2

1

1

sqrt(2)

2 sqrt(2)

sqrt(2)

sqrt(2)

2

The problem:
 From these three polygons can create a mirror-symmetric shape.  
(The three pieces should not have common interior points, thus the shapes can not be placed
 on top of each other.)
 

Hiroshi Yamamoto's three-element puzzle

May be show the solution?

sqrt(2)

2

2-sqrt(2)

1

1+sqrt(2)

The find shape is a mirror symmetric octagon.

http://www.johnrausch.com/designcompetition/2010/default.htm#24

g = 0 (slider from 0 to 10, step 0.1)
B = Point(xAxis)
C = Point(xAxis)
A = Intersect(xAxis, yAxis)
B'_1 = Rotate(B, 90*°, A)
A'_1 = Mirror(A, B'_1)
A'_2 = Rotate(A, 270*°, B)
poligon1, a'_1, a, b, a' = Polygon(A'_1, A, B, A'_2)
u = Vector(A'_2, C)
D = Rotate(A'_1, 315*°, A'_2)
D' = Translate(D, u)
D'' = Rotate(D', 270*°, C)
C'_1 = Rotate(C, 270*°, D'')
C'_2 = Mirror(C, D')
poligon2, c', c, d'', c'_1 = Polygon(C'_2, C, D'', C'_1)
E = Point(xAxis)
s = false
v = Vector(C, D'')
E'_1 = Translate(E, v)
z = Vector(A, B)
E''_1 = Rotate(E'_1, 90*°, E)
d = Vector(B, A'_2)
E'''_1 = Translate(E''_1, d)
F = Translate(E'''_1, z)
G = Translate(E'''_1, v)
F'_1 = Rotate(F, 90*°, G)
poligon3, e, e', f'_2, f_1, e''' = Polygon(E, E'_1, F'_1, F, E'''_1)
H = Mirror(A'_1, a'_1)
A' = Mirror(A, a'_1)
B' = Mirror(B, a'_1)
I = Mirror(A'_2, a'_1)
poligon1', h, a'_2, b', i = Polygon(H, A', B', I)
w: g * Vector[A', C] / 10
H' = Translate(H, w)
A'' = Translate(A', w)
B'' = Translate(B', w)
I' = Translate(I, w)
f, h', a'', b'', i' = Polygon(H', A'', B'', I')
E' = Mirror(E, e')
K = Mirror(E'_1, e')
F''_1 = Mirror(F'_1, e')
F' = Mirror(F, e')
M = Mirror(E'''_1, e')
poligon3', e'_1, k, f'', f'_1, m = Polygon(E', K, F''_1, F', M)
l: g * Vector[K, C'_1] / 10
E'' = Translate(E', l)
K' = Translate(K, l)
F'''_1 = Translate(F''_1, l)
F'' = Translate(F', l)
M' = Translate(M, l)
poligon3'', e'', k', f''', f''_1, m' = Polygon(E'', K', F'''_1, F'', M')
E''' = Rotate(E'', 45° * g / 10, K')
K'' = Rotate(K', 45° * g / 10, K')
P = Rotate(F'''_1, 45° * g / 10, K')
F''' = Rotate(F'', 45° * g / 10, K')
M'' = Rotate(M', 45° * g / 10, K')
poligon3''', e'''_1, k'', p, f'''_1, m'' = Polygon(E''', K'', P, F''', M'')
j = OrthogonalLine(H', c')
n = Vector(A', H)
C'_3 = Translate(C, n)
C'' = Dilate(C'_3, 1.5, C'_1)
J = Dilate(C'_1, 1.5, C'_3)
q = Segment(C'', J)
L = Rotate(C'_2, -((g * 67.5)°), C)
C'_4 = Rotate(C, -((g * 67.5)°), C)
D'''_1 = Rotate(D'', -((g * 67.5)°), C)
N = Rotate(C'_1, -((g * 67.5)°), C)
O = Rotate(C'_2, -((g * 6.75)°), C)
C' = Rotate(C, -((g * 6.75)°), C)
D''' = Rotate(D'', -((g * 6.75)°), C)
Q = Rotate(C'_1, -((g * 6.75)°), C)
poligon2', o, c'_2, d''', q_1 = Polygon(O, C', D''', Q)
H'' = Rotate(H', -((g * 6.75)°), C)
A''' = Rotate(A'', -((g * 6.75)°), C)
B''' = Rotate(B'', -((g * 6.75)°), C)
I'' = Rotate(I', -((g * 6.75)°), C)
f', h'', a''', b''', i'' = Polygon(H'', A''', B''', I'')
R = Rotate(E''', -((g * 6.75)°), C)
K''' = Rotate(K'', -((g * 6.75)°), C)
P' = Rotate(P, -((g * 6.75)°), C)
S = Rotate(F''', -((g * 6.75)°), C)
M''' = Rotate(M'', -((g * 6.75)°), C)
poligon4, r, k''', p', s_1, m''' = Polygon(R, K''', P', S, M''')
C''' = Rotate(C'', -((g * 6.75)°), C)
J' = Rotate(J, -((g * 6.75)°), C)
q' = Segment(C''', J')
poligon5, d'''_1, k'''_1, r_1, m'''_1, s_2, h''_1, i''_1, b'''_1 = Polygon(D''', K''', R, M''', S, H'', I'', B''')