Vol. 128 (2015)
No. 2-B
ACTA PHYSICA POLONICA A
Special issue of the International Conference on Computational and Experimental Science and Engineering (ICCESEN 2014)
Analytical Derivation of the Fuss Relations
for Bicentric Hendecagon and Dodecagon
M. Orlića,∗ and Z. Kalimanb
University of Applied Sciences, Zagreb, Croatia
University of Rijeka, Department of Physics, Rijeka, Croatia
a
b
In this article we present analytical derivation of the Fuss relations for n = 11 (hendecagon) and n = 12
(dodecagon). We base our derivation on the Poncelet closure theorem for bicentric polygons, which states that if
a bicentric n-gon exists on two circles then every point on the outer circle is the vertex of same bicentric n-gon.
We have used Wolfram Mathematica for the analytical computation. We verified results by comparison with earlier
obtained results as well as by numerical calculations.
DOI: 10.12693/APhysPolA.128.B-82
PACS: 02.70.Wz
1. Introduction
• There is no bicentric n-gon whose circumcircle is
C1 and incircle C2 .
The bicentric n-gon is polygon with n sides that
are tangential for incircle and chordal for circumcircle.
The connection between the radii of the circles and the
distance of their centers represents the Fuss relation for
the certain bicentric n-gon. The problem of finding relations between R, r and d (where R is a radius of circumcircle, r is a radius of incircle, and d is distance between their centers) is considered in Ref. [1] as one of
100 great problems of elementary mathematics. Swiss
mathematician Nicolaus Fuss (1755–1826) was the first
who has found these relations for the bicentric quadrilateral, pentagon, hexagon, heptagon and octagon. The corresponding formula for triangle was already given by Euler. Since that time, mathematicians proved the Fuss relation for these and some other bicentric polygons using
various methods based on geometry, trigonometric transformation (Steiner, Jacobi, Dörrie, Chaundy, etc.), or the
Jacobi elliptic function and a complete elliptic integral of
the first kind (Richelot) [1]. Radić [2–5] used different
methods based on elementary geometry or by introducing adequate functions in obtaining the Fuss relations for
bicentric polygons which can be multiply circumscribed.
Also he found connections between the Fuss relations for
bicentric polygons with incircle and with ones with excircle. In Refs. [6, 7] we have given analytical derivation
of the Fuss relations for pentagon and hexagon.
Here we derive the Fuss relations for hendecagon and
dodecagon. Our derivations are based on the Poncelet
closure theorem for bicentric polygons [8]. This theorem
can be stated as follows:
Theorem 1. Let C1 and C2 be any given two circles
in a plane such that one is completely inside of the other.
Then only one of the following two assertions is true:
• There are infinitely many bicentric n-gons whose is
circumcircle C1 and incircle C2 . For every point A1
in C1 there is bicentric n-gon A1 ... An whose circumcircle is C1 and incircle C2 .
We use coordinate system with x-axes defined by centers of the circles, and starting point of our bicentric polygon is on those axes. In actual derivation of the Fuss relations we use only upper half-plane (y ≥ 0) because such
polygons are symmetric with respect to x-axes. Slightly
different derivation is for odd and for even sided polygons [6], as is also described later in this article.
2. Hendecagon
Polygon with eleven sides is called hendecagon.
The procedure is general for all odd sided bicentric polygons. Let us denote outer circle with radius R by C1 , and
chose coordinate system with origin in the center of C1 .
Inner circle C2 has radius r and center in point I(d, 0)
(Fig. 1a).
Using the Poncelet closure theorem we choose first tangent t1 on C2 in point P (r + d, 0). Tangent intersect
circle C1 in vertices A1 (r + d, y1 ) and A11 (r + d, y11 ).
(2) (2) (2)
From vertex A2 we get next vertex A3 (x3 , y3 ) where
abscissa is calculated as intersection of tangent t3 and
(2)
circle C1 . Obtained expression for x3 is
(2)
x3 =
r
(1)
2
2 (p3 q 3 +p3 q 2 −p3 q−p3 −p2 q 3 +2p2 q 2 −p2 q−pq 3 +pq 2 +q 3 )
× p7 q 6 + 2p7 q 5 − p7 q 4 − 4p7 q 3 − p7 q 2 + 2p7 q + p7
−p6 q 7 + 6p6 q 6 − p6 q 5 − 10p6 q 4 + 5q 4 + 5p6 q 3 + 2p6 q 2
−3p6 q + 2p6 + 2p5 q 7 + p5 q 6 − 12p5 q 5 + 6p5 q 4 + 6p5 q 3
+p5 q 2 − 4p5 q + p4 q 7 − 10p4 q 6 − 6p4 q 5 + 12p4 q 4 + 5p4 q 3
∗ corresponding
author; e-mail: [email protected]
−2p4 q 2 − 4p3 q 7 + 5p3 q 6 + 6p3 q 5 − 5p3 q 4 + 8p3 q 3 +p2 q 7
(B-82)
Analytical Derivation of the Fuss Relations. . .
