The family of pentagons considered by Dr. Crater is rather easy to describe.  There is one pentagon corresponding to any angle D, where  36o < D < 180o.  For D=84.3o,  a picture of the corresponding pentagon can be found here.  This is the pentagon which has been the subject of my experiment. To see how the shape changes for different values of D, here are some more pictures corresponding to D = 110o,  70o, and 50o.  The Torun Model for the base of the D&M pyramid corresponds to D=69.4o, which is quite close in shape to the pentagon for D=70o.  In the Torun Model,  the special interior point, which is defined below,  is supposed to represent the point in the base which lies directly below the apex of the D&M pyramid.

Description of the infinite family of pentagons:

1. Three of the five vertices of the pentagons are always the same (independently of the value of D). They are the three vertices of an isosceles triangle whose base angles are 72o. This triangle is pictured here.  Two of the vertices are the endpoints of the base. They will be denoted by v1 and v2. The apex of this isosceles triangle will be another vertex of the pentagon, which will be denoted by v3.   An equivalent way to describe this isosceles triangle is to require that the ratio  d/b is equal to the Golden Section   phi = (1+\/5)/2  .  The equivalence results from the fact that cos(72o) = 1/2phi.

2.  Suppose that D is an angle between 36o and 180o. On the same base (of length b), construct another isosceles triangle whose apex is within the previous triangle and such that the angle at that apex is D.  The apex of this new, smaller triangle will be referred to as the "special interior point."  Here is a picture. The new triangle is in red, the special interior point is called  w.   The base angles of this new triangle are equal to  (180o - D)/2  .

3.  We now describe how to get the remaining two vertices of the pentagon. Consider the line segment joining w and v3.  Using this line segment as a base, construct two equilateral triangles.  Here is a picture .  The two equilateral triangles are in purple. The remaining two vertices of the pentagon which corresponds to a given angle D will be called v4 and v5. They are the indicated vertices of the two equilateral triangles.

4. The five vertices v1, . . . , v5  are the vertices of the pentagon.