The curvilinear shape (A) is equidecomposable to
2 squares and the cross-shaped figure (B) to
a larger square. We can then demonstrate thanks
to the Pythagorean
Theorem that they are of the same area,
as shown in the figure below. During this operation
no pieces are superimposed nor placed side
by side!
Wasan
puzzles
During the Edo period (l603-1867), when Japan was almost
completely cut off from the western world, a distinctive
style of mathematics, called Wasan (和算; "native
Japanese mathematics" in contrast to yosan, "Western
mathematics"), was developed.
Results and theorems were originally displayed in the
form of problems, sometimes with answers but with no
solutions, inscribed on wooden boards and accompanied
by beautiful coloured figures. These problems dealt
predominantly with Euclidian geometry and, true to
wasan preferences, mostly dealt with circles and ellipses. These
boards (see an example below), known as Sangaku(算額; "mathematical tablet"),
were hung under the eaves in shrines and temples. Later,
books appeared, either handwritten or printed from
hand-carved wooden blocks, containing collections of
sangaku problems with solutions. The earliest known
Sangaku tablet was created in 1683.
The samurai remained the dominant creators of sangaku,
consistent with their status of the educated and artistic
caste in Japan. A majority of sangaku are inscribed
in Kambun(漢文), an archaic Japanese
dialect related to Chinese. Kambun was the equivalent
of Latin in
Europe, used during the Edo period for scientific works
and
known predominantly by only the most educated castes.