1 Introduction and basic definitions

Impurity of the corner angles in certain special families of simplices

Mowaffaq Hajja and Mostafa Hayajneh

Abstract. Focusing on the fact that the sum of the angles of any Euclidean triangle is constant and equals π for all triangles, M. Hajja and H. Martini raised, in [19, Problem 9], the question whether an analogous statement holds for higher dimensional d-simplices. An interesting answer was given by M. Hajja and I. Hammoudeh in [20], where they proved that for the measure arising from what is known as the polar sine, the sum of measures of the corner angles of an orthocentric tetrahedron is constant and equals π. A cruicial ingredient in that treatment is the fact that orthocentric d-simplices are pure, in the sense that the planar subangles of every corner angle are all acute, all obtuse, or all right. In this article, it is shown that this property is not shared by any of the three other special families of d-simplices that appear in the literature, namely, the families of circumscriptible, isodynamic, and isogonic (or rather tetra-isogonic) d-simplices, thus answering Problem 3 of [19]. Specifically, it is proved that there are d-simplices in each of these families in which one of the corner angles has an acute, an obtuse, and a right planar subangle. The tools used are expected to be useful in various other contexts. These tools include formulas for the volumes of d-simplices in these families in terms of the parameters in their standard parametrizations, simple characterizations of the Cayley-Menger determinants of such d-simplices, embeddability of a given d-simplex belonging to any of these families in a (d + 1)-simplex in the same family, formulas for some special determinants, and a nice property of a certain class of quadratic forms. Mathematics Subject Classification (2010). Primary 52B11; Secondary 52B12, 52B15, 51M20, 52B10. Keywords. Cayley-Menger determinant, Cayley-Menger matrix, circumscriptible simplex, corner angle, isodynamic simplex, isogonic simplex, tetraisogonic simplex, orthocentric simplex, pure polyhedral angle, quadratic form.

1

Introduction and basic definitions

A characteristic feature of plane Euclidean geometry is the fact that the sum of measures of angles in any triangle is constant, i.e., is the same for all triangles. With this pleasant property in mind, the authors of [19] posed, as Number 9 in their list 1

of 14 problems, the question whether this has a higher dimensional analogue. Thus they asked if there is a suitable measure for which the sum of measures of corner angles of a d-simplex, d > 2, is constant, where a d-simplex S ⊂ Rn , n ≥ d, is the convex hull of d affinely independent vectors, called the vertices of S. As they felt this to be too good to be true, they suggested that one may have to restrict oneself to certain dimensions d and to d-simplices of certain special type. Even with heavy restrictions of these sorts, one still would not expect a constant angle-sum property for higher dimensional d-simplices. It was thus an extremely pleasant surprise when the authors of [20] proved that if one restricts oneself to d = 3, to orthocentric tetrahedra, i.e., tetrahedra whose altitudes are concurrent, and to the measure µpolsin arising from what is known as the polar sine, then the sum of measures of corner angles is constant, and again equals π. It also turns out that the three restrictions are, in some sense, necessary. The polar sine is one of the three major measures that have appeared often in the literature, and especially in the recent investigations pertaining to higher dimensional analogues of the pons asinorum theorem, made in [15], [16], and [17], and of the open mouth theorem, made in [1], and [18]. If Θ = ⟨A; A1 , · · · , Ad ⟩ is the d-dimensional polyhedral (or simply, the d-polyhedral) angle having vertex A and arms AAi , 1 ≤ i ≤ d, then the polar sine of Θ is defined by ( d ) ∏ 1 vol(S) = ∥A0 − Ai ∥ polsin Θ, (1) d! i=1 where S is the d-simplex [A, A1 , · · · , Ad ], and where vol(S) is its d-volume (or ddimensional Lebesgue measure). This definition is taken from [10, §1, §7], and it is one of two legitimate generalizations of the ordinary sine. It also has the property that 0 ≤ polsin Θ ≤ 1 for all Θ, as can be deduced from the product formula (8) in [10]. Motivated by the definition of the measure µ(Θ) of an ordinary (planar) angle Θ given by   if Θ is acute or right,   arcsin(sin Θ), µ(Θ) = (2)   π − arcsin(sin Θ), if Θ is obtuse, one is led to wonder whether d-polyhedral angles, d > 2, can be classified into acute, right, and obtuse in a natural and useful manner. In this context, it must come as pleasant news that corner angles of orthocentric d-simplices are always pure in the sense that the planar subangles of any such corner angle Θ are either all acute, all obtuse, or all right. One can then refer to a corner angle of an orthocentric d-simplex as acute, obtuse, or right, accordingly. It is also pleasant to know that at most one of the corner angles of an orthocentric d-simplex can be non-acute, making it possible to classify orthocentric d-simplices as acute, obtuse, or right. Using the definition of polsin(Θ) given in (1), it is now clear how to use (2) to define the corresponding measure µpolsin (Θ) of any pure d-polyhedral angle Θ, and 2

hence for any corner angle of an orthocentric d-simplex. The theorem in [20] alluded to earlier states that if Θ1 , · · · , Θd+1 are the corner angles of a d-simplex S, and if ν is a measure on polyhedral angles, then the equation d+1 ∑

ν(Θi ) is constant

(3)

i=1

holds if the conditions (i) S is orthocentric, (ii) ν = µpolsin , and (iii) d = 3

