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On Relating Computation On relating computation Fran¸coise Chatelin ∗ CERFACS Technical Report TR/PA/11/37 Abstract : The report shows how the seminal ideas of Fourier, Poincar´e and Einstein blend together harmoniously to explain many features of computation in Nature and in the human mind. 1)The Fourier analysis of complex signals leads to the Fourier transform whose complex spectrum lies on the unit circle, with 4 eigenvalues +1, 1, i, i which are the 4 units of the Gaussian ring of complex integers. − − 2)The Poincar´eapproach to relativity bears on the Lorentz transformations in the field H of quaternions, using a noncommutative . Relative significant computation evolves from H to G2, where G is the alternative× algebra of octonions. 3)The Einstein perspective on relativity is based on a noncommutative + , yield- ing a geometric 2-fold information potential with 2 types of geodesics. The potential◦ lies in a R3-framed metric cloth, a non-euclidean space which is a computational construct exhibiting some features attributed to axiomatically defined geometries, either hyperbolic or elliptic. By mixing these 3 views with the quadratic logistic iteration and hypercompu- tation, we uncover an algorithmic mechanism which underlies the law of organic causality that we experience in Nature and mind. Keywords : Einstein, Poincar´e, Fourier, noncommutative +, nonassociative , quaternions, octonions, sedenions, relativity, causal space, logistic iteration. × ∗Ceremath, Universit´eToulouse 1 and CERFACS,42 avenue G. Coriolis 31057 Toulouse Cedex 1, France. E-mail: [email protected] 1 1 Introduction 1.1 Presentation In the report TR/PA/11/27 we studied the computational principle of relativity from the point of view of a noncommutative addition + creating a metric cloth ◦ framed into a normed linear vector space. In this report we investigate the alterna- tive point of view on relativity stemming from a noncommutative multiplication which defines recursively the quadratic Dickson algebras A of dimension 2k, k ×2 k ≥ (Chatelin 2011a, Chapter 2). The two approaches to relativity have emerged con- currently in 1905, presented respectively by Einstein and Poincar´e. In (Chatelin 2011b) we described the key role of the nonstandard Einstein addition + for weav- ◦ E ing information processing (WIP) by geometric means in the metric cloth WE. Below we present the equally important role of the multiplication when it × becomes noncommutative over the quaternions in A2 = H and nonassociative over the octonions in A3 = G, the two largest real quadratic division algebras. The rel- ativistic computational evolution in H and G is sustained by the dynamics of the logistic iteration (Chapter 6). By means of the organic logic derived from cloth ge- ometry, it wraps up the perspectives of Fourier, Poincar´eand Einstein on Relativity in computation into a holistic context whose number of dimensions can vary from 1 to 14. 1.2 A review of Dickson algebras The reader is referred to Chapter 2 in (Chatelin 2011a). The Dickson doubling process in Ak = Ak 1 Ak 1, A0 = R defines recursively two operations: − − conjugation: (x, y×)=(¯x, y), • − multiplication: • (x, y) (x′,y′)=(x x′ y¯ y),y′ x + y x¯ ). × × − ′ × × × ′ For k 1, x A satisfies the quadratic relation x2 = xx¯ +(x +x ¯), and A ≥ ∈ k − k represents a complexified version of Ak 1 : Ak = Ak 1 Ak 1 1˜k where 1˜k = − − ⊕ − × (0, 1k 1) Ak 1 is the complex unit in Ak, k 1. − ∈ − ≥ C C For k = 1, i = (0, 1) is the complex unit of . For k 2, the 2D algebra 1˜ ˜ ˜ C ≥ C spanned by 1 = 1k and 1= 1k is isomorphic to . We write Ak = Ak 1 ⋆ 1˜ , k 2. − k ≥ For k = 2, 1˜2 = (00, 10) = e2 = j in H = C⋆Cj, and for k = 3, 1˜3 = (0000, 1000) = e4 G H C in = ⋆ e4 . 2 We recall that all Ak, k 4, have zerodivisors and an algebraic depth da(k) = k 3 ≥ 2 − 2 (Chapter 3, Chatelin 2011a). For k 3, the four algebras admit division ≥ ≤ and euclidean rings of integers (Chapter 9). Moreover, the algebraic depth of Ak is 1, 2 for k = 2, and 1 for k =0, 2, 3. { } Unless otherwise stated, all references are to the book Qualitative Computing (Chatelin 2011a). 