SECRETS OF A CERTAIN PENTAGON

This page is about another numerical experiment.  In 1997 Horace Crater sent me a study that he had done concerning the D&M pyramid in which he considered a certain infinite family of pentagons. One of these pentagons was the hypothetical model for the base of the D&M pyramid proposed by Erol Torun.  For a description of this infinite family of pentagons, click here.  Each pentagon is determined completely by choosing one specific angle D.  For example,  D = 69.4 corresponds to Torun's model.  If one changes the choice of the angle D, the shape of the pentagon changes, although it will still share some specific properties with the pentagon in Torun's Model.
In the Fall of 1997, I asked Dr. Crater to choose another angle D completely at random so that I could see what kinds of numerical relationships could be discovered in the corresponding pentagon.  Since there were so many possibilities for the kinds of relationships that might exist, I felt quite confident that I would find lots of striking relationships.  The angle he chose was D=84.3.   Here is a rough picture of that pentagon,  which is the subject of this page. This pentagon, which we will refer to by the letter P, is a precisely specified pentagon. All of the angles, lengths, and areas associated with the pentagon could, in principle, be computed to any desired accuracy.    Over a period of many months, mostly on Sundays, I devoted quite a bit of time to searching for numerical relationships. It was actually a lot of fun, perhaps something like a mathematical analogue of doing the Sunday New York Times Crossword Puzzle. Instead of a dictionary, all that I needed was my hand calculator.  I started with lengths and areas, and never got around to considering relationships involving the angles of the above pentagon.  I looked exclusively for relationships where the percentage error was less than .1% and found about 100 relationships which struck me as rather elegant and natural,  many of which were impressively accurate. On this page, we will give a selection of those relationships.  To indicate the accuracy of the relationships, I will use the following notation. If x and y are two positive numbers, then I will write x ~ y  to mean that x and y are approximately equal.  In parentheses, the error in the approximation will be indicated by both the absolute value |y-x|  of the difference between y and x  and the percentage error  (in comparison to the number y).  For example, if  x=10.2  and   y=10,  then I will  write 10.2 ~ 10   and put in parentheses  ( .2  ,  2%)  since  |10- 10.2| = .2  and  .2  is  2% of the number  10.

Relationships Involving the Golden Section.

The Golden Section is the number (1+\/5)/2, which is often called  phi.  There is one relationship involving phi which is valid for all of the pentagons in the family considered by Crater, no matter what the angle D is chosen to be.  This relationship states that d/b = phi , where b is the length of the base of the pentagon as pictured above, and d is the length of the diagonal of the pentagon joining the vertex at the top to either of the endpoints of the base. In the illustration which is found here, d is the length of either one of the green line segments, i.e., the distance between the vertex A and either of the vertices B or B', and b is the distance between B and B'.  This relationship is intentional and is implicit in part 1of the description of Crater's family of pentagons given here.  But for the specific pentagon pictured above (with D=84.3o), there is another interesting relationship involving  phi which is simply an unintentional consequence of the randomlychosen value of D. It involves the length f of either one of the two red line segments in the illustration.  The point M is the midpoint of the side of the pentagon joining B to C (and M' is the midpoint of the side B'C'). Thus f is the distance between M and C' (or between M' and C).

f /b ~ phi          (.00018 ,     .012%)

Thus, if b is taken as the unit of measurement, then the length d is exactly equal to phi and the length f is very close to phi.

Relationships Involving Areas and the Number e.

There are many rather striking numerical relationships involving the areas of various parts of this pentagon. The relationships that I will give here were chosen because they are quite accurate, they seem rather elegant to me, and they show some thematic unity in that they all involve areas coming from the above pentagon and areas of squares with sides of length  2/e,  3/e, 4/e, 5/e, and 6/e.

1.   Let's denote the above pentagon by the letter  P.  There is another pentagon  which has precisely the same perimeter as  P,  namely the pentagon formed by flipping the two top edges (which have length c)  inward.  (See the picture below.)  Then not only are the perimeters equal,  but the areas have a ratio which is approximately equal to the area of a square with sides of length  2/e .  The two pentagons P and Q share four vertices. The remaining fifth vertex of Q is precisely the "special interior point" which plays a role in the definition of the pentagon P.

Area(Q) /Area(P) ~   (2/e)2                        ( .000002  ,   .004%)

2.  The base of the pentagon has length b . The two lower sides of the pentagon each have length  a .  This relationship involves the areas of three squares: one with sides of length b, one with sides of length a, and one with sides of length  3/e.

a2/ b2   (3/e)2                                        ( .0008  ,     .07% )

3.  This relationship involves the areas of two bilaterally symmetric subregions of the pentagon and the area of a square with sides of length  4/e.  Note that the four vertices of the quadrilateral T are precisely specified points: the apex of the pentagon, the two endpoints of the base, and the special interior point .  (See the picture below.)  I was quite excited when I discovered this extremely accurate relationship one Sunday.  But the following Sunday, I realized (with a little disappointment) that all of the pentagons in Crater's family satisfy this very same relationship.  Even though the quantities Area(S) and Area(T) will change, the ratio Area(S)/Area(T)  turns out to always equal the largest of the four roots of the polynomial  x4+2x3-6x2-7x+1.  It just so happens that  x = (4/e)2  is an extremely good approximation to this root.

Area(S) /Area(T) ~ (4/e)2                      ( .000013  ,    .0006% )

4.   Now we give a Pythagorean type relationship involving the areas of the squares on the five sides of the pentagon P and a square whose sides have length 5/e .

a2 + a2 + c2 + c2 - b2   ~ (5/e)2                 ( .0005  ,   .01% )

5.    This relationship is a double one involving the area and the perimeter of the pentagon  P  and the area and perimeter  (which is usually called the circumference) of a certain circle  C.  This circle C is tangent to the base and the upper two sides of the pentagon, but not tangent to the lower two sides.  (See the picture below. The picture is slightly deceptive because it almost makes it appear that the circle C is tangent to all five sides of the pentagon P.)  The areas of squares with sides of length 2 and of length 6/e are also involved.

Area(C) / Area(P)    +   22 ~   (6/e)2                        ( .0000047   ,   .0006% )

Peri(C) / Peri(P~ \/3 / 2                                         (.000098 ,     .01% )

COMMENTARY:   This experiment is a good illustration of  "The Power of Randomness."  Coincidences abound!! They are there to be found if one spends enough time looking for them.  And it seems that some people spend a lot of time looking for them in various contexts, and, sadly,  fool themselves and others into thinking that they have discovered something significant.