Pre-kites: simplices having a regular facet

Abstract

Abstract. The investigation of the relation among the distances of an arbitrary point in the Euclidean space R n to the vertices of a regular n-simplex in that space has led us to the study of simplices having a regular facet. Calling an n-simplex with a regular facet an n-pre-kite, we investigate, in the spirit of [14], [10], [9], and [15], and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if n ≥ 3, then the intersection of the family of n-pre-kites with any of the four known special families is the family of n-kites, thus extending the result in [18]. A basic tool is a closed form of a determinant that arises in the context of a certain Cayley-Menger determinant, and that generalizes several determinants that appear in [9], [15], and [16]. Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry.Mathematics Subject Classification (2010). Primary 52B11; Secondary 52B12, 52B15, 51M20, 52B10.

Distances from the Vertices of a Regular Simplex

“…The situations when the field R, appearing in the previous theorem, is replaced by an arbitrary field, and when n is not restricted to the values n ≥ 3 are treated in full detail in the aforementioned reference [9]. The relation of the polynomial in (7) to the Cayley-Menger determinant is exhibited in [10].…”

“…Feedbacks on Gardner's article appear in [5], [8], and [14], and in possibly other places. A proof of the general case can be found in [2], and another proof that uses the Cayley-Menger formula for the volume of a simplex is given in [10]. The relation (1) can also be derived using linear algebra.…”

Irreducibility of the Cayley–Menger determinant and of a class of related polynomials

“…However, the problem appears in [16], where the given numbers are 5, 7, and 8, and where the fourth number is found to be 3. One wonders whether there are integral quadruplets other than (3,5,7,8). This naturally leads to the Diophantine equation…”

“…However, the problem appears in [16], where the given numbers are 5, 7, and 8, and where the fourth number is found to be 3. One wonders whether there are integral quadruplets other than (3,5,7,8). This naturally leads to the Diophantine equation (w 2 + x 2 + y 2 + z 2 ) 2 − 3(w 4 + x 4 + y 4 + z 4 ) = 0, (38) which is studied in [15], where many (but not all) of its integer solutions are found.…”

“…The polynomial obtained from g by replacing n+1 with n and t with n is essentially the Cayley-Menger determinant of a special n-simplex, named a prekite, that was introduced and studied in [8], and calculated in Theorem 4.1 there. If we put a = 0 in g, then the instances t = 0, t = n, t = n − 1, and t = n − 2 are very closely related to the Cayley-Menger determinants, with respect to certain parametrizations, of orthocentric, isodynamic, circumscriptible, and tetra-isogonic n-simplices; see Theorem 5.2 of [7].…”