The International Society for the Interdisciplinary Study of Symmetry

116 Pages·2008·4.31 MB·English

Symmetry:
The Quarterly of the
International Society for the
Interdisciplinary Study of Symmetry
(ISIS-Symmetry) Volume 8, Number 1, 1997
INTERNATIONAL SOCIETY FOR THE
INTERDISCIPLINARY STUD Y OF SYMMETRY
(1SIS-SYMMETRY)
President Member.s of the Board
D~nes Nag.v, Institute of Applied Phystcs, Fred I~ Book’,lein. Ceater for Human Growth
Umversity of Tsukuba, Tsukuba Science C~ly and Development, University of Miclugan,
305. Japan: nagy. @bukko bk tsukuba.ac.jp 300 North Ingalls Building, Ann Arbor,
[Geometry.. Crystallography, HistoO, of Mmhigan 4810%0406, U.S.A ;
Scmnce and Technology.,. klngmstics] [email protected] u [Bmlogy,
SlaliMics]
Chairman of the A dvtsoO, Board
Arlhnr I- Ix, eb, Carpenter Center for the Siglind Brul,m, Department of Mus~colog2(cid:128) and
V~sual Arts, Harvard Umvers~ty, Cambridge. lnshtute for the Humamties, Universfly of
MA 02138, U.S.A.; Mmhigan, 1308 Broadway, Ann Arbor,
mhatpenn@husc harvard edu [Visual Arts. MI 48105. U.S A.; [email protected] [Music,
Choreography, Musm, Custallography} Sen’uottcs, L~terature}
Honorary Prestdents Belly Coiling,,, 1991 1 hlls~de Drive, Columbus,
Konst~nlin V. Frolov. Russmu Academy of ()|t 43221. U.S A : ted+@osu edu [Sculpture]
Sciences, Mechanical Engiueermg Research
Liz EAward,,, Visual Arls Program, Academy
Inshtute, ul Gnboedova 4, 101830 Moscow.
of the Arts. Queensland University of
Ruxsm [Mecb~mcal Engineering]
"l’echnolo~,3,, P.O. Box 362. Redhfll. Bmsbane,
and
QLD 4059, Australia: [email protected]
Yuval Ne’eman, R. and B. Sackler Facully of
Exact Sciences, Tel-Avtv Univers~, TcI-Av~v, [Fme Art, History of ArtI
Israel 69978; matflda@novel~.tau.ac il, Ru,,lan 1. Ko,,to,v, Faculty of Geology and
and Center for Particle Physic, University Prospecting, Umvcrs~ly of M~nlng and Geology.
of Texas, Austin, TX 78712, U.S.A [Part icle "’SI. Ivan R~lski", Sofia 1100. [3ulgana;
Physms} [email protected] bg
M mearolog2,.,, I listol.w of Scmnce]
Secretary General
Gy6rga" Da~’as, Symmelnon - "l’l~e Inmilute Sergei V. Pehtkhov. Mechamcal Engineering
[or Advanced Symmet~3, Studms, Research Institute, Russmn Academy of
and Institute for Research Orgamsatton of the Scmnces, ul. Gnboedova 4. 101830 Moscow,
Hungarian Academy of Sciences, Nfidor u 18. Russm; [email protected] [Mechamcal
P O Box 994, Budapest, H-1245 Hungary, Engineering, Bmmechanics|
[email protected] [Philosophy of Scmnce,
Theoretical Physics] htfi)nnation Nem,ork A dvlsor
M ih~ly Szoboszlai, Facully of Architecture,
Techmcal Universtty of Budapest, Mtlegy. elem
Marlha Pardavi-llorvath, Department of
Electrmal Engineering and Computer Science, rkp. 3, Budapest, I 1-1111 Hungary;
szoboszlai@arch brae hu [Architecture,
The George Washington University,
Geometry, Computer Aided Arclutectural
Washington, D.C. 20!)52, U.S.A.;
Dcsiga]
pardavi@scas gwu.edu [Matermls Scmncel
and Associate Ednor
Janusz Reblelak, Department of Architecture, (of .S):mmetry: Culture and Science)
Faculty. of Building Structures, Techmcal John IIosack. Department of Mathematics
University of Wroclaw, ul. B. Prusa 53/55, and Computing Science, University of the
PL-50-317 Wrocl.aw, Poland, South Pacflm, P.O. Box 1168, Suva, F@
jan [email protected] pl j [email protected]! [Mathematical Aimlysis,
IArchitecture, Building Engineering] Philosophy]
See theA,h,~o~), Boc~rd ms,de back cover
Symmetry:
The Quarterly of the
International Society for the
Interdisciplinary Study of Symmetry
(ISiS-Symmetry)
Temporarily edited by the curatory of the International Symmetry Foundation
Published by the International Symmetry Foundation
Volume 8, Number 1, 1-112, 1997
SPECIAL ISSUE: WASAN
Guest Editor: D~nes Nagy
CONTENTS
EDITORIAL
SYMMETRY: SCIENCE & ART
¯ Circle patterns arising from results in Japanese geometry,
4
Hiroshi Okumura
24
¯ Symmetry in traditional Japanese Mathematics, Hidetoshi Fukagawa
55
¯ An incorrect sangaku conjecture, John Rigby
58
=Symmetry properties of tangent circles, Jun Ozone, Yoshimasa Michiwaki
68
¯ Circle problems arising from wasan, John Rigby
SYMMETROSPECTIVE
¯ Golden section(ism): From mathematics to the theory of
74
art and musicology, Part 2, D~nes Nagy
SYMMETRY: CULTURE AND SCIENCE is edited by the Board of the International Society
for the Interdisciplinary Study of Symmetry (ISIS-Symmetry), and published by the
International Symmetry Foundation. The views expressed are those of individual
authors, and not necessarily shared by the Society or the editors.