2 The hypercomputation approach to relativity 2.1 Poincar´eon relativity Let q = α + X represent a quaternion in H with real part q = α R and ℜ ∈ imaginary part q = X H = R3. One computes q2 = α2 X 2 +2αX with ℑ ∈ ℑ ∼ − q2 = α2 X 2 and q2 =2αX colinear with q. ℜ According− to (Poincar´e1905)ℑ the group of Lorentzℑ transformations leaves q2 = ℜ f R invariant when q H. This is the aspect of the mathematical content of∈ the relativity principle which∈ is related to multiplication rather than addition. Phrased into physical terms, this principle explains why it is impossible to exhibit experimentally an absolute movement of the earth by mechanical or electromagnetic means. 2.2 Significant evolutions in relativity We observe that the Lorentz invariance can be interpreted as ( q)2 = X 2 + f, which fixes q2 at f R. Among the possible evolutions of q satisfyingℜ the relativity condition weℜ choose∈ specific ones by enforcing a complementary relation between ( q)2 and q2 2 according to the ℜ ℑ Definition 2.1 The significant relative evolutions of q = α + X in H, X 0 ≥ are such that f = q2 is a real constant and at least one of the two conditions (i) q = q2 , (ii) ℜ( q)2 = q2 2 is satisfied. |ℜ | ℑ ℜ ℑ The real parameter α in q will be allowed to become octonionic (α2 < 0). We first assume that X 2 > 0 (X H). ∈ℑ 3 2.3 The linear constraint (i) q = q2 |ℜ | ℑ R 1 The condition (i) tells us that α = 2 α X for any α , hence X = 2 for α = 0. | | | | ∈ Proposition 2.1 Under the linear constraint (i), the evolution of q = α + X takes place in H with (α + X, α + X) H2 iff f 1/4. When f < 1/4, α = β1˜ with β = α , β2 = α2 >−0 and the∈ evolution≥ of − g = X + β1˜ takes− place in G with ± − (X + α 1˜,X α 1)˜ in G2. − Proof. For X = 1/2, α2 = f +1/4 0 iff f 1/4. For f < 1/4, 2 2 ˜ ≥G ≥ − α α − α = α < 0, we set the complex unit 1 in to be either α or α , that is − − α = β1˜ with β = α . The number β1+˜ X is an octonion in G = H H 1˜ which can be written g ±=(X, α ). ⊕ × ± ± 2 We observe for future reference that, for f = 0, x = α equals 1 + 1/4f, f = 0. f 1 G R Proposition 2.2 For f 4 , g = (q, q¯) satisfies g = √4f +1 . For 1 ≥ − 2− ∈ 4 2 1± ∈ f < , g′ =(X, α ) G satisfies g′ = f and g′ = f > . − 4 ∈ℑ 16 1 2 1 Proof. 1) f 4 , q = α + X with α = fx = f + 4 0, 2α = ε√4f + 1, 2 ≥ −1 1 2 2 2 ≥ ε = 1, q = f + 2 4 , g =(q + q , q¯(q +¯q)) = ±2f + 1 + ε√4f +1≥ X , ε√4f +1¯ q =− ε√4f + 1(q, q¯). 2 − − Because G is a division algebra we conclude that g = ε√4f + 1, together with g 2 =2 q 2 =2f + 1. 1 2 2 G 2) f < 4 , α = α , α + X belongs to in the form g′ = (X, α ) such 2 − 2 − 2 ℑ that g′ = X + α = f > 0. − We have learnt in Chapter 3 that computation in G can be paradoxical by lack of associativity (Section 3.9). Proposition 2.2 provides a striking example of such 1 2 G paradoxes. For f 4 the squared norm g is computed in with two values 1 and 4f + 1 ≥0 which− differ for f = 0 (i.e. x finite). The transformation ≥ T : q H g = T (q)=(q, q¯) G triggers a remarkable reduction: the octonion∈g →G is converted into− the real∈ numbers g = √4f +1 = 2 q with g = g =∈ √2f + 1 unless f = 0. The reduction in dimensions± from± 8ℜ to 1 is | | metamorphosed into a metric expansion (f > 0) or contraction ( 1 f < 0). The 2 2 4 2 g g 2f − ≤ 1 | | − rate of change for g is ρ(f) = g 2 = 2f+1 , 1 ρ < 1 for f 4 . If − ≤ ≥ − 1 1 g f = , g = (q, q¯) = and g is computed at 0 when f , | | √2. 4 √2 g − − | | →∞ → 4 1 Proposition 2.3 For f = , the sedenion s =(g+,g ) A4 is a zerodivisor with 2 − norm 1. − ∈ 1 1 2 Proof. For f = 2 , X = α = f = x = 2 , hence < g+,g >= X 2 − − 1 − − α = 0. Together with g+ = g (= ), this is a necessary and sufficient − √2 condition for s =(g+,g ) to be a zerodivisor in Zer (A4) = Aut (G) (Chapter 4). − ∼ 3 The quadratic constraint (ii) ( q)2 = q2 2 ℜ ℑ We first assume that X H. ∈ℑ 3.1 Exact solutions We write (ii) α2 =4α2 X 2 =4α2(α2 f) under the form − x = 4fx(1 x) (3.1) 2 − − α 2 where x = f is the ratio of the real numbers α and f. This is the quadratic fixed-point equation known as the logistic (Section 6.8 in Chapter 6). The equation (3.1) has the two solutions a) f x = 0, b) 0 =→f x =1+ 1 , → 4f the second solution is the hyperbola solution of (i) on which X = 1 . 2 Proposition 3.1 a) The evolution of q = X remains in H for f 0 only under the condition X 2 0.
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