Tokyo Polytechnic University Repository / 東京工芸大学学術リポジトリ

14KJ00004761218論文Paperdepartmental bulletin pape

満, 柏

2783

MAN, Bai

2784

A Study of Balance in Two-Dimensional Art

4KJ00004878593論文Articledepartmental bulletin pape

中根, 静男

ナカネ, シズオ

Nakane, Shizuo

1174

Turning deformation of cubic polynomials with
critical orbit relations
Shizuo Nakane ∗
In this note, turning deformation of cubic polynomials is investigated. Especially, turning
deformation of maps with critical orbit relations is studied. It is described via the Dehn twist and
transitivity of the monodromy is shown.
1 Introduction
Consider the family of cubic polynomials of the form :
P (z) = Pb,λ(z) = z
3 + bz2 + λz, b ∈ C,
where λ ∈ D \ {0}. Thus the origin is an attracting ﬁxed point of multiplier λ. We
consider the central hyperbolic component A of this family, that is, the subset of
maps both of whose critical points belong to the immediate basin BP of the origin.
The critical points of P are expressed by c± =
−b±√b2 − 3λ
3
. Note that these
functions c± are multi-valued and interchanged through some paths. We ﬁx their
branches soon. Put Aλ = {b ∈ C; (b, λ) ∈ A}.
Consider the Kænigs coordinate ϕP of the origin. It is a conformal map around
the origin satisfying ϕP (0) = 0 and ϕP (P (z)) = λϕP (z). It is analytically contin-
ued by this functional equation until it meets a critical point. It is deﬁned up to
multiplication by non-zero constants. Thus we normalize it so that ϕP extends to a
conformal isomorphism ϕP : UP → D, where UP is a simply connected domain, ∂UP
contains c+ and ϕP (c+) = 1. This normalization uniquely determines ϕP . This also
ﬁxes the branch of c±.
For a complex number s ∈ H+ = {s ∈ C,Re s > 0}, put s(z) = z|z|s−1 and we
deﬁne a P -invariant almost complex structure σs by
σs =
{
(s ◦ ϕP )∗σ0 on UP ,
σ0 on C \K(P ).
Then, by the Measurable Riemann Mapping Theorem, there exists a unique quasi-
conformal map hs such that σs = h
∗
sσ0, hs(0) = 0 and Ps := hs ◦P ◦ h−1s ∈ A. Thus
we deﬁne a holomorphic map WP : H+ → A by WP (s) = Ps. This quasiconformal
∗
Received Sept. 20, 2007
Professor, General Education and Research Center, Tokyo Polytechnic University
19 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
deformation is deﬁned in Petersen-Tan Lei [PT] and is called the Branner-Hubbard
deformation. Note that such a deformation was originally deﬁned for Bo¨ttcher co-
ordinate and developed in Branner-Hubbard [BH1, BH2]. They called it wringing.
See also Branner [B].
It easily follows that ϕPs = s ◦ ϕP ◦ h−1s and P ′s(0) = s(P ′(0)). Thus s(λ) = λ
for s = s(k) := 1 +
2kπi
log |λ| , k ∈ Z. Thus we can deﬁne the monodromy map
Mλ : Aλ → Aλ by Mλ(P ) = Ps(1). Note that 1+ti ◦ 1+t′i = 1+(t+t′)i. Hence it
follows that ks(1) = s(k) and M
k
λ (P ) = Ps(k).
We call the curve {P1+ti; t ∈ R} the turning curve through P and the corre-
sponding deformation turning deformation.
In this note, we investigate the monodromy map Mλ. The monodromy map is the
ﬁrst return map to Aλ along the turning curve. Put Aλ(k,m) = {b ∈ Aλ;P k(c−) =
Pm(c+), P
k−1(c−) = Pm−1(c+)}. These relations are kept invariant under turning
deformation. We will show that the monodromy map acts transitively on Aλ(k,m).
This extends the result in Yazawa [Y], where the transitivity on Aλ(k, 0) is shown.
2 Dehn twists
In this section, we describe the monodromy map Mλ via the Dehn twist on an
annulus. We may assume 0 < λ < 1.
Let A1 = {z ∈ C; 1 < |z| < λ−1} be an annulus. The Dehn twist D1 on A1 is
just deﬁned by
D1(r exp(2πit)) = r exp(2πi(t− log r
log λ
)).
Put s = 1 +
2πi
log λ
. Then the map s(z) = z|z|s−1 is a qc-map A1 → A1 of the
form :
s(re
2πit) = r exp(2πi(t +
log r
log λ
)).
Hence its inverse −1s (re
2πit) = r exp(2πi(t− log r
log λ
)) is just the Dehn twist D1 on the
annulus A1. See Figure 1. It spirals once around the origin.
Consider the inverse ψP of the map ϕP . It is univalent on D but it has multiple
branches on C\{1}. Put ψP,0 be the univalent extension of ψP to C\[1,∞). Let ψP,n
be the univalent extension of ψP to UˆP,n = 
−1
s(n)(C \ [1,∞)) and UP,n = ψP,n(UˆP,n).
Note that
hs(n)(UP,n) = hs(n) ◦ ψP,n ◦ −1s(n)(C \ [1,∞))
= (s(n) ◦ ϕP ◦ h−1s(n))−1(C \ [1,∞))
= ψPs(n),0(C \ [1,∞)).
The following Figures 2 and 3 give the maps ψP,0 and ψP,1 : UˆP,1 → UP,1 re-
spectively in case P ∈ Aλ(1, 0). Note that the set Aλ(1, 0) consists of 4 points,
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 20 
1 λ−1
Figure 1: Dehn twist
