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, Poincaré 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 Poincaréapproach 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 + , yielding 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 hypercomputation, 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 alternative 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 Poincaré. 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 relativistic 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 geometry, it wraps up the perspectives of Fourier, Poincaréand 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. ℑ ≤ ≥ b) The evolution of q = α + X for x = 1+ 1 , f = 0 is in H (resp. G2) for 4f f 1 , (resp. f < 1 ). ≥ − 4 − 4 Proof. a) x = α = 0, q = X and q2 = X 2 = f. For f 0, X 2 0; for − ≤ ≥ f > 0, X < 0 is impossible by assumption. 2 b) x = 1+ 1 , f = 0 is such that 0 < x = α < 1 for f < 1 . Necessarily 4f f − 4 f < α2 < 0 and α = β1˜ in G with β = α , X 2 > 0 hence g = X + β1=(˜ X, β). 1 1 ± 2 For f 4 , x 0 for 4 f < 0 and x> 1 for f > 0. In both cases, α > 0 implies α ≥R − and ≥X 2 =−α2 ≤f > 0. ∈ −

5 We observe on Figure 3.1 that the situations (i) and (ii) agree on the hyperbola ( X = 1 ). 2

1 1

1/4 0 1/4 0 −

X = 1 (i) X = 1 (ii) 2 2 α =0 G2 H2 H

Figure 3.1: Signiﬁcant evolution in H, H2 or G2 in the plane (f, x) R2, X H ∈ ∈ℑ

3.2 Approximation by successive iteration We suppose that the resolution of (3.1) is realised by the iteration : 1 x0 = , xn = 4fxn 1(1 xn 1), n 1. (3.2) 2 − − − − ≥ 1 When n , the iterates xn do not escape to for f in [ 1, 2 ]. The inter- pretative context→∞ stemming from (f, x ) is ruled by the±∞ 5 values f − 1, 1 , 0, c, 1 n ∈ {− − 4 2 } where c is the value for the confluence of the iterates on the exact solution x = 0. 2 A According to (Nagashima and Baba 1999, p. 135), 3c = A + 2 1 where A = 1/3 − 19+3√33 , c 0.419. ≈ Lemma 3.2 c satisfies the relation 4c2(1 + c)=1. Proof. Simple check (computer algebra). The importance of the confluence parameter value c is that no odd period exists for f c. They appear in decreasing order for c < f 1 1 0.457. ≤ ≤ √2 − 4 ≃

Corollary 3.3 For f c (resp. f

6 Proof. Set Lf (x)= 4fx(1 x). In the intervals where exact convergence does − − 1 not take place, the iterates xn are conﬁned by the two waveguides Lf ( 2 )= f and 2 1 2 2 − Lf ( 2 )= Lf ( f)=4f (1+f). Because 4c (1+c) = 1, it follows readily that xn > 1 is possible for− f>c. For f c, c x 1. ≤ − ≤ n ≤ Theorem 3.4 The context for the quaternion α + X, X H, stemming from the ∈ℑ computation of xn at f by (3.2) varies generically according to the table

f 1 1 0 c 1 − − 4 2 @ H2 for x > 1 algebra G2 H @ n ℑ @ impossible for xn < 1

α α 1˜ 0 α real for x 0 ± n ≥

Proof. For f = 0, α2 = xf and X 2 = f(x 1) > 0 X H. 1) f [ 1, 1 [= 0 < x 1 = f α−2 < 0. The⇐⇒ context∈ℑ is G2, but for ∈ − − 4 ⇒ n ≤ ⇒ ≤ f = 1, x = 0 deﬁnes q = X H, X = 1. 2)−f [ 1 , 0] = x = α =0=∈ℑ q2 = X 2 = f < 0. The context is H. ∈ − 4 ⇒ ⇒ − ℑ 3)f ]0, 1 ]= x = α =0= X 2 = f < 0: this is impossible. 4 2 ∈ 1 ⇒ α ⇒ 2 − 2 4)f ] 4 ,c[= xn = f < 1= X = α f < 0: impossible. ∈ 1 ⇒ ⇒ − 2 5)f [c, 2 ]= xn > 1 is possible by Corollary 3.3, hence X > 0. Therefore q H whenever∈ ⇒x > 1. In particular, if x = 1, α2 = f and q= √f is real. ∈ n n ±

