HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON

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1 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON B.D.S. BLUE MCCONNELL MATH@DAYLATEANDDOLLARSHORT.COM Abstract. This living document will serve as an ever-expanding resource of results in hyperbolic (tetra)hedronometry. To maximize information density, I minimize discussion and proof (at least in early drafts), but the reader can readily verify the geometry with basic plane trigonometry and the algebra with standard identities; the only truly sophisticated notions are the Schalfli and Derevnin-Mednykh formulas for volume. (Refer to Appendix A for notation used throughout.) 1. Fundamentals: The Laws of Cosines At the core of hedronometry lie three Laws of Cosines that epitomize the field as dimensionally-enhanced trigonometry by relating, not the edges and planar angles of a triangle, but the faces (W, X, Y, Z) and dihedral angles (A, B, C, D, E, F ) of a tetrahedron. The Second Law, in particular, also formally defines the figure s geometricallyamorphous 1 but algebraically-rewarding pseudo-faces (H, J, K). Theorem 1.1 (The First Law of Cosines). (1) Ẅ = ẌŸ Z X Y Z 4[ A, B, C ] + ẌY Z Ä + X Ÿ Z B + X Y Z C Theorem 1. (The Second Law of Cosines, and the Definition of 0 H, J, K π). () Ÿ Z + Y Z Ä = Ḧ = ẄẌ + W X D Z Ẍ + Z X B = J Theorem 1.3 (The Third Law of Cosines). = ẄŸ + W Y Ë Ẍ Ÿ + X Y C = K = Ẅ Z + W Z F (3) 0 = 1 Ẅ Ẍ Ÿ Z 4 Ẅ Ẍ Ÿ Z Ḧ J K Ḧ J K + Ḧ ÄẄ Ẍ + Ÿ Z ä + J ÄẄ Ÿ + Z ä ä Ẍ + K ÄẄ Z + ẌŸ 1 In Euclidean space, a pseudo-face is a quadrilateral determined by projection of the tetrahedron into a plane parallel to a pair of opposite edges; no geometric interpretation of a general hyperbolic pseudo-face is known. (See Section 6 for some special cases.) Nevertheless, we declare that the element s area satisfies the Second Law of Cosines, and that the area is bounded as if the element were a hyperbolic quadrilateral. I doubt that the significance of pseudo-faces in catalyzing tetrahedral analysis can be overstated. 1

2 B.D.S. BLUE MCCONNELL These give rise to symbolic abbreviations 3 that we ll find helpful: (4a) (4b) W := Ẅ ẌŸ Z + ḦẌ + J Ÿ + K Z 0 H := Ḧ + J K ẄẌ Ÿ Z 0 We also note this consequence of () (5) H, W, X = 1 4 W X D 0 and this Law-of-Cosines-like equality that follows upon reduction modulo (3): (6) J + K Ḧ J K = 16 H, W, X H, Y, Z. From Lengths to Areas, and Back Again Standard formulas give the area of a tetrahedron s face from lengths of bounding edges (7) Ẅ = d + ë + f 1 d, e, f d W = ë f d ë f and we can determine pseudo-face areas from lengths of edges in conspicuous pairs î Ḧ = a d + b ë + c f a d, b ó ë, c f (8a) b H = ë c f b ë c f a Ä ä d + b ë + c f a Ä ä d b ë c f (8b) Ḧ 4 = 4 b H 4 = ë c f 4 b ë c f From these, we can verify (9) W ä b c = X ä ë f = Y d b f = Z d ë c = H ä d = J b Êg ë = K c f = 64ä b c d ë f Consequently, we have hedronometric formulas 4 for edge lengths: (10a) (10b) ä = J K W X J K d = Y Z a = 4 H H Y Z W X 4 H H W X d = Y Z â = 4 H H Y Z J K 4 H H W X d = J K 3 Non-negativity follows from (9). 4 The sine formulas derive from the cosine formulas, reduced modulo the Third Law of Cosines (3).

3 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 3 and, in turn, a symmetric equality involving opposing pairs of edges 5 (11) a d H Y Z H W X = b e J Z X J W Y = c f = Ç 16 H J K W X Y Z å K X Y K W Z 3. Gram Relations Consider the face-indexed ( angle ) Gram matrix, G, and conjugate ( edge ) Gram matrix, g, detailed in Appendix A.3. Their determinants and the conjugation ratio Γ have the following hedronometric forms: (1a) (1b) ÁG := det G = 4 H J K W 0 Êg := det g = 43 H 3 J 3 K 3 X Y Z W X Y Z 0 G Γ := Á Ç å Êg = W X Y Z 4 H J K W X Y Z Cofactors of entries along the diagonals of the matrices have these forms: (13a) (13b) M W = 4[ A, B, C ] = W X Y Z Ç å 4 H J K m W = 4 d, e, f = 16W d ë f = W X Y Z while, via (70b), the remaining cofactors of G and g have these forms: (14a) (14b) M W X = ä M W M X = Ä ä + a ä M W M X = J K + 4 H H Y Z W X Y Z m W X = D m W m X = Ä ä 4 H J K Ḧ ẄẌ W X Y Z One can readily verify identities 6,7 presented by Mednykh and Paskevich in [3], such as: 5 A cleaner version of this result appears in (15b). 6 (15a) This follows from our expressions for Γ, MW, m W. M&P s equation () prior to Lemma 1 asserts equality between the ratios and positive Γ; M&P s derivation includes factors such as (in our notation) m W, but, by (69b), m W 0. Someone among us has a sign error. (15b) M&P s Theorem, which follows from (5) and (11). (15c) M&P s Theorem 4. (15d) M&P s Corollary 1, which follows from (15b) and (15c). The ± i s match for a given i, but are otherwise independent. 7 M&P s result relating altitude lengths to Γ appears to be in error; see the footnote for equation (19).