(4)

are satisfied. The questions regarding how necessary the conditions in (4) are to guarantee the conclusion (3) are addressed and treated in [20]. One of these questions has to do with Condition (i) of (4) and with whether orthocentric d-simplices can be replaced by any of the well known special families that have appeared in the literature, namely, the circumscriptible, the isodynamic, and the isogonic tetrahedra and their d-dimensional analogues. Admitting that we see no sensible way for defining µpolsin (Θ) when Θ is not pure, we are forcibly led to ask whether tetrahedra (and more generally d-simplices, d ≥ 3) that belong to the three afore-mentioned special families are pure, in the sense that their corner angles are. This paper is concerned mainly with this question, and it answers it in the negative by showing that there are, in any of these special families, d-simplices that have some impure corner angles, i.e., polyhedral angles whose planar subangles are of mixed types. This emphasizes the special role that orthocentric d-simplices have among the other families, and shows how indispensible Condition (i) of (4) is for ensuring (3). We add here that even if we restrict ourselves to the pure tetrahedra that belong to each of these special families, the result (3) will still not hold. The paper is organized as follows: In Section 2, we describe the Cayley-Menger matrix of a d-simplex S, and how its determinant is related to the volume of S. We also describe those matrices that can serve as Cayley-Menger matrices of some d-simplices, and how the Cayley-Menger matrix of a d-simplex S can be enlarged to a Cayley-Menger matrix of a (d + 1)-simplex obtained from S by adding an extra vertex. In Section 3, we find formulas for certain families of determinants, and we use these formulas in Section 4 to give a short proof of the d-dimensional Pythagorean theorem, and in Section 5 to derive usable formulas for the volumes of orthocentric, circumscriptible, isodynamic, and isogonic (or rather tetra-isogonic) d-simplices. In Section 6, we establish some useful properties of certain quadratic forms, and we use these properties in Section 7 to characterize the Cayley-Menger matrices of d-simplices in the four special families. In Section 8, we show how to embed a given d-simplex of one of these four special types in a (d + 1)-simplex of the same type. In Section 9, we construct circumscriptible, isodynamic, and isogonic tetrahedra that are impure. Combined with the results in Section 8, this shows how to construct impure circumscriptible, isodynamic, and tetra-isogonic d-simplices for all d ≥ 3. 3

2

The Cayley-Menger matrix and determinant

By a (non-degenerate) d-simplex S, we mean the convex hull of d + 1 (affinely independent) position vectors (or simply points) A1 , · · · , Ad+1 in the Euclidean space Rn for some n ≥ d. The d-simplex S is then denoted by S = [A1 , · · · , Ad+1 ]. Unless explicitly specified, all d-simplices throughout this paper are assumed to be non-degenerate. If S = [A1 , · · · , Ad+1 ] is a d-simplex, then the Cayley-Menger matrix Γ of S is defined to be the d + 2 by d + 2 symmetric matrix Γ = (xi,j )0≤i,j≤d+1 whose top row is [0 1 1 · · · 1] and whose other entries are given by xi,j = ∥Ai − Aj ∥2 , 1 ≤ i, j ≤ d + 1.

(5)

Thus      Γ =     

0 1 1 1 0 x1,2 1 x2,1 0 ··· ··· ··· ··· ··· ··· 1 xd,1 xd,2 1 xd+1,1 xd+1,2

··· 1 1 · · · x1,d x1,d+1 · · · x2,d x2,d+1 ··· ··· ··· ··· ··· ··· ··· 0 xd,d+1 · · · xd+1,d 0

     .    

(6)

The determinant of Γ is called the Cayley-Menger determinant of S, which appears in the context of finding a formula for the volume vol(S) of S. This formula is given by 2d (d!)2 V 2 = (−1)d+1 det(Γ),

(7)

and seems to have been discovered for d = 3 (i.e., for the tetrahedron) by Tartaglia; see [24, Problem 1.18, p. 29], [27, p. 125], and [22, Formula (1)]. It is convenient to number the rows (and columns) of Γ by the numbers 0, 1, 2,· · ·, d + 1. Thus the top row of Γ is referred to as the zero-th (or 0-th) row, and so on. The j-th leading principal minor submatrix of Γ, 0 ≤ j ≤ d + 1 is the (j + 1) by (j + 1) matrix obtained from Γ by deleting the i-th row and i-th column for all i > j. It is denoted by Γj . Thus Γ = Γd+1 . Note that Γj+1 is the Cayley-Menger matrix of the j-simplex [A1 , · · · , Aj+1 ] for 0 ≤ j ≤ d. The properties that characterize the Cayley-Menger matrix of a d-simplex are given in the next theorem. Part (a) is Theorem 9.7.3.4 (p. 239) of [4], and it also appears as Problem 1.22 (p. 30) (with a solution on pp. 215–216) in [24]. Part (b) is a trivial consequence of Part (a). We shall see, in Theorem 7.1, that when the matrix Γ given in (6) is of a special type, then Conditions (a) and (b) reduce to the condition (−1)d+1 det(Γ) > 0.

4

Theorem 2.1 Let xi,j , 1 ≤ i, j ≤ d + 1, be given real numbers with xi,j = xj,i for 1 ≤ i, j ≤ d + 1. Let Γ be as defined in (6), and let Γi , 0 ≤ i ≤ d + 1, denote the i-th leading principal minor submatrix of Γ. Then Γ is the Cayley-Menger matrix of some d-simplex if and only if any of the following two conditions holds: (a) (−1)i det (Γi ) > 0 for 1 ≤ i ≤ d + 1. (b) Γd is the Cayley-Menger matrix of a (d − 1)-simplex, and (−1)d+1 det (Γ) > 0. The next theorem shows how the Cayley-Menger matrix of a d-simplex S can be enlarged to the Cayley-Menger matrix of a (d + 1)-simplex obtained from S by adding an extra vertex. This will be used in Section 7 to show how to embed a given d-simplex of one of the four special types in a (d + 1)-simplex of the same type. Theorem 2.2 Let xi,j , 1 ≤ i, j ≤ d + 1, be given real numbers with xi,j = xj,i for 1 ≤ i, j ≤ d + 1. Let Γ be as defined in (6), and let Γi , 0 ≤ i ≤ d + 1, denote the i-th leading principal minor submatrix of Γ. If Γd is the Cayley-Menger matrix of a (d − 1)-simplex S = [A1 , · · · , Ad ], then there exists Ad+1 such that Γ is the CayleyMenger matrix of the d-simplex [A1 , · · · , Ad+1 ] if and only if (−1)d+1 det(Γ) > 0. Proof. It follows from Theorem 2.1 that Γ is the Cayley-Menger matrix of a dsimplex T = [B1 , · · · , Bd+1 ] if and only if (−1)d+1 det(Γ) > 0. For such a T , the d-simplices T ′ = [B1 , · · · , Bd ] and S have the same Cayley-Menger matrix. It follows that ∥Ai −Aj ∥ = ∥Bi −Bj ∥ for 1 ≤ i, j ≤ d. Therefore there is, by Theorem 9.7.1 (p. 236) of [4], an (affine) isometry ϕ : Rd−1 → Rd−1 such that ϕ(Bi ) = Ai for 1 ≤ i ≤ d. Let ψ : Rd → Rd be an (affine) isometry that extends ϕ and let Ad+1 = ψ(Bd+1 ). Then Ad+1 thus defined satisfies the desired properties. Regarding ψ, one defines it by taking e = (0, 0, · · · , 0, 1) ∈ Rd and setting ψ(e) = e + ϕ(0, 0, · · · , 0). 

