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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
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.