3.3 X2 > 0: X is real We relax the condition X 2 > 0 which defines q as a quaternion. The case X 2 < 0 which makes no sense for X R3 can be reinterpreted as X2 > 0, henceX R : ∈ ∈ X2 = f α2 > 0 yields X , two real numbers in R, and the nature of the numbers α X −depends on α2±|(α2|< 0 is possible for x < 0). When both X and α are real,±| one| may consider that (i) they represent two different variables or (ii) they are different instances of the same unique variable. We complement Theorem 3.4 by the

1 Theorem 3.5 When X is real for 0 f 2 the evolution of α + X can take place in (i) R2, (ii) R or C. It is described≤ generically≤ by the table

7 f 0 1/4 1/2 α = 0 real if 0 x < 1 ⇒ • ≤ n algebraic 0 X 1/2 = R2 ≤|2 | ≤ ⇒ context = R (i) or R(ii) α i if xn < 0 ⇒ • ±| |= C ⇒

Proof. For 0 α2 > 0, X2 = f α2 > 0 yield four points ε X + ε α when (i) X = α at the vertices of a rectangle− in R2 (Figure 3.2.a) { 1| | 2| |} | | | | below). X2 = α2 = f α2 implies α2 = f = xf x = 1 . When X = α = f − 2 ⇐⇒ 2 | | | | q 2 (that is x = 1 ) the four points reduce in case (ii) to three, namely √2f, 0 on 2 {± } R √5 1 1 √5 1 (Figure 3.2.b)). This is possible iﬀ 4− f 2 , 4− 0.3090. When x < 0, 2 ≤ ≤ ≃ α < 0 and α = ε2 α i yielding ε1 X + iε2 α in C, whether X = α or not (see Figure 3.2.c)). | | { | | | |} | | | | If x = 0, α = 0, f = X2 = X = √f for f > 0. If x = 1, X = 0, ⇒ ± f = α2 = α = √f for c f 1 . ⇒ ± ≤ ≤ 2

α i α | | | |

X X | | √2f 0 √2f | | −

f (X, α) R2 X = α = X + iα C ∈ | | | | q 2 ∈ x ]0, 1[ x =1/2 1/2 x< 0 ∈a) b) − ≤c)

Figure 3.2: Reduction when X is real, 0 = x< 1

3.4 The alternative viewpoint of geometric algebra It is instructive to compare the interpretations of f = α2 X 2 either as q2 with − ℜ q = α + X in H (see above) or as a quadratic form in 4 real or complex variables. This latter approach, classically known as geometric algebra, is based on Cliﬀord

8 algebras related to the signature (1, 3) of the quadratic form. Such algebras are associative and play an important role in theoretical physics (Baez 2001). They provide an alternative perspective (in mathematics and physics) which diﬀers for 1 G2 f < 4 from the computational perspective given in Theorem 3.4, because is not associative.− We add that the interpretation shift from X H ( X 2 > 0,x> 1) to X R (X2 > 0,x< 1) which occurs for f in ]c, 1/2] and∈ℑ induces a reduction in the number∈ of dimensions is not a matter of consideration in the current presentation of Special Relativity.

