16 June 1999 Version
Using analytic geometry and determinants yields the following expansion of
Adjoining Two Kinds of Pyramids, i.e., the material that begins:
The following problem appeared in the 1980 PSAT. The answer believed to
be correct by the Educational Testing Service experts was in fact
Construct two pyramids with every edge of equal length, L. The first
has a square base and four equilateral triangle faces. The second
an equilateral triangle base. It has three more equilateral triangles
as faces. This object is a Tetrahedron. The task
is to glue any triangular face of the first pyramid to one of the
second, the tetrahedron. In doing this, completely match the two glued
together. This creates a single solid object. How many faces does the
ng solid have? Helpful Diagrams.
Reasoning from visual images can be tricky. Hence the following
discussion. It was sent by electronic mail by a UCLA mathematician.
To avoid fractions assume that the triangles all have side 2. Place the
square base of the pyramid on the x-y plane with its center at the origin
and the vertex pointing in the positive z direction.
[This means that
the base edges of the pyramid are parallel to the coordinate axes.]
Then the coordinates
of the four base points
are (-1, -1, 0), (-1, 1, 0), (1, -1, 0), (1, 1, 0) and the coordinate of the vertex
is (0, 0, sqrt 2). You can check this by computing the distance between the
various points using the Pythagorean theorem. If you place the tetrahedron
on the face which contains the vertices (1, -1, 0), (1, 1, 0) and (0, 0, sqrt 2),
then you can check that the other vertex is (2, 0, sqrt 2). Again use the
Pythagorean theorem to verify all distances.
Next you must verify that the four points (1, -1, 0), (0, 0, sqrt 2), (1, 1, 0),
(2, 0, sqrt 2)
all lie in a plane. Call these points, in order, P1, P2, P3, P4. There are
many ways to verify that these lie in a plane. For example, check that the
three vectors P1-P4, P2-P4,
P3-P4 are linearly dependent. These are the
vectors from the origin to the points (1, 1, sqrt 2), (0, 0, sqrt 2), (-1, -1, sqrt 2). To check, look at the
3 by 3 determinant whose rows are these three vectors. It is immediate that it
is 0, completing everything.