The Laws of Cosines for Non-Euclidean Tetrahedra

THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA

B.D.S. MCCONNELL

Darko Veljan’s article “The 2500-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.

c2 = a2 + b2

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 2), the extensions that apply arbitrary triangles (Figure 3), and the generalization to Euclidean tetrahedra (Figure 4(a)). The “Law of Cosines” extensions of the last (Figure 4(b)), 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/2 = 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, 2005; updated 30 January, 2006, to include the Second Law of Cosines. 1Mathematics Magazine, Vol. 73, No. 4. (October, 2000.) 1 2 B.D.S. MCCONNELL

(a) cos c = cos a cos b (b) cosh c = cosh a cosh b

Figure 2. The Pythagorean Theorems for (a) spherical and (b) hyperbolic geometry

(a) c2 = a2 + b2 − 2ab cos γ

(b)(c) cos c = cos a cos b + sin a sin b cos γ cosh c = cosh a cosh b − sinh a sinh b cos γ cos γ = − cos α cos β + sin α sin β cos c cos γ = − cos α cos β + sin α sin β cosh c

Figure 3. Assorted Laws of Cosines THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 3

(a) (b) W 2 = X2 + Y 2 + Z2 − 2YZ cos DA W 2 = X2 + Y 2 + Z2 ∠ −2ZX cos ∠DB − 2XY cos ∠DC

2 2 W + X − 2WX cos ∠BC 2 2 = Y + Z − 2YZ 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 B.D.S. MCCONNELL

(a) Spherical Space (b) Hyperbolic Space

W X Y Z W X Y Z cos 2 = cos 2 cos 2 cos 2 cos 2 = cos 2 cos 2 cos 2 X Y Z X Y Z + sin 2 sin 2 sin 2 S − sin 2 sin 2 sin 2 S X Y Z X Y Z + cos 2 sin 2 sin 2 cos ∠DA + cos 2 sin 2 sin 2 cos ∠DA X Y Z X Y Z + sin 2 cos 2 sin 2 cos ∠DB + sin 2 cos 2 sin 2 cos ∠DB X Y Z X Y Z + sin 2 sin 2 cos 2 cos ∠DC + sin 2 sin 2 cos 2 cos ∠DC √ S := 1 − 2 cos ∠DA cos ∠DB cos ∠DC − cos2 ∠DA − cos2 ∠DB − cos2 ∠DC

W X W X W X W X cos 2 cos 2 − sin 2 sin 2 cos ∠BC cos 2 cos 2 + sin 2 sin 2 cos ∠BC Y Z Y Z Y Z Y Z = cos 2 cos 2 − sin 2 sin 2 cos ∠AD = cos 2 cos 2 + sin 2 sin 2 cos ∠DA

Figure 5. The Laws of Cosines for Non-Euclidean Tetrahedra THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 5

(a) Spherical Space (b) Hyperbolic Space

W X Y Z W X Y Z cos 2 = cos 2 cos 2 cos 2 cos 2 = cos 2 cos 2 cos 2 X Y Z X Y Z + sin 2 sin 2 sin 2 − sin 2 sin 2 sin 2

Figure 6. The Pythagorean Theorems for Right-Cornered Non-Euclidean Tetrahedra 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 byx ¨, 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 Sidesc ¨ =a ¨¨b ± abγ¨

c¨ − a¨¨b (solved for the angleγ ¨ = ± ) ab Area of Triangle ± (α + β + γ) ∓ π

x Also, we’ll reference half-angles (and half-sides) often; we’ll write x2 for 2 to save some typesetting space. THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 7

1.2. 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, W (± (α + β + γ) ∓ π) cos = cos 2 2 π = cos ± (α + β + γ ) ∓ 2 2 2 2 = sin (α2 + β2 + γ2)

= α2β¨2γ¨2 +α ¨2β2γ¨2 +α ¨2β¨2γ2 − α2β2γ2 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

2 1 γ¨2 = 2 (1 +γ ¨) = a2 + b2 + c2 a2 + b2 − c2 /ab = s (s − c) /ab 2 1 γ2 = ± 2 (1 − γ¨) = −a2 + b2 + c2 a2 − b2 + c2 /ab = (s − a) s − b /ab with “semi-perimeter” s := a2 + b2 + c2 = (a + b + c) /2.