4 The relative nature of the evolution of q = α+X sustained by (3.2)

4.1 The critical value x =1/2 When f < 0, the critical value 1/2 can occur for 1 f 1/2. − ≤ ≤ −

f 1 Lemma 4.1 =0 X = α = x = . − ⇐⇒ q− 2 ⇐⇒ 2 2 2 2 2 Proof. = X α =0 X = α . Thus X = α f − = α2 f =2α2 =2 −xf x =1⇐⇒/2 for all f = 0. − − ⇐⇒ ⇐⇒ Given q = α + X, we denote g =(X, α ) in G. ± ±

Corollary 4.2 The sedenion s = (g+,g ) in A4 is a zerodivisor for f 1/2 − ≤ − f 1 1 whenever X = α = − , X √ . q 2 2 ≤ ≤ 2

f Proof. x =1/2 X = α = −2 by Lemma 4.1. Then s has alternative ⇐⇒ q 2 2 orthogonal parts g+ and g in G with equal norm: X + α = f, hence − s 2 = 2f. − − Corollary 4.2 unravels an unsuspected link between the critical value x = 1/2 s and the construction of sedenions s in A4 such that s′ = √ 2f , s′ = 1 belong to − the zeroset Zer (A4) which is classically homeomorphic to Aut (G) (Theorem 4.7.1). This computational result is beyond the current analysis of relativistic evolution which is set in associative Cliﬀord algebras. We believe that it plays the role of a meta-rule in the ubiquitous “self-organised” phenomena found in experimental

9 sciences. The conventional disregard for ﬂexible (nonassociative) quadratic Dickson algebras may explain why the phenomenon has resisted mathematical analysis to the present day (Atlan 2011).

√5 1 1 1 f Lemma 4.3 For − f , x = X = α = . 4 ≤ ≤ 2 2 ⇐⇒ | | | | q 2 Proof. x = 1 α2 = f = X2 X = α = f with α and X real. See 2 ⇐⇒ 2 ⇐⇒ | | | | 2 Theorem 3.5 and Figure 3.2.a) (i) and b) (ii). q

4.2 A review of the three exceptional values f 1, 1/2, 1/2 in (3.2) ∈ {− − } At the endpoints f = 1/2 and f = 1, the dynamics of (3.2) is complex, ruled by z3 = 1 and z6 = 1 respectively, and− it is real for f = 1/2. At these three values − − (3.1) can be transformed into the equation t(u)= εt(2u), with ε = 1, where t is a real or complex exponential ±

f 1 1 1 − − 2 2 t sin2 exp cos ε +1 +1 1 −

Moreover, the iteration (3.2) can be solved in closed form as a function of n and x [0, 1]. The formulae are listed below. 0 ∈

f iterations 2 π 1 2x = cos2ψ, x = sin ψ, ψ [0, 2 ] − n 1 ∈ 1 1 2xn = cos(2 cos− (1 2x0)), n 0 − − n − 1 ≥ 2 or ψn =2 ψ0, ψ0 = sin− √x0, xn = sin ψn. 1 π 2 x = cos θ = cos( 3 ξ), θ [0, π] 1 − π −n π ∈ 1 1/2 2 xn = cos 3 ( 2) 3 cos− (1/2 x0) , n 0 − n − − π − − ≥ or ξn =( 2) ξ0, ξ0 = θ0 − 3 − 1 2x = ey, y< 0 | − | 2n 1/2 1 2xn = (1 2x0) , x0 ]0, 1[, x0 =1/2, n 0 − or− y =2ny−, y = ln 1 ∈2x < 0 ≥ n 0 0 | − 0| 10 For f = 1, we use the deﬁnition of the Tchebychef polynomial of degree k 1: − ≥ T (d) = cos(kArcos d), d 1 to write, with d = 1 2x, d = T n (d ), n 0. We k | | ≤ − n 2 0 ≥ recall that the quantity min(max t 1 p(t) , p is a monic polynomial of degree k) is k 1 | |≤ | | achieved for p(t)= Tk(t)/2 − . With obvious changes of variable the dynamics of (3.2) takes the three remark- ably simple forms for n 0: ≥

f iteration

1 ψ =2ψ mod π − n+1 n 2 1 ξ = 2ξ mod π 2 n+1 − n 1 y =2y in R− − 2 n+1 n Some remarks are in order. For f = 1, the angle ψ refers to the real axis R; π − 1 π it is uniformly distributed on [0, 2 ]. For f = 2 , the angle ξ = 3 θ refers to the rotated axis eiπ/3R in C; ξ and θ run in opposite directions. − We consider the change of notation r = 4f, δ = 1 x = d . Then the iteration − 2 − 2 (3.2) takes for r 4, 2 one of the three recursive forms listed in Table 4.1. In ∈ { ± } 1 each case h is a continuous bijective function and h− is the inverse function.