4 4 B.D.S. BLUE MCCONNELL (15a) (15b) M W m W = M X m X = M Y m Y = M Z m Z = Γ AD ad = BE be = CF cf = Γ (15c) Ä D BË ä d bë = BË C F bë c f = C F Ä D c f ä d = Γ (15d) (A ± 1 D) (B ± E) (a ± 1 d) (b ± e) = (B ± 3 E) (C ± 4 F ) (b ± 3 e) (c ± 4 f) = (C ± 5 F ) (A ± 6 D) (a ± 5 d) (b ± 6 e) = Γ We can often (as in Section 4) use the conjugation ratio to catalyze the conversion between equations in edge-related lower-case symbols (a, m, g, etc.) and ones in angle- or area-related upper-case symbols (A, M, G, etc.), effectively doubling the utility of each result. We call these conjugate relations. For example, write P := M W M X M Y M Z and p := m W m X m Y m Z. Then (16) P = Ç å W X Y Z p = W X Y Z Ç 4 4 H 4 J 4 K 4 å W X Y Z W 3 X 3 Y 3 Z 3 and, again, one can readily verify these conjugate identities 8 from [3, Proposition 5]: (17) ÊgP = Ë G 3 Á Gp = Ág 3 P 3 p = Ë G 8 p 3 P = Ág 8 4. Altitudes and Pseudo-altitudes Let w, x, y, z be altitudes to respective faces W, X, Y, Z. We have conjugate relations 9 (18a) (18b) w M W = x M X = y M Y = z M Z = Á G w m W = x m X = y m Y = z m Z = Êg 8 Given the interplay of factors S := WX Y Z, T := W X Y Z, U := 4 H J K, one can show that any vanishing product of integer powers of G = U/S, g = U 3 /T, P = T /S 6, p = S U 8 /T 6, can be written ( Ê Gp/Êg 3 ) α (P p 3 /Êg 8 ) β = 1 for integers α, β. Abusing terminology, the relations Ê Gp = Êg 3 and P p 3 = Êg 8 have all possible relations of this kind in their integer span. Likewise, their conjugates. For one relation and its conjugate to span all relations in this way, however, requires non-integer exponents. 9 Equation (18b) agrees with Mednykh and Pashkevich [3, Proposition 4(ii)], which appears in this note as equation (70b). The second set of equations arises from dividing the first set through by Γ. See (15a).

5 whence 10 (19) wxyz = HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 5 ÁG MW M X M Y M Z = By (14a) and (10), we can write Êg mw m X m Y m Z = 4 H J K W X Y Z W X Y Z (0) Also, w x M W X ä = w y M W X b = w z M W X c = y z M W X d = z x M W X ë (1) w W W = Êg W W = Êg W X Y Z m W 4 H J K = x y M W X f () w W W = w W X Y Z ( 4[ A, B, C ]) Let h, j, k be pseudo-altitudes 11 between respective edge pairs (a, d), (b, e), (c, f); and let τ h, τ j, τ k be the corresponding twists about those pseudo-altitudes. Equation (74a) from Appendix B gives these conjugate relations (3a) adhτ h = bejτ j = cfkτ k = Êg (3b) ADhτ h =BEjτ j =CF kτ k = G Á Recall now from (69b) that m is a Heronic sine product; invoking the product s sideangle-side form (63), we see that equalities (18b) and (3a) have an identical structure, giving rise to a result that neatly accommodates both standard and pseudo elements. 1,13 Theorem 4.1 (Law of Side-Angle-Side-Altitude Sines). The product (4) sinh ( edge ) sinh ( edge ) sin ( angle ) sinh ( altitude ) is a metric invariant for hyperbolic tetrahedra, with value Êg for any two distinct edge s and the angle and altitude they determine This disagrees with Mednykh and Paskevich [3, Proposition 5(iv)], which claims that the product is ÊG /M W /M X/M Y /M Z = 1/Γ. 11 A pair of skew lines in hyperbolic space admits an orthogonal transversal. (See [].) A pseudo-altitude is such a transversal, joining lines through opposing edges of a tetrahedron. The twist is the (supplementarily ambiguous) angle between an edge and the plane of the opposing edge and their mutual pseudo-altitude. 1 Mednykh and Pashkevich [3, Proposition 6] invoke the area-based form of the Heronic sine product (69b) to arrive at this alternative result sin ( half-face ) cosh ( half-edge ) cosh ( half-edge ) cosh ( half-edge ) sinh ( altitude ) = Êg/4 for any (standard) face, the edge s surrounding it, and the altitude to it. No pseudo elements appear. 13 The analogous product for Euclidean tetrahedra is edge edge sin(angle) altitude = 6 volume, which is valid for both standard and pseudo elements. 14 For adjacent edges, the angle lies between them, and the altitude drops to the face containing them; for opposite edges, the altitude is the pseudo-altitude joining them, and the angle is the corresponding twist. Note that taking the sine of a twist renders the supplementary ambiguity of the twist moot.

