Tutorial on Geometer's Sketchpad--Dudeney's Decomposition
This paper appears in the
Proceedings of ATCM 1999:
Tsing Hua University
Hsinchu, Taiwan 300
This is the first paragraph of the first section entitled "Triangles,
Squares and Games" in the famous book "Mathematical
Snapshots" written by H. Steinhaus:
From these four small boards
(1) we can compose a square or an equilateral triangle, according as we
turn the handle up or down. The proof is given by sketch (2).
Shown below are the accompanying figures in the book:
Question: Is it possible to give a mathematically correct proof from
Steinhaus acknowledged that the idea of the decomposition was taken
from p.27 of H.E. Dudeney's "Amusements in Mathematics."
As we checked against the original source, there was no mention of this
particular decomposition in the book at all! A figure included in Martin
Gardner's book "More Mathematical Puzzles and Diversions",
however, does enlighten us on the decomposition that Dudeney invented.
We now convert the procedure into a series of drawing steps with Geometer's
Construct the equilateral triangle together with the three midpoints.
Construct an outward semi-circle by taking one side as diameter.
Draw the axis of symmetry.
Draw a circle by taking the axis as diameter and find its intercept with
the extension of the side of triangle. This way, the length of one side
of the required square is found.
The position of the second vertex of the square is located.
This is one side of the square.
Locate a point with a distance one-half the length of the equilateral triangle
from the point found in Step 6.
Drop perpendiculars to the last segment. The decomposition is now complete.
Fill the interiors of the four regions with distinct colors.
In order to re-assemble the four pieces into a square, place an arbitrary
point on the semi-circle. The transformation from triangle to square is
to be performed by moving this arbitrary point along the semi-circle. This
is the first of three intermediate steps in the construction: rotate the
three pieces with respect to the midpoint.
The second intermediate step: rotate the two pieces with respect to the
The third intermediate step: rotate the last piece with respect to the
This way we may perform a continuous transformation by turing the decomposition
of the triangle from
into the decomposition of the square.
Using the software we find the ratio BH/BL = 0.255 approximately.
The complete Geometer's Sketchpad file of this demonstration is located
H.M. Cundy and A.P. Rollett, Mathematical
Models, Tarquin, p. 24.
H.E. Dudeney's, Amusements
in Mathematics, Dover (1970).
Howard Eves, A
Survey of Geometry, Vol. One, pp. 260-261.
Martin Gardner, More
Mathematical Puzzles and Diversions, p. 26.
Martin Gardner: The
Second Scientific American Book of Mathematical Puzzles and Diversions,
H. Steinhaus, Mathematical
Snapshots, pp. 3-4.