say ±b and ±b¯. Since P−b(−z) = −Pb(z) holds, there exist essentially two maps.
The dynamics in Figure 3 above is deformed by hs(1) to the dynamics in Figure 3
below, which turns out to be of Mλ(Pb) = Pb¯. Thus the monodromy is transitive on
Aλ(1, 0).
ψP,0
c−
c+
1 λ−10
0
Figure 2: the map ψP,0, the shaded region is UP,0
The following ﬁgures express the parameter space A1/2 and the basin for Pb ∈
A1/2(1, 0) with b  −1.25− 0.464i.
To show the transitivity of the monodromy on Aλ(k, 0), we need to know the
pattern of the regions UP,j, which is possible via the Dehn twists. Let An be the
annulus in BP corresponding to the equipotential λ−n+1 < r < λ−n for 0 ≤ n ≤ k.
The critical point c− must be one of the 2k preimages of c+ on the outer boundary
∂outAk of Ak. The outer boundary of An consists of 2
n preimages of ∂UP , which
we denote γ(n, j (mod 2n)) anticlockwise. Here we put γ(n, 0) the one contained in
UP,0. See Figure 4 in case n = 2.
Since the map P : An → An−1 is of degree two, the inner boundary of An
Turning deformation of cubic polynomials with critical orbit relations 21 
ψP,1
c−
c+
1 λ−10
0
1 λ−1
c+
c−
ψPs(1),0
0 0
Figure 3: the map ψP,1 (above) and the map ψPs(1),0 (below)
is mapped by −1s(1) to the second half of its outer boundary. This implies that
UP,1 contains γ(n, 2
n − 1) for each n. The same argument says that UP,j contains
γ(n, 2n − j) for each n. Thus this covers all possible locations of the critical point
c−. This proves the transitivity of Mλ on Aλ(k, 0). See [Y].
3 Monodromy of cubic polynomials
We will show the transitivity of the monodromy on Aλ(k,m) with m ≥ 1.
Proposition 3.1. Aλ(k,m) ∪ Aλ(m, k) = ∅ if k,m ≥ 2.
proof. The proof is a modiﬁcation of that of Proposition 2 in Yazawa [Y].
Here we use his notations. Let f = fa,b = z
3 − 3(a + b)z2/2 + 3abz be a cubic
polynomial with a ﬁxed point at the origin and distinct critical points a, b. Put
Fk,m(a, b) = f
k(b) − fm(a), Sk,m = {(a, b) ∈ C2;Fk,m(a, b) = 0} and Rk,m =
{(a, b) ∈ C2; Fk,m(a, b)
Fk−1,m−1(a, b)
= 0}. Since h(z)− h(w)
z − w
∣∣
z=w
= h′(w), it follows
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 22 
c+0
γ(2, 3)
γ(2, 2)
γ(2, 1)
γ(2, 0)
γ(1, 0)
γ(1, 1)
c+0
γ(2, 3)
γ(2, 2)
γ(2, 1)
γ(2, 0)
γ(1, 0) γ(1, 1)
Figure 4: Shaded regions indicate UP,0 (left) and UP,1 (right)
Rk,m∩Sk−1,m−1 = {(a, b) ∈ C2; fk−1(b) = fm−1(a) equals to a critical point a or b}.
The following lemma was proved in [Y] in case m = 0. Then since Rk,m = {(a, b) ∈
C
2; z2 + zw+w2−3(a+ b)(z+w)/2+3ab = 0, z = fk−1(b), w = fm−1(a)}, it follows
that (0, 0) ∈ Rk,m.
Lemma 3.1. (cf. [Y] Lemma 4) The point (0, 0) is isolated in Rk,m ∩ ({0} × C).
proof. Since Rk.m ∩ {a = 0} = {(0, b) ∈ C2; (fk−10,b (b))2 − 3bfk−10,b (b)/2 = 0}, we
have only to show that (fk−10,b (b))
2 − 3bfk−10,b (b)/2 ≡ 0. It is easy to see, by induction
on n, that fn0,b(b) is a polynomial of b of degree 3
n. Thus the lemma is true. 
Note that Rk,m is an algebraic set and has no isolated points. Take a small
r > 0. Lemma 3.1 assures the existence of points (a, b) ∈ Rk,m \ {(0, 0)} with
0 < |a|, |b| < r. Then 0 < |f ′a,b(0)| < 1 and |fa,b(z)| < |z| for |z| < r. Hence
both critical points belong to Bfa,b . Now suppose (a, b) ∈ Rk,m ∩ Sk−1,m−1. By the
argument above, it follows that fk−1a,b (b) = f
m−1
a,b (a) = a or b. That is, a or b is a
periodic point. Then it does not belong to Bfa,b , a contradiction. Thus we conclude
that (a, b) ∈ Sk,m \ Sk−1.m−1. Since we cannot distinguish a and b, we can just
say (a, b) ∈ Aλ(k,m) ∪ Aλ(m, k) with λ = f ′a,b(0). This completes the proof of
Proposition. 
Lemma 3.2. Aλ(1,m) = ∅ for m ≥ 2.
proof. The argument above does not work if k = 1 or m = 1 because, in
Rk,m ∩ Sk−1,m−1, fk−1(b) = fm−1(a) is automatically b or a respectively. Instead,
we use the real subfamily P (x) = x3 + bx2 + λx, b, λ ∈ R. Suppose 0 < λ < 1 and
b ∈ [−2√λ,−√3λ]. Then, P has three real ﬁxed points and both critical points c±
are real and belong to BP and satisfy 0 < c+ ≤ c−. It is easy to see that c+ = c− at
b = −√3λ and P (c−) = 0 at b = −2
√
λ. For any b ∈ (−2√λ,−√3λ), there exists
a unique m such that Pm+1(c+) < P (c−) ≤ Pm(c+). Thus, for any m ≥ 2 there
exists b ∈ (−2√λ,−√3λ) such that P (c−) = Pm(c+). Now we have shown that
Aλ(1,m) = ∅ for m ≥ 2. 
Lemma 3.3. Aλ(k, 1) = ∅ for k ≥ 1.
Turning deformation of cubic polynomials with critical orbit relations 23 
proof. We deﬁne the equipotential curves in BP by the level curves of the func-
tion |ϕP (z)|. Note that the level curves are pinched only at preimages of the critical
