People who haven't had the right opportunity to learn math have
something in common with people who were born before math was invented.
Perhaps some of the techniques of Euclidean geometry could be revived in a format
that is more accessible today. Specifically, the ancient tools of straight-edge and compass, could be reinvented to suit today's teaching needs. Take this geometry problem: *What is the largest area of the semi-circle that can be inscribed in a square of edge length 1 unit?* That was the question posted by harpreet in the topic " maxima & minima" on a Mathematics forum on Orkut.com. Some mathematics enthusiasts took a team approach to this problem on the Mathematics24x7 social network, Christian drew a diagram of the solution and Steve calculated the radius of the semicircle. We calculated the radius as 2-sqrt(2) and the area as* *pi*(3-2*sqrt(2))
*.*(sqrt() represents the square root.) Danny proposed a less abstract expression of the problem, which would be more interesting to students. *"I have been asked to help paint a mural on the outside wall of a
grocery store in my neighborhood. My task is to create the background
for the mural. The instructions are to create the largest possible
semicircle on the wall, with the semicircle touching all 4 sides. The
wall is square with 10 feet on each side. I need to find out how to
position the semicircle to satisfy the instructions. I also need to
know the radius and center of the semicircle. How can I figure this out
with the basic math that I know?" *In the discussion Straight Edge and Compass Construction For Developmental Math, I described a method of drawing the mural using
a long plank, lengths of rope, a few pegs and some chalk. *Use the rope to extend the base of the wall, to the right, by its
width, and mark the point with a peg. This defines an imaginary square
that is side by side with, and on the right of the square wall. Draw a
diagonal on the original square, because we know that the solution is
symmetrical about the diagonal. Mark a diagonal on the imaginary square
by stretching a rope from the peg to the top right corner of the square
wall. This is a way of calculating the square root of two. Using the
peg as the center point and the rope as a radius, follow an arc down to
the base of the wall, and mark the point where the arc intersects the
base. The distance of that point from the left wall is 2-sqrt(2), which
is the radius of the semicircle. Draw a perpendicular from that point,
and where it meets the diagonal, peg the center point of the semicircle.
Attaching a length of rope to the peg, stretch it to the furthest wall,
and with chalk held fixed on the rope, draw the semicircle.* I also described the construction in more abstract terms, and used the GeoGebra geometry software to demonstrate.(See the diagram, where the biggest semicircle that fits in the square is positioned diagonally in the top left of the diagram.) *Draw the square as a 4 sided regular polygon. Draw
an identical square to its right Draw the diagonals of the right hand
square and use them as radii of arcs that intersect the left hand
square. Create points where the arcs intersect the base and top side of
the left hand square. Join these two points with a vertical line. Its
intersection with the diagonal of the left hand square defines the
centre of the semicircle. Draw the semicircle through any one point on
the square and notice that it touches or intersects the square at 4 points. Draw the
base of the semicircle through the two intersection points.* (I mistakenly drew the wall as 11 feet instead of 10, but that does not detract from the construction.) Constructive geometry involves no measurement of length, except use of
the compass as a tool to copy a length and duplicate it somewhere else
on the plane. It also involves no algebra. We multiply a length
by extending it with additional equal lengths. We draw the diagonal of a
square without realising that we are calculating the square root of
two. Constructive geometry is available to people who don't do algebra. They can solve ancient problems, rediscover the
history of mathematics and apply it to their own environment. Straight-edge and compass problems range in difficulty from simply drawing a hexagon to the more complex procedure for drawing a regular pentagon.

Two other challenging constructions are Napoleon's problem and Compass construction of Pappus chain. A useful reference is Geometry Construction Reference.

Posted by: Colin McAllister | 08/24/2009 at 12:29 AM