r = 4f recursion, n 0 h − ≥ d = h r h 1(d ) h(2ψ) = cos2ψ = d 4 n+1 2 − n d 1 0 x 1 2ψ [0, π] | | ≤ ⇐⇒ ≤1 ≤ ∈ y d = h (rh− ( d ) h(y)= e = d 2 | n+1| | n| | | d < 1 0

1 Table 4.1.: The iteration (3.2) on h− ( ) where the variable is either d, d or δ | | Remark 4.2.1 The transcendental context (a real or complex exponential function) reduces to an algebraic one (a polynomial of degree 2) at the exceptional values.

11 This essential property has been observed for long (Schr¨oder 1870) as a curiosity not worthy of serious computational consideration (Chapter 6).

1 4.3 At the end point f = 2 x = 2x(1 x) cos θ = cos2θ with x = 1 cos θ. The point (f, x)=( 1 , 3 ) − − ⇐⇒ − 2 − 2 2 denoted P is the only point on the hyperbola in the context H2 which is reachable by iteration.

Proposition 4.4 P corresponds to θ = π and q = ε √3 + X, ε = 1, X = 1 , 2 ± 2 satisfy q6 = 1, q12 =1 in H. − 3 1 Proof. The exact solutions 0 and 2 for f = 2 correspond, by the change of variable, to cos θ = 1 and cos θ = 1 respectively. Hence θ = π for x = 3 . The 2 − 2 complex dynamics of (3.2) corresponds to z3 = 1. − Now X 2 = 1 = α2 1 , α = √3 , and q = ε √3 + X, q = 1, q2 = 1 + ε√3X, 4 − 2 ± 2 2 2 q3 =2X, q6 =4X2 = 1. − √3 1 The quaternions q = 2 + X, X = 2 are two instances of the 12th roots of √3 ± 1 such that q = 2 . The complete set contains in addition the unit quaternions with the real|ℜ part| q 1, 1 , 0 . ℜ ∈ {± ± 2 } Since q6 = 1 and z3 = 1 , q2 = 1 + ε √3 X, X = 1 in H can be identified − 2 − 1 2 2 1 √3 C H with z = 2 i 2 in : the imaginary unit sphere X , X =1 is reduced to the two complex± units i on iR in the process.{ ∈ℑ } The remarkable structural{± } simplification which takes place by computation at P may be partially explained by the fact that P on the upper branch of the hyperbola (f > 0 and x > 1) satisfies the two constraints (i) linear: q = q2 and (ii) quadratic: q 2 = q2 2, defined in Section 2 above. These|ℜ constraints| ℑ enforce a |ℜ | ℑ H C √3 2 1 reduction in complexity from to when α = q = 2 and f = q = z = 2 . This is another example of simplexity. ℜ ± ℜ ℜ

For q = α + X H, αX = 0, c < f 1 , x> 1, q 2 = α2 + X 2 = f(2x 1) ∈ ≤ 2 − with 1 < x 4f 2(1 + f) hence f < q 2 f(8f 2(1 + f) 1). We go back≤ to the map T : q H ≤g = T (q)=(q, −q¯) G. ∈ → − ∈ Lemma 4.5 g2 =2αg G g =2α R. ∈ ⇐⇒ ∈