points. Put Γn = {z ∈ BP ; |ϕP (z)| = |ϕP (P−n(c−))|} and let Un(z) be the connected
component of BP \ Γn containing z for n ∈ Z. Then P : Un(z) → Un−1(P (z)) is a
proper holomorphic map of some degree. We have that Un(z)∩Γn−1 is connected. In
fact, otherwise, an equipotential curve must be pinched between level n and n− 1,
which is impossible since there is no critical level there. Here we use the critical
orbit relation.
Now suppose that P ∈ Aλ(k, 1). Then since P k(c−) = P (c+) and P k−1(c−) =
c+, we have P
k−1(c−) = c˜+, the co-critical point of c+. Consider the components
Un,± := Un(c±) and Vn := Un(c˜+). We will show by induction that these are distinct
for any n ≥ 1. Then the open set U := ∪n≥0Un(c+) is a proper subset of BP and
any point on its boundary has level ∞, a contradiction. This ﬁnishes the proof.
First we show the case n = 1. Suppose U1,+ = U1,−. Then P (U1,+) = U0,+.
Hence c˜+ = P
k−1(c−) ∈ U0,+ and P : U0,+ → U−1(P (c+)) = U−1,+ is of degree 3,
which contradicts c− ∈ ∂U0,+. Next suppose U1,+ = V1. Then P : U1,+ → U0,+ is
of degree 3, which is impossible since c− /∈ U1,+. If U1,− = V1, then P k−1(U1,−) =
V2−k ⊂ V1 = U1,−, which implies 0 ∈ U1,−, hence c+ ∈ U1,−. This is impossible since
c+ /∈ U1,−.
Now we consider the case n > 1. If Un,+ = Un,−, then P : Un,+ → Un−1,+ is of
degree 3, hence c˜+ = P
k−1(c−) ∈ Un−1,+. This contradicts the induction hypothesis.
If Un,+ = Vn, then P : Un,+ → Un−1,+ is of degree 3, which contradicts c− /∈ Un,+. If
Un,− = Vn, then P k−1(Un,−) = Vn−k+1 ⊂ Un,−, which is impossible since c+ /∈ Un,−.
Thus we conclude that Aλ(k, 1) = ∅ for k ≥ 2.
Finally we show that Aλ(1, 1) = ∅. In fact, P (c−) = P (c+) and c− = c+ implies
c− = c˜+, which implies the ﬁber P−1(z) of a point z close to P (c+) consists of 4
points, a contradiction. This completes the proof. 
If we take a P ∈ Aλ(k,m), it follows P k(c−) ∈ UP , hence the component Uj
containing P j(c−) touches a point zj ∈ P j−k+1(c+)∩ ∂outA−j+k−1 for 0 ≤ j ≤ k− 1.
Especially, Uk−1  c− touches ∂outAk−1. Then the same argument as for Aλ(k−1, 0)
works if we replace c− by z0. Put A˜(k,m) = A(k,m)/ ∼affine.
Theorem 3.1. The monodromy Mλ on A˜λ(k,m) is transitive for any (k,m).
Figure 5 expresses the dynamics of P = Pb ∈ Aλ(2, 2) with b  −1.28 − 0.46i.
Compare Figure 3. We have Ps(1) = Pb¯. The set A˜λ(2, 2) consists exactly of these
two maps.
As a corollary, we get the number of parameters in A˜λ(k,m).
Theorem 3.2. We have #A˜λ(k, 0) = 2k and #A˜λ(k,m) = 2k−1 for m ≥ 1.
Put A(k,m) = {(b, λ) ∈ A;P k(c−) = Pm(c+), P k−1(c−) = Pm−1(c+)}. Theorem
3.1 says that the Branner-Hubbard deformation WP : H+ → A˜(k,m) gives a locally
analytic parametrization (hence a universal covering) of A˜(k,m). Consider the
multiplier map λ on A˜(k,m). It is a proper holomorphic map from A˜(k,m) onto D.
We will calculate its degree.
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 24 
ψP,1
c−
c+
1 λ−10
0
U1
Figure 5: the map ψP,1
Theorem 3.3. The map λ : A˜(k,m) → D is of degree 2k if m = 0 and 2k−1
otherwise.
proof. Fix λ0 ∈ D. Then since λ−1(λ0) = A˜λ0(k,m) consists of 2k or 2k−1
elements if m = 0 or m ≥ 1, respectively. This completes the proof. 
Figure 6 is the connectedness locus of the subfamily with λ = 1/2. Figures 7, 8
and 9 present the immediate basins of the maps in A1/2(1, 0),A1/2(2, 0) and A(2, 2),
respectively. They express the euipotential curves and correspond to Figures 2, 4
and 5, respectively.
Figure 6: Parameter space for A1/2 Figure 7: Dynamics of P ∈ A1/2(1, 0)
Turning deformation of cubic polynomials with critical orbit relations 25 
Figure 8: Dynamics of P ∈ A1/2(2, 0) Figure 9: Dynamics of P ∈ A1/2(2, 2)
References
[B] B. Branner: Turning around the connectedness locus. In: “Topological methods
in modern Mathematics.” pp. 391–427. Houston, Publish or Perish, 1993
[BH1] B. Branner and J. Hubbard: The iteration of cubic polynomials. Part I: The
global topology of parameter space. Acta Math. 160, (1988), pp. 143–206.
[BH2] B. Branner and J. Hubbard: The iteration of cubic polynomials. Part II:
Patterns and parapatterns. Acta Math. 169, (1992), pp. 229–325.
[PT] Petersen, C. & Tan, L.: Branner-Hubbard motions and attracting dynamics.
Proceedings of Bodilfest, ed. P. Hjorth and C. L. Petersen. (2006), pp. 45–70.
[Y] H. Yazawa: Monodromy on the central hyperbolic component of polynomials
with just two distinct critical points. To appear in Tokyo J. Math., 2007.
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 26 