Curatory of the International Symmetry Foundation:
Szaniszl6 B6rczi (head), J~inos F. B6hm, Gyt~rgy Darvas, S~indor Kabai
Any correspondence should be addressed to the
International Symmetry Foundation
c/o Symmetrion
P.O. Box 994, Budapest, H-1245 Hungary
Phone: [phone] Fax: [phone]
E-mail: [email protected], [email protected], [email protected]
http://isis-symmetry.org
Assistant Editor: Csaba Mezei
Subscription:
Annual membership fee of the Society:
Benefactors, US$780.00;
Ordinary Members, US$78.00 (includes the subscription to the quarterly),
Student Members, US$63.00 (includes the subscription to the quarterly);
Institutional Members, please contact the Secretary General.
Annual subscription rate for non-members: US$100.00.
Make checks payable to ISIS-Symmetry and mail to G. Darvas, Secretary General, or
transfer to the following account number: ISIS-Symmetry, International Symmetry
Foundation, 504-00004-2100-4015, Hungarian Foreign Trade Bank, Budapest, Szt.
Istv~in t~r 11, H-1821 Hungary (Telex: Hungary 22-6941 extr-h; Swift MKKB HU HB).
© ISIS-Symmetry. No part of this publication may be reproduced without written
permission from the Society.
ISSN 0865-4824
Cover layout: Gunter Schmitz
Images on the front and back cover:
Student works from the Basic Design Studio by William S. Huff
(Department of Architecture, SUNY at Buffalo, N.Y.)
Anabigram on the back cover: Douglas R. Hofstadter
24826 Akaprint Kft. F. v." Freier Lhszl6
Symmetry: Culture and Science
Vol. 8, No.l, 3, 1997
EDITORIAL
4 years have gone since the last volume of Symmetry: Culture and Science has been
published. Considering the wide demand for a printed journal of the Society, the Board
of ISIS-Symmetry made the necessary measures to restart the publication in 2001. It
suspended the president and editor of the journal, appointed a new curatory for the
Symmetry Foundation, and entrusted the new curatory to finish the editorial work and
publish the missing issues. The team took over the manuscripts and started the work in
February 2001. Four years’ delay is a strong stroke in the life of a journal. Therefore,
while accepting the necessity for certain renewal, the new editors decided to emphasize
the continuity of the periodical with unchanged format.
The new, temporary editorial team expresses its apologies to the authors of the submit-
ted papers, and to the subscribers of the journal for the long delay in the publication,
caused by the previous editor. Thanks to all for your patience.
The editorial team decided to publish the submitted and accepted papers as soon as
possible. The accelerated technical edition, the co-operation in a new team may result in
more possible errors in the first time, for which the editors apologise again, in advance.
During the years, several people contributed with a piece of their work to the edition,
technical preparation, and publication of this journal: D. Nagy, then M. Szoboszlai, and
finally Sz. Bdrczi, G. Darvas, S. Kabai, as well as Cs. Mezei.
This issue contains a thematic collection of papers on the traditional Japanese mathe-
matics: wasan. Reading the papers one needs no clarification how this subdiscipline is
related to symmetry and a national cultural tradition. Papers to this issue were invited by
the former editor. The new editorial team continued the correspondence with the authors
on shaping the papers.
The last part of the issue continues to publish the paper on Golden section(ism), started
in the previous (Vol. 7, No. 4) issue - symbolically linking the past and the renaissance
of the journal. This paper contains also a chapter on wasan, which gives some cultural
background explanation to those, who have not been familiar with this mathematics
developed originally, on Chinese roots, in Japan (cf. Okumura, pp. 4-5; Fukagawa, pp.
24-26, Nagy, Chap. 3.3).
Symmetry: Culture and Science
Vol 8, No.I, 4-23, 1997
SYMMETRY." SCIENCE AND ART
CIRCLE PATTERNS ARISING FROM RESULTS
IN JAPANESE GEOMETRY
Hiroshi Okumura
Mathematician, (b. Gunma, Japan, 1950).
Address: Department of Information Engineering, Maebashi Institute of Technology, 460-1 Kamisadori
Maebashi Gunma 371-0816, Japan. E-mail: [email protected].
Fields ofmterest: Geometry, Wasan Geometry (also omamental arts).
Publications: Fujita configurations, J. Recreational Math., 21 (1989); Configurations arising from the three
circle theorem, Math. Mag., 63 (1990); A five-circle problem, Crux Mathemattcorum, 20 (1994); Circle
patterns arising from a seven-circle problem, Crux Mathematicorum, 21 (1995).
