The Possibility of Impossible Pyramids

In his article "The Possiblity of Impossible Pyramids", Tom Sibley wants to determine if Euclidean space is pyramid complete, in addition to being triangle complete.
He begins his article by proving that Euclidean space is indeed triangle complete. He first defines a metric space as a non-empty set X together with distance d such that for any elements P, Q, R \in X,

(i) d(P,Q)$\geq$0;
(ii) d(P,Q)=d(Q,P);
(iii) d(P,Q)=0 $\Leftrightarrow$ P=Q;
(iv) d(P,Q)+d(Q,R)$\geq$d(P,R).

He next defines triangle complete as a metric space that for any three non-negative real numbers a, b, and c satisfying the triangle inequality, the fourth property above, in any order. To prove that Euclidean space is triangle complete, the author uses the real numbers a, b, and c, and sets P=(0,0) and Q=(a,0). Using the distance formula and the Pythagorean Theorem, the author shows that there exist values b and c such that a,b,and c fulfill the triangle inequality and are non-negative.

Next, the author defines pyramid complete, and shows that Euclidean space is not pyramid complete. The author's definition of a pyramid complete metric space:

• A metric space (X,d) is pyramid complete if, for any sextuple of non-negative real numbers (a,b,c,d,e,f) such that each of the triples {a,d,e}, {b,d,f}, and {c,e,f} satisfies the triangle inequalities in any order, there are four points P,Q,R, and S in X so that d(P,q) =a, d(P,R)=b, d(Q,R)=c, d(P,S)=d, d(Q,S)=e, and d(R,S)=f.

In order to prove that Euclidean Space is not pyramid complete, the author uses the counter-example (14,8,8,8,8,8). When this sextuple is used, and d(P,Q)=14, all other distances must be equal to 8. Four of the five remaining distances can be set to 8 with no problems. However, the final distance value, which the author chooses as d(R,S), must also be equal to 8 for the space to be pyramid complete. The author next fixes the points P,Q, and R and rotates the triangle so that P,Q,R, and S are all in the same plane. This orientation gives the largest possible value for d(R,S). Defining the point M as the midpoint of the line RS, by definition d(P,M)=7 and d(P,S)=8. Therefore, at its largest, d(M,S)=3.873. This means the largest possible value for d(R,S) is 7.75. This counterexample proves that Euclidean Space is not pyramid complete.

By proving that Euclidean Space is not pyramid complete, the author answered one of his initial questions; might there be lengths for the sides of a pyramid that form four triangular faces even though no Euclidean pyramid exists with those lengths? Because of the counterexample above, it is clear that there are lengths for the sides of a pyramid that form four triangular faces and no Euclidean pyramid exists with those lengths.

After answering his initial questions regarding Euclidean space's pyramid completeness, the author goes on to give an example of a metric space that is pyramid complete; the taxicab metric in $\mathbb{R}^3$. The author uses a lengthy proof to show that it is possible to have the triangle inequality holds for every set of triplets of a sextuple in the taxicab metric in $\mathbb{R}^3$.

The author's intent behind the article was to determine whether or not Euclidean space is pyramid complete, and whether or not there are "impossible pyramids." The author alos explores other metric spaces such as the taxicab metric in $\mathbb{R}^3$. At the end of the article, the author suggests that an interested reader can use ideas from the article to explore higher dimension metrics, and gives an example suggestion from a professor at Moorehead State Univeristy. Also, searching "pyramid completeness" on any of the journal databases yields articles that will allow more exploration in similar topics.

• Material for this summary taken from

Sibley, Tom. "The Possiblity of Impossible Pyramids." Mathematics Magazine June 2000: 185-193