THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA

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1 THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA B.D.S. MCCONNELL Darko Veljan s article The 500-Year-Old Pythagorean Theorem 1 discusses the history and lore of probably the only nontrivial theorem in mathematics that most people know by heart, depicted in Figure 1. The article motivated the author to put his own few-years-old results on public display. c a + b Figure 1. The Pythagorean Theorem for Euclidean Geometry Veljan mentions a large number of relatives of the Theorem in various contexts. Of interest here are the direct analogues governing right triangles in spherical and hyperbolic geometry Figure, the extensions that apply arbitrary triangles Figure 3, and the generalization to Euclidean tetrahedra Figure 4a. The Law of Cosines extensions of the last Figure 4b, while perhaps not universally known, are known, are readily proven, and have straightforward analogues in all higher dimensions. The purpose of this note is to formally present non-euclidean counterparts of the two tetrahedral Laws of Cosines. The formulas appear below in Figure 5, with the rather elegant Pythagorean versions for right-cornered tetrahedra in which m BDC m CDA m ADB π/ m DA m DB m DC in Figure 6. The author has posted these results a time or two to online discussion lists; the results seem to have been unknown to the mathematical community before then. Date: 1 January, 005; updated 30 January, 006, to include the Second Law of Cosines. 1 Mathematics Magazine, Vol. 73, No. 4. October,

2 B.D.S. MCCONNELL a cos c cos a cos b b cosh c cosh a cosh b Figure. The Pythagorean Theorems for a spherical and b hyperbolic geometry a c a + b ab cos γ b cos c cos a cos b + sin a sin b cos γ cos γ cos α cos β + sin α sin β cos c c cosh c cosh a cosh b sinh a sinh b cos γ cos γ cos α cos β + sin α sin β cosh c Figure 3. Assorted Laws of Cosines

3 THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 3 a b W X + Y + Z W X + Y + Z Y Z ZX cos DB XY cos DC W + X W X cos BC Y + Z Y Z cos AD Figure 4. The a Pythagorean Theorem and b Laws of Cosines for Euclidean Tetrahedra. X, Y, Z, and W are face areas; DA, etc., are the dihedral angles between faces.

4 4 B.D.S. MCCONNELL a Spherical Space cos W cos X cos Y cos Z b Hyperbolic Space cos W cos X cos Y cos Z + sin X sin Y sin Z S sin X sin Y sin Z S + cos X sin Y sin Z + cos X sin Y sin Z + sin X cos Y sin Z cos DB + sin X cos Y sin Z cos DB + sin X sin Y cos Z cos DC + sin X sin Y cos Z cos DC S : 1 cos DB cos DC cos DA cos DB cos DC cos W cos X sin W sin X cos BC cos W cos X + sin W sin X cos BC cos Y cos Z sin Y sin Z cos AD cos Y cos Z + sin Y sin Z Figure 5. The Laws of Cosines for Non-Euclidean Tetrahedra

5 THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 5 a Spherical Space cos W cos X cos Y cos Z + sin X sin Y sin Z b Hyperbolic Space cos W cos X cos Y cos Z sin X sin Y sin Z Figure 6. The Pythagorean Theorems for Right-Cornered Non-Euclidean Tetrahedra

6 6 B.D.S. MCCONNELL 1. Preliminaries Proof of these identities begins with the remarkably simple formulas of non-euclidean area. See Figure 7. The area of a triangle in spherical geometry is equal to its angular excess the amount by which sum is its radian angle measures exceeds π. The area of a triangle in hyperbolic geometry is equal to its angular defect the amount by which the sum of its radian angle measures falls short of π. Area α + β + γ π Area π α + β + γ Figure 7. Formulas for non-euclidean Triangle Area 1.1. Notation. To streamline the formulas, we ll employ a notational Morse Code dots for cosines and dashes for sines that merges the spherical and hyperbolic cases. We will denote cos θ for angle θ simply by θ; complementarily, sin θ appears as θ. For side x, we ll denote both cos x and cosh x by ẍ, while both sin x and sinh x become x. There is no ambiguity here. The notation will indicate circular functions when representing the spherical case, and hyperbolic functions for the hyperbolic case. To account for the occasional sign change, we agree that, in ± and, the top sign applies to the spherical case, and the bottom one to the hyperbolic case. Then we have formulas such as these Law of Cosines for Sides solved for the angle Area of Triangle c ä b ± ab γ c ä b γ ± ab ± α + β + γ π Also, we ll reference half-angles and half-sides often; we ll write x for x to save some typesetting space.

