This picture shows my experiments with semicircles inscribed in squares, using open source geometry software. I used the method of compass and straightedge construction to generate recursively smaller inscribed semicircles, each half the area of the previous semicircle. There are interesting coincidences
in the diagram; the diameter of the second semicircle lies on the side of the first
square, and the corner of the third semicircle is the centre of the
square. The figure also contains a circle inscribed in an isosceles right triangle. The blue and red areas are self similar
shapes, related by translation, rotation and scale. The construction is
reproducible at any magnification or reduction that is a multiple of
the square root of two. A simpler case of self similar recursion in the ratio of the square root of two is squares inscribed diagonally in squares. The self similarity resembles that of the Sierpinski triangle. My construction involves exchange of ideas with Linda Fahlberg-Stojanovska, who published a recent paper on compass and
straightedge construction, which includes a simple proof and procedure
for inscribing a semicircle in a square. The topic is also being discussed by teachers and other mathematics enthusiasts on http://mathematics24x7.ning.com/, a social network on Ning.com.

Dr. Geo is interactive geometry software for the GTK desktop on Linux. It saves the drawings in XML format. I had to edit the XML file of the square to set the corner points, to 0,0; 10,0; 10,10 and 0,10, as I use an older version of Dr. Geo that doesn't support coordinate entry. After exporting the drawing as a PNG picture, I used the open source Inkscape drawing software to colour the segments for clarity.

Dr. Geo is interactive geometry software for the GTK desktop on Linux. It saves the drawings in XML format. I had to edit the XML file of the square to set the corner points, to 0,0; 10,0; 10,10 and 0,10, as I use an older version of Dr. Geo that doesn't support coordinate entry. After exporting the drawing as a PNG picture, I used the open source Inkscape drawing software to colour the segments for clarity.

## Recent Comments

Colin McAllister:A movie by Frans Pretorius at Princeton Univers... | more »Colin McAllister:Linda Fahlberg-Stojanovska has published a pape... | more »devmobdev:well worth the read. thank you very much for ta... | more »