3

Some useful determinants

In this section, we derive formulas for the determinants of some matrices that will be useful later. These are the matrices J, K, and L introduced in Lemmas 3.1, 3.3, and 3.4 below. Note that Lemma 3.1 is used in the proof of Lemma 3.3, which in turn is used in the proof of Lemma 3.4. We start by summarizing the notation and the terminology that will be used. If M is a square n by n matrix, then its i-th row and i-th column, 1 ≤ i ≤ n, are denoted by Ri = Ri (M ) and Ci = Ci (M ), respectively. Sometimes (as in the case of the Caley-Menger matrix), we find it more convenient to call the top row the zero-th row and the leftmost column the zero-th column, and we denote them by R0 and C0 . For 1 ≤ i, j ≤ n, the (i, j)-th minor submatrix of M , i.e., the matrix obtained from M by deleting the i-th row and the j-th column, is denoted by Mi,j . 5

Its determinant det (Mi,j ) is what is usually referred to as the (i, j)-th minor of M . The operation of interchanging Ri = Ri (M ) with Rj = Rj (M ) will be denoted by Ri ↔ Rj . The operation of moving Ri so that it lies just after Rj , where j ≥ i, is denoted by Ri 7→ Rj,+ . Since this is equivalent to performing successively the j − i operations Ri ↔ Ri+1 , Ri+1 ↔ Ri+2 , · · · , Rj−1 ↔ Rj , it follows that if M ′ is the result of performing the operation Ri 7→ Rj,+ on M , then det(M ′ ) = (−1)j−i det(M ). The operations Ci ↔ Cj and Ci 7→ Cj,+ are defined similarly. These operations can also be performed on determinants. Lemma 3.1 Let J(n; a, b) be the n by n matrix whose entries mi,j are given by mi,j = b if i = j, and mi,j = a if i ̸= j, and let Ji,j (n; a, b) be the (i, j)-th minor submatrix of J. Then   b a ··· ··· a  a b ··· ··· a     det (J(n; a, b)) := det  · · · · · · · · · · · · · · ·    ··· ··· ··· ··· ···  a a ··· ··· b = ((n − 1)a + b) (b − a)n−1 ,

(8)

(9)

 n−2 if i = j,  ((n − 2)a + b) (b − a) det (Ji,j (n; a, b)) =



(10) (−1)

j−1−i

(b − a)

n−2

a

if i ̸= j.

Proof. Replacing the first column by the sum of all columns, then replacing every i-th row Ri , i ≥ 2, by Ri − R1 , we obtain a diagonal matrix, and the result follows. It remains to prove (10). If i = j, then Ji,j is nothing but J(n − 1; a, b), and the first line of (10) follows. If i ̸= j, we may clearly assume that i < j. In this case, Ji,j is an n − 1 by n − 1 matrix whose i-th column Ci and (j − 1)-th row Rj−1 have a in each entry. If we perform the operation Ci 7→ Cj−1,+ , we obtain a matrix D∗ which differs from J(n − 1; a, b) in the (j − 1, j − 1)-th entry only, where D∗ has a in place of b. The (j − 1)-th row R of D∗ (which consists of a’s) is the sum of the rows R1 , R2 shown in the table R R1 R2

1 a a 0

2 a a 0

··· ··· ··· ···

j−1 a b a−b

6

··· ··· ··· ···

n−1 a . a 0

Letting A1 and A2 be the matrices obtained from D∗ by replacing R with R1 and R2 , respectively, we obtain det (D∗ ) = = = det(Ji,j ) =

det(A1 ) + det(A2 ) = det (J(n − 1; a, b)) + (a − b) det (J(n − 2; a, b)) [ ] ((n − 2)a + b) (b − a)n−2 + (a − b) ((n − 3)a + b) (b − a)n−3 (b − a)n−2 a, (−1)j−1−i det(D∗ ) = (−1)j−1−i (b − a)n−2 a.

This proves the second line of (10), and completes the proof of the theorem.



Remark 3.2 Formula (9) appears as Problem 192 (p. 35) (with hint on p. 154 and answer on p. 187) in [11]. It also appears, in stronger forms, as an example on pp. 135–136 of [12] and as Fact 2.12.11 on p. 59 of [5]. Lemma 3.3 Let x = (x1 , · · · , xn ) and let K(x; a, b) be the n + 1 by n + 1 symmetric matrix whose 0-th row is [0, x1 , · · · , xn ] and whose other entries mi,j , i ̸= 0, j ̸= 0, are given by mi,j = b if i = j, and mi,j = a if i ̸= j. Then



0 x1 x2 · · · · · · xn−1 xn  x1 b a ··· ··· a a   x2 a b · · · · · · a a det (K(x, a, b)) := det   ··· ··· ··· ··· ··· ··· ···   ··· ··· ··· ··· ··· ··· ··· xn a a · · · · · · a b [ ] = (b − a)n−2 aM2 − ((n − 1)a + b)N ,

       

(11)

(12)

where M=

n ∑

xi , N =

i=1

n ∑

x2i .