6 6 B.D.S. BLUE MCCONNELL Moving to cosines, we expand Appendix B s equation (74b) into conjugate relations c f bë = adḧ τ h ä d c f = be j τ j bë ä d = cf k τ k (5a) (5b) C F BË = ADḧ τ h Ä D C F = BE j τ j BË Ä D = CF k τ k Isolating either of ḧ or τ h in (3a) and (5a) gives the same result; that is, we find that u = ḧ and u = τ h are the two roots in each of these conjugate quadratic equations: u a d u (a d + Ä c f bë ä ) Êg + Ä c f bë ä (6a) = 0 u A D ( u A D + Ä C F BË ä G Á ) + Ä C F BË ä (6b) = 0 As ḧ 1 τ h for hyperbolic cosine ḧ and circular cosine τ h, we can write specifically ḧ = a d + Ä c f bë ä Êg + δ a d = A Ä D + C F BË ä G Á + (7a) A D τ h = a d + Ä c f bë ä Êg δ a d = A Ä D + C F BË ä G Á (7b) A D for non-negative discriminants, δ and, which a convenient imaginary unit allows us to express as Heronoic products: [ (7c) δ = ad, Ä c f bë ä ] ï, i Êg = AD, Ä C F BË ä ò, i G Á Note that the above implies î ad, Ä bë ó c fä, i Êg (8) ḧ τ h = a d = ï AD, Ä BË C F ä ò, i G Á A D 5. Orthogonal Twists and Perfect Tetrahedra A right-angle twist about a pseudo-altitude is naturally called orthogonal. A tetrahedron is called perfect 15 when all three pairs of opposing edges are orthogonally twisted about their pseudo-altitudes. For example, a right-corner tetrahedron (A = B = C = π/) is orthogonal, as is a regular tetrahedron whose faces are equilateral triangles. By (5a) and (5b), a perfect tetrahedron s opposite elements combine symmetrically: (9) ä d = bë = c f Ä D = BË = C F Using the Second Law of Cosines () to re-write the dihedral cosines above, we derive this hedronometric characterization of perfection: (30) Ḧ H = J J = K K 15 Also orthocentric or orthogonal. Perfect is my own term, coined decades ago, so I ll keep it.

7 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 7 An orthogonal twist simplifies the quadratics in (6), providing concise expression of the length of the corresponding pseudo-altitude; for example, (31) τ h = π Êg = h = ad = G Á AD The pseudo-altitudes of a perfect tetrahedron, then, have this symmetric property (3) h j k = = Êg 3 a b c d e f = G Á3 A B C D E F H 3 J 3 K 3 H W X H Y Z J W Y J Z X K W Z K X Y The dependencies of perfection reduce the degrees of freedom in a tetrahedron s elements from six to four. That is to say, up to symmetry, a perfect tetrahedron is uniquely determined by the areas of its faces. In theory, we can remove all references to pseudo-faces in our formulas by solving for H, J, K in terms of W, X, Y, Z via the system of equations comprising (30) and (3); in practice, the results seem symbolically intractable, as with this degree-1 polynomial equation for Ḧ: = 16Ḧ1 64 Ḧ + 4 Ḧ Ä 4s X 3 ä 9 Ä 8Ḧ 8s 3 ä (33a) X 7s Y s Z 4s X + + s Ä s + s ä Ä ä Ä Y s + s Z s + s Y + s ä Z where (33b) s := 1 Ẅ Ẍ Ÿ Z 4 Ẅ Z Ÿ Z s X := ẄẌ + Ÿ Z s Y := ẄŸ + Z Ẍ s Z := Ẅ Z + ẌŸ This equation and its counterparts may be helpful numerically, but what their sprawling symbolic structure is trying to say about the tetrahedron remains a mystery Pseudo-Faces: Shadows and the Search for Substance For a Euclidean tetrahedron, the pseudo-face associated with pair of opposite edges is a shadow: the quadrilateral projection of the remaining edges into any plane parallel to that pair. Projection and parallelism are tricky concepts in hyperbolic space, so the appropriate analog of a pseudo-face shadow is non-obvious. We consider here a seemingly-promising construction. Take a tetrahedron having vertices P, Q, R, S, with a := P Q and d := RS. Suppose the pseudo-altitude h meets P Q and RS at M and N, respectively, and define p := MP, q := MQ, r := NR, s := NS. Let O be the midpoint of the pseudo-altitude. 16 Much the same is true in Euclidean space, which has its own expansive polynomials for pseudo-face areas, but at least perfect Euclidean tetrahedra admit a quartic formula for volume in terms of face areas.

8 8 B.D.S. BLUE MCCONNELL Emulating the notion of a shadowy projection into a plane parallel to a and d, let P, Q, R, S be the feet of perpendiculars dropped from respective points P, Q, R, S into the plane perpendicular to h at O. Write p := OP, q := OQ, r := OR, s := OS. We consider the candidacy of P R Q S for the geometric realization of pseudo-face H. For simplicity, we will assume that edges a and d are orthogonally twisted about pseudo-altitude h, and that p = r (so that also p = r ). Observe that P MOP is a Lambert quadrilateral with legs of length h and p, and with a co-leg of length p. A little trigonometry gives p = p ḧ 1 + p p p = h 1 + p h Moreover, P OR is an isosceles right triangle with leg length p and area given by sin( P OR ) = p p 1 + p p = p 1 + p ḧ We consider three cases. M and N are midpoints of P Q and RS. This tetrahedron is symmetric about pseudo-altitude h, and we have that p = q = r = s = a and b = c = ë = f = p ḧ = ä ḧ, and that P R Q S comprises four identical copies of P OR. Consequently, sin( 1 4 P R Q S ) = sin( P OR ) = a // b = H4 by (8b). Our mid-plane shadow is a geometric realization of pseudo-face H! M = Q and N = S. This tetrahedron is an orthoscheme, with q = s = 0 and p = a = d = r; assign b := P R, c := P S, e := QS = h, f := QR, so that b = ä ḧ and c = f = äḧ. The points Q and S coincide at O, so that P OR is P R Q S, with area given by sin( P OR ) = a // b. On the other hand, as the reader can verify, H via (8) has an incompatible, far-less-concise expression, so that this mid-plane shadow is not a realization of pseudo-face H. h = 0. This tetrahedron is degenerate, with coplanar vertices. We can abandon the orthogonal twist condition, and the supposition p = r, but let us assume that edges a and d themselves not merely the lines determined by them intersect as diagonals of convex quadrilateral P RQS. (Non-intersecting circumstances are handled similarly.) Then the quadrilateral is the union of the tetrahedron s faces in two ways: P RQS = Y + Z = W + X. But also, A = D = π, so that Ḧ = cos (Y + Z ) = cos (W + X ). The quadrilateral arguably, the shadow of the degenerate tetrahedron into a plane perpendicular to pseudo-altitude h realizes pseudo-face H. Moreover, the bow-tie quadrilateral P SRQ, with diagonals (say) b and e, has area 17 P SRQ = Z X = W Y, whereas B = E = 0 and J = cos (Z X ) = cos (W Y ); the bow-tie realizes pseudo-face J. Bow-tie 17 We take a bow-tie s area as the absolute value of the difference of its triangular regions, as is consistent with tracing the four edges of the quadrilateral in order, traversing the triangles in different orientations.