12 Proof. g2 = (q2 + q 2, q¯(q +¯q)) = (2α2 +2αX, 2αq¯)=2α(q, q¯)=2αg. − − − Because G is a division algebra g2 = 2αg g = 2α, with √c < α √3 and ⇐⇒ | | ≤ 2 g 2 =4α2 =2 q 2 +2f. Lemma 4.5 shows that the domain of validity f 1/4 for Proposition 2.2 and 1 ≥ − (3.1) becomes c < f 2 for the iteration (3.2). The transformation T creates a similar reduction in≤ dimensions from 8 to 1 together with a metric expansion: g 2 = 2 q 2 = 2f(2x 1) grows by 2f 1, with growth ratioρ ˜(x) = 1 , − ≤ 2x 1 1 ρ˜ < 1. The minimum ratioρ ˜( 3 ) = ρ( 1 ) = 1 is achieved at P = ( 1 , 3 ) where− 2 ≤ 2 2 2 2 2 g 2 increases from 2 to 3. We encounter again the computational tension between 2 and 3 which underlies the logistic iteration (Chapter 10).

1 1 For f = 2 , the change of variable x = 2 cos θ entails a threefold real representation for cos θ, θ [0, π] given by the − ∈ 1 1 Proposition 4.6 At f = 2 , cos θ = 2 x yields three possible quadratic forms: 2 2 1 − (i) X α 0 < cos θ 2 0 1, X = 2 (1 x) for x < 1. For x = 1 , the form equals 0(X2= α2 − cos θ = 0). − 2 ⇐⇒

Example 4.1 For f = 1 , we can check easily the table below where r R, z C, q H: 2 ∈ ∈ ∈

x 1 0 1 1 3 − 2 2 2 π π 2π θ 0 3 2 3 π

r – 1 1, 0 1 – ± √2 ± ± √2 1 √ z 2 (ε1 3+ iε2)––––

q –––– √3 + X, X = 1 . ± 2 2

2 1 2 2 1 We observe a curious symmetry between r and x : x = 2 r 0, 1 and r = 2 x 1 12 ⇐⇒ ∈ { } 3 ⇐⇒ ∈ 0, 1 . For x = 2 , we get 4 out of 10 complex roots of z = 1. The case x = 2 has already been discussed.{ } − △

13 4.4 An octonionic evolution for f < 0 Lemma 4.7 For f < 0, f = X 2 + α 2 = q 2, and α 2 X 2 = f(2x 1). − − − − Proof. Use α2 = xf 0 and X 2 = f(x 1) 0. ≤ − ≥

1 2 Corollary 4.8 For f < 4 , g = (X, α ) satisfy g = f and s = (g+,g ) is 2 − ± ± ± − such that s =2f. Moreover (g+ +g ) (g+ g )=4 α X 1˜ =0 iff x(1 x) =0. − × − − × − 2 2 2 2 2 2 Proof. g =(X α , 0) = q = f and s =(g+ + f, 0)=2f. ± − − The octonions g and the sedenion s are pure imaginary in G and A4 with ± 4 2 ℑ 4 ℑ 4 invariant norm √ f and √ 2f respectively: g = f > 0 in G, s = 4g in A4 when α = 0, and q−4 = f 2 > 0− in H when α = 0. 2 2 2 g+ g = X + α 1˜ X α 1˜ = X α 1˜ + α 1˜ X X 1˜ , − × × − − 2 2 × − × with 1˜ X = X 1˜ = X˜ (say). Then g+ g = X + α 2 α X˜ with 2 × 2 − ×2 − 2 × − − X + α = X (f + X ). And g g+ = X α 1˜ X + α 1˜ = − − − × − × g+ g . × − 2 2 Thus 0 = g+ g implies that (g+ + g ) (g+ g ) = (2X, 0) (0, 2 α ) = − − − × − − × 2 (g+ g )=4 α X˜. The product is zero iff α X = 0 thus g+ g = − ℑ × − × − f 2 X 2 = f(1 2x). Observe that α 2 X 2 = f 2x(x 1)=0 iff f =0 or −x −0,1 . For f < −0, the condition is satisfied for f =− 1 only.− ∈{ } −