Turning deformation of cubic polynomials with critical orbit relations

8KJ00004878597論文Articledepartmental bulletin pape

大嶋, 正人

1264

オオシマ, マサト

Oshima, Masato

Murao, Takahiro

3288

Sakurai, Yumi

3291

Sasaki, Yukio

3292

Nanbu, Noritoshi

3293

1196

1197

1198

1199

1200

*1 Associate Professor, Nanochemistry, Tokyo Polytechnic University  *2 Bachelor Student.   
*3  Professor, Nanochemistry, Tokyo Polytechnic University  *4 Lecturer, Nanochemistry, Tokyo Polytechnic University 
 Received Sept. 25, 2007 E-mail:oshima@nano.t-kougei.ac.jp 
 
Theoretical Study of Electronic Substituent Effect  
in Fluorine-Containing Sydnones 
 
Masato Oshima*1 Takahiro Murao*2 Yumi Sakurai*2  
Yukio Sasaki*3 and Noritoshi Nanbu*4 
Abstract: Calculation of several sydnone derivatives were performed with density functional theory 
(DFT) at the B3LYP levels to understand the substituent effects that had been introduced to sydnone ring.  
Sydnone ring can be introduced two substituents that are R1 on nitrogen and R2 on carbon.  Sydnone 
derivatives that had alkyl groups as both of R1 and R2 were studied as electrolyte of lithium rechargeable 
batteries and expected to improve the properties by changing the combination of R1 and R2.  Potential 
window and resistance against oxidation that are important electrolyte properties can be estimated from 
Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) energy 
levels; and then DFT calculations of sydnone derivatives having several combination of R1 and R2 were 
performed.  Although, the substituent effects of sydnone were still partial understanding, we proposed 
three types of R1 and R2 combination, such as R1 = CH3 and R2 = CF3, R1 = (CH3)3Si and R2 = CH3, and R1 
= (CH3)3Si and R2 = CF3, for the improved sydnone derivatives. 
Introduction 
Lithium rechargeable battery is expected as useful power source in the next generation and one of the 
important technical breakthroughs toward the practical use is finding out of the good electrolyte.  The 
electrolyte is required to have wide electrical potential window, resistance to oxidation, good solubility of 
electrolyte salt, low-viscosity, and good electric constant.  Sydnone derivatives, which is a 
room-temperature ionic liquid, was investigated as a candidate of electrolyte and Sasaki et al reported that 
sydnones have good properties and potentials for electrolyte [1].  General synthetic scheme of sydnones is 
shown in Scheme 1 and substituent R1 and R2 can be altered to suit required electrolyte properties by the 
selection of the amino acid and the amine moieties.  Among the required electrolyte properties, the 
electrical potential window and resistance against oxidation can be estimated from the calculated energy 
levels of HOMO and LUMO by computational chemistry.  According to Koopman’s theorem, a 
correlation is expected between the HOMO energy level and the oxidation potential, because the negatives 
of the orbital energies in the ground state are equal to the ionization potentials [2].  The electron affinity 
of the electrolyte is also correlated to the LUMO energy level; therefore, the electrical potential window 
can be estimated from the energy gap (Eg) between HOMO and LUMO energy level.  In this paper, we 
report the analyses of the substituent effects between the electrical potential window and the combination 
of substituent R1 and R2 by means of the density functional theory (DFT) calculations. 
59 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
R2 CO2R
NH2
R1X R
2 CO2R
NH
N2O4
CH3CO2Na
hydrolysis
R1
R2 CO2R
N
R1 NO
R2 CO2H
N
R1 NO
Ac2O
heat N
O
NR1
R2
O-
sydnone  
Scheme 1 
Computational Methods 
All geometry optimizations, population analyses, and energy calculations were carried out with density 
functional theory (DFT)[3] at Becke’s three parameters (B3) exchange functional along with the 
Lee-Yang-Parr (LYP) non-local correlation functional 
(B3LYP) level [4, 5] with 6-311+G(d, p) basis set [6] 
and verified to be either minima or transition states from 
their Hessian matrices for all reactions: minima for all 
positive eigenvalues.  Natural population analysis and 
natural charge analysis were performed at the same level 
using natural bond orbital (NBO) program [7].  The 
DFT and the NBO calculations were performed using 
the Gaussian98 program package [8]. 
Results and Discussions 
The sydnone derivatives that non-substituted at N 
position (R1 = H) are expected too reactive for 
electrolyte usage then we selected 1 (R1 = CH3, R2 = H) 
as a reference compound.  Sydnone 1 was also used in 
experimental studies and the target of our computational 
study was to explore sydnone derivatives that have 
good properties than 1.  The calculation of geometry 
optimization for 1 was performed and optimized 
structure was shown in Figure 1.  The five atoms, O1, 
N2, N3, C4, and C5, were coplanar and these atoms 
constructed a distorted five membered ring because of 
O1-N2 and C4-C5 lengths were slightly longer than 
other three bonds.  The energy levels of LUMO, 
HOMO, and Eg (energy gap) of 1 were obtained by the 
population analyses as –2.01 eV, –6.54 eV, and 4.53 eV, 
respectively.  The Eg is corresponding to the range of 
electrical potential window and we explored the 
compound that has lower HOMO level as well as 
   
 HOMO LUMO 
Figure 2: The shapes of HOMO and LUMO of 1 
 
Figure 1: Selected bond lengths(Å) and angles(o) 
of 1 were as follows; O1-N2 = 1.4539, N2-N3 = 
1.3030, N3-C4 = 1.3458, C4-C5 = 1.4187, C5-O6 
= 1.1999, C5-O1-N2 = 111.33, O1-N2-N3 = 
105.00, N2-N3-C4 = 115.14, N3-C4-C5 = 
106.89, C4-C5-O1 = 101.64, O1-C5-O6 = 
121.34, C4-C5-O6 = 137.03. 
60 
Theoretical Study of Electronic Substituent Effect in Fluorine-Containing Sydnones 
higher LUMO level by changing the combination of R1 and R2. 
It is expected that the substituent R1 and R2 have a direct electronic influence on N3 and C4, 
respectively.  The shapes of HOMO and LUMO of 1 obtained by the population analyses were shown in 
Figure 2.  Focus on N3 and C4 atoms, large HOMO coefficient and middle LUMO coefficient were found 
on C4.  On the other hand, there was no HOMO coefficient on N3 and middle LUMO coefficient was 
found on N3.  From above, it was expected that substituent R2 to make an effect on both of HOMO and 
LUMO and R1 make an effect on LUMO.   
The substituent R1 was fixed to CH3 by way of 
experiment and geometry optimizations and population 
analyses were performed for several kinds of R2 as 
shown in Table 1.  The energy levels of HOMO and 
LUMO were increased by introducing alkyl (CH3, iPr, 
tBu) or trimethylsilyl ((CH3)3Si) groups that have 
electron-donating property as R2.  By introducing 
halogens (F, Cl, Br, I) that have electron-withdrawing 
property as R2, both of HOMO and LUMO levels were 
lowered.  The reduced value of the HOMO levels were 
proportional to electronegativity of halogen atoms, the 
HOMO levels of fluorine substituted 6 was lowest and of iodide substituted 9 was highest among halogen 
substituted sydnones 6-9.  The HOMO levels were greatly lowered by acetyl (10) and methoxycarbonyl 
(11) substituents that have strong electron-withdrawing property.  Unfortunately, the LUMO levels of 10 
and 11 also lowered about the same value of that of HOMO, electrical potential windows (Eg) had not 
expanded.  The carbonyl moieties of acetyl and methoxycarbonyl groups were located on the same plane 
as sydnone structure and these geometries were made available to resonance stabilization.  As shown in 
Figure 3, both of HOMO and LUMO have expanded to R2 substituent, which contributed to stabilize 
molecule by delocalization.  However, the aromatic groups such as phenyl (12) and 2-pyridyl (13) 
substituents were also electron-withdrawing group, LUMO energy were increased as well as no enough 
effect for decreasing HOMO energy.  Especially about 12, the phenyl and sydnone rings of the optimized 
structure were not coplanar.  For this reason, there was no resonance stabilization and both of HOMO and 
LUMO energy level were raised.  From the results described above, introducing electron-withdrawing 
group can be expected desired effects.  Then, the combination R1 = CH3 and R2 = CF3 was calculated and 
the sydnone derivative 14 gave low HOMO level (-7.28 eV) and large Eg (4.56 eV) as we had expected. 
Table 1: HOMO and LUMO energy levels and Eg of the substituted sydnones having CH3 group as R1. 
Sydnone 1 2 3 4 5 6 7 8 9 10 11 12 13 14 
R1 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 
R2 H CH3 iPr tBu (CH3)3Si F Cl Br I –COCH3 –CO2CH3 –C6H5 –2-Py CF3 
LUMO [eV] -2.01 -1.80 -1.79 -1.71 -1.88 -2.22 -2.22 -2.21 -2.11 -2.73 -2.56 -2.11 -2.32 -2.72 
HOMO [eV] -6.54 -6.20 -6.16 -6.13 -6.28 -6.62 -6.58 -6.55 -6.43 -6.96 -6.98 -6.05 -6.25 -7.28 
Eg [eV] 4.53  4.40  4.37 4.42  4.40  4.40 4.36 4.34 4.32 4.23 4.42  3.94  3.93  4.56 
  