The terms in the area formula then expand: r (s−b)(s−c) s(s−b) s(s−c) α β¨ γ¨ = · · = s s − b (s − c) /abc 2 2 2 bc ca ab

α¨2β2γ¨2 = = s (s − a)(s − c) /abc α¨2β¨2γ2 = = s (s − a) s − b /abc −α2β2γ2 = = − (s − a) s − b (s − c) /abc

Combining the terms, and canceling a denominator rewritten as 8a ¨2a2b¨2b2c¨2c2, the sum simpliﬁes to 2 W a¨ 2 + b¨ +c ¨ 2 − 1 1 +a ¨ + ¨b +c ¨ cos = 2 2 2 = 2 2a ¨2b¨2c¨2 4a ¨2b¨2c¨2 1.3. The Heron-like Formula in Tetrahedral Context. Taking our triangle, 4ABC, 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 a¨ =y ¨z¨ ± yzθ¨ ¨b =z ¨x¨ ± zxφ¨ c¨ =x ¨y¨ ± xyψ¨ 8 B.D.S. MCCONNELL

The Heron-like formula for W then becomes ¨ ¨ ¨ W 1 + y¨z¨ ± yzθ + z¨x¨ ± zxφ + x¨y¨ ± xyψ cos = 2 4a ¨2b¨2c¨2 1 +y ¨z¨ +z ¨x¨ +x ¨y¨ ± yzθ¨ ± zxφ¨ ± xyψ¨ = 4a ¨2b¨2c¨2 or in terms of half-sides W x¨ 2y¨ 2z¨ 2 ± x 2y 2z 2 ± y y¨ z z¨ θ¨ ± z z¨ x x φ¨ ± x x¨ y y ψ¨ cos = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a¨2b¨2c¨2

2. Verifying the Laws 2.1. 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 W cos = cos X cos Y cos Z ± sin X sin Y sin Z S 2 2 2 2 2 2 2 X Y Z + cos 2 sin 2 sin 2 cos ∠DA X Y Z + sin 2 cos 2 sin 2 cos ∠DB X Y Z + sin 2 sin 2 cos 2 cos ∠DC reduces to the formula at the end of the preceding subsection, thereby proving this Law. Expanding the bits of the expression gives X 1 +y ¨ +z ¨ +a ¨ 1 +y ¨ +z ¨ +y ¨z¨ ± yzθ¨ cos = = 2 4y ¨2z¨2a¨2 4y ¨2z¨2a¨2 (1 +y ¨) (1 +z ¨) ± yzθ¨ 4y ¨ 2z¨ 2 ± 4y y¨ z z¨ θ¨ = = 2 2 2 2 2 2 4y ¨2z¨2a¨2 4y ¨2z¨2a¨2 y¨ z¨ ± y z θ¨ = 2 2 2 2 a¨2 Y z¨ x¨ ± z x φ¨ cos = 2 2 2 2 2 b¨2 Z x¨ y¨ ± x y ψ¨ cos = 2 2 2 2 2 c¨2

and also X y z θ Y x z φ Z y z ψ sin = 2 2 sin = 2 2 sin = 2 2 2 a¨2 2 b¨2 2 c¨2

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, THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 9

θ¨ − φ¨ψ¨ φ¨ − θ¨ψ¨ ψ¨ − θ¨φ¨ cos ∠DA = cos ∠DB = cos ∠DC = φψ θψ θφ and s 1 − 2 cos ∠DA cos ∠DB cos ∠DC S = 2 2 2 − cos ∠DA − cos ∠DB − cos ∠DC 1 + 2θ¨φ¨ψ¨ − θ¨2 − φ¨2 − φ¨2 = θφψ The terms of the proposed First Law of Cosines expansion are as follows: ¨ ¨ ¨ X Y Z y¨2z¨2 ± y2z2θ z¨2x¨2 ± z2x2φ x¨2y¨2 ± x2y2ψ cos cos cos = 2 2 2 a¨2b¨2c¨2  2 2 2 2 2 2  x¨2 y¨2 z¨2 ± x2 y2 z2 θ¨φ¨ψ¨  2 ¨ 2 ¨ ¨   +y ¨2y2z¨2z2 ±x¨2 θ + x2 φψ     2 ¨ 2 ¨¨   +x ¨2x2z¨2z2 ±y¨2 φ + y2 θψ   2 2  +x ¨2x2y2y¨2 ±z¨2 ψ¨ + z2 θ¨φ¨ = a¨2b¨2c¨2