Abstract: Japanese mathematics in 17th-19th CC, called wasan, left many results. Those
results sometimes can be generalized and also be used to construct interesting patterns
or configurations. Several such examples will be given.
1 INTRODUCTION
The mathematics we will present here is one developed in Japan during the Edo period
(1603-1867). Its root is in Chinese mathematics and it developed rapidly during the Edo
era. It is called wasan (wa means "Japan" and san means "mathematics", respectively).
At the beginning of the Meiji period (1868-1911), the new government adopted Western
mathematics into its new school system, bringing a sudden end to the short life of wasan.
But the wasan tradition was still able to survive.
Results of wasan geometry belong to two- or three-dimensional Euclidean geometry.
These results concern elementary figures such as triangles, quadrangles, circles, spheres,
etc., where tangent circles are in the majority. The results are stated as problems with
CIRCLE PATTERNS ARISING FROM RESULTS IN JAPANESE GEOMETRY
their final answers, but in most cases there is no explanation how such answers were
achieved.
There were two ways to publish the results in those days: One was in a book and the
other was on a wooden board called a sangaku. (The characters for san and gaku mean
"mathematics" and "framed board" respectively in Japanese in this case.) When people
found interesting problems, they would often write them down on a framed wooden
board and dedicated it to a shrine or a temple. Then the board would be hung under the
roof on the grounds. Most such problems were of a geometric nature and the figures
were beautifully drawn in color. But since there are so many of these wasan problems,
many still remain that have yet to be confirmed as correct.
Sometimes we can generalize about many of those problems. Furthermore, patterns or
configurations consisting of figures in these problems can often be discovered. In this
article we will demonstrate some of these patterns in the plane arising from problems
involving tangent circles in the old Japanese geometry.
2 A PATTERN BY FUJITA CONFIGURATIONS
Let ABCD be a parallelogram, E a point on the segment CD, F and G points on the
segments CE and DA respectively, and let FG meet BE at H. Suppose the quadrangles
ABHG, BCFH and DGHE have incircles ~a, ~’b, ~’d of radii a, b, d respectively. Also we
denote the incircle and its radius of the triangle FEH by ~’c. and c (see Figure 1). This
figure seems to have been first considered by Fujita in his book Seiy6 Samp6 (1781) in
the case where ABCD is a square. It seems appropriate to call this figure a Fujita
configuration. The problem is as follows:
Problem 1. In a Fujita configuration, where ABCD is a square, 2a+2b+2d+BE+FG+AB
be given, find AB.
The answer is AB = 12(2a+2b+2d+BE+FG+AB)/55. Though several problems
involving Fujita configuration where ABCD is a rhombus or a rectangle can be found in
some wasan books, they do not demonstrate any interesting properties of the
configuration. But there is one remarkable property (Okumura, 1987 and 1989b, see
Figures 2 and 3.):
Theorem 1. In a Fujita configuration a+c = b+d holds. If ABCD is a square then
c:d:b:a= 1:2:3:4.C onversely, from a triangle with the inradius c and two exradii b and
d, we can get two Fujita configurations by setting the radius of the fourth circle equal to
btd-c.
Using Figure 2, we can construct a pattern in the entire plane so that the segments BE
and FG are portions of some lines (see Figure 4). In this pattern we can superpose
similar ones. Two similar patterns which are double and four times the size are drawn on
the figure.
Figure 1 Figure 2

H. OKUMURA
3 PATTERNS OBTAINED BY UNIFYING SEVERAL PROBLEMS
Our patterns in this section are rather trivial. But it is worth-while considering them here
because they contain several figures that can be found in several wasan problems. A
typical problem is as follows:
Problem 2. Let ao be a circle of radius s with center O and a diameter AB, al and a2
circles with diameters AO and BO respectively. Then let 71, ?2, ?3 be distinct circles with
a radii r contained in ao tangent to ao and lying on the same side of AB, and ill, f12
distinct circles tangent to cq at O such that ?a touches a~ externally, fll (fl~ (cid:128) aa) touches
?~, 72 touches ill, f12 touches ?2, ?3 touches f12 and az externally. Show that s = 5r (see
Figure 5).
The problem is easily generalized as the following theorem (Okumura 1995a).
Theorem 2. Let a0, ax, a2, 71 and fl~ be as in Problem 2. Suppose that two families of
distinct circles ?z, ?3 ..... ?, with a radii r and f12, ]33 ..... 13, have been drawn such that ?~,
?i are tangent to ao and lying in ao on the same side of AB, and for any integer j (2
<j < i) ?j touches flj.~ and flj touches ?j. and ~z~ at O. Then define circles ?,+a ((:9i) with a
radius r and fl~+~ (~bi) such that ?,+~ touches 13, and ao internally and lies in the same side
of AB with ?a, and fli+l touches ?,+~ and al at O. Suppose that ?1, ?z ..... ?n and ill, 132 .....
13n have been drawn in this procedure and 13, = az, then we have s = (n+2)r.
B
Figure 5 Figure 6