9 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 9 P RSQ, likewise, realizes K. Note that, while the bow-ties may be considered shadows of the tetrahedron, and in a plane (degenerately) parallel to pairs of opposing edges, they are not in a plane perpendicular to the pseudo-altitudes corresponding to J and K. The symmetric case demonstrates that a straightforward construction can realize a pseudo-face as a shadow, and the degenerate case further correlates pseudo-faces with shadows; the orthoscheme case, however, indicates that there is work yet to do in devising a general construction of pseudo-face shadows, if indeed pseudo-faces are shadows in general. 7. Special Tetrahedra 7.1. Regular Tetrahedra, A = B = C = D = E = F. If the dihedral angles of a hyperbolic tetrahedron match (so that the figure is necessarily perfect, by (9)), then its side lengths match (a = b = c = d = e = f), its face areas match (W = X = Y = Z), and its pseudo-face areas match (H = J = K). Moreover, any one metric angle measure A, side length a, face area X, pseudo-face area H completely determines the tetrahedron. A sampling of relations among these metrics follows. The most straightforward route to relating A and X passes through the realms of hyperbolic and spherical trigonometry. We recall that, by the definition of hyperbolic area, the plane angle θ at any vertex of our equilateral faces satisfies X = π 3θ, so that θ = π 3 X 3. The dihedral angle A relates to face angle θ by the spherical Law of Cosines for Sides: (34) Ä = θ θ θ θθ = θ 1 + θ = cos(π 3 X 3 ) 1 + cos (π 3 X 3 ) = Ü Ü3 where U := π X. By the Second Law of Cosines (), (35) Ḧ = Ẍ + XÄ = 1 We can also derive (36a) (36b) (36c) (36d) ä = ( ) 1 + 4Ü3 4Ü3 = H 4 = 1 Ä Ü 3 1 ä X Ä ä = H4 3 + H 4 Ä ä Ä ä Ẍ = 1 + H4 1 H4 3 X = 4H Ä ä H 4 / 1 H 4 3 H = 4H Ä ä H 4 4 H, X, X = H Ä ä H H 4 a = H 4 1 H 4 â = H 4 = Ü3 1 = 3Ä 1 1 Ä 7.. Isohedral Right-Corner Tetrahedra, A = B = C = π/, D = E = F. Here, three areas match (X = Y = Z), all pseudo-face areas match (H = J = K), and triples of side lengths match (a = b = c and d = e = f, with d = ä ). The First Law of Cosines relates hypotenuse-face W and leg-face X (37) Ẅ = Ẍ X = Ä + X ä 1 X W = 1 4 X Ä 3 + X ä

10 10 B.D.S. BLUE MCCONNELL but the spherical Law of Cosines applied to plane angle θ := π 3 W 3 in hypotenuse W and (acute) plane angle φ := π 4 X in leg X gives something more direct: (38) 0 = Ä = θ φ φ φφ = φ = θ = X = (π 3 W 3 ) 1 From the Second Law of Cosines (), (39) Ẍ = Ḧ = ẄẌ + W X D so that (40) D = T 1 + T = Ü3 Ü6 D = T = Ü6 where T := π X and U := π W. Also, (41a) H = ẌX 3 W = X 3 X = ẌX 3 ( Ẍ X ) (41b) H X X = 1 4 X 4 H W X = 1 Ẍ X 4 (41c) (41d) ä = Ẍ Ẍ X a = d = 1 1 X d = X Ẍ X â = X X 1 X d = X = Ü 3 1 = 3 D 1 D 8. Volume Volume seems the least-scrutable measurement in hyperbolic space, even for tetrahedra. The principle formula is a differential equation from Schlafli [4] in 1950: (4) dv = a da b db c dc d dd e de f df Derevnin and Mednykh s 005 solution [1] takes the form of a monolithic integral: (43) V = 1 4 where α+β α β log cos A+B+C+θ cos sin A+D+B+E+θ sin α := atan p α q α A+E+F +θ cos B+E+C+F +θ sin β := atan p β q β D+B+F +θ cos D+E+C+θ C+F +A+D+θ sin θ dθ