X Y Z y z θ x z φ x y ψ 1 + 2θ¨φ¨ψ¨ − θ¨2 − φ¨2 − ψ¨2 ± sin sin sin S = ± 2 2 · 2 2 · 2 2 · 2 2 2 a¨2 b¨2 c¨2 θφψ 2 2 2 2 2 2 ±x2 y2 z2 1 + 2θ¨φ¨ψ¨ − θ¨ − φ¨ − ψ¨ = a¨2b¨2c¨2

X Y Z y¨2z¨2 ± y2z2θ¨ x2z2φ x2y2ψ θ¨ − φ¨ψ¨ cos sin sin cos ∠DA = · · · 2 2 2 a¨2 b¨2 c¨2 φψ 2 x2 y2z2 y¨2z¨2 ± y2z2θ¨ θ¨ − φ¨ψ¨ = a¨2b¨2c¨2 2 2 2 2 2 2 y¨2y2z¨2z2 x2 θ¨ − x2 φ¨ψ¨ x2 y2 z2 θ¨ − θ¨φ¨ψ¨ = ± a¨2b¨2c¨2 a¨2b¨2c¨2 2 ¨ 2 ¨¨ 2 2 2 ¨2 ¨¨ ¨ X Y Z x¨2x2z¨2z2 y2 φ − y2 θψ x2 y2 z2 φ − θφψ sin cos sin cos ∠DB = ± 2 2 2 a¨2b¨2c¨2 a¨2b¨2c¨2 2 ¨ 2 ¨¨ 2 2 2 ¨2 ¨¨ ¨ X Y Z x¨2x2y¨2y2 z2 ψ − z2 θφ x2 y2 z2 ψ − θφψ sin sin cos cos ∠DC = ± 2 2 2 a¨2b¨2c¨2 a¨2b¨2c¨2 10 B.D.S. MCCONNELL

Therefore, combining the terms and invoking the trigonometric identity

x¨ ± x2 = cos2 x + sin2 x = cosh2 x − sinh2 x ≡ 1 gives

 2 2 2 2 2 2  x¨2 y¨2 z¨2 ± x2 y2 z2 ¨ 2 2  ±y¨2y2z¨2z2θ x¨2 ± x2  ¨ Total =  ¨ 2 2  /a¨2b2c¨2  ±x¨2x2z¨2z2φ y¨2 ± y2  2 2 ±x¨2x2y¨2y2ψ¨ z¨2 ± z2 2 2 2 2 2 2 x¨2 y¨2 z¨2 ± x2 y2 z2 = /a¨2b¨2c¨2 ±y¨2y2z¨2z2θ¨ ± x¨2x2z¨2z2φ¨ ± x¨2x2y¨2y2ψ¨ W = cos 2 as desired.

2.2. The Second Law of Cosines. Now we will verify the Second Law of Cosines for Non- Euclidean Tetrahedra W X W X Y Z Y Z cos cos ± sin sin cos BC = cos cos ± sin sin cos DA 2 2 2 2 ∠ 2 2 2 2 ∠ by exposing the symmetric nature of one side of the equation. Before doing so, we observe an immediate consequence of the relation.

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

Y Z Y Z 1 +x ¨ +z ¨ + ¨b 1 +x ¨ +y ¨ +c ¨ xzφ xyψ θ − φψ cos cos + sin cos cos ∠DA = + 2 2 2 2 4x ¨2z¨2b¨2 4x ¨2y¨2c¨2 4x ¨2z¨2b¨2 4x ¨2y¨2c¨2 φψ (1 +x ¨ +z ¨ + ¨b)(1 +x ¨ +y ¨ +c ¨) + x2yz(θ¨ − φ¨ψ¨) = 2 16x ¨2 y¨2z¨2b¨2c¨2 Now, using the (triangular) Law of Cosines in appropriate faces, we eliminate references to θ, φ, and ψ.