Corollary 4.9 The quadratic form α 2 X 2 in 6 real variables equals cos2ψ for f = 1 and 1 ey for f = 1 . − − − − 2 − 2 2 2 π 1 Proof. 1) If f = 1, α X = 2x 1 = cos2ψ, ψ [0, 2 ]; x = 2 iﬀ π − 2 − − − ∈ ψ = 4 . Equivalently x = sin ψ, α = sin ψ, X = cos ψ, 2 α X = sin 2ψ. 1 y 1 2) For f = 2 , 2x 1 = e , y < 0; x = 2 iﬀ y = . Alternatively, α 2 = x = 1 (1 −ey), X− 2 = 1 (1− + ey). In the limit y −∞, α and X tend 2 4 − 4 → −∞ to 1/2, g (X, 1 ), X = 1 (see Proposition 2.3). ± → ± 2 2 For f = 1 and αX =0(x(1 x) = 0) the unit octonion g+ = X + α 1˜ can − − X ˜ be represented in eulerian form, if we choose the unit 1X = X , X = 0 and 1 in the plane spanned by X and 1˜ when α and X = 0. uψ Set u = 1X 1˜ = 1˜X (say), and e = cos ψ + u sin ψ in the plane lin(1,u) (Chapter 2). Clearly× 0 <ψ< π α X = 0. 2 ⇐⇒ uψ uψ Lemma 4.10 When α X =0, g+ = 1X e− = e 1X = g . × × −

14 Proof. For f = 1, α 2 = x and X 2 = 1 x. g = X 1 + α 1˜ = 1 − − + X X × (cos ψ + 1¯X 1˜ sin ψ) = (cos ψ + 1˜ 1¯X sin ψ)1X with u = 1˜ 1X = 1X 1.˜ × uψ uψ × × × Hence g+ = 1X e− = e 1X . Similarly, g = X 1X α 1˜ = (cos ψ × uψ × uψ − − − 1˜ 1¯X sin ψ) 1X , g = e 1X = 1X e− = g+. × × − 2 × 2 × uψ Therefore g+ g = g+ = g = 1 = 1(1 2x) for x = 0. e = 1X uψ ×uψ− uψ − − − − − × e− 1 , 1 = e 1 e with u = 1 1.˜ See Figure 4.1.a). × X X × X × X × Lemma 4.10 tells us that at f = 1, for x(1 x) = 0, the distinct numbers − − uψ g+ = X + α 1˜ and g = X α 1˜ share the same eulerian representation e 1X . − However one could choose a− diﬀerent representation by means of ϕ = π ψ, ×1˜ and 2 − euϕ = cos ϕ + u sin ϕ.

uϕ uϕ uϕ Lemma 4.11 When α X = 0, g+ = 1˜ e = e− 1˜; and g = e 1˜ = uϕ × × − − × 1˜ e− . − × ¯ uϕ Proof. g+ = sin ϕ1X + cos ϕ1=˜ 1˜ (cos ϕ + 1˜ 1X sin ϕ)= 1˜ e = (cos ϕ + ¯ uϕ × × uϕ× uϕ 1X 1˜ sin ϕ) 1=˜ e− 1;˜ g = cos ϕ1˜ + sin ϕ1X = 1˜ e− = e 1.˜ See − Figure× 4.1 b).× × − − × − ×

u u euϕ 1˜ euψ ψ 1ϕ 1 1 u X uψ 1X e− 1˜ uϕ e−

0 <ψ< π ϕ = π ψ 2 2 − a) b)