7 THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA A Heron-like Area Formula. Our target formulas involve the cosine of the half-area of a triangle. This is easy to compute in terms of angles. For example, cos W ± α + β + γ π cos cos ± α + β + γ π sin α + β + γ α β γ + α β γ + α β γ α β γ The Law of Cosines for Sides allows us to convert this to a formula in terms of the triangle s half-sides, just as the Heron formula for Euclidean triangles computes area from sides. The reader can verify the following γ γ a + b + c a + b c /ab s s c /ab γ ± 1 1 γ a + b + c a b + c /ab s a s b /ab with semi-perimeter s : a + b + c a + b + c /. The terms in the area formula then expand: s bs c α β γ ss b bc ca ss c s s b s c /abc ab α β γ s s a s c /abc α β γ s s a s b /abc α β γ s a s b s c /abc Combining the terms, and canceling a denominator rewritten as 8ä a b b c c, the sum simplifies to cos W ä + b + c 1 ä b c 1 + ä + b + c 4ä b c 1.3. The Heron-like Formula in Tetrahedral Context. Taking our triangle, ABC, to be the face of a tetrahedron with fourth vertex D see Figure 4, we can express the cosines of the triangle s half-edges and ultimately its half-area in terms of the lengths of half-edges and measures of angles of elements meeting at D: x : DA y : DB z : DC θ : m BDC φ : m CDA ψ : m ADB The triangular Law of Cosines for Sides gives us ä ÿ z ± yz θ b zẍ ± zx φ c ẍÿ ± xy ψ

8 8 B.D.S. MCCONNELL The Heron-like formula for W then becomes cos W 1 + ÿ z ± yz θ or in terms of half-sides cos W + zẍ ± zx φ + ẍÿ ± xy ψ 4ä b c 1 + ÿ z + zẍ + ẍÿ ± yz θ ± zx φ ± xy ψ 4ä b c ẍ ÿ z ± x y z ± y ÿ z z θ ± z z x x φ ± x ẍ y y ψ ä b c. Verifying the Laws.1. The First Law of Cosines. Here, we will demonstrate that the right-hand side of our proposed First Law of Cosines for Non-Euclidean Tetrahedra cos W cos X cos Y cos Z ± sin X sin Y sin Z S + cos X sin Y sin Z + sin X cos Y sin Z cos DB + sin X sin Y cos Z cos DC reduces to the formula at the end of the preceding subsection, thereby proving this Law. Expanding the bits of the expression gives cos X cos Y cos Z 1 + ÿ + z + ä 1 + ÿ + z + ÿ z ± yz θ 4ÿ z ä 4ÿ z ä 1 + ÿ 1 + z ± yz θ 4ÿ z ± 4y ÿ z z θ 4ÿ z ä 4ÿ z ä ÿ z ± y z θ ä z ẍ ± z x φ b ẍÿ ± x y ψ c and also sin X y z θ ä sin Y x z φ b sin Z y z ψ c Finally, the formulas for the dihedral angles at point D come from spherical trigonometry, with face-angles θ, φ, and ψ as the sides, and dihedral angles DA, DB, and DC as the angles. Thus,

9 THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 9 and θ φ ψ φψ S cos DB φ θ ψ θψ 1 cos DB cos DC cos DA cos DB cos DC 1 + θ φ ψ θ φ φ θφψ cos DC ψ θ φ θφ The terms of the proposed First Law of Cosines expansion are as follows: cos X cos Y cos Z ÿ z ± y z θ z ẍ ± z x φ ẍ ÿ ± x y ψ ä b c ẍ ÿ z ± x y z θ φ ψ +ÿ y z z ±ẍ θ + x φ ψ +ẍ x z z ±ÿ φ + y θ ψ +ẍ x y ÿ ± z ψ + z θ φ ä b c ± sin X sin Y sin Z S ±y z θ xz φ xy ψ 1 + θ φ ψ θ φ ψ ä b c θφψ ±x y z 1 + θ φ ψ θ φ ψ ä b c cos X sin Y sin Z ÿ z ± y z θ xz φ xy ψ θ φ ψ ä b c φψ x y z ÿ z ± y z θ θ φ ψ ä b c ÿ y z z x θ x φ ψ θ x y z ± ä b c ä b c sin X cos Y sin Z ẍ x z z y φ y θ ψ x y z φ cos DB ± ä b c ä b c sin X sin Y cos Z ẍ x ÿ y z ψ z θ φ x y z ψ cos DC ± ä b c ä b c θ φ ψ θ φ ψ θ φ ψ