(13)

i=1

Proof. Let D = K(x, a, b). Notice that the minor submatrix D0,0 of D is nothing but the matrix J(n; a, b) defined in Theorem 3.1. Let J(i) (n; a, b) denote the matrix obtained from J(n; a, b) by replacing the i-th column by the column [x1 x2 · · · · · · xn ]t , (i) where t denotes the transpose. Notice that Jk,i (n; a, b) = Jk,i (n; a, b) for 1 ≤ k ≤ n, since J(n; a, b) and J(i) (n; a, b) differ only in the i-th column. This observation will be freely used. Expanding det(D) along the 0-th row, and letting Di,j , 0 ≤ i, j ≤ n, be the (i, j)-th minor submatrix of D, we obtain det(D) =

n ∑

(−1)i xi det (D0,i ) .

i=1

7

(14)

On D0,i , we perform the operation C0 7→ Ci−1,+ to obtain ( ) det (D0,i ) = (−1)i−1 det J(i) (n; a, b) . ( ) We now expand det J(i) (n; a, b) along the i-th column to obtain n ( ) ∑ ( ) (i) det J(i) (n; a, b) = xk (−1)k+i det Jk,i (n; a, b)

(15)

k=1

=

n ∑

xk (−1)k+i det (Jk,i (n; a, b))

k=1 n ∑

= xi det (Ji,i (n; a, b)) +

xk (−1)k+i det (Jk,i (n; a, b))

k=1, k̸=i

[

= (b − a)n−2 xi ((n − 2)a + b) − a

]

n ∑

xk .

k=1, k̸=i

Therefore det(D) = =

n ∑ i=1 n ∑

(−1)i xi det (D0,i ) [

n ∑

(−1)i xi (−1)i−1 (b − a)n−2 xi ((n − 2)a + b) − a

i=1

= −(b − a)n−2

n ∑

[

n ∑

x2i ((n − 2)a + b) − a

i=1

= −(b − a)

xk

k=1, k̸=i

] xi xk

k=1, k̸=i

[ n−2

]

((n − 2)a + b)

n ∑ i=1

x2i

− 2a

]

∑

xi xk

1≤i 0 with ∥Ai − Aj ∥ = βi + βj , (30) 2 S is isodynamic ⇐⇒ ∃ βi > 0 with ∥Ai − Aj ∥ = βi βj , (31) 2 2 2 S is tetra-isogonic ⇐⇒ ∃ βi > 0 with ∥Ai − Aj ∥ = βi + βi βj + βj , (32) where the indices in the right hand sides run over all i, j ∈ {1, · · · , d + 1} with i ̸= j. Also, the parameters βi , 1 ≤ i ≤ d + 1, appearing in the right hand sides are unique. Theorem 5.2 Let S (o) , S (c) , S (y) , S (g) be the orthocentric, circumscriptible, isodynamic, and tetra-isogonic d-simplices S = [A1 , · · · , Ad+1 ] defined by (29), (30), (31), and (32), and let their Cayley-Menger matrices be Γ(o) , Γ(c) , Γ(y) , Γ(g) , respectively. Let M, N , and P be defined as in (18). Then det(Γ(o) ) det(Γ(c) ) det(Γ(y) ) det(Γ(g) )

= = = =

(−1)d+1 (−1)d+1 (−1)d+1 (−1)d+1

2d P M, 22d−1 P 2 (M2 − (d − 1)N ), P 2 (M2 − dN ), 3d−1 P 2 (M2 − (d − 2)N ).

(33) (34) (35) (36)

Proof. The last three formulas follow immediately from Lemma 3.4 by taking (s, t) to be (1, 2), (0, 1), and (1, 1), respectively. Note that (30) can be written as ∥Ai − Aj ∥2 = (βi2 + βj2 ) + 2βi βj . The first formula also follows from Lemma 3.4 by taking (s, t) to be (1, 0) and by replacing βi2 by βi . Note that if we substitute βi for βi2 in N and in P 2 , we obtain M and P, respectively. 

6

Properties of certain quadratic forms

In this section, we exhibit a nice property of a certain type of quadratic form. This will be used later in the proof of Theorems 7.1 and 8.1. (s)

Lemma 6.1 Let s ∈ R. For k ∈ N, let the quadratic form Qk = Qk : Rk → R be defined by Qk (z1 , · · · , zk ) = (z1 + · · · + zk )2 − (k − s)(z12 + · · · + zk2 ).

(37)

(a) If Qk+1 (t1 , · · · , tk+1 ) > 0 for some t1 , · · · , tk+1 > 0, then Qj (t1 , · · · , tj ) > 0 for 1 ≤ j ≤ k + 1. (b) If Qk (t1 , · · · , tk ) > 0 for some t1 , · · · , tk > 0, then there exists an open interval W in (0, ∞) such that Qk+1 (t1 , · · · , tk+1 ) > 0 for all tk+1 ∈ W . Proof. To prove (a), we let t1 , · · · , tk+1 > 0 be such that Qk+1 (t1 , · · · , tk+1 ) > 0, and we prove that Qk (t1 , · · · , tk ) > 0. Set m = t1 + · · · + tk , n = t21 + · · · + t2k . 12

(38)

Then Qk+1 (t1 , · · · , tk+1 ) = (t1 + · · · + tk+1 )2 − (k + 1 − s)(t21 + · · · + t2k+1 ) = (tk+1 + m)2 − (k + 1 − s)(t2k+1 + n) = (−k + s)t2k+1 + 2mtk+1 + m2 − (k + 1 − s)n.