11 and (44) HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 11 p α := sin(a + B + C + D + E + F ) + sin(a + D) + sin(b + E) + sin(c + F ) + sin(d + E + F ) + sin(d + B + C) + sin(a + E + C) + sin(a + B + F ) q α := (cos(a + B + C + D + E + F ) + cos(a + D) + cos(b + E) + cos(c + F ) + cos(d + E + F ) + cos(d + B + C) + cos(a + E + C) + cos(a + B + F )) p β := p α + qα p β = det G q β := (sin A sin D + sin B sin E + sin C sin F ) We note that p α + qα = p β + qβ = 8 Ä 1 + Ä B C + ÄË F + D B F + DË C +Ä D BË + BË C F + C F Ä D + ADBE + BECF + CF AD ä Also, with considerable trigonometric effort, one finds that the difference of the numerator and denominator in the logarithm s argument reduces thusly: (45) cos( ) cos( ) cos( ) cos( ) sin( ) sin( ) sin( ) sin( ) = q β (p α sin θ + q α cos θ) The values θ = α ± β are roots of this expression and thus are also roots of the integrand. Unfortunately unlike in the Euclidean case 18 these formula, in general, resist direct hedronometric re-parameterization in terms of face and pseudo-face areas. (Below, we examine a couple of less-resistant special cases, and we discuss the complexity of the general case.) For the most part, the best we can do currently is invoke the Second Law of Cosines () to convert areas into dihedral angle measures, and substitute those measures into the Derevnin-Mednykh formula (43) Regular Tetrahedra, A = B = C = D = E = F. Since a = b = c = d = e = f and A = B = C = D = E = F, we have dv = 6 a da. We can express this differential in terms of angle A, face area X, or pseudo-face area H: (46a) dv 6 = atanh 3Ä 1 1 Ä da = atanh Ü3 1 da dx dx = atanh da H 4 dh dh (46b) da dx = U Ü3 da dh = L 8 Ä 3 + L 4 ä 18 The volume, V, of a Euclidean tetrahedron with faces W, X, Y, Z and pseudo-faces H, J, K satisfies a formula bearing a striking resemblance to the Third Law of Cosines in hyperbolic space (3). 81V 4 = W X Y + W Y Z + W Z X + X Y Z + H J K H (W X + Y Z ) J (W Y + Z X ) K (W Z + X Y )

12 1 B.D.S. BLUE MCCONNELL with U := π X and L := π H. So (47a) V = 6 A acos 1 3 atanh 3 cos θ 1 1 cos θ dθ (47b) = X 0 tan(π 6 ξ 6 ) cos(π3 ξ 3 ) atanh cos(π 3 ξ 3 ) 1 dξ (47c) = 3 H 0 sin(π 4 η 8 ) 3 + sin η 4 atanh sin η 4 dη 8.. Isohedral Right-Corner Tetrahedra, A = B = C = π/, D = E = F. Limiting ourselves (and our paths of integration) to the universe of right-corner tetrahedra with A = B = C = π/, we can assert da = db = dc = 0; then, with D = E = F, we have dd = de = df, so that dv = 3 d dd. We can write: (48a) (48b) dv 3 = atanh 3 D 1 dd = atanh X dd D dx dx = atanh Ü3 1 dd dw dw dd dx = T ( T ) dd dw = U 6 6Ü3 with T := π X and U := π W. So D 3 cos V = 3 atanh θ 1 (49a) dθ sin θ (49b) (49c) = 3 = 1 1 acos 3 X 0 W 0 sin(π 4 ξ ) 1 + cos (π 4 ξ ) atanh sin ξ dξ tan(π 6 ω 6 ) cos(π3 ω 3 ) atanh cos(π 3 ω 3 ) 1 dω 8.3. General Tetrahedra. A key complication in deriving a face-based formula for tetrahedral volume is that our seven hedronometric parameters faces areas W, X, Y, Z and pseudo-face areas H, J, K aren t independent (and shouldn t be, as there are only six degrees of freedom in determining a tetrahedron). Nevertheless, the differentialized Second Law of Cosines () converts the Schlafli angular differentials to hedronometric ones: (50a) 4 H Y Z da = H dh Z ŸḦ Y dy Ÿ Z Ḧ Z 4 H W X dd = H dh Ẍ ẄḦ dw Ẅ ẌḦ W X dz dx

13 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 13 while the differentialized Third Law of Cosines (3) expresses the dependency among the hedronometric differentials: (50b) W W dw + X X dx + Y Y dy + Z Z dz = H H dh + J J dj + K K dk While we hope for a monolithic integral in the spirit of Derevnin-Mednykh that respects the inherent symmetries of the face and pseudo-face area parameters, for now we investigate an asymmetric solution Using an auxiliary parameter. Consider the universe of tetrahedra with constant W, X, Y, Z, H. Variable pseudo-face areas J and K have a dependency via the Third Law of Cosines (3). We can interpret the Law as a quadratic in J and K that describes an ellipse (rotated by π/4 from the coordinate axes), allowing us to parameterize the values in terms of a single (and remarkably well-suited) polar angle, θ. Translating and rotating the ellipse, we arrive at these relations, where we indicate dependence of J and K on θ with a superscript: J θ H = ẄŸ + Z ä (51a) Ẍ ÄẄ Ḧ Z + ẌŸ + 4 cos(θ H 4 ) H W X H Y Z K θ H = Ẅ Z + ẌŸ ÄẄ Ḧ Ÿ + Z ä (51b) Ẍ 4 cos(θ + H 4 ) H W X H Y Z Corresponding formulas for J θ and Kθ reduce nicely (5a) J θ H = 4 sin(θ + H 4 ) H W X H Y Z (5b) K θ H = 4 sin(θ H 4 ) H W X H Y Z and provide the insight that, given the non-negative nature of each other factor in the products, we must have sin(θ ± H 4 ) 0, so that we can take H 4 θ π H 4. Abusing notation slightly, we define (53) H θ := Ḧ + K θ ẄẌ Ÿ Z J θ and observe that H θ does not appear to admit insightful simplification, so that its behavior with respect to θ remains unclear. Numerical experiments in Mathematica suggest that prudent (re-)labeling of face areas can guarantee H θ to be strictly positive across the domain determined by J θ and Kθ in the preceding paragraph. The validity of this phenomenon would help streamline Theorem 8.; unfortunately, proof is elusive, so we state it here as a conjecture: Conjecture 8.1. Given a hyperbolic tetrahedron, one can assign W, X, Y, Z to its face areas and H to the corresponding pseudo-face area in such a way that H θ > 0 for all H 4 < θ < π H 4.

