Reassembling triangular puzzle
pieces, induces always to paradoxical conclusions. The squares
of the fig. 2.c) and 2.d) have extra triangular elements. Is
then their area larger than the one in the fig. 2.a)? As before,
we have to consider the angles of each right triangle which
form these squares. By doing that, we will easily notice that
the hypotenuse slopes of the small and of the large right triangles
are slightly different (a difference of approx. 0.8 degrees,
visually unnoticeable). So, the 8 right triangles do not form
exactly a square and the sum of all these tiny fitting errors
(grey zones in the fig. 5) is equal to the area of the protruding
triangular elements. In short, space apparition is only illusion!
Paradoxical
missing square puzzle
(called 'Fehlendes-Quadrat-Puzzle', in German;
and 'wigparadox', in Dutch)
We can even enhance the 'vanishing
area paradox' effect by adding 4 squares of 6 units
per side to the Circea's puzzle. When you will rearrange the
puzzle pieces a square space appears or disappears (see examples
below).
Both
squares are formed with the same 12 pieces.
The second one, however, needs an extra piece!
Our
puzzles Quadrix and Geometrex are
based on this vanishing area principle:
when several pieces of the puzzles are permuted, a small
square hole magically appears...
You can purchase these hands-on games, useful for your
classroom demonstrations, online (Europe).
We
are presently actively looking for manufacturers interested
to produce and market Quadrix under
licence for specific countries on exclusive or non-exclusive
basis. Please contact
us if interested for details.
Italy
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