(39)

Notice that if k − s ≤ 0, then Qk (t1 , · · · , tk ) = m2 − (k − s)n > 0, and there is nothing to prove. Thus we may assume that k − s > 0.

(40)

Then the parabola (in U ) defined by f (U ) = (−k + s)U 2 + 2mU + m2 − (k + 1 − s)n

(41)

opens down. Also, its value at U = tk+1 is Qk+1 (t1 , · · · , tk+1 ) by (39), and is hence positive. Therefore its discriminant ∆ is positive. But [ ] ∆ = 4 m2 − (−k + s)(m2 − (k + 1 − s)n) [ ] = 4 (1 + k − s)m2 + (−k + s)(k + 1 − s)n [ ] = 4(1 + k − s) m2 + (−k + s)n [ ] = 4(1 + k − s) (t1 + · · · + tk )2 − (k − s)(t21 + · · · + t2k ) = 4(1 + k − s)Qk (t1 , · · · , tk ). (42) Since ∆ > 0 and 1+k −s > 0 (by (40)), it follows that Qk (t1 , · · · , tk ) > 0, as desired. This proves (a). To prove (b), we let t1 , · · · , tk > 0 be such that Qk (t1 , · · · , tk ) > 0, and we prove that Qk+1 (t1 , · · · , tk+1 ) > 0 for all tk+1 in some open interval W in (0, ∞). Let m, n be as in (38), and consider the parabola (in U ) defined in (41). By (39), Qk+1 (t1 , · · · , tk+1 ) = f (tk+1 ). If s − k > 0, then the parabola in (41) opens up, and f (tk+1 ) > 0 for all tk+1 large enough. If s − k < 0, then the parabola in (41) opens down. Its discriminant, given by (42), is positive, since 1 + k − s > 0. Also the sum of its zeros is s = −2m/(−k + s), and hence f (s/2) = f (−m/(−k + s)) > 0. Therefore f (tk+1 ) > 0 for all tk+1 in some neighborhood of −m/(−k + s). If s − k = 0, then f (U ) = 2mU + m2 − n. This is positive for all U > 0, because ∑ 2ti tj > 0. m2 − n = (t1 + · · · + tk )2 − (t21 + · · · + t2k ) = 1≤i 0 for all tk+1 > 0. Thus in all cases, there is an open interval W in (0, ∞) such that f (tk+1 ) > 0 for all tk+1 ∈ W . This completes the proof of (b).  13

7

Restrictions on the values of βi in Theorem 5.1

The next theorem describes the restrictions on the values of β1 , · · · , βd+1 that appear in (29), (30), (31), and (32). It is a stronger version of Theorem 2.1 that holds for the special matrices appearing as Cayley-Menger matrices of orthocentric, circumscriptible, isodynamic, and tetra-isogonic d-simplices. The part pertaining to circumscriptible d-simplices provides another proof of Theorem 4.10 of [14]. Theorem 7.1 Let β1 , · · · , βd+1 be given real numbers, and let Γ(o) , Γ(c) , Γ(y) , and Γ(g) be the matrices obtaind from the matrix Γ defined in (6) by setting xij equal to βi + βj , (βi + βj )2 , βi βj , βi2 + βi βj + βj2 , respectively. Thus Γ(o) , Γ(c) , Γ(y) , and Γ(g)  0 1 1  1 0 β1 + β2   1 β + β 0 2 1 Γ(o) =   ··· ··· ···   1 βd + β1 βd + β2 1 βd+1 + β1 βd+1 + β2 

Γ(c)

   =   

0 1 1 1 0 (β1 + β2 )2 2 1 (β2 + β1 ) 0 ··· ··· ··· 1 (βd + β1 )2 (βd + β2 )2 1 (βd+1 + β1 )2 (βd+1 + β2 )2 

Γ(y)



Γ(g)

   =   

   =   

0 1 1 1 0 β1 β2 1 β2 β1 0 ··· ··· ··· 1 βd β1 βd β2 1 βd+1 β1 βd+1 β2

are given by ··· 1 1 · · · β1 + βd β1 + βd+1 · · · β2 + βd β2 + βd+1 ··· ··· ··· ··· 0 βd + βd+1 · · · βd+1 + βd 0

   ,   

··· 1 1 2 · · · (β1 + βd ) (β1 + βd+1 )2 2 · · · (β2 + βd ) (β2 + βd+1 )2 ··· ··· ··· ··· 0 (βd + βd+1 )2 · · · (βd+1 + βd )2 0 ··· 1 1 · · · β1 βd β1 βd+1 · · · β2 βd β2 βd+1 ··· ··· ··· ··· 0 βd βd+1 · · · βd+1 βd 0

0 1 1 2 1 0 β1 + β1 β2 + β22 0 1 β22 + β2 β1 + β12 ··· ··· ··· βd2 + βd β2 + β22 1 βd2 + βd β1 + β12 2 2 2 1 βd+1 + βd+1 β1 + β1 βd+1 + βd+1 β2 + β22

Let M, N , and P be defined as in (18). Then 14



(43)

     , (44)   

    ,   

(45)

··· 1 2 2 · · · β1 + β1 βd+1 + βd+1 2 · · · β22 + β2 βd+1 + βd+1 ··· ··· 2 2 · · · βd + βd βd+1 + βd+1 ··· 0

     .(46)   