14 14 B.D.S. BLUE MCCONNELL The relations (5) provide compact representations of the trigonometric functions of θ: 19 (54a) (54b) (54c) 4θ H W X H Y Z = H 4 ( J θ + K ) θ 4 θ H W X H Y Z = Ḧ4( J θ K ) θ θ = H 4 J θ + Kθ J θ Kθ With most face areas constant, the differentials of J θ and K θ are straightforward: 0 J θh (55a) dj θ = 8 sin(θ H 4 ) H W X H Y Z dθ K θh (55b) dk θ = 8 sin(θ + H 4 ) H W X H Y Z dθ Moreover, the various angular differentials (as in (50a)) simplify. Of course, da and dd vanish outright; otherwise, we have, for instance, J θ (55c) db = dj θ 4 J θz X = H W X H Y Z sin(θ H 4 ) H J θ Z X dθ We may therefore give this formula for the Schafli volume differential (4): (56a) Ñ é b θ sin(θ H 4 ) H W X H Y Z dv = J θ Z X + e θ J θ W Y Ñ é H c θ + sin(θ + H 4 ) K θ X Y + f θ dθ K θ W Z for non-negative b θ, c θ, e θ, f θ satisfying (56b) b θ J θ = 4 Z X sin(θ + H 4 ) H θ sin(θ H 4 ) e θ J θ = 4 W Y sin(θ + H 4 ) H θ sin(θ H 4 ) c θ K θ = 4 X Y sin(θ H 4 ) H θ sin(θ + H 4 ) f θ K θ = 4 W Z sin(θ H 4 ) H θ sin(θ + H 4 ) Now, note that our constant areas make the variable aspect of G, Á the tetrahedron s Gram determinant (see (1a)), proportional to H θ J θ Kθ, which is in turn proportional to H θ sin(θ + H 4) sin(θ H 4 ). Consequently, G Á vanishes indicating a degenerate tetrahedron at the endpoints H 4 and π H 4 of the bounds on θ imposed after (5); in the absence of proof of Conjecture 8.1, we recognize the possibility of degenerate tetrahedra determined by some θ = θ between these endpoints. Interpreting the above appropriately, we find that we can compute the volume of a tetrahedron via a hedronometric formula that integrates through our universe of tetrahedra 19 Squaring and adding the sine and cosine relations re-confirms equation (6). 0 With (5), these are evidently consistent with the differentialized Third Law of Cosines (50b).

15 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 15 with constant W, X, Y, Z, H, from the degenerate form (corresponding to θ = H 4, or possibly θ = θ ) to our target (corresponding to the particular θ satisfying (54), with J θ = J and K θ = K). Theorem 8.. A hyperbolic tetrahedron with face areas W, X, Y, Z and corresponding pseudo-face areas H, J, K has volume, V, given by (57) V = H W X H Y Z H Θ Φ sin(θ H 4 ) + sin(θ + H 4 ) Ñ Ñ b θ J θ Z X + e θ J θ W Y c θ K θ X Y + f θ K θ W Z é é dθ with non-negative b θ, c θ, e θ, f θ satisfying (56b); H 4 Θ π H 4, such that θ = Θ satisfies (54) with J θ = J and K θ = K; Φ is the largest root of H θ J θ Kθ = 0 such that H 4 Φ Θ What is the Pythagorean Theorem for Right-Corner 4-Simplices? In Euclidean any-dimensional space, our understanding of the Pythagorean Theorem for right-corner simplices is complete: always, the square of the content of the hypotenuse is equal to the sum of the squares of the contents of the legs. In hyperbolic geometry, our knowledge seems embarrassingly limited: Dimension : cosh hyp = cosh leg 1 cosh leg Dimension 3 : cos hyp = cos leg 1 Dimension 4+ :??? =??? cos leg cos leg 3 sin leg 1 sin leg sin leg 3 Consider the case of a right-corner simplex in hyperbolic 4-space: as Section 8 indicates, any relation between the volume of the simplex s hypotenuse-tetrahedron and the volumes of its leg-tetrahedra involves comparing complicated integrals. Even the most-special of special cases (below) defies this author s attempts at comprehension. 1 By Conjecture 8.1, appropriate face and pseudo-face labeling would allow us to take Φ = H4.