HOMO              LUMO 
Figure 3: The shapes of HOMO and LUMO of 10 
61 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
For the screening of R1, the substituent R2 was fixed 
as CH3, then geometry optimization and population 
analyses were performed.  The results that the energy 
levels of HOMO and LUMO and Eg of sydnones 15-27 
were shown in Table 2.  Introducing electron-donating 
groups as R1, the HOMO energy levels of sydnone 15 and 
18 were lower than that of 16 and 17.  It was notable that 
the LUMO level of 18 was not lowered, therefore 18 gave 
largest Eg (4.80 eV) among all calculations.  Sydnones 
19-22 had halogen group elements as R1 and the Eg of 
these sydnones became reduced as increasing atomic 
number.  The HOMO energy levels were increased 
gradually as well as the LUMO energy levels were 
decreased considerably in 19-22.  The shapes of HOMO 
of 19-22 were similar to that of 1, however, the LUMO were spread around the halogen atoms except 19.  
As an example, the shapes of HOMO and LUMO of 22 were shown in Figure 4.   The LUMOs of 20, 21, 
and 22 were corresponding with the anti-bond orbital between nitrogen and halogen atoms.  The energy 
levels of σ* orbital were calculated using NBO program to give -0.81 (N-Cl σ* bond of 20), -2.29 (N-Br σ 
* bond of 21), and -2.35 (N-I σ * bond of 22) eV, respectively, and these order of energy levels was the 
same the order of LUMO levels of 20 (-2.95 eV), 21 (-3.73 eV), and 22 (-4.06 eV).  By introducing 
electron-withdrawing groups, the LUMO energy levels of 23-27 were reduced and Eg also became lower.  
Table 2: HOMO and LUMO energy levels and Eg of the substituted sydnones having CH3 group as R2. 
Sydnone 15 16 17 18 19 20 21 22 23 24 25 26 27 
R1 H iPr tBu (CH3)3Si F Cl Br I –COCH3 –CO2CH3 –C6H5 –2-Py CF3 
R2 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 
LUMO [eV] -2.08 -1.68 -1.60 -1.64 -2.37 -2.95 -3.73 -4.06 -2.78 -2.84 -2.10 -2.53 -2.78 
HOMO [eV] -6.39 -6.06 -6.01 -6.44 -6.95 -6.68 -6.59 -6.41 -6.45 -6.38 -6.08 -6.08 -6.80 
Eg [eV] 4.31  4.38 4.41 4.80 4.59  3.73 2.86 2.35 3.67 3.54 3.98  3.54 4.02 
 
For the detail analyses of substituent effect, several natural charges were calculated by NBO program 
and sum of natural charge of R1, R2, and sydnone rings moieties were shown in Table 3.  Make a 
comparison between 2, 10, and 14, all the natural charges of sydnone ring were about -0.3 and CH3(R1) 
group of 2 was almost neutral.  Introducing electron-withdrawing group to R2, the natural charges of 
CH3(R1) groups of 10 and 14 were changed to ca +0.3.   
  
 
 HOMO  LUMO 
Figure 4: The shapes of HOMO and LUMO of 22 
62 
Theoretical Study of Electronic Substituent Effect in Fluorine-Containing Sydnones 
Then the natural charges of 
each atoms in CH3(R1) group of 2 
and 14, electronic populations and 
energy levels of valence atomic 
orbital were shown in Table 4 for 
the analyses of detail electron 
configuration.  By changing R2 
from CH3 to CF3, the electron 
density on CH3(R1) was lowered.  
Most of the lost electrons from R1 were on carbon atom that mainly belonging to 2px and 2py atomic 
orbitals.  The energy levels of all atomic orbitals of carbon and hydrogen atoms of 14 were decreased than 
that of 2.  By the population analyses, there were no HOMO and LUMO coefficients on carbon and 
hydrogens in CH3 group.  It was considered that these energy decreases of atomic orbitals not induced 
ether decreasing HOMO or LUMO energy levels but stabilizing whole molecule 14.  Therefore, the 
energy levels of HOMO and LUMO showed parallel change to give similar Eg.  
Table 4: Natural Charge, Electronic Population and Energy levels of AO on CH3 (R2)group. 
Sydnone 2 14 
 Natural Charge 
Electronic Population 
(Energy Level of AO [eV])
Natural Charge
Electronic Population 
(Energy Level of AO [eV]) 
C -0.614  
2s: 1.086 (-7.59) 
2px: 1.243 (-3.61) 
2py: 1.064 (-3.33) 
2pz: 1.211 (-3.49) 
-0.369 
2s: 1.093 (-9.06) 
2px: 1.184 (-4.86) 
2py: 0.853 (-4.17) 
2pz: 1.225 (-4.93) 
Ha +0.225 1s: 0.774 (+0.30) +0.232 1s: 0.766 (-0.76) 
a Natural Charge, Population, and Energy level of AO were average of three hydrogen atoms. 
 