(a) Γ(o) is the Cayley-Menger matrix of an orthocentric d-simplex ⇐⇒ (i) βi + βj > 0 for 1 ≤ i < j ≤ d + 1, and (ii) (−1)d+1 det(Γ(o) ) > 0 ⇐⇒ (i) βi + βj > 0 for 1 ≤ i < j ≤ d + 1, and (ii) PM > 0. (b) Suppose that βi > 0 for 1 ≤ i ≤ d + 1. Then Γ(c) is the Cayley-Menger matrix of a circumscriptible d-simplex ⇐⇒ (−1)d+1 det(Γ(c) ) > 0 ⇐⇒ M2 − (d − 1)N > 0, Γ(y) is the Cayley-Menger matrix of an isodynamic d-simplex ⇐⇒ (−1)d+1 det(Γ(y) ) > 0 ⇐⇒ M2 − dN > 0. Γ(g) is the Cayley-Menger matrix of a tetra-isogonic d-simplex ⇐⇒ (−1)d+1 det(Γ(g) ) > 0 ⇐⇒ M2 − (d − 2)N > 0. Proof. We start by giving a combined proof of the three parts of (b). We let M stand for Γ(c) , Γ(y) , or Γ(g) , as the case may be. We also let Mi , 1 ≤ i ≤ d + 1, be the i-th leading principal minor submatrix of M . Then ⇐⇒ ⇐⇒

⇐⇒ ⇐⇒

M is the Cayley-Menger matrix of a d-simplex (−1)i det(Mi ) > 0 for 1 ≤ i ≤ d + 1 (by Theorem 2.1) M2i − (i − s)Ni > 0 for 1 ≤ i ≤ d + 1, where s = 2 if M = Γ(c) , s = 1 if M = Γ(y) , s = 3 if M = Γ(g) , by (34), (35), (36) M2d+1 − (d + 1 − s)Nd+1 > 0 (by Lemma 6.1 (a)) (−1)d+1 det(M ) > 0,

as desired. It remains to prove (a). We again let Mi , 1 ≤ i ≤ d + 1, be the i-th leading principal minor submatrix of M , and we put γi = 1/βi for 1 ≤ i ≤ d + 1. The “only if” part is trivial by the definition of the Cayley-Menger matrix and by (2). For the “if” part, we take two cases. Case 1. If βi > 0 for all i, then the assumptions (i) and (ii) are redundant, since ( j )( j ) ∏ ∑ det(Mj ) = βi γi i=1

i=1

is obviously positive for all j, 1 ≤ j ≤ d + 1. So we are done by Theorem 2.1. Case 2. If βk ≤ 0 for some k, 1 ≤ k ≤ d + 1, then βi > 0 for all i ̸= k, because βk + βi > 0. In this case, we have d+1 ∏ i=1

βi < 0,

d+1 ∑

γi < 0, by (ii).

i=1

15

If j < k, then j ∏

j ∑

βi > 0,

i=1

γi > 0,

i=1

and hence det(Mj ) > 0. If j ≥ k, then j ∏ i=1

βi < 0,

j ∑

γi
0 again. Thus det(Mj ) > 0 for all j, 1 ≤ j ≤ d, and again we are done by Theorem 2.1. as desired. 

8

Embedding a d-simplex of a certain type in a (d + 1)-simplex of the same type

In this section, we prove that every orthocentric (respectively, circumscriptible, isodynamic, tetra-isogonic) d-simplex can be embedded in a (d+1)-simplex of the same type. Theorem 8.1 Let S = [A1 , · · · , Ad ] be an orthocentric (respectively, circumscriptible, isodynamic, tetra-isogonic) (d−1)-simplex in the Euclidean space Rd−1 , where d ≥ 3. Then there exist points Ad+1 in Rd such that the d-simplex [A1 , · · · , Ad , Ad+1 ] is orthocentric (respectively, circumscriptible, isodynamic, tetra-isogonic). Proof. We give a unified proof for the circumscriptible, isodynamic, and tetraisogonic cases. By Theorem 5.1, there exist β1 , · · · , βd > 0 such that  2 if S is circumscriptible   (βi + βj ) 2 βi βj if S is isodynamic (47) ∥Ai − Aj ∥ =   β 2 + β β + β 2 if S is tetra-isogonic i j i j for 1 ≤ i < j ≤ d. By Theorem 2.2, it is sufficient to find βd+1 > 0 for which the respective matrix Γ defined in (44), (45), or (46) has the property that (−1)d+1 det(Γ) > 0. Let    2 if S is circumscriptible 1 if S is isodynamic s = ,   3 if S is tetra-isogonic and let γi = 1/βi for 1 ≤ i ≤ d, as usual. Using (34), (35), (36), and defining the quadratic form Qk , k ∈ N, as in (37) by ) ( Qk (z1 , · · · , zk ) = (z1 + · · · + zk )2 − (k − s) z12 + · · · + zk2 , 16

we see that ( ) 2 (−1)d+1 det(Γ) > 0 ⇐⇒ (γ1 + · · · + γd+1 )2 − (d + 1 − s) γ12 + · · · + γd+1 ⇐⇒ Qd+1 (γ1 , · · · , γd+1 ) > 0. But we know that Qd (γ1 , · · · , γd ) > 0, since β1 , · · · , βd define a (d − 1)-simplex. Therefore it follows from Part (b) of Lemma 6.1 that there is an open interval W in (0, ∞) for which Qd+1 (γ1 , · · · , γd+1 ) > 0 for all γd+1 in W . This open interval corresponds to an open interval containing βd+1 . This proves the cases when S is circumscriptible, isodynamic, or tetra-isogonic. It remains to deal with the case when S is orthocentric. In this case, there exist, by Theorem 5.1, β1 , · · · , βd ∈ R such that ∥Ai − Aj ∥2 = βi + βj for 1 ≤ i < j ≤ d.