16 16 B.D.S. BLUE MCCONNELL 9.1. Isosceles Right-Corner 4-Simplices. Consider right-corner 4-simplex such that each of its four congruent legs are isohedral right-corner tetrahedra determined by dihedral angles (A, B, C, D, E, F ) = ( π, π, π, P, P, P ); its hypotenuse is a regular tetrahedron with congruent dihedral angles (Q, Q, Q, Q, Q, Q). One can verify that 1 (58) 3 cos P = χ = cos Q 1 for a parameter χ that we use to write tantalizingly-similar formulas for the volume H of the hypotenuse (via (47a)) and volume L of each leg (via (49a)): (59a) (59b) H := 3 acos χ acos 1 3 atanh acos χ L := 6 atanh acos cos θ 1 1 cos θ dθ = 3 χ t (1 t) atanh 3t 1 1 t dt 3 cos θ 1 χ 1 cos θ dθ = 6 1 atanh 3t t 1 t dt Although Pythagoras beckons, a connection between these volumes remains elusive. Indeed, the rather exotic but not unfamiliar values for χ = 1/ (defining a simplex with a quadrupally-asymptotic hypotenuse and triply-asymptotic legs) (60) H := k=1 1 πk sin k 3 = L := 1 k=1 1 πk sin k = suggest that the corresponding Pythagorean Theorem is quite non-trivial. L is half of Catalan s constant, and H is known, e.g., as the maximum of the Clausen function, Cl.

17 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 17 A. Standard Notation This note (in the tradition of most notes on hedronometry by this author) reserves much in fact, just-over half of the Latin alphabet for specific notational use, as described below. Generally, lower-case letters indicate trigonometric elements (sides and planar angles, and their measures), while upper-case letters indicate hedronometric elements (faces and dihedral angles, and their measures); the lower- and upper-case counterparts of a letter indicate elements in a natural correspondence. We avoid the use of integer subscripts as indices; rather, we reserve them for compact representation of fractions in formulas. (See Section A..) Note. Without fear of confusion, a symbol represents both a geometric element and its measure. (E.g., edge a has length a; angle B has measure B; face Z has area Z.) A.1. Tetrahedral Elements (a, b, c, d, e, f; W, X, Y, Z; H, J, K). A tetrahedron s edges a, b, c concur at a vertex and oppose respective edges d, e, f. The dihedral angles along the edges are, respectively, A, B, C and D, E, F. The tetrahedron s faces are W := def X := dbc Y := aec Z := abf When discussing right-corner tetrahedra, we take edges a, b, c to be mutually-orthogonal, and angles A, B, C to be right angles. This makes W the figure s hypotenuse-face and X, Y, Z its right-triangular leg-faces. We sometimes (as in Section 4) assign symbols w, x, y, z to the altitudes to faces W, X, Y, Z, but other times revert using these symbols for generic quantities and coordinates in the traditional way. With regard to pseudo-elements 3 (and their measures), we make the following associations between various pairs of traditional elements and trios comprising a pseudo-face (H, J, K), a pseudo-altitude (h, j, k), and a twist angle (τ h, τ j, τ k ): a, d A, D (Y, Z), (W, X) H, h, τ h b, e B, E (Z, X), (W, Y ) J, j, τ j c, f C, F (X, Y ), (W, Z) K, k, τ k A.. Trigonometry. To remove visual clutter from trigonometric expressions, we adopt this Morse Code (and hat) notation: ẍ := cosh x and x := sinh x and x := tanh x, for lengths x θ := cos θ and θ := sin θ and θ := tan θ, for angles θ Ẍ := cos X and X := sin X and X := tan X, for areas X Context should make clear whether hyperbolic or circular functions are intended. 3 See footnotes 1 and 11 for definitions of pseudo-face and pseudo-altitude.

18 18 B.D.S. BLUE MCCONNELL Half-measures (half-angles, half-lengths, half-areas) and even quarter-measures appear so often in formulas that writing (and reading) them as fractions can be a strain. For typographic simplicity, we write x n for x/n. Finally, we define the Heronic sine and cosine products 4 via (61a) (61b) x, y, z := (x + y + z ) ( x + y + z ) (x y + z ) (x + y z ) = 1 Ä ẍÿ z ẍ ÿ z + 1 ä 4 [ x, y, z ] := (x + y + z ) ( x + y + z ) (x y + z ) (x + y z ) = 1 Ä ẍÿ z + ẍ + ÿ + z 1 ä 4 These take their name and symbology from the so-called and -symbolized (by me) algebraic Heronic product (6) [x, y, z] := (x + y + z ) ( x + y + z ) (x y + z ) (x + y z ) that gives Heron s formula for the square of the area of a Euclidean triangle with edges x, y, z. This note occasionally uses (6) to abbreviate the characteristic four-part product. 5 We ll note the side-angle-side form of the Heronic (hyperbolic) sine product of edges x, y, z of a hyperbolic triangle, with angles θ, φ, ψ opposite respective sides x, y, z: (63) x, y, z = 1 4 y z θ = 1 4 z x φ = 1 4 x y ψ This equation inspires the Law of Side-Angle-Side-Altitude Sines (Theorem 4.1). A.3. Matrices, Determinants, Conjugation Ratio, and Minors (G, M). Define the traditional ( angle ) Gram matrix, G, and conjugate ( edge ) Gram matrix, g, with rows and columns indexed by faces W, X, Y, Z (in order). 6 (64) G := 1 D Ë F D 1 C B Ë C 1 Ä F B Ä 1 g := 1 ä b c ä 1 f ë b f 1 d c ë d 1 4 The equivalence of factored and expanded forms holds for both circular and hyperbolic functions. 5 See, for instance, (8a) and (7c). Note that, because I incorporate the halving of arguments in building the factors (to provide uniformity with the sine and cosine products), I must double arguments or else multiply the result by 16 when I simply want to use the [ ] notation to compress these four-fold products. 6 The angles in a row (column) of G surround the index-face, while remaining angles oppose that face; conjugately, the edges in a row (column) of g oppose the index-face, while the remaining edges surround it.