 
Compare tBu group with 
(CH3)3Si group using 17 and 18 in 
Table 3, (CH3)3Si group had large 
electron-donating ability than tBu 
group.  The electron density of 
sydnone ring of 18 has risen and it 
may be caused lowering HOMO 
level.  Among the sydnone ring, 
the electron density of nitrogen 
atom that was substituted by R2 was raised.  The donated electrons were belonged 2s and 2px atomic 
orbitals and the energy levels of these electronic population increased atomic orbital were decreased as 
shown in Table 5. 
Table 3: Sum of Natural Charge of R1, R2, and sydnone ring moieties.
Sydnone 1 2 6 10 14 17 18 
R1 
+0.308 
(CH3) 
+0.060 
(CH3)
+0.318 
(CH3)
+0.335  
(CH3) 
+0.327 
(CH3)
+0.337 
(tBu)
+0.653 
((CH3)3Si)
R2 
-0.233  
(H) 
-0.304 
(CH3) 
-0.306 
(F) 
+0.027  
(–COCH3) 
+0.007 
(CF3)
+0.056 
(CH3)
+0.051 
(CH3) 
Sydnone ring -0.542 -0.364 -0.012 -0.308 -0.334 -0.393 -0.704 
Table 5:  Natural Charge, Electronic Population and Energy levels 
  of AO on the nitrogen that was substituted R1. 
Sydnone 17 18 
 
Natural 
Charge 
Electronic Population 
(Energy Level of AO [eV]) 
Natural 
Charge 
Electronic Population 
(Energy Level of AO [eV])
N -0.144  
2s: 1.185 (-15.30) 
2px: 1.344 (-8.73) 
2py: 1.195 (-8.09) 
2pz: 1.392 (-8.34) 
-0.435 
2s: 1.293 (-16.64) 
2px: 1.530 (-9.71) 
2py: 1.183 (-7.60) 
2pz: 1.394 (-8.13) 
63 
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
Similar analyses were 
performed for sydnone 1 and 6.  
Sydnone 1 had CH3 group as R1 and 
hydrogen as R2.  On the other hand, 
6 had fluorine as R2 instead of 
hydrogen.  In this case, the 
electronic perturbation by R2 was 
much influenced to the electronic 
density of sydnone ring as shown in 
Table 3.  Generally, hydrogen has 
strong electron-donating ability; and then the electron density of sydnone ring of 1 was raised to -0.542.  
In contrast, fluorine has strong electron-withdrawing ability and the electron density of sydnone ring of 6 
was considerably less than that of 1.  Contrary to expectation, there were not so much the difference of 
HOMO and LUMO levels between 1 and 6.  Among the sydnone ring, electron densities of the carbon 
that was substituted R2 were changed significantly, then natural charges, electronic population and energy 
levels of AO on the carbon of 1 and 6 were collected up in Table 6.  Compare 6 with 1, the energy levels 
of 2px and 2pz were decreased but the populations of these atomic orbitals were similar.  The populations 
of 2s and 2py were decreased but the energy levels of these atomic orbitals were similar.  By the reason of 
these behaviors as above, the electronic perturbation that caused by changing R2 were denied.  
Although, the substituent effects of sydnone were still partial understanding, we had designed sydnone 
19 that has (CH3)3Si as R1 and CF3 as R2.  All calculations were performed to 19 and following results, 
HOMO energy level = -6.89 eV, LUMO energy level = -2.39 eV, and Eg = 4.50 eV, were obtained. 
Conclusion 
In our computational study, we can propose some sydnone derivatives that were expected to have good 
properties than 1.  Sydnone 14 and 19 had lower HOMO level and similar Eg than 1.  So 14 and 19 were 
expected to have excellent oxidation resistance and similar electrical potential window with 1.  Sydnone 
18 can be expected good oxidation resistance and wide potential window than 1, because HOMO level and 
Eg of 18 were suitable for the desired properties than that of 1.  The study of the substituent effects was 
not enough but we got some clue as to design the desired molecules.  We have planned to synthesize the 
proposed sydnone derivatives, then measuring the properties of them in the further investigation.  
 
References 
[1] (a) Y. Sasaki and Y. Ishibashi, Bull. Chem. Soc. Jpn. 63, 1422 (1990).  (b) Y. Sasaki, I Komiya, and 
M. Handa, Bull. Chem. Soc. Jpn. 67, 2571 (1994).  (c) M. Handa, T. Suzuki, H. Fukuda, and Y. 
Sasaki, Chem. Lett. 457 (1998).  (d) Y. Sasaki, M. Ohashi, S. Okahata, H. Kaido, T. Mori, and M. 
Handa, Nippon Kagaku Kaishi 1217 (1983).  (e) Y. Sasaki, K. Sekine, and M. Handa, Prog. Batteries 
& Battery Materials 17, 239 (1998).  (f) S. Iinou, M. Handa, and Y. Sasaki, Electrochem. 67, 22 
(1999). 
[2] T. A. Koopmans, Physical 1, 104 (1968). 
[3] (a) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).  (b) W. Kohn and L. J. Sham, Phys. Rev. 
Table 6:  Natural Charge, Electronic Population and Energy levels
 of AO on the carbon that was substituted R2. 
Sydnone 1 6 
 