(48)

By Theorem 7.1, we know that βi + βj > 0 for 1 ≤ i ≤ d, ) (β1 · · · βd ) γ1 + · · · + Γ(y) > 0, (

(49) (50)

where γi = 1/βi for 1 ≤ i ≤ d. We are to find βd+1 ∈ R such that (

(β1 · · · βd βd+1 ) γ1 + · · · + Γ

(y)

βi + βj > 0 for 1 ≤ i ≤ d + 1. ) + γd+1 > 0,

(51) (52)

where γd+1 = 1/βd+1 . If β1 , · · · , βd > 0, then we choose βd+1 to be any positive number. If βk < 0 for some k, 1 ≤ k ≤ d, then ρ := −(γ1 + · · · + Γ(y) ) > 0, by (50) and (49), and we can choose γd+1 to be any number in (0, ρ). For such a choice, γd+1 < −(γ1 + · · · + Γ(y) ) < −γk ,

(53)

and hence γd+1 + γk < 0. Therefore βd+1 + βk > 0. The remaining sums βi + βj , 1 ≤ i < j ≤ d + 1, are positive because of (49) and because βj > 0 for j ̸= k. As for (52), it follows immediately from (53). 

9

Constructing impure special simplices

In this section, we show how to construct circumscriptible, isodynamic, and tetraisogonic d-simplices, d ≥ 3, each having a corner angle with an acute, right, and obtuse planar subangle. This shows that d-simplices of these three types are not necessarily pure, and contrasts heavily with the case of orthocentric d-simplices. In view of Theorem 8.1, it is sufficient to restrict ourselves to d = 3, i.e., to tetrahedra. Thus we write “isogonic” for “tetra-isogonic”. 17

9.1

Constructing impure circumscriptible tetrahedra

We shall construct a circumscriptible tetrahedron XY ZW in which one of the corner angles consists of one acute, one right, and one obtuse planar angle. Let XY ZW be a general circumscriptible tetrahedron, and let x, y, z, w be the positive numbers, guaranteed by Theorem 5.1, for which ∥X − Y ∥ = x + y, ∥Y − Z∥ = y + z, ∥Z − X∥ = z + x, ∥W − X∥ = w + x, ∥W − Y ∥ = w + y, ∥W − Z∥ = w + z.

(54)

By Theorem 7.1, the only condition that (the positive numbers) x, y, z, w have to satisfy is ( )2 ( ) 1 1 1 1 1 1 1 1 Q := + + + −2 + + + > 0. (55) x y z w x2 y 2 z 2 w2 Let ξ = ∠Y W Z, η = ∠ZW X, and ζ = ∠XW Y , as shown in Figure 1. Without loss of generality, we assume that w = 1. We also set 1+

1 1 1 = L, 1 + = M, 1 + = N. x y z W ηξ ζ

X

Y

Z Figure 1. Illustrating the planar subangles ξ, η, and ζ

of the corner angle W Using the law of cosines, we see that ( )( ) 1 1 cos ξ > 0 ⇐⇒ 1 + y + z − yz > 0 ⇐⇒ 1+ 1+ > 2. y z

18

(56)

Thus we have cos ξ > 0 ⇐⇒ M N > 2, cos ξ = 0 ⇐⇒ M N = 2, cos ξ < 0 ⇐⇒ M N < 2. Similar statements hold for cos η and cos ζ. To arrange for ξ to be right, η acute, and ζ obtuse, we look for ρ ∈ (0, 1) for which M N = 2, N L = 2 + ρ, and LM = 2 − ρ. For this, we have to take √ √ √ (2 + ρ)(2 − ρ) 2(2 − ρ) 2(2 + ρ) L= , M= , N= . (57) 2 2+ρ 2−ρ The requirement x, y, z > 0 is equivalent to L, M, N > 1, which in turn is equivalent to ρ < 2/3. The requirement (55) simplifies, using (56) and (57) and routine calculations, to the condition ρ4 + 16ρ2 − 16 < 0, or equivalently to the condition √ ρ2 < 4 5 − 8 ≈ 0.944. This ( 2 )is obviously satisfied if ρ < 2/3. Thus taking any ρ ∈ 0, 3 , and letting x, y, z be defined by (56) and (57), we see that the corner angle at W of the circumscriptible tetrahedron XY ZW defined by (54) consists of an acute angle, a right angle, and an obtuse angle, as desired.

9.2

Constructing impure isodynamic tetrahedra

Let us now assume that XY ZW is isodynamic. Then there exist positive numbers x, y, z, w such that ∥X − Y ∥2 = xy, ∥Y − Z∥2 = yz, ∥Z − X∥2 = zx, ∥W − X∥2 = wx, ∥W − Y ∥2 = wy, ∥W − Z∥2 = wz.

(58)

By Theorem 7.1, the only condition that the positive numbers x, y, z, w have to satisfy is ( )2 ( ) 1 1 1 1 1 1 1 1 Q := + + + −3 + + + > 0. (59) x y z w x2 y 2 z 2 w2 We again define ξ, η, ζ as shown in Figure 1, and we again assume w = 1. This time we set 1 1 1 = L, = M, = N. x y z

(60)

Then cos ξ > 0 ⇐⇒ y + z − zy > 0 ⇐⇒

1 1 + > 1. y z

Thus cos ξ > 0 ⇐⇒ M + N > 1, cos ξ = 0 ⇐⇒ M + N = 1, cos ξ < 0 ⇐⇒ M + N < 1. 19

Similar statements hold for cos η and cos ζ. To arrange for ξ, η, ζ to be right, acute, obtuse, respectively, we need to find ρ ∈ (0, 1) for which M + N = 1, N + L = 1 + ρ, and L + M = 1 − ρ. Thus we have to take 1 1 1 L = , M = − ρ, N = + ρ, 2 2 2

(61)

showing that we need to take ρ such that ρ < 1/2. By (60), the condition (59) simplifies into ρ2 < 1/6. Once this is satisfied, the previous condition ρ < 1/2 is also satisfied. ( √ ) Thus taking any ρ ∈ 0, 1/6 , and letting x, y, z be defined by (60) and (61), we see that the corner angle at W of the isodynamic tetrahedron XY ZW defined by (58) consists of an acute angle, a right angle, and an obtuse angle.