10 10 B.D.S. MCCONNELL Therefore, combining the terms and invoking the trigonometric identity ẍ ± x cos x + sin x cosh x sinh x 1 gives Total ẍ ÿ z ± x y z ±ÿ y z z θ ẍ ± x ±ẍ x z z φ ÿ ± y ±ẍ x ÿ y ψ z ± z /ä b c ẍ ÿ z ± x y z ±ÿ y z z θ ± ẍ x z z φ ± ẍ x ÿ y ψ /ä b c cos W as desired... The Second Law of Cosines. Now we will verify the Second Law of Cosines for Non- Euclidean Tetrahedra cos W cos X ± sin W sin X cos BC cos Y cos Z ± sin Y sin Z by exposing the symmetric nature of one side of the equation. immediate consequence of the relation. Before doing so, we observe an Corollary 1. In an ideal hyperbolic tetrahedron that is, one with each of its vertices at infinity dihedral angles along opposite angles are congruent. Proof. Simply note that each of the faces of the tetrahedron is an ideal triangle, with angles measuring 0 and hence areas W, X, Y, and Z measuring π. Each instance of the Second Law of Cosines then reduces to an equality between the cosines of opposing dihedral angles. Formulas from preceding sections give us the following cos Y cos Z + sin Y cos Z 1 + ẍ + z + b 1 + ẍ + ÿ + c + xzφ xyψ θ φψ 4ẍ z b 4ẍ ÿ c 4ẍ z b 4ẍ ÿ c φψ 1 + ẍ + z + b1 + ẍ + ÿ + c + x yz θ φ ψ 16ẍ ÿ z b c Now, using the triangular Law of Cosines in appropriate faces, we eliminate references to θ, φ, and ψ. x yz θ φ ψ x yz θ xz φxy ψ ±x ä ÿ z b ẍ z c ẍÿ 1 ẍ ä ÿ z b ẍ z c ẍÿ

11 THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 11 so that 1 + ẍ + z + b1 + ẍ + ÿ + c + x yz θ φ ψ 1 + ẍ ẍ b + c + ÿ + z + ÿ z + c z + bÿ + b c +1 ẍ ä ÿ z b c + bẍÿ + cẍ z ẍ ÿ z 1 + ẍ1 + b + c + ẍ + ÿ + z + bÿ + c z + 1 ẍä ÿ z + ÿ z ẍ 1 + ä + b + c + ẍ + ÿ + z äẍ + bÿ + c z Therefore, cos Y cos Z + sin Y sin Z 1 + ä + b + c + ẍ + ÿ + z äẍ + bÿ + c z 8ÿ z b c The right-hand side of this formula is symmetric with respect to switching pairs of edges a x b y c c z z which implies that the left-hand side must be symmetric with respect to switching its elements as well. Y xbz X ayz W abc Z xyc DA BC This proves the result..3. Pseudo-Faces. As in the Euclidean case, the Second Law of Cosines invites the formal definition of pseudo-faces with areas H, J, and K satisfying the relations cos W cos X + sin W sin X cos BC cos H cos Y cos Z + sin Y sin Z cos W cos Y + sin W sin Y cos CA cos J cos Z cos X + sin Z sin X cos DB cos W cos Z + sin W sin Z cos AB cos K cos X cos Y + sin X sin Y cos DC Unlike with the Euclidean case, I have not yet determined if the pseudo-faces have a geometric interpretation. Whether they might provide any insights with respect to a Heron-like formula for volume is not at all clear. With the help of pseudo-faces, we can re-write First Law of Cosines in a symmetric, face-agnostic form that, as a bonus, avoids reference to the quantity S or its counterparts in different orientations: 0 1 Ẅ Ẍ Ÿ Z 4 Ẅ Ẍ Ÿ Z Ḧ J K Ḧ J K +ḦẄẌ + Ÿ Z + J ẄŸ + Z Ẍ + K Ẅ Z + ẌŸ We arrive at this formula by isolating S in the W -centric instance of the First Law of Cosines. X Y Z S Ẅ + ẌŸ Z + ẌY Z + X Ÿ Z cos DB + X Y Z cos DC

12 1 B.D.S. MCCONNELL After squaring both sides of the equation, we use the Second Law of Cosines to express, cos DB, and cos DC in terms of the face and pseudo-face areas, and then simplify. A computer algebra system comes in handy during this process. The instances of the First Law of Cosines can be recovered by making appropriate substitutions for H, J, and K, based on the Second Law of Cosines, and solving the resulting quadratic. Indeed, many additional formulas for a total of 6, 44 arise from such substitutions. 3. Remark Infinitesimally, the Laws of Cosines for Non-Euclidean Tetrahedra become the Laws of Cosines for Euclidean Tetrahedra. This is demonstrated by expressing each half-area term with power series, and ignoring quantities with degree higher than two. For the First Law, we have 1 W X Y 1 Z X Y Z 8 X + 1 Y Z 8 X Y + 1 Z 8 1 X 8 Y 8 Z ± 0 S 8 + Y Z 4 + XZ 4 ± cos DB cos DC cos DB + XY 4 X Y cos DC Z S whence W X + Y + Z Y Z XZ cos DB XY cos DC Likewise, for the Second Law, Arithmetic check: There are 3 different ways to substitute into Ḧ using BC in both factors, using DA in both factors, or using each of these angles once; likewise, there are 3 ways each to substitute into J, K, Ḧ, J, K. There are 36 ways to substitute into Ḧ J K. Thus, the total number of formulas is , 44.

13 THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 13 1 W 1 X W 8 X 8 + W X cos BC + W X cos BC 4 1 Y 1 Z Y 8 Z 8 + Y Z + Y Z 4 W + X W X cos BC Y + Z Y Z B.D.S. McConnell

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