Sangaku

Intro to CalculusSangaku 1

No visitor to a foreign country has failed to experience the fascination and unease that accompanies an encounter with unknown traditions and customs. Some visitors attempt to overcome their fears, while the majority quickly retreats to familiar shores, and in this lies a distinction: Those who embrace culture shock are travelers; those who do not are tourists.

The most profound culture shock comes about when one is confronted by a different way of thinking. Most of us can hardly imagine walking into a Western church or cathe-dral to encounter a stained glass window covered by equations and geometrical figures. Even if we can conceive of it, the thought strikes us as alien, out of place, perhaps sac-rilegious. Yet for well over two centuries, Japanese mathematicians— professionals, amateurs, women, children— created what was essentially mathematical stained glass, wooden tablets adorned with beautiful geometric problems that were simultaneously works of art, religious offerings, and a record of what we might call "folk mathematics." The creators of these sangaku— a word that literally means "mathematical tablet"— hung by the thousands in Buddhist temples and Shinto shrines throughout Japan, and for that reason the entire collection of sangaku problems has come to be known as "temple geometry", sacred mathematics.

— Tony Rothman, Sacred Mathematics

Problem 1— Suanfa Tong Zong (Systematic Tretise on Mathematics) by Chang Da-Wei, 1592

Find the radius of a circle that is inscribed in a right triangle whose short sides are 36 and 27.

Name ________________ ! Date _______! Class # ______! Block ______

Critical and Informed Thinkers • Effective Communicators • Collaborative Workers

Goals

Appreciate mathematics

as a human activity with a

deep and complex his-

tory.

Improve ability to formu-

late and solve problems.


Problem 1

A square is inscribed in a right triangle whose short sides are in the ratio of 1:2. What is the length of the side of the square in terms of the length of the shortest side of the circumscribed triangle?

Problem 2

A semi-circle is inscribed in a right triangle such that its diameter lies on the shortest side of the triangle. If the short sides of the triangle are 3 and 4, what is the difference between the length of the diameter and the length of the shortest side of the triangle?



Goals




tory.




Fujita Sadasuke (1734-1807)Two circles of radii a and b are tangent to each other as well as tan-

gent to line l as shown in the diagram below. Show that DE = 2 ab .

Justify with clear and complete work.

l

a

b

ED



Goals




tory.




Sangaku of the Katayamahiko ShrineThe sangaku below is one of 16 puzzles on a dragon-framed tablet which was dedicated by Irie Shinjun in 1873 to the Katayamahiko shrine of Mu-rahisagun Okayama city.

Two circles of radius r are tangent to the line l. A square of side t touches both circles. Find t in terms of r.



Goals

Appreciate mathemat-

ics as a human activ-

ity with a deep and

complex history.

Improve ability to for-

mulate and solve

problems.

Scoring GuideDefine variables and la-

bel drawing. (4 pt.s)

Clearly and convincingly

guide the reader to the

solution. (12 pt.s)

Correctly answer the

question. (4 pt.s)

算額Intro to CalculusSangaku 51

The following problem was originally proposed in 1800 by Kobata Atsukuni, a student of the Aida school, and presented on a tablet to the Kanzeondo temple of Toba castle town.

A big circle of diameter 2R = 100 inscribes a large and small equilateral triangle. Find

the side q in terms of R of the small equilateral triangle ABC if A is the midpoint of one side of the large triangle.

R

q

B C

A



1 This problem is based on material from Sacred Geometry, F. Hidetoshi and T. Rothman,

Goals




tory.







solution. (12 pt.s)


question. (4 pt.s)


This problem is from the Katayamahiko shrine Sangaku in Mura-hisagun Okayama city. It was dedicated by Irie Shinjun in 1873.

A circle of radius r inscribes three circle of radius t. Find t in terms of r.



Goals




tory.







solution. (12 pt.s)


question. (4 pt.s)


Kobayashi Syouta proposed this problem on a table that was hung in the Shimizu shrin, Nagano prefecture, in 1828.

A big square of side a, encloses a smaller square of side 2r, and a circle of radius r . The circle is tangent to two sides of the big square and is tangent to the small square at a corner (as shown in the diagram). Find r in terms of a.

Extra ChallengeUse Geogebra to make this sangaku so that all of the objects move cor-rectly as the result of changing the length of side a.



1 Based on the work of F. Hidetoshi and T. Rothman

Goals




tory.







solution. (12 pt.s)


question. (4 pt.s)


Watanabe Kiichi proposed this problem, which is on the Sangaku of teh Abe no Monjyuin temple in the Fukushima prefecture. The tablet contains 21 problems. It was hung in 1877 by the students of the

Sakuma Y!ken.

An equilateral triangle with side t , a square of side s , and a circle touch each other in a right triangle ABC with vertical side a. Find t in terms of a.

t

a

s

A

C

B

Extra ChallengeUse Geogebra to make this sangaku so that all of the objects move correctly as the result of changing the length of side a.




