Mathematical Beadwork
愛知県立春日井東高等学校
Aichi Prefectural
Kasugai-Higashi Senior High School
堀部 和経
HORIBE
Kazunori
URL http://horibe.jp
Introduction to Japanese Geometry
SANGAKU
CAN YOU SOLVE THIS ?
MATCH WITS AGAINST
JAPANESE GEOMETRY
（日本の幾何学と知恵比べ）
Scientific American
May 1998
The article on Scientific American begins with
Of the world’s countless customs
and traditions, perhaps none is
as elegant, nor as beautiful, as
the tradition of sangaku,
Japanese temple geometry.
世界中の数え切れない習慣や伝統の中で、日本独自の幾何学
を記した「算額」ほど優雅で美しいものは他に見当たらない。
An illustration on the
cover page of Scientific
American
Sangaku problem
It asks for the radius
of the n-th largest blue
circle in terms of r,
the radius of the green
circle.
Sangaku (mathematical wooden tablet)
1788 in Edo(Tokyo) Prefecture.
Note that the red circles are identical, each
with the same radius.
Hint ? : The radius of
the 5-th largest blue
r
circle is
.
95
rn 
r
 2n  1
2
 14
r
2
Other problems in sangaku
URL http://horibe.jp
HD
HD
Sangaku(replica) W45cm-H30cm 1841 Atsuta Shrine
Sangaku(Replica) W240cm-H60cm
dated on 1844
Atsuta Shrine in Nagoya City of Aichi Prefecture
Sangaku(replica)
at Atsuta Shrine 2013
with Sonoda, Ono, and Fukagawa
Atsuta Shrine
Sangaku W330cm-H132cm
1830 (the genuine tablet)
Iwaifudou Temple in Chiba Pref.
Steiner Chain
HD
Iwaifudou Temple in Chiba Pref.
Wasan-Juku a private school for mathematics in the city
The students were not only
Samurai but also children,
women and chandlers.
W119cm-H37cm
1877
Ishibe Shrine
in Fukui Pref.
Private mathematics school
Studying the method
of an equation
Studying
arithmetic
Studying how to do soroban,
a Japanese abacus
Another Sangaku Problem
It asks for the ratio of
the radius of the
inscribed large sphere
in terms of the radius of
the small sphare for
pentagone.
Dodecahedron with regular pentagons
Including an inscribed sphere
Please take a look
at the model in motion.
(gif animation)
算法助術 Sanpo-Jojutsu
1841 by Hiromu Hasegawa
The collection of
mathematical formulae
of the Edo period.
About 100 formulae
are contained.
Reprinted edition 2005
The problem appeared as one of the
applied problems at the end of the book.
30-ball problem in Sanpo- Jojutsu
The description of the question
As shown in
the figure,
the big ball is
surrounded by 30 small balls.
The small balls touch each other,
and are tangent to the surface of
the inner big ball as well.
If the diameter of
Each small ball is
305 sun,
what is the diameter
of the large ball?
The Answer is 682.000 sun.
Sun (寸): a unit of the old Japanese length
Modeling
Cut along an equatorial plane
正１０角形
断面図
Regular decagon
正５角形
Regular pentagon
R
Ratio 

r
5
additional lines
How to solve the problem
( Formula No.3 of Sanpo  Jojutsu )
角中径 1  5

面
2
diagonal 1  5

edge
2
r  R 1 5

2r
2
R
1
 1 5
r
R
Therefore
 5
r
r
r
r
R
r
r
r
R
Very strange
d  2r , D  2 R
small d  305 寸
LARGE D  682.000  寸
寸( sun ) ≒ 30.3 mm（ 日本）
d  9 m 24 cm
D  20 m 66 cm
They are too big as a model.
Nagoya City Science Museum
The planetarium dome has a diameter of 35 m.
682
5  2.236068  ,
 2.236066 
305
Actually
The large numbers 305 & 682 are chosen.
Why such large numbers ?
682
5≒
305
 High accuracy in

 rational number approximation 



682  465,124 , 305 5
2

2
 465,125

( Extension ) Pell's equation x  y 5
2
Solution  x , y   (305, 682)

2
 1
Next question
How to find
682
≒
5　305
Set x  5
1
1
thus x  2  5  2 

2 5 2 x
1
 2
x  2
2 x
1
1 

2  2 

2 x

  
1
1
1
x  2
 2
   2 
1
1
2 x
4
4
1
2 x
4
4
零約術( Reiyaku  jutsu )
set :   0
1
1
1
x≒2 
 2
 2
1
1
17
4
4
4
1
4
72
4
4
1
17
4
4
72 682
 2

305 305
1
4  
0
Notice:
We know this problem which was carried in
the 1830 book Sanpo-Kisho by Baba Seitoku
(1801-1860). Accoding to the book, the
problem was written on a mathematical tablet.
In the book, Baba recorded thirty-six
sangaku collected from shrines in Tokyo.
The problem was originally proposed by
Ishikawa Nagamasa, a student of the school
of Baba Seito (1777-1840), who was Seitoku’s
father. It was written on a tablet, which was
hung in 1798 in Gyuto Temple Shrine, Tokyo.
Therefore, it had already existed as
a mathematical problem in 1798.
Personal Memorandum
算額（Sangaku） 1798
東京都四谷区牛頭天王社 馬場正督の門人・石川永政
算法奇賞(Sanpo-kishyo)1830
正督の息子・馬場正統
算法助術(Sanpo-jyojutsu)1841
長谷川弘閲
Main Subject
Mathematical Beadwork
My Work is
mathematical work???
or
hobby work???
N=6
N=12
N=30
N=90
N=30
N=120
N=270
Semiregular
polyhedron
N=210
N=120
N=90
N=30
Straight shape
Too simple !
Helical shape
Y shape
Torus shape
Other Torus
Red coral
Tricolor ring
Orthogonal coordinate system shape
3D-hashtag character shape 「3D井の字」
Tetrapod shape
3D continuous tetrapod shape
Regular dodecahedron shape
作品（１５）
作品（１６）
This model’s description
Diameter = 38 mm
was the former size of a ping-pong ball.
This wooden ball diameter is about 17 mm.
5 ≒2 
1
1
4
40
4
38
 2

17 17
零約術( reiyaku  jutsu ) 
Diameter = 40 mm is the present size.
Mathematical Beadwork
model making of 30 ball problem
愛知県立春日井東高等学校
Aichi Prefectural
Kasugai-Higashi Senior High School
堀部 和経
HORIBE
Kazunori
URL http://horibe.jp
http://horibe.jp/PDFBOX/Manual_B30.pdf
Now, I will show you how to beadwork.
Steps １，２
Steps ３，４
Steps ５，６
Steps ７，８
Steps ９，１０
Steps １１，１２
Steps １３，１４
Steps １５，１６
Steps １７，１８
URL http://horibe.jp
HD
Trivia
The company name, Nittaku,
is not printed on the ball.
Because this is a custom-made ball
by NIPPON TAKKYU Co.,Ltd.
I think that no-logo balls are beautiful.
I wish that the balls are former size.
My treature
No logo ping-pong balls
This work made with no logo ping-pong balls
was displayed in the lobby of Nittaku Co. Bld.
Ballstructure.com
WB_MakingB.htm
Back to the interrupted place
Personal Memorandum
Personal
Memorandum
•
•
•
•
•
•
•
Personal Memorandum
That’s the point.
This is the point I want to emphasize here.
Please remember this point.
Please pay attention to the following.
Let’s turn to the ・・・.
Let me see.