Figure 4.1: Two eulerian representations in G at f = 1, 0

uϕ uϕ Moreover g+ 1=˜ e− and 1˜ g+ 1=˜ 1˜ e− = g since [1˜,g+, 1]=0˜ × − × × − × − by ﬂexibility. Hence 1˜ g+ = g 1.˜ If g+ = g , g+ and 1˜ anticommute, which − − contradicts < g , 1˜ >=× α =− 0. The× exponential representations by means of (ψ, + 1 ) and of (ϕ, 1)˜ are at odds for α = 0. X At f = 1, g = 1 and both g+ and g describe the manifold of G deﬁned by − ± − Σ = g; g2 = 1 a 3D-variety embedded in R8 for 0

g+ = i 1(= 1)˜

0 0 1X = g+ = g 1(= 1X ) −

g = i − − x = α =0 x = 1, X =0 ψ =0 ϕ =0

Figure 4.2: x 0, 1 ∈{ }

We observe a discontinuity: g+ = g iff α = 0, ψ = 0 and g+ = g iff − − X = 0, ϕ = 0 which reflects each of the two different eulerian represent−ations. When X = 0, g+ = g = i according to Lemma 4.11 with ϕ = 0, and when α = 0, − − g+ = g = 1X according to Lemma 4.10 with ψ = 0. This discontinuity reflects − the existence of two conflicting representations for g based either on (ψ, 1X when ± X = 0) or on (ϕ, 1˜ with α = 0). In the limit case when the angle ψ (resp. ϕ)= 0, that is α(resp. X)= 0, the reason for the discontinuity vanishes when one ignores the complementary angle ϕ (resp. ψ) at the value π/2. Continuous contradiction for 0 x 1 is perceived as discrete discontinuity for x 0, 1 . Moreover x = 1 (resp.≤ 0) entails≤ a transformation to i (resp. q; q2 = ∈{1 ). } {± } { − }

16 4.5 A logistic-based dynamics in division algebras 1 1 1 The iteration (3.2) converges for f + 4 2 to exact values on x =0( f 4 ) and 1 | | ≤ 1 3 | | ≤ x =1+ 4f . In addition, it can give the exact isolated point P ( 2 , 2 ) on the hyperbola and R( 1, 0) on the real axis. The points P and R are connected by computation. − At R, q2 = 1, we get back the identity z3 = q6 = 1 which was found valid at P in Section 4.3− when z is any of the two complex third− roots of 1. In the present connection, z = 1 is the real root of z3 = 1 and q describes− the imaginary unit − − sphere of H (q2 = 1). The computational connection between the isolated extreme points R and P is−global: it is based on all three roots of z3 = 1 corresponding to 1 − z = 2 = f and z = 1 = f. Moreover R and P lie on the second waveguide w2 ℜ 2 − 2 defined by Lf (1/2) which is the cubic 4f (1 + f) (Corollary 3.3 and Figure 4.3). The first waveguide w defined as L (1/2) = f is linear: it links the points 1 f − Q( 1, 1) and S( 1/2, 1/2) which correspond to a complex context: Q α = − − ⇐⇒ i α2 = 1 and S z = √3 i 1 4 of the 12 roots of z12 = 1. ± ⇐⇒ − ⇐⇒ ± 2 ± 2 ⇐⇒ w P • Q 1 •

R f •1 1/2 0 1/2 − − •S w1 w2

Figure 4.3: The waveguides w1 and w2 and the four extreme points P,Q,R,S

This discussion uncovers that a global dynamics for (3.2) takes place in the four division algebras, complementing the local real and complex dynamics already presented in Chapter 6. The holistic view blurs the neat distinctions z3 = 1 af f = 1/2 and z6 =1 at f = 1. The evolution is quadratic over G (possibly− H, C) for f < 0, and at − ℑ ℑ ℑ the highest point P (1/2, 3/2), the rule is z3 = 1 in C and q6 = 1 in H. We note for future reference that the waveguides− intersect at 0,− but are tangent at f = 1/2 which deﬁnes the real dynamics for (3.2) where the context G is not − ℑ reduced. On the other hand, it is arbitrary for f =0(α = X = 0).