19 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 19 The determinants of these matrices are as follows: â 1 Ä B C D Ë F ì (65a) (65b) ÁG := det G = Êg := det g = Ü Ä B C ÄË F D B F DË C +Ä D + B Ë + C F Ä D BË BË C F C F Ä D 1 ä b c d ë f + d b c + äë c + ä b f + dë f +ä d + b ë + c f ä d bë ä d c f bë c f and we define (what I call) the conjugation ratio 7 thusly: (66) Γ := Á G Êg For face-indices P and Q, define the (P Q)-th cofactors, M P Q and m P Q : (67) M P Q := ( 1) P +Q minor P Q G m P Q := ( 1) P +Q minor P Q g Thus, (68a) (68b) M W X = A D + Ä Ä BË + C F ä + B F + CË mw X = äd + d Ä bë ä + c fä Ä b f + cë When P = Q, we use a single index. For instance, (69a) (69b) M W = Ä B C Ä B C + 1 = 4[ A, B, C ] mw = dë f + d + ë + f 1 = 4 d, e, f = 16W d ë f We note that Mednykh and Pashkevich [3, Proposition 4] provide cofactor formulas for the lengths of a tetrahedron s edges and altitudes: (70a) ä = M W X M W M d = M Y Z X M Y M Z Likewise, we have cofactor formulas for dihedral angles: (70b) D = m W X m W m X Ä = m Y Z m Y m Z w = Á G M W = Êg m W ê Note the use of upper-case gamma as a reminder that the numerator of the conjugation ratio contains the upper-cased that is, angle-based determinant, G. Ê I m tempted to introduce γ to denote the reciprocal ratio, just for the sake of completeness.

20 0 B.D.S. BLUE MCCONNELL B. Coordinates In coordinatized hyperbolic space, 8 let P (x p, y p, z p ) be such that z p measures the signed length of a perpendicular dropped from P to a point P xy in the xy-plane; y p measures the signed length of a perpendicular dropped from P xy to a point P x on the x-axis; and x p measures the signed distance from P x to the origin. Every coordinate system has a distance formula, and here s the one for this context: Theorem B.1 (The Distance Formula in Coordinatized Hyperbolic Space). The distance, AB, from point A(x a, y a, z a ) to point B(x b, y b, z b ) is given by (71) cosh AB = z a z b ( ÿ a ÿ b ( ẍ a ẍ b x a x b ) y a y b ) z a z b We wish to describe a coordinatization of a tetrahedron. Let the tetrahedron have elements with standard 9 edge, (pseudo)face, and pseudo-altitude labels, and let vertices P, Q, R, S be opposite respective faces W, X, Y, Z. We align pseudo-altitude h with the x-axis and an adjacent edge P Q with the y-axis; say that the pseudo-altitude separates P Q into segments of length a P and a Q, and RS into segments of length d R and d S. We write P (0, a P, 0) Q(0, a Q, 0) R(h, y R, z R ) S(h, y S, z S ) where tanh y R = tanh d R cos τ h tanh y S = tanh d S cos τ h sinh z R = sinh d R sin τ h sinh z S = sinh d S sin τ h for τ h the angular twist of edge RS out of the xy-plane. This arrangement, and the Distance Formula, provide these formulas for edge lengths. 30 (7) ä = cosh P Q = a P a Q + a P a Q b = cosh P R = ap dr ḧ a P d R τ h c = cosh P S = a P ds ḧ + a P d S τ h d = cosh RS = dr ds + d R d S ë = cosh SQ = a QdS ḧ a Q d S τ h f = cosh QR = aq dr ḧ + a Q d R τ h With these, the conjugate Gram determinant, Êg, (see (65b)) collapses from over a dozen terms to just one: (73) Êg = a d h τ h 8 Here, we enlist x, y, z as symbols for standard coordinates instead of tetrahedral altitudes. 9 See Appendix A. 30 For instance, ( (ḧ ) ) b = zr ap y R 0 ap y R 0 = ap ( y R z R)ḧ ap (yr zr) = ap d R ḧ a P d R where the simplifications y R z R = d R and y R z R = d R τ h follow from the trigonometry of the right triangle with hypotenuse d R, legs y R and z R, and angle τ h. τ h

21 HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 1 whence, all un-squared sines being non-negative while Êg being non-positive, (74a) a d h τ h = Êg Also, one readily verifies this identity: (74b) a d ḧ τ h = c f bë where the supplementary ambiguity of τ h necessitates the use of absolute values. With a bit of work, one can extract the sub-segment lengths a P, etc, writing (75a) sinh a P = a (m W X + ä m X ) î ad, Ä bë c fä, i Êg ó (75b) so that (76) sinh a Q = a (m W X + ä m W ) î ad, Ä bë c fä, i Êg ó sinh a P sinh a Q = m W X + ä m X m W X + ä m W References [1] Derevnin, D. A., and Mednykh, M. G. A formula for the volume of a hyperbolic tetrahedron. Communications of the Moscow Mathematical Society, Vol. 60, No., pp , 005. http: //mathlab.snu.ac.kr/~top/articles/mednykhderevninhyperbolictetrahedron.pdf [] Horvath, Akos. Skew Lines in Hyperbolic Space. Periodica Polytechnica Ser. Mech. Eng., Vol. 47, No. 1, pp. 530, [3] Mednykh, A. D., and Pashkevich, M. G. Elementary Formulas for a Hyperbolic Tetrahedron. Siberian Mathematical Journal, Vol. 47, No. 4, pp , Elementary_Formulae_for_tetrahedron.pdf [4] Schlafli, L. Theorie der vielfachen Kontinuität. Gesammelte mathematishe Abhandlungen, Basel: Birkhäuser, 1950.