x2yz(θ¨ − φ¨ψ¨) = x2(yzθ¨) − (xzφ¨)(xyψ¨) = ±x2(ä − y¨z¨) − (¨b − x¨z¨)(¨c − xÿ¨) = (1 − x¨2)(ä − y¨z¨) − (¨b − x¨z¨)(¨c − xÿ¨) THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 11 so that (1 +x ¨ +z ¨ + ¨b)(1 +x ¨ +y ¨ +c ¨) + x2yz(θ¨ − φ¨ψ¨) = (1 +x ¨)2 + (1 +x ¨)(¨b +c ¨ +y ¨ +z ¨) +y ¨z¨ +c ¨z¨ + ¨by¨ + ¨bc¨ +(1 − x¨2)(ä − y¨z¨) − ¨bc¨ + ¨bxÿ¨ +c ¨x¨z¨ − x¨2y¨z¨ = (1 +x ¨)(1 + ¨b +c ¨ +x ¨ +y ¨ +z ¨ + ¨by¨ +c ¨z¨ + (1 − x¨)(ä − y¨z¨ +y ¨z¨)) 2 = 2x ¨2 (1 +a ¨ + ¨b +c ¨ +x ¨ +y ¨ +z ¨ − a¨x¨ + ¨by¨ +c ¨z¨) Therefore, Y Z Y Z 1 +a ¨ + ¨b +c ¨ +x ¨ +y ¨ +z ¨ − a¨x¨ + ¨by¨ +c ¨z¨ cos cos + sin sin cos ∠DA = 2 2 2 2 8y ¨2z¨2b¨2c¨2 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 (= 4xbz) ↔ X(= 4ayz) W (= 4abc) ↔ Z(= 4xyc) ∠DA ↔ ∠BC This proves the result.

2.3. Pseudo-Faces. As in the Euclidean case, the Second Law of Cosines invites the formal deﬁ- nition of “pseudo-faces” with areas H, J, and K satisfying the relations W X W X H Y Z Y Z cos cos + sin sin cos BC = cos = cos cos + sin sin cos DA 2 2 2 2 ∠ 2 2 2 2 2 ∠ W Y W Y J Z X Z X cos cos + sin sin cos CA = cos = cos cos + sin sin cos DB 2 2 2 2 ∠ 2 2 2 2 2 ∠ W Z W Z K X Y X Y cos cos + sin sin cos AB = cos = cos cos + sin sin cos DC 2 2 2 2 ∠ 2 2 2 2 2 ∠ 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 orienta- tions): 2 2 2 2 0 = 1 − W¨2 − X¨2 − Y¨2 − Z¨2 − 4W¨2X¨2Y¨2Z¨2 2 2 2 −H¨2 − J¨2 − K¨2 − 2H¨2J¨2K¨2

+2H¨2(W¨2X¨2 + Y¨2Z¨2) + 2J¨2(W¨2Y¨2 + Z¨2X¨2) + 2K¨2(W¨2Z¨2 + X¨2Y¨2) We arrive at this formula by isolating S in the W -centric instance of the First Law of Cosines. ¨ ¨ ¨ ¨ ¨ ¨ ¨ X2Y2Z2S = −W2 + X2Y2Z2 + X2Y2Z2 cos ∠DA + X2Y2Z2 cos ∠DB + X2Y2Z2 cos ∠DC 12 B.D.S. MCCONNELL

After squaring both sides of the equation, we use the Second Law of Cosines to express cos ∠DA, 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 26, 244— arise from such substitutions.2

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

W 2 X2 Y 2 Z2 X Y Z 1 − ∼ 1 − 1 − 1 − ± S 8 8 8 8 2 2 2 X2 Y Z + 1 − cos DA 8 2 2 ∠ X Y 2 Z + 1 − cos DB 2 8 2 ∠ X Y Z2 + 1 − cos DC 2 2 8 ∠ X2 Y 2 Z2 ∼ 1 − − − ± (0) S 8 8 8 YZ XZ XY + cos DA + cos DB + cos DC 4 ∠ 4 ∠ 4 ∠

whence

2 2 2 2 W ∼ X + Y + Z − 2YZ cos ∠DA − 2XZ cos ∠DB − 2XY cos ∠DC

Likewise, for the Second Law,

2 ¨ 2 Arithmetic check: There are 3 diﬀerent ways to substitute into H2 (using ∠BC in both factors, using ∠DA in 2 2 both factors, or using each of these angles once); likewise, there are 3 ways each to substitute into J¨2 , K¨2 , 2H¨2, 6 2J¨2, 2K¨2. There are 36 ways to substitute into 2H¨2J¨2K¨2. Thus, the total number of formulas is 3 · 36 = 26, 244. THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA 13

W 2 X2 W X Y 2 Z2 Y Z 1 − 1 − + cos BC ∼ 1 − 1 − + cos DA 8 8 2 2 ∠ 8 8 2 2 ∠ W 2 X2 WX Y 2 Z2 YZ 1 − − + cos BC ∼ 1 − − + cos DA 8 8 4 ∠ 8 8 4 ∠

2 2 2 2 W + X − 2WX cos ∠BC ∼ Y + Z − 2YZ cos ∠DA

B.D.S. McConnell [email protected]