9.3

Constructing impure isogonic tetrahedra

Let us now assume that XY ZW is isogonic. By Theorem 5.1, there exist positive numbers x, y, z, w such that ∥X − Y ∥2 = x2 + xy + y 2 , ∥Y − Z∥2 = y 2 + yz + z 2 , ∥Z − X∥2 = z 2 + zx + x2 , ∥W − X∥2 = w2 + wx + x2 , ∥W − Y ∥2 = w2 + wy + y 2 , ∥W − Z∥2 = w2 + wz + z 2 .

(62)

By Theorem 7.1, the only condition that (the positive numbers) x, y, z, w have to satisfy is ( )2 ( ) 1 1 1 1 1 1 1 1 (63) Q := + + + − + + + > 0. x y z w x2 y 2 z 2 w 2 This condition is vacuous, since the right hand side of (63) is a sum of positive terms. We again define ξ, η, ζ as shown in Figure 1, and we again assume w = 1. This time we set x − 1 = L, y − 1 = M, z − 1 = N. Using the law of cosines, we see that cos ξ > 0 ⇐⇒ 2 + y + z − yz > 0 ⇐⇒ (y − 1)(z − 1) < 3. Thus cos ξ > 0 ⇐⇒ M N < 3, cos ξ = 0 ⇐⇒ M N = 3, cos ξ < 0 ⇐⇒ M N > 3. Similar statements hold for cos η and cos ζ. 20

(64)

To arrange for ξ, η, ζ to be right, acute, obtuse, respectively, we need to find ρ > 0 for which M N = 3, N L = 3 − ρ, and LM = 3 + ρ. Thus we have to take √ √ √ (3 + ρ)(3 − ρ) 3(3 + ρ) 3(3 − ρ) L= , M= , N= , (65) 3 3−ρ 3+ρ showing that we need to take ρ such that ρ < 3. Thus taking any ρ ∈ (0, 3), and letting x, y, z be defined by (64) and (65), we see that the corner angle at W of the isogonic tetrahedron XY ZW defined by (62) consists of an acute angle, a right angle, and an obtuse angle.

References [1] S. Abu-Saymeh, M. Hajja, and M. Hayajneh, The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra, J. Geom. 103 (2012), 1–16. [2] C. Alsina and R. Nelsen, Charming Proofs: A Journey into Elegant Mathematics, Dolciani Math. Expo. No. 42, M. A. A., Washington, D. C., 2010. [3] R. Bhatia, A letter to the editor, Linear Multilinear Algebra 30 (1991), 155. [4] M. Berger, Geometry I, Springer, N. Y., 1994. [5] D. S. Bernstein, Matrix Mathematics, Princeton University Press, Princeton and Oxford, 2005. [6] P. Costabel, Descartes. Exercises pour la G´eom´etrie des Solides, Presses Universitaires de France, Paris, 1987. [7] N. A. Court, Modern Pure Solid Geometry, Chelsea Publishing Company, N. Y., 1964. [8] A. L. Edmonds, M. Hajja, and H. Martini, Orthocentric simplices and their centers, Results Math. 47 (2005), 266–295. [9] L. Eifler and N. H. Rhee, The n-dimensional Pythagorean theorem via the divergence theorem, Amer. Math. Monthly 115 (2008), 456–457. [10] F. Eriksson, The law of sines for tetrahedra and n-simplices, Geom. Dedicata 7 (1978), 184–87. [11] D. Faddeev and L. Sominsky, Problems in Higher Algebra, MIR, Moscow, 1968. [12] R. Gelca and T. Andreescu, Putnam and Beyond, Springer, N. Y., 2007.

21

[13] F. G-M., Exercises de G´eom´etrie, Coprenant l’expos´e des m´ethodes G´eom´etrique et 2000 questions r´esolues, 6i`eme ´edition, J. Gabay, Paris, 1920; r´e´edit´e par les Editions Jacque Gabay en 1991. [14] M. Hajja, Coincidences of centers in edge-incentric, or balloon, simplices, Results Math. 49 (2006), 237–263. [15] M. Hajja, The pons asinorum for tetrahedra, J. Geom. 93 (2009), 71–82. [16] M. Hajja, The pons asinorum in higher dimensions, Studia Sci. Math. Hungarica 46 (2009), 263–273. [17] M. Hajja, The pons asinorum and other related theorems for tetrahedra, Beit. Algebra Geom. 53 (2012), 487–505. [18] M. Hajja and M. Hayajneh, The open mouth theorem in higher dimensions, Linear Algebra Appl. 437 (2012), 1057–1069. [19] M. Hajja and H. Martini, Orthocentric simplices as the true generalizations of triangles, Math. Intelligencer 35, no. 3 (2013), 16–28. [20] M. Hajja and I. Hammoudeh, The sum of measures of the angles of a simplex, Beit. Algebra Geom., to appear. [21] M. Hajja and others, Various characterizations of certain special families of simplices, preprint. [22] V. F. Ivanoff, The circumradius of a simplex, Math. Mag. 43 (1970), 71–72. [23] S-Y. Lin, Y-F. Lin, The n-dimensional Pythagorean theorem, Linear Multilinear Algebra 26 (1990), 9–13. [24] V. V. Prasolov and V. M. Tikhomirov, Geometry, Translations of Mathematical Monographs, vol. 200, Amer. Math. Soc., R.I., 2001. [25] V. V. Prasolov, Problems in Plane and Geometry, v. 2 (solid Geometry). [26] J.-P. Quadrat, J.-P. Lasserre, and J.-P. Hiriari-Urruty, Pythagoras’ theorem for areas, Amer. Math. Monthly 108 (2001), 549–551. [27] D. M. Y. Sommerville, An Introduction to the Geometry of N Dimensions, Dover, N. Y., 1958.

22

Mowaffaq Hajja Department of Mathematics Yarmouk University Irbid – Jordan [email protected], [email protected] and Mostafa Hayajneh Department of Mathematics Louisiana State University Baton Rouge - LA 70803 USA [email protected]

23