B-83
+4608d18 r8 R4 − 17408d16 r10 R4 + 24576d14 r12 R4
−16384d12 r14 R4 − 546d24 rR5 + 6480d22 r3 R5
−16032d20 r5 R5 + 2304d18 r7 R5 + 4608d16 r9 R5
−4096d14 r11 R5 + 8192d12 r13 R5 + 455d24 R6
+936d22 r2 R6 + 3696d20 r4 R6 + 5760d18 r6 R6
−34560d16 r8 R6 + 71680d14 r10 R6 − 45056d12 r12 R6
+2184d22 rR7 − 20768d20 r3 R7 + 30080d18 r5 R7
+21504d16 r7 R7 − 2048d14 r9 R7 + 24576d12 r11 R7
−1365d22 R8 − 3432d20 r2 R8 − 11440d18 r4 R8
Fig. 1. (a) Symmetric bicentric hendecagon A1 . . . A11 .
(b) Symmetric bicentric dodecagon A1 . . . A12 . They are
inscribed in the circle C1 and outscribed to C2 . Circle C1 with the radius R is centered at the origin, while
C2 is centered at I with radius r. Distance between
centers is d.
−34560d16 r6 R8 + 91392d14 r8 R8 − 104448d12 r10 R8
−6006d20 rR9 + 44440d18 r3 R9 − 24480d16 r5 R9
−59136d14 r7 R9 − 19968d12 r9 R9 − 55296d10 r11 R9
−24576d8 r13 R9 − 32768d6 r15 R9 + 3003d20 R10
2 6
2 5
2 4
7
6
5
7
+2p q − p q − 2p q + 2pq + 3pq − 4pq − q +2q
6
,
where p and q are given by
R+d
R−d
p=
, q=
.
(2)
r
r
Because of symmetry, vertex A6 is on x-axes and has coordinate (−R, 0). From this point we draw tangent on C2
in the upper half-plane t6 whose intersection with C1 is
vertex A5 . In the same way we get vertices A4 , and
(4) (4)
(4)
A3 (x3 , y3 ) with abscissa
r(p + q)
(4)
x3 =
2
2
4
2
2
2p (p + 2p q − 2p4 q 2 − 3q 4 + 2p2 q 4 +p4 q 4 )
× −2p8 + p10 + 8p6 q 2 − 4p8 q 2 − 4p10 q 2 − 12p4 q 4 +6p6 q 4
+6p10 q 4 + 8p2 q 6 − 4p4 q 6 − 20p6 q 6 + 20p8 q 6 − 4p10 q 6
−2q 8 + p2 q 8 − 8p4 q 8 + 22p6 q 8 − 14p8 q 8 + p10 q 8 .
(3)
(2) (2)
(2)
(4) (4)
(4)
A3 (x3 , y3 ) and A3 (x3 , y3 )
A3 , and the coordinates must be the
Of course, vertices
are the same vertex
same. We subtract values of abscissas and equate them
with zero and after inserting (2) we get
(d + r + R) F11 (R, r, d) = 0,
(4)
where only factor F11 (R, r, d) given by relation (5) can
vanish and it gives the Fuss relation for hendecagon, i.e.
F11 (R, r, d) = 0.
