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]. You may wish to visit [url]https://sangaku-archive.org/[/url].
目錄
- 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 #6
- Interesting examples
- Rakumanji #4
- Plum Blossom
- Myōjōrinji #1
- Gion shrine problem
- Kitamuki Kannon
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. Here is its entry at the [url=https://sangaku-archive.org/works/1864_Tokyo_Konno_273]Sangaku Archive[/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.
Rakumanji #4
This problem was posted at Rakumanji temple in Chiba Prefecture in 1886. Here is its entry at the [url=https://sangaku-archive.org/works/1886_Chiba_Rakumanji_224]Sangaku Archive[/url].[br][br]A right triangle with legs of length 3 and 4 has an inscribed ellipse with its major axis parallel to the long leg of the triangle, as shown below. The minor axis of the ellipse has length one. Find the length of the major axis.
Let [i]u[/i] refer to this desired length. First, consider a modern approach to this problem, which might go as follows. Begin by placing the diagram on a Cartesian coordinate system.[br][br]Using techniques that many of us learned in secondary school, we can write down equations that describe the line containing the hypotenuse,[br][center][br][math]\large[br]y=3+3x/4[br][/math][br][/center][br]as well as the ellipse:[br][center][br][math]\large[br]\frac{(x - u/2)^2}{u^2} + \frac{(y-1/2)^2}{1^2}=1.[br][/math][br][/center][br]Since the ellipse and the hypotenuse are tangent at a single point (D), solving the two equations above simultaneously for [i]x[/i] must yield a single value. This means that the discriminant of the resulting quadratic equation must be equal to zero:[br][center][math]\large 120u-384 = 0.[/math][/center][br]Hence, the length of the major axis of the ellipse is 3.2.
[center][icon]/images/ggb/toolbar/mode_parallelplane.png[/icon][/center]
While wasanka would have been perfectly capable of the algebraic moves described above, including the "discriminant must equal zero" observation, there is a more fundamental reason why traditional Japanese mathematicians would not have solved the problem like this: the Cartesian coordinate system was not in the wasan toolkit. First introduced by René Descartes in 1637, [i]analytic geometry[/i] (also known as [i]coordinate geometry[/i]) allows planar shapes to be realized as loci of points---ordered pairs ([i]x[/i], [i]y[/i])---that satisfy algebraic expressions, like those for the line and ellipse above. Linking geometry and algebra in this way was a powerful innovation in mathematics, but due to Edo Japan’s policy of national seclusion wasanka never incorporated it.[br][br]Instead, Edo-period mathematicians had their own set of tools for working with ellipses. Unlike the ancient Greeks, who viewed an ellipse as a conic section, wasanka thought of an ellipse as a section of a circular cylinder.
Looking at the cylinder, it’s easy to observe that the ellipse is, in a sense, a stretched-out version of the circle. An analysis of similar triangles shows that this circle-to-ellipse transformation preserves ratios along the direction of the stretch. This means we can view an ellipse as a [i]linear[/i] (also [i]affine[/i], since there is no preferred origin) [i]transformation[/i] of a circle. Linear transformations are especially useful when doing planar geometry, since a line that undergoes such a transformation is still a line.[br][br]We can use this idea to transform the original picture in our problem into something simpler (i.e., with no ellipses). Just perform a horizontal stretch by a scale factor equal to [math]\large 1/u[/math] (the effect of which is a horizontal compression). The transformed picture now has a circle of diameter 1 inscribed in a right triangle with legs of length 3 and [math]\large 4/u[/math].
A wasanka might now be in a position to apply one final theorem to obtain an equation, whether a statement imported from the Chinese (who studied right triangles extensively) or a homegrown geometric result. For example, it was likely known in Japan that for any circle of radius [i]r[/i] inscribed in a right triangle with legs [i]a[/i] and [i]b[/i],[br][center][math]\large a=\frac{2r(b-r)}{b-2r}.[/math][/center][br]Plugging in our values, we find again that[br][center][math]\large u = 3.2.[/math][/center][br]It's worth mentioning that this tablet was dedicated well into the Meiji period (1868--1912), an era in which Western scientific methods were flooding into Japan. It is not impossible that a Japanese mathematician would have applied the tools of analytic geometry to a sangaku problem around this time. However, the technique given on the tablet is much more aligned with the second approach.