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Mathematical Beadwork

愛知県立春日井東高等学校

堀部 和経HORIBE Kazunori

URL http://horibe.jp

Aichi Prefectural Kasugai-Higashi Senior High School

CAN YOU SOLVE THIS ?

MATCH WITS AGAINST

JAPANESE GEOMETRY（日本の幾何学と知恵比べ）

Introduction to Japanese Geometry SANGAKU

Scientific AmericanMay 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 (mathematical wooden tablet)1788 in Edo(Tokyo) Prefecture.

It asks for the radiusof the -th largest blue circle in terms of ,the radius of the green circle.

nr

Sangaku problem

Hint ? : The radius of the 5‐th largest blue circle is           .

Note that the red circles are identical, each with the same radius.

r

2

95

r

22 1 14

n

rr

n

Other problems in sangaku

URL http://horibe.jp HD HD

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Sangaku(replica) W45cm‐H30cm 1841 Atsuta Shrine

Sangaku(Replica) W240cm‐H60cm dated on  1844

Atsuta Shrine in Nagoya City of Aichi Prefecture

with Sonoda, Ono, and Fukagawa

Sangaku(replica)at Atsuta Shrine 2013

Atsuta Shrine

Sangaku W330cm‐H132cm1830  (the genuine tablet)

Iwaifudou Temple in Chiba Pref. Iwaifudou Temple in Chiba Pref.

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W119cm‐H37cm1877

Ishibe Shrinein Fukui Pref.

The students were not only Samurai but also children, women and chandlers.

Wasan-Juku a private school for mathematics in the city

Private mathematics school

Studying the method of an equation

Studyingarithmetic

Studying how to do soroban, a Japanese abacus

Another Sangaku Problem 

Dodecahedron with regular pentagons Including an inscribed sphere 

It asks for the ratio of the radius of the inscribed large spherein terms of the radius of the small sphare for pentagone.

Please take a lookat the model in motion.(gif animation)

算法助術 Sanpo‐Jojutsu1841  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. 

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30‐ball problem in Sanpo‐ JojutsuThe 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 is305 sun, what is the diameterof the large ball?

Sun (寸): a unit of the old Japanese length

The Answer is 682.000 sun.

Modeling

Cut along an equatorial plane

断面図

このイメージは、現在表示できません。

5R

Ratior

正１０角形Regular decagon

正５角形Regular pentagon

additional lines

How to solve the problem

( ormula No.3 of )

1 5

2

F Sanpo Jojutsu

角中径

面

1 5

2 2

r R

r

Therefore 5R

r

diagonal 1 5

edge 2

r

r

r

R

r

r

r

R 1 1 5

R

r

5

Very strange 

9 24d m cm　

682.000LARGE D 寸 30.3 mm 寸( )≒ （日本）

305small d 寸

20 66D m cmThey are too big as a model.

2 , 2d r D R

sun

Nagoya City Science Museum

The planetarium dome has a diameter of 35 m.

6825

305≒

22682 465,124 , 305 5 465,125

22( ) Pell 's equation 5 1Extension x y

Solution , (305, 682)x y

High accuracy in

rational number approximation

682Actually 5 2.236068 , 2.236066

305

The large numbers 305 & 682 are chosen.   Why such large numbers ?

Next question 682

How to find 5305

　≒

5

1 12 5 2

22 5

Set x

thus xx

12

2x

x

1

21

2 22 x

1 1 12 2 2

1 1 174 4 4

1 4 724 41 1744

72 6822

305 305

x

≒

1 1 12 2 2

1 12 4 412 4

14

4

xx

x

( )

: 0

Reiyaku jutsu

set

零約術 0

Notice: 

We know this problem which was carried inthe 1830 book Sanpo-Kisho by Baba Seitoku(1801-1860). Accoding to the book, theproblem was written on a mathematical tablet.In the book, Baba recorded thirty-sixsangaku collected from shrines in Tokyo.The problem was originally proposed by

Ishikawa Nagamasa, a student of the schoolof Baba Seito (1777-1840), who was Seitoku’sfather. It was written on a tablet, which washung in 1798 in Gyuto Temple Shrine, Tokyo.

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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

Therefore, it had already existed as a mathematical problem in 1798.

Main Subject 

Mathematical Beadwork

My Work is 

mathematical work???or 

hobby work???

N=6N=12

N=30

N=30N=90

N=120

Semiregular polyhedron

N=30

N=90N=120

N=210N=270 Straight shape 

Too simple !

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Helical shape  Y shape

Torus shape Other Torus 

Red coral 

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Tricolor ring Orthogonal coordinate system shape

3D‐hashtag character shape  「3D井の字」

Tetrapod shape

3D continuous tetrapod shape

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Regular dodecahedron shape 

作品（１５）

作品（１６）

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This model’s description

Diameter = 38 mm

was the former size of a ping‐pong ball. 

This wooden ball diameter is about  17 mm.

15 2

14

4 0

≒

Diameter = 40 mm is  the present size. 

( )reiyaku jutsu 零約術

4 382

17 17

Mathematical Beadworkmodel making of 30 ball problem

愛知県立春日井東高等学校

堀部 和経HORIBE Kazunori

URL http://horibe.jp

Aichi Prefectural Kasugai-Higashi Senior High School

http://horibe.jp/PDFBOX/Manual_B30.pdf

Now, I will show you how to beadwork.

Steps １，２

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Steps ３，４ Steps ５，６

Steps ７，８ Steps ９，１０

Steps １１，１２Steps １３，１４

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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.

No logo ping‐pong balls This work made with no logo ping‐pong ballswas displayed in the lobby of Nittaku Co. Bld.

My treature

Back to the  interrupted place 

Ballstructure.comWB_MakingB.htm

Personal Memorandum

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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.