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Elementary Geometric Constructions
Exercises
Pasts and Presences
5 November 1997

In all the constructions that follow we begin with a line on which two points A, B are marked. The points serve to define a unit distance relative to which all subsequent distances are measured. So, for example, if we speak of a distance 3/2 we mean a distance which is half again as great as the distance with which we began. In making constructions we require a straightedge on which there are no markings (or, if markings exist, we ignore them) and a compass (the modest type favored by schools or just a piece of string with which we can swing arcs).

The exercises tend to build upon one another, so that we begin with very simple things and add complexity as we proceed.

Exercise A. Construct a point P equidistant from A and B so as to obtain the vertices of an equilateral triangle.

Exercise B. Construct a second point Q equidistant from A and B so that line through P, Q is perpendicular to the line through A, B.

Exercise C. Construct a line through A which is perpendicular to the line through A,B.

Exercise D. Construct the vertices of a square.

Exercise E. Construct an octagon whose edge length is the same as the distance between A and B.

Exercise F. Let P be any point not on the line through A, B. Construct a parallelogram whose vertices include P, A, B. In this way we find a line through P which is parallel to the line through A, B.

Exercise A. Draw a second line through A (it doesn't matter much which one; an angle of about 45_ between the lines would be good) and mark off on both lines integral multiples of the unit distance given by the separation between A, B. Note how the drawing of parallels to lines joining these integral points on the two lines creates new points on these lines whose distances to A are fractions.

Exercise A. Construct six contiguous squares which fold into a cube.

Exercise B. Here is a thought experiment: mark the midpoints of the six faces of a cube. If these are joined in as many ways possible with line segments, what kind of polyhedron is formed? What is the nature of the faces and how many faces are there? This new polyhedron is called the dual of the original. What is the dual of the dual?

Exercise C. Construct a square on the sement AB and swing and set the compass with one end at the midpoint of AB and the other end at the vertex of the square above B. Keeping one end of the compass at the midpoint, swing an arc down until the line through A, B is intersected at a point G, whose distance to A is called the golden ratio (in reference to the disance between A and B). The distance to B is called the golden mean.

Exercise D. Using Pythagoras's theorem in order to compute the distance between the midpoint of the line segment AB and the vertex above B of the square on AB, find algebraic expressions for the golden ratio and mean and compute their product. Show that your result implies that the rectangle whose edge lengths are the distance between A, B and the golden ratio is similar (i.e., proportional) to the rectangle whose edge lengths are the distance between A, B and the golden mean.

Exercise A. Form a hexagon by constructing six contiguous equilateral triangles.

Exercise B. Form a plane tiling using a hexagonal motif.

Exercise C. Cut a hexagon from a sheet and remove two of its component triangles. Fold the remainder into a tetrahedron, a regular polyhedron that has four triangular faces. What is the dual of the tetrahedron?

Exercise D. Again, cut a hexagon from a sheet and this time remove only one of the component triangles. Then crease along the lines of the remain triangles and fold about the central point to form a cone-like surface.

Exercise E. Paste together four copies of the above surface to obtain a closed surface having 12 vertices and 20 faces. This is the icosohedron, one of the so-called "Platonic Solids," known to the Pythagoreans long before Plato.

Exercise F. Study the nature of the dual of the icosohedron.

Exercise A. Construct an isoscoles triangle on AB whose equal sides have length equal to the golden ration (relative to the distance between A and B).

Exercise B. Make 12 contiguous copies of a pentagon in such a way that you can fold the configuration into a closed surface (keeping each of the pentagons planar. Count the edges and vertices.

Exercise C. Study the dual of the preceeding closed surface. Count the number of faces, edges and vertices of the dual and relate them to the corresponding numbers for the original surface.

Exercise D. Make lots of copies of a pentagon and experiment with the possbilities for tiling a floor using a pentagonal motif. Make the gaps between tiles as regular as possible. Such tilings were studied by Albrecht Durer ca. 1500.

Exercise A. Construct a circle with radius AB and locate the point where the perpendicular bisector of AB intersects the circle in points P and Q. Verify that the distance between the midpoint of AB and either of the points P, Q serves as a good approximation to the length of a side of a heptagon inscribed in the circle.