F11 (R, r, d)=−d30 −6d28 rR + 104d26 r3 R − 512d24 r5 R
+896d22 r7 R − 512d20 r9 R + 15d28 R2 + 12d26 r2 R2
+64d24 r4 R2 − 192d22 r6 R2 +256d20 r8 R2 +84d26 rR3
−1216d24 r3 R3 + 4416d22 r5 R3 − 4096d20 r7 R3
−105d26 R4 −156d24 r2 R4 −720d22 r4 R4 +320d20 r6 R4
+8580d18 r2 R10 + 23760d16 r4 R10 + 94080d14 r6 R10
−112896d12 r8 R10 +56320d10 r10 R10 +28672d8 r12 R10
+16384d6 r14 R10 + 12012d18 rR11 − 66528d16 r3 R11
−14976d14 r5 R11 + 64512d12 r7 R11 + 33792d10 r9 R11
+57344d8 r11 R11 + 16384d6 r13 R11 − 5005d18 R12
−15444d16 r2 R12 − 34848d14 r4 R12 − 153216d12 r6 R12
+48384d10 r8 R12 − 5120d8 r10 R12 − 4096d6 r12 R12
−18018d16 rR13 + 70752d14 r3 R13 + 63168d12 r5 R13
−21504d10 r7 R13 − 12800d8 r9 R13 − 24576d6 r11 R13
+6435d16 R14 + 20592d14 r2 R14 + 36960d12 r4 R14
+161280d10 r6 R14 + 37632d8 r8 R14 + 12288d6 r10 R14
+20592d14 rR15 − 52800d12 r3 R15 − 78336d10 r5 R15
−21504d8 r7 R15 − 12288d6 r9 R15 − 6435d14 R16
−20592d12 r2 R16 − 28512d10 r4 R16 − 11360d8 r6 R16
−57600d6 r8 R16 − 20480d4 r10 R16 − 4096d2 r12 R16
−18018d12 rR17 + 26136d10 r3 R17 + 54720d8 r5 R17
+26496d6 r7 R17 + 12288d4 r9 R17 + 2048d2 r11 R17
+5005d12 R18 + 15444d10 r2 R18 + 15840d8 r4 R18
+48960d6 r6 R18 + 27648d4 r8 R18 + 7168d2 r10 R18
B-84
M. Orlić, Z. Kaliman
inserting (2) we get
−2R d4 −2d2 r2 −2d2 R2 −2r2 R2 +R4 F12 (R, r, d)=0, (8)
where factor F12 (R, r, d) is given as
F12 (R, r, d) = d32 − 16d30 r2 + 16d28 r4 − 16d30 R2
+12012d10 rR19 − 7040d8 r3 R19 − 22720d6 r5 R19
−11264d4 r7 R19 − 3072d2 r9 R19 − 3003d10 R20
−8580d8 r2 R20 − 6160d6 r4 R20 − 12480d4 r6 R20
−4864d2 r8 R20 − 6006d8 rR21 − 176d6 r3 R21
+208d28 r2 R2 + 224d26 r4 R2 − 1792d24 r6 R2
+5088d4 r5 R21 + 1792d2 r7 R21 + 1365d8 R22
+5888d22 r8 R2 − 8192d20 r10 R2 + 4096d18 r12 R2
+3432d6 r2 R22 + 1584d4 r4 R22 + 1408d2 r6 R22
+120d28 R4 − 1232d26 r2 R4 − 3920d24 r4 R4
+2184d6 rR23 +864d4 r3 R23 −384d2 r5 R23 −455d6 R24
+16128d22 r6 R4 − 40960d20 r8 R4 + 28672d18 r10 R4
−936d4 r2 R24 − 240d2 r4 R24 − 546d4 rR25
−4096d16 r12 R4 − 560d26 R6 + 4368d24 r2 R6
−280d2 r3 R25 − 32r5 R25 + 105d4 R26 + 156d2 r2 R26
+23744d22 r4 R6 − 62720d20 r6 R6 + 121600d18 r8 R6
+16r4 R26 + 84d2 rR27 + 32r3 R27 − 15d2 R28
−20480d16 r10 R6 +57344d14 r12 R6 −65536d12r14 R6
(5)
−12r2 R28 − 6rR29 + R30 .
−82544d20 r4 R8 +134400d18 r6 R8 −204800d16 r8 R8
3. Dodecagon
Polygon with twelve sides is called dodecagon.
The procedure is general for all even side bicentric polygons. Let us denote outer circle with radius R by C1 and
choose coordinate system with origin in the center of C1 .
Inner circle C2 has radius r and center in point I(d, 0)
(Fig. 1b).
For all even side bicentric polygons the starting point
is A1 (R, 0), which also represents the first vertex of polygon. The first tangent t1 on C2 in vertex A1 intersect
circle C1 in vertex A2 (x2 , y2 ). By analogous procedure,
(3) (3) (3)
we calculated vertex A4 (x4 , y4 ) as intersection of tangent t4 (pulled from vertex A3 ) and circle C1 . The ab(3)
scissa of vertex A4 are given by
−r (p+q)
(3)
x4 =
2
2
4
2
2
2q (−3p +2p q +2p4 q 2 +q 4 −2p2 q 4 +p4 q 4 )
× −2p8 + 8p6 q 2 + p8 q 2 − 12p4 q 4 − 4p6 q 4 − 8p8 q 4 +8p2 q 6
+8p2 q + 6p4 q 6 − 20p6 q 6 + 22p8 q 6 − 2q 8 − 4p2 q 8 +20p6 q 8
−14p8 q 8 + q 10 − 4p2 q 10 + 6p4 q 10 − 4p6 q 10 + p8 q 10 . (6)
Using the symmetry reason, we introduce substitution with R to −R in the coordinates of the vertex
(3) (3) (3)
A4 (x4 , y4 ) and we get the coordinates of the vertex
(5) (5) (5)
A4 (x4 , y4 ), where the abscissa is given by
(p + q) r
(5)
x4 =
2
2
4
2
2
2p (p + 2p q − 2p4 q 2 − 3q 4 + 2p2 q 4 + p4 q 4 )
× −2p8 + p10 + 8p6 q 2 − 4p8 q 2 − 4p10 q 2 − 6p6 q 4 +6p10 q 4
+8p2 q 6 − 4p4 q 6 − 20p6 q 6 + 20p8 q 6 − 4p10 q 6 − 2q 8 + p2 q 8
−8p4 q 8 + 22p6 q 8 − 14p8 q 8 + p10 q 8 .