17 4.6 An epistemological pause on computational complexity The complexity of relative computation evolves in two opposite ways ruled by the sign of f = q2. Either fℜ 1 and the number of dimensions may increase to 14 whereas the ≤ − 4 complex dynamics z6 = 1 for f = 1 becomes the octonionic dynamics g4 = f 2 with 1 − degree 4 < 6. Or f 4 and the number of dimensions is at most 4 but the powers ≥ H 12 1 involved can be as high as 12 in : q = 1(f = 2 ). Moreover the behaviour at 3 2 P (f =1/2, x = 2 ) is paradoxical and leads to a case of simplexity: q = z (the unit sphere in H can be identiﬁed to i on iR). For f = 1 , cos θ = 1 x represents ℑ {± } 2 2 − the distance to the critical value.

Based on g , the computation yields zerodivisors in Ak at x = 1/2 for f ± [ 1, 1/2]. We posit that the intellect interprets only the true zeros corresponding∈ − − to f = 1, x =0(g+ g )or1(g+ +g ); it ignores the sedenions which are another − − − − source for 0 iﬀ x = 1/2. It also ignores the conﬂicting exponential representations for g which arise when 0

18 of spatial separation, computationally connected by the waveguides w1 (linear) and w2 (cubic) respectively.

The above study of the computational evolution of q = α+X under the relativity condition q2 = α2 x 2 = f R has put on central stage the paradoxical ℜ − ∈ computation which takes place in the nonassociative algebras G and A4. Quite unexpectedly it unravels a spectacular consequence of nonassociativity in G which amounts to a metamorphosis of the octonion g = T (q)=(q, q¯) into the pair of real numbers 2 q (for c < f 1/2). − ± ℜ ≤ 5 Poincar´emeets Einstein at f 1, 1, 1 ∈{2 − −2} 5.1 A review of regular polygons Pv in cloth geometry

We consider the regular polygon Pv with v vertices, v 2 lying in a hyperbolic plane in a V -framed metric cloth associated with a non standard≥ addition + (Chatelin ◦ 2011b). Let θ,0 θ < π denote the common angle between any two adjacent sides in Pv, ≤ v 2 then 0 θ ϕ(v)= π −v , v 2 (Section 8.1 in TR/PA/11/27). For θ [0,ϕ(v)], ≤ ≤ ≥ a ∈ v 2, let ρ (θ) denote the relative side length for P , where represents the ≥ v λ v metric d˚associated with + . ◦ a π We ﬁnd in (Ungar 2008) the computation of ρ (θ) = for v =3 (0 θ ) v λ ≤ ≤ 3 and v =4(0 θ π ) and for the metric d˚derived from the additions + (Poincar´e) ≤ ≤ 2 ◦ P as in Example 7.1, and + E (Einstein) as in Example 2.2. The equivalence between ◦ 1 the two models of hyperbolic geometry is a simple homothety: xP = 2 xE P ◦× ⇐⇒ E ρ3 xE =2 xP . Moreover ρ3 = P 2 and we get the table ◦× (ρ4 )

v 3 4

d˚ √2 cos θ 1 √cos θ P − ˚ √2 cos θ 1 √cos θ dE − 2 θ cos θ cos 2 (see Ungar’s formulae (8.188), (8.260), (12.30) and (12.39)).

19 5.2 Organic causality in cloth geometry

Let two geodesics G◦ and G◦ ′ associated with the metric d˚intersect at x under the angle θ,0 <θ<π. The causal space at x has exactly v dimensions when ϕ(v 1) < π − θ ϕ(v) for v 3. For an acute angle θ, 0 < θ 2 , the causal space may have ≤ ≥ π ≤ 3 or 4 dimensions. But an obtuse angle θ, 2 <θ<π, would require more than 2 4 dimensions because 1 < cos θ < 0. This would entail negative values for ρ4(θ) ˚ ˚− (computed by dP or dE).

Example 5.1 We sketch on Figure 5.1 the polygons corresponding to v =2, 3, 4, 6, . For v< , they deﬁne an organic logic with v 1 causes, 1 v 1 5. ∞ ∞ − ≤ − ≤

x x x × v =2 v =3 v =4 θ =0 0 <θ π π <θ π ≤ 3 3 ≤ 2

x x v = ∞ v =