Sangaku Resources
This is a collection of resources for traditional Japanese geometry problems. Such problems began appearing on colorful tablets called [i]sangaku[/i] during Japan's Edo period. The broader discipline of traditional Japanese math is known today as [i]wasan[/i].
Tabla de Contenidos
- Theorems
- Law of Cosines
- Secants, Chords, and Tangents
- Six Segment Theorem
- Descartes' Circle Theorem
- Bōshajutsu
- Common power of two circles
- Sangaku problems for beginners
- Konnō Hachimangū #1
- Myōjōrinji Temple #6
- Interesting examples
- Gion Shrine Problem
Law of Cosines
This theorem, which applies to oblique (non-right) triangles, is a generalization of the Pythagorean Theorem.[br][br][i]Try dragging the vertices of the triangle below.[/i]
Trigonometry was not used widely in Edo Japan (see Unger); [i]wasan[/i] practitioners would have known a version of the theorem that avoids reference to angle measure. Note, however, that a change in sign is necessary for obtuse triangles.
Konnō Hachimangū #1
This problem was posted at Konnō Hachiman shrine in Tokyo in 1824. It is located on the Sangaku Archive at [URL][color=rgba(0, 0, 0, 0.87)].[/color]
Statement
Three circles and a line are mutually tangent, as shown below. Find an equation that relates the radii of the three circles.
Gion Shrine Problem
This problem was posted at the Gion (now Yasaka) shrine in Kyoto in 1749. It is located on the Sangaku Archive at [URL].
Statement
In the circular segment below, the chord has length [i]a[/i] and its perpendicular bisector has length [i]m[/i]. Consider the square of side [i]s[/i] and circle of diameter [i]d[/i] as shown. Given the quantities[br][center][math]p=a+m+s+d[/math][/center]and[br][center] [math]q=\frac{m}{a}+\frac{d}{m}+\frac{s}{d}[/math][/center]find [i]a [/i]in terms of [i]p [/i]and [i]q[/i].
Discussion
The values of [i]p [/i]and [i]q[/i] depend on geometrical aspects of figure. You can try adjusting the radius of the big circle and moving the red triangle point in the middle of the chord to see how they change [i]p[/i] and [i]q[/i]. [br][br]Notice also that the value of [i]a[/i] varies with [i]p[/i] and [i]q, [/i]but in a non-obvious way. In 1774, mathematician Ajima Naonobu presented this relationship as a tenth degree polynomial in [i]a[/i] with coefficients in terms of [i]p[/i] and [i]q[/i]. This polynomial sufficed as a solution, since any specific values given for [i]p[/i] and [i]q[/i] would yield a polynomial in the variable [i]a[/i] that could be solved numerically; however, Ajima did not address the issue of choosing which of the potentially ten distinct roots would be the actual length of the chord. As of this writing, no one has found a polynomial solution with lower degree.[br][br]Ajima's solution is also notable for its use of a 4x4 determinant calculation using a form of cofactor expansion, which Japanese mathematicians developed independently from their counterparts in Europe. For all its algebraic challenges, however, the Gion shrine problem is fairly simple geometrically: there are three right triangles--with sides composed of the five basic variable lengths--that completely describe the construction. You can display these above.
Classroom Notes
This problem is algebraically very difficult as stated. For a more approachable version, change the two given quantities to simpler ones and use numerical values. For example, "given that [math]m=10[/math] and [math]d=9[/math], find [i]a[/i] and [i]s[/i]."
For Further Reading
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