(7)
(3)
(3)
(3)
(5)
+65536d10 r16 R6 + 1820d24 R8 − 10192d22 r2 R8
(5)
(5)
The vertices A4 (x4 , y4 ) and A4 (x4 , y4 ) are the
same, i.e. must have the same coordinates. We subtract
values of abscissas and equate them with zero and after
−28672d14 r10 R8 −258048d12 r12 R8 +65536d10 r14 R8
−65536d8 r16 R8 − 4368d22 R10 + 16016d20 r2 R10
+189728d18 r4 R10 − 161280d16 r6 R10
+232960d14 r8 R10 + 28672d12 r10 R10
+401408d10 r12 R10 + 65536d8 r14 R10
+65536d6 r16 R10 + 8008d20 R12 − 16016d18 r2 R12
−306768d16 r4 R12 + 75264d14 r6 R12
−229376d12 r8 R12 + 28672d10 r10 R12
−258048d8 r12 R12 − 65536d6 r14 R12 −11440d18 R14
+6864d16 r2 R14 + 359040d14 r4 R14 +75264d12 r6 R14
+232960d10 r8 R14 −28672d8 r10 R14 +57344d6 r12 R14
+12870d16 R16 + 6864d14 r2 R16 − 306768d12 r4 R16
−161280d10 r6 R16 −204800d8 r8 R16 −20480d6 r10 R16
−4096d4 r12 R16 − 11440d14 R18 − 16016d12 r2 R18
+189728d10 r4 R18 +134400d8 r6 R18 +121600d6 r8 R18
+28672d4 r10 R18 + 4096d2 r12 R18 + 8008d12 R20
+16016d10 r2 R20 − 82544d8 r4 R20 − 62720d6 r6 R20
−40960d4 r8 R20 − 8192d2 r10 R20 − 4368d10 R22
−10192d8 r2 R22 + 23744d6 r4 R22 + 6128d4 r6 R22
+5888d2 r8 R22 + 1820d8 R24 + 4368d6 r2 R24
B-85
Analytical Derivation of the Fuss Relations. . .
−3920d4 r4 R24 − 1792d2 r6 R24 − 560d6 R26
Acknowledgments
−1232d4 r2 R26 + 224d2 r4 R26 + 120d4 R28
This work was supported by the University of Rijeka,
project number 13.12.1.4.07.
+208d2 r2 R28 + 16r4 R28 − 16d2 R30
−16r R + R
(9)
can vanish and it gives the Fuss relation for dodecagon
i.e. F12 (R, r, d) = 0.
We got required relations for hendecagon and dodecagon by subtracting values of abscissas of appropriate
vertices. Also we got the same relation by subtracting
values of ordinates of same vertices.
2
30
References
32
4. Conclusion
Throughout history mathematicians proved the Fuss
relations for bicentric polygons using different methods
based on geometry, trigonometric transformations or the
Jacobi elliptic function and a complete elliptic integral
of the first kind. Until now we have derived the Fuss
relations using analytical geometry for n from 4 to 12
by method described for hendecagon and dodecagon in
this article. Our goal is analytical derivation of the Fuss
relation for n = 13.
[1] H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, New York 1965.
[2] M. Radić, Z. Kaliman, Rad HAZU, Matematičke
znanosti 503, 21 (2009) (in Croatian).
[3] M. Radić, Rad HAZU, Matematičke znanosti 519,
145 (2014) (in Croatian).
[4] M. Radić, Math. Commun. 19, 139 (2014).
[5] M. Radić, Compt. Rend. Math. 348, 415 (2010).
[6] Z. Kaliman, M. Orlić, in: 6 Int. Conf. Aplimat 2007,
Vol. 3, 2007, p. .
[7] M. Orlić, Z. Kaliman, N. Orlić, in: 6 Int. Conf. Aplimat 2007, Vol. 3, 2007, p. 83.
[8] http://mathworld.wolfram.com/PonceletsPorism.html.