Sangaku

The Sangaku or San Gaku (算額; literally mathematical shelves) are Japanese geometrical enigmas in the Euclidean Géométrie engraved on wood shelves, appeared during the Period Edo (1603-1867) and manufactured by members of all the social classes.

History

During this Period Edo, the Japan was completely isolated from the rest of the world, so that the shelves were created by using Japanese Mathematics ( wasan ), without influence of the Western mathematical thought. For example fundamental connection between an Integral and its Dérivée was unknown, so that the problems of Sangaku on the surfaces and volumes were solved by the expansion of infinite series and calculation term by term. It was one period of intense cultural creation, in the broad sense, with the appearance of other deeply original forms of art: the theater Kabuki, the Bunraku (puppet theater), the Ukiyo-e (prints). The Japanese benefitted from the cultural heritages Chinese brought back continent, of which incomprehensible works mathematical at the beginning, that they assimilated quietly to make theirs.

Sangaku were painted color on wood shelves which were suspended on the entry of Temple S and furnace bridges shintoïstes (Jinja) in offering with the local divinities (votive shelves). Many of these shelves was lost after the period modernization which succeeded the Period Edo, but approximately 900 could be preserved. Sangaku were published for the first time in 1989 by Hidetoshi Fukagawa, a professor of Mathématiques of college and by Daniel Pedoe in one entitled book: Japanese Temple Geometry Problems .

Types of problems

The shelves sangaku often present simple figures where the esthetics of the forms is determining in the choice of the problems. One finds particularly there Polygone S and Polyèdre S simple or regular, Cercle S, ellipse S, Sphère S and Ellipsoïde S. the different Paraboloïde and the Conique S make their appearance there too. The Cylindre intervenes especially to create the ellipse by intersection with the Plan. The transformations Affine S are used to pass from the circle to the ellipse. Problems relate to for example several mutually tangent circles or several tangent circles with an ellipse.

One of the beautiful problems, that found on a shelf of the Prefecture of Tokyo in 1788 and which made the cover of the Scientific American , brings into play the disc or the circle of the entireties , where, in a circle of radius 1, one wedges two discs of ray 1/2 (or Courbure 2, curve being the reverse of the ray), the interstices being filled discs of curve 3, thus creating other interstices, which in their turn will be filled by smaller discs with whole curves (6, 11,27, etc. ) This remarkable construction, which utilizes an infinity of quadruplets of mutually tangent circles (satisfactory thus the Théorème of Descartes), contains only circles with the whole curves. The problem required simply which was the radius of a circle of one of the intersticielles series.

See too

the mathematician Seki Kowa

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Sangaku

History

Types of problems

See too

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