Natural 
Charge 
Electronic Population 
(Energy Level of AO [eV]) 
Natural 
Charge 
Electronic Population 
(Energy Level of AO [eV])
C -0.208  
2s: 0.955 (-6.15) 
2px: 0.925 (-2.77) 
2py: 1.114 (-2.81) 
2pz: 1.192 (-4.69) 
0.424 
2s: 0.789 (-5.81) 
2px: 0.949 (-3.39) 
2py: 0.654 (-2.62) 
2pz: 1.152 (-5.45) 
64 
Theoretical Study of Electronic Substituent Effect in Fluorine-Containing Sydnones 
140, A1133 (1965).  (c) The Challenge of d and f Electrons, Ed. D. R. Salahub and M. C. Zerner 
(ACS, Washington, D.C., 1989).  (d) R. G. Parr and W. Yang, Density-functional theory of atoms and 
molecules (Oxford Univ. Press, Oxford, 1989). 
[4] A. D. Becke, J. Chem. Phys. 98, 5648 (1993). 
[5] (a) A. D. Becke, Phys. Rev. A38, 3098 (1988).  (b) C. Lee, W. Yang and R. G. Parr, Phys. Rev. B37, 
785 (1988).  (c) B. Miehlich, A. Savin, H. Stoll, and H. Preuss, Chem. Phys. Lett. 157, 200 (1989). 
[6] For valence triple zeta with polarization: (a) R. Krishnan, J.S. Binkley, R. Seeger, and J.A. Pople, J. 
Chem. Phys. 72, 650 (1980).  (b) J-P. Blaudeau, M. P. McGrath, L.A. Curtiss, and L. Radom, J. 
Chem. Phys. 107, 5016 (1997).  (c) L. A. Curtiss, M. P. McGrath, J-P. Blaudeau, N. E. Davis, R. C. 
Binning, and Jr., L. Radom, J. Chem. Phys. 103, 6104 (1995).  (d) M. N. Glukhovtsev, A. Pross, M.P. 
McGrath, and L. Radom, J. Chem. Phys. 103, 1878 (1995).  For diffuse exponents: (e) T. Clark, J. 
Chandrasekhar, and P.v.R. Schleyer, J. Comp. Chem. 4, 294 (1983). 
[7] E. D. Glendening, A. E. Reed, J. E. Carpenter, F. Weinhold, NBO Version 3.1. 
[8] Gaussian 98, Revision A.11.3,  M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. 
Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. 
Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. 
Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. 
Ayala, Q. Cui, K. Morokuma, N. Rega, P. Salvador, J. J. Dannenberg, D. K. Malick, A. D. Rabuck, K. 
Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, A. G. Baboul, B. B. Stefanov, G. Liu, A. 
Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, 
C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, J. L. 
Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle, and J. A. Pople, Gaussian, Inc., Pittsburgh PA, 
2002. 
65 


Theoretical Study of Electronic Substituent Effect in Fluorine-Containing Sydnones.

7KJ00004878585論文Articledepartmental bulletin pape

菅田, 圭次

1773

スガタ, ケイジ

小沢, 一仁

1774

オザワ, カズヒト

滝沢, 利直

1775

タキザワ, トシナオ

Sugata, Keiji

1776

Ozawa, Kazuhito

1777

Takizawa, Toshinao

1778

A Study of the Relationship between Students' Self-understanding and their Social Recognition(4): Necessity and Possibility supporting Engineering Education of Formation of "the Power of Living"