Goals




tory.







solution. (12 pt.s)


question. (4 pt.s)


This problem is from the Shimizu shrine from the sangaku presented in 1828 by Kobayashi Nobutomo.

As shown in the figure a small circle of radius b sits on the point of contact between two squares of side 2b that in turn sit on a line. A big circle of radius a is tangent to line l, the small circle, and the corner of the nearest square. Find a in terms of b.

Extra ChallengeUse Geogebra to make this sangaku so that all of the objects move correctly as the result of changing the radius of the small circle.




Goals

Appreciate

mathematics as a

human activity with a

deep and complex

history.

Improve ability to

formulate and solve

problems.





solution. (12 pt.s)


question. (4 pt.s)


This is another problem from the sangaku of the Katayamahiko shrine dedicated by Irie Shinjun in 1873. Murahisagun Okayama City.

On a circular field of diameter 2r = 100m, we make four lines of length t

such that they divide the circle into five equal areas, S, one of which is a square of side d. Find approximate numerical values for t and d using ! = 3.16 (the

common value used for ! during this period in Japan.)

Extra ChallengeUse Geogebra to make this sangaku so that all of the objects move correctly as the result of changing radius of the circle.



1 Adapted from the work of F. Hidetoshi and T. Rothman in Japanese Temple Geometry

Goals

Appreciate

mathematics as a


deep and complex

history.

Improve ability to

formulate and solve

problems.




guide through your rea-

soning (12 pt.s)


question. (4 pt.s)

Sangaku 11



The tablet from which this problem was taken was hung in 1874 in the Akahagi Kannon temple in Ichinoseki city. Its size is 188 cm by 61 cm.

The problem itself was proposed by Sat! Naosue, a thirteen-year-old boy.

Two circles of radius r and two of radius t are inscribed in a square, as shown. The square itself is inscribed in a large right triangle and, as illustrated, two circles of radii R and r are inscribed in the small right triangles outside the square. Show that R = 2t .

R

r

tt

r

r

Extra ChallengeUse Geogebra to make this sangaku so that all of the objects move correctly as the result of changing radius of the circle.




Goals

Appreciate

mathematics as a


deep and complex

history.

Improve ability to

formulate and solve

problems.





soning. (12 pt.s)


question. (4 pt.s)

R

r

tt

r

r

Sangaku 12



The problem below is originally from the Sangaku of the Sugawara shrine of Ueno city, Mie prefecture. It was hung by Hojiroya Sh!emon in 1854.

As shown in the figure below, a square of side c is inscribed in an equi-lateral triangle of side k. Two smaller squares of sides a and b are inscribed between the equilateral triangle and square c. A smaller equilateral triangle of side d is inscribed within square c and a circle of radius r is inscribed within this triangle. Find b, c, d, k, and r in terms of a.

a

b

c

k dr

Extra ChallengeUse Geogebra to make this sangaku so that all of the objects move correctly as the result of changing the length of the side of the smallest square.

Name ________________ " Date _______" Class # ______" Block ______



Goals

Appreciate

mathematics as a


deep and complex

history.

Improve ability to

formulate and solve

problems.





soning. (12 pt.s)


question. (4 pt.s)

a

b

c

k dr

Sangaku 13



Here we have a rare example of a problem proposed by a woman, Okuda Tsume. Hung in 1865 at the Meiseirinji temple in Ogaki city, Gifu.

In a circle of diameter AB = 2R , draw two arcs of radius R with centers

A and B, respectively, and ten inscribed circles. Show that the eight

small circles all have equal radii, t, and show that t =R

6.

A B

Extra ChallengeUse Geogebra to make this sangaku so that all of the objects move correctly as the result of changing AB .




Goals

Appreciate

mathematics as a


deep and complex

history.

Improve ability to

formulate and solve

problems.





soning. (12 pt.s)


question. (4 pt.s)

A B

Sangaku 14


算額Intro to CalculusSangaku X+1

This problem has survived on an 1824 tablet from the Gumma Pre-fecture. !A and !C are tangent to each other at one point and are

tangent to the same line. !B is tangent to both !A and !C and is

also tangent to the same line. How are the radii of the three circles related?

B

AC



Goals




tory.



算額Intro to CalculusSangaku X

From a 1803 Sangaku found in Gumma Prefecture. The base of an isosceles triangle sits on a diameter of the large circle. This diameter also bisects the circle on the left, which is inscribed so that it just touches the inside of the container circle and one vertex of the trian-gle. The top circle is inscribed so that it touches the outsides of both the left circle and the triangle, as well as the inside of the container circle. A line segment connects the center of the top circle and the intersection point between the left circle and the triangle. Show that this line segment is perpendicular to the drawn diameter of the container cir-cle1.



1 When solving this problem I found myself being drawn to making assumptions based on the drawing

which, on deeper inspection, turned out to be false.

Goals




tory.



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