which the ratio of
the length of the diagonal to the side of the polygon is equal to the Bronze Mean.
References
REDONDO BUITRAGO, Antonia. 2006. Algunos resultados sobre Números Metálicos.
Journal of
Mathematics & Design
6, 1: 29-45.
SPINADEL, Vera W. de. 1998.
From the Golden Mean to Chaos.
2nd ed. Buenos Aires: Nueva
Librería
———. 1999. The family of metallic means.
Visual Mathematics
1, 3.
http//members.tripod.com/vismath1/spinadel/
About the author
Antonia Redondo Buitrago teaches Mathematics in a high school of Albacete (Spain). She is a doctor
in Applied Mathematics by University of Valencia (Spain). Her research interests and her
contributions in international journals and congresses include works about the fractional powers of
operators, continued fractions and the Metallic Means. At the present, in the domain of mathematics
and design, her collaborations with Vera W. Spinadel in the research of new properties of Metallic
Number Family are the most relevant.
... Alguns estudos mais recentes mostram a relação entre os números metálicos e as diagonais de polígonos regulares [3,12]. O número de ouro, por exemplo, está associado à medida de uma das diagonais do pentágono regular [10]. ...
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... design aspects of the metallic means. A notion of an associated polygon with the metallic means is analyzed in [10,13]. In a recent article [14], the author relates how different taffy (a candy made by pulling sticky sugar base) pullers are related to golden and silver ratios. ...
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In this article, we study the connection between Pythagorean triples and metallic means. We derive several interconnecting identities between different metallic means. We study the Pythagorean triples in the three-term recurrent sequences corresponding to different metallic means. Further, we relate different families of primitive Pythagorean triples to the corresponding metallic means.
... In a PBS Infinite Series broadcast, Perez-Giz raised the question of whether or not the same statement is true for φn for any n ≥ 3 [PG18]. This question was in fact answered negatively by Buitrago for the bronze ratio φ3 several years previously [Bui07]. The methods used were entirely numeric, and did not extend to n ≥ 4. Our first contribution is the following classification, which in fact allows an even more general definition of metallic mean. ...
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By viewing the regular N-gon as the set of Nth roots of unity in the complex plane we transform several questions regarding polygon diagonals into when a polynomial vanishes when evaluated at roots of unity. To study these solutions we implement algorithms in Sage as well as examine a trigonometric diophantine equation. In doing so we classify when a metallic ratio can be realized as a ratio of polygon diagonals, answering a question raised in a PBS Infinite Series broadcast. We then generalize this idea by examining the degree of the number field generated by a given ratio of polygon diagonals.
... No associated tiling is known nor can it be constructed in the same way as Penrose and Ammann-Beenker tilings because there exists no regular polygon with a diagonal-toedge ratio equal to β (ref. 21). ...
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The most striking feature of conventional quasicrystals is their non-traditional symmetry characterized by icosahedral, dodecagonal, decagonal or octagonal axes. The symmetry and the aperiodicity of these materials stem from an irrational ratio of two or more length scales controlling their structure, the best-known examples being the Penrose and the Ammann-Beenker tiling as two-dimensional models related to the golden and the silver mean, respectively. Surprisingly, no other metallic-mean tilings have been discovered so far. Here we propose a self-similar bronze-mean hexagonal pattern, which may be viewed as a projection of a higher-dimensional periodic lattice with a Koch-like snowflake projection window. We use numerical simulations to demonstrate that a disordered variant of this quasicrystal can be materialized in soft polymeric colloidal particles with a core-shell architecture. Moreover, by varying the geometry of the pattern we generate a continuous sequence of structures, which provide an alternative interpretation of quasicrystalline approximants observed in several metal-silicon alloys.
... This dynamic proportion appears when in a regular octagon of side L, we draw the diagonal D, as in Figure 1, and we calculate the ratio D : L. Many examples of this number involved in the geometry of the octagon can be found in [5]. The Silver Mean is the positive solution of the quadratic equation x 2 −2x−1=0, and this irrational number is one of the Metallic Numbers [24], [16], [13] , [14]. The second important proportion in the regular octagon is the Cordovan Proportion. ...
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... El Número de Plata es la solución positiva de la ecuación cuadrática x 2 −2x−1=0. Este número irracional es uno de los Números Metálicos ( [24], [16], [13], [14]). La segunda proporción importante en el octógono regular es la Proporción Cordobesa. ...
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Recently, higher-order topology has been expanded to encompass aperiodic quasicrystals, including those with eightfold or twelvefold rotational symmetry. The underlying mechanism for these high-order topological phases is generally protected by CnMz symmetry, resulting in the presence of n corner states. However, this mechanism is not applicable to other C2N quasicrystals when N is an odd number. In this work, we propose the realization of a second-order topological superconductor (SOTSC) within a sixfold symmetric bronze-mean hexagonal quasicrystal with six Majorana zero-energy modes. This SOTSC emerges from the combination of vertical and horizontal mirror symmetries, which flips the mass-term sign along the horizontal mirror-invariant line and produces Majorana zero-energy modes at each corner of the quasicrystal sample. Moreover, this mechanism can extend to quasicrystals with C4N+2 and C4N rotational symmetries, namely encompassing systems with C2N symmetry. Our findings provide useful guidance for achieving SOTSC in quasicrystals featuring C2N rotational symmetry and introduce bronze-mean hexagonal quasicrystals as a promising platform for exploring quasicrystal SOTSC.
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The article focuses on point spirals derived mainly from Fermat and Archimedean spirals. The notion of golden angle is extended to the set of metallic angles as an analogy to the set of metallic diameters introduced by Vera de Spinadel.
From the Golden Mean to Chaos Buenos Aires: Nueva Librería ———. 1999. The family of metallic means
  • Spinadel
  • W Vera
  • De
SPINADEL, Vera W. de. 1998. From the Golden Mean to Chaos. 2nd ed. Buenos Aires: Nueva Librería ———. 1999. The family of metallic means. Visual Mathematics 1 1, 3
Algunos resultados sobre Números Metálicos
  • Redondo Buitrago
REDONDO BUITRAGO, Antonia. 2006. Algunos resultados sobre Números Metálicos. Journal of Mathematics & Design 6, 1: 29-45.
Buenos Aires: Nueva Librería-. 1999. The family of metallic means
  • Vera W Spinadel
  • De
SPINADEL, Vera W. de. 1998. From the Golden Mean to Chaos. 2nd ed. Buenos Aires: Nueva Librería-. 1999. The family of metallic means. Visual Mathematics 1, 3.