裏表紙

10KJ00004878599論文Articledepartmental bulletin pape

植野, 義明

1207

ウエノ, ヨシアキ

Ueno, Yoshiaki

1208

√
2の数学
植野義明∗
概要
高大連携が 1998年前後からはじまり, 最近ではかなり定着してきた。高校側は進路指導の一環として
この制度を利用しているらしいが, 大学教員にとっても, 高校の雰囲気を知ることができ, 意味のある活
動だと思う。√
2は, 数学を習ったことのある者なら誰でも知っている数である。素朴な題材からどこまで数学を深
められるか, 受講する生徒やその親, 講師役の教員にとっても興味が湧く。ただ黙って話を聞くだけでな
く, 作業しながら楽しめる演題として, 今後の定番にしたい数学である。
1 はじめに,
√
2の思い出
小川洋子の小説 [1] には, ルートという名の少年
が登場する。
彼のことを, 私と息子は博士と呼んだ。そし
て博士は息子を, ルートと呼んだ。ルート記
号の中に数字をはめ込むとどんな魔法が掛
かるか, 三人で試した日のことはよく覚えて
いる—。
これは冒頭部分からの引用だが, 小説全体の語り
部である「私」は, ここに引用したシーンのような
時間を後の方で, 「とっても大切な時間」と言って
いる。ここに書かれているのは, 初めてルート記号
を習ったときの, ときめくような体験であり, 誰も
がもっている貴重な記憶である。
中でも
√
2 は親しみのある数である。ルートの中
に, 数を 1, 2, . . . と順番に入れていくとき, 初めて
表れるトリビアルでない数—無理数と呼ばれる数が√
2である。
また,
√
2 は正方形の対角線の長さのイメージと
も結びついている。それは, ピタゴラスが教会の敷
∗Associate professor, General Education and Research Center, Tokyo Polytechnic University
Received Sept.28, 2007
石を見て「三平方の定理」を思いついたというエピ
ソードを呼び覚ます。
2 身近な数
√
2
一方,
√
2 は現代を生きる私たちにとって案外身
近な数でもある。日本人である私たちは語呂合わせ
という得意技によって,
√
2 の値が約 1.41421356で
あることを暗記している。そんなに多くの桁数を暗
記しなくても, 1.4ぐらいでも, 経済活動には十分に
役に立つ。つまり, 4 割り増しを 2 度続けると, 近
似的に 2倍になるということである。
街のコンビニ・ストアにあるコピー機で拡大コ
ピーをするとき, 「141% に拡大」というボタンの
お世話になる。これは, 長方形の紙の縦・横を同時
に 1.41 倍すれば, 面積は 2 倍になるということで
ある。面積比は相似比の 2乗になるからである。
3
√
2の近似値
開平という方法は, 和算では算木を並べて行った
そうである。最近はあまり教えることがなくなった
のかもしれないが, 実際に開平計算をしてみると,
縦書きの割り算のアルゴリズムとよく似ていること
に驚く。また, √ という記号も縦書きの割り算の
記号に似ている。√
2 の近似値を得るもうひとつの考え方は, 分数
(有理数) でこれに近いものを探すことである。い
74 
東京工芸大学工学部紀要 Vol. 30 No.1（2007） 
ま, 1.4 = 14/10 = 7/5が良い近似値であることは,
1.42 = 1.96としても分かるが,
(
7
5
)2
=
49
25
= 2− 1
25
としても分かる。
より一般に, a/bが
√
2 に等しいか近いとすると,
a/b =
√
2 の両辺を 2乗して
a2 = 2 b2
となる。そこで, 2 つの平方数であって, 比が 2 に
等しいか近いペアを見つければ,
√
2 に等しいか近
い分数 (有理数)が得られることになる。
これは, 想像したよりも面白い作業である。なぜ
なら, 平方数を求めることは暗算でどんどんできる
からである。平方数 (の数列)の階差数列をとると,
奇数の数列となるので, 奇数を順に足していくと平
方数がすべて求まる。他にもいろいろな方法があ
る。最近は「インドの数学」が流行っているらしく,
書店に行くと 1 つのコーナーが「インド式計算術」
の本で埋まっていることがある。
こうして求めた平方数 n2 の表と, その 2 倍の表
を比べる。
n n2 2n2 n n2 2n2
1 1 2 11 121 242
2 4 8 12 144 288
3 9 18 13 169 338
4 16 32 14 196 392
5 25 50 15 225 450
6 36 72 16 256 512
7 49 98 17 289 578
8 64 128 18 324 648
9 81 162 19 361 722
10 100 200 20 400 800
四角で囲んだところで「ニアミス」が起こってい
る。それらを抜き出すと,
12 = 1 = 2− 1 = 12 × 2− 1
32 = 9 = 8 + 1 = 22 × 2 + 1
72 = 49 = 50− 1 = 52 × 2− 1
172 = 289 = 288 + 1 = 122 × 2 + 1
図 1 ニュートン法で
√
2を求める。
となるので,
√
2 の近似分数として
1
1
,
3
2
,
7
5
,
17
12
が得られる。
この分数の列に何か規則性はあるだろうか。分子
だけ, あるいは分母だけの並びを見ていたのではな
かなか規則性に気付かない。分子と分母を同時に見
ていると, きれいな規則に従っていることに気付く。
4 ニュートン法
√
2 の値を計算するのに, ニュートン法を用いる
という方法がある。方程式 x2 − 2 = 0 を考える。
この方程式の二つの根のうち, x > 0の範囲にある
のが
√
2 である。もっと詳しく, 1 <
√
2 < 2 であ
ることが分かる。
そこで, y = x2 − 2のグラフに x = 2のところで
接線を引き, x軸との交点を x1 とする。
一般に, y = f(x) の x = a で引いた接線の方程
式は y = f(a) + f ′(a) (x− a) なので, これと x軸
との交点の座標は a− f(a)/f ′(a) となる。
そこで, 数列 xn を漸化式
xn+1 = xn − f(xn)
f ′(xn)
で定義する。出発点 x0 を適当に定めれば, 目標と
する方程式 f(x) = 0の根に収束することが期待さ
れる。
f(x) = x2− 2の場合は, f ′(x) = 2xとなるので,
漸化式は
xn+1 = xn − x
2
n − 2
2xn
=
1
2
(
xn +
2
xn
)
75 東京工芸大学工学部紀要 Vol. 30 No.1（2007） 
となる。
x1 = 1から始めると, x2 = 3/2 = 1.5となるが,
それ以降この数列は単調減少となる。
x2 = 32 = 1.5
x3 = 1712 = 1.41
x4 = 577408 = 1.41421
x5 = 665857470832 = 1.41421356237
x >
√
2 の範囲で f(x) > 0 かつ f ′(x) > 0 だか
らである。また, 下に有界であることもすぐにわか
るので, 収束することが分かる。漸化式で n を無
限に大きくした極限を考えれば, 収束する先の数は√
2でなければならないことも直ちにわかる。
以上は, 数列が収束することの証明であるが, こ
れは定性的な証明, 極限の存在証明である。しかし,
x > 0の範囲で f ′′(x) > 0であることにも注意する
と, この収束は意外に速いことも分かる。実際, 上
の計算例が示すように, 十進小数展開で一致する桁
数は 2倍, 2倍に増えてゆく。
図 1を見ると, 収束の様子がわかる。極限に近づ
けば近づくほど, 小さな区間の中では関数のグラフ
と接線 (という直線)は区別ができなくなっていく。
視覚的には, 初期値から数えて 3番目の点を見るの
がやっとである。
5 メソポタミアの粘土板に見られる
√
2
の近似値
メソポタミアで紀元前 1900年に書かれたと同定
されている粘土板に正方形とその対角線の図ととも
に
√
2の値を刻んだものが見つかっているそうであ
る [4]。
60進法で表されたその楔形文字の数字を読むと,
それがニュートン法で計算し, ある桁で丸めたもの
と一致することが確かめられている。その粘土板を
見ると, 古代メソポタミアの人たちは, 小数点以下
も 60進法を使用したことが分かる。
ただし, メソポタミアの人たちがニュートン法を
知っていたはずはないので, 近似値どうしを平均す
るとよりよい近似値が得られるというような直観に
従って, 上述の漸化式で示されている計算を行った
のだろうと推測される。
それにしても, その時代に「実用の範囲を超えた
精度で」求めようとした人がいたことをその粘土板
は語っている。
6 連分数との関係
√
2については連分数展開
√
2 = 1 +
1
2 + 1
2+ 1
2+ 1
...
も有名で, そこから導かれる近似分数 pk/qk は真の
値
√
2 を上と下から交互に「絞り込む」ようにし
て近づいてゆく。ニュートン法で得られる有理数列
は, これらの近似分数を「とびとびに」取る。その
「とびとび」さがどんどん加速していく。これは, 収
束する数列の収束速度を加速する「加速法」の好例
となっている。
ところで, 連分数展開の近似分数
1
1
,
3
2
,
7
5
,
17
12
,
41
29
,
99
70
, . . .
もことのほか重要で, これらの分母 q と分子 pは
p2 − 2 q2 = ±1
の整数解の全体を網羅している。整数論の問題 (不
定方程式)のルーツが解析学にあるという典型的な
例である。
これらの分数は実は第 3節で示したものと一致し
ている。詳しくは, [2]を参照されたい。
参考文献
[1] 小川洋子『博士の愛した数式』新潮社,
2003/08/29
[2] David Flannery, The square root of 2, Praxis
Publ. Ltd., New York, 2006
[3] 足立恒雄『
√
2の不思議』筑摩書房 (2007/02)
[4] 数学セミナー 2000 年 9 月号「数学をすること
の楽しさ」（中村滋）
[5] 植野義明,
√
2の数学, 2007.7.1数学教育学会夏
季研究会 (早稲田大学教育学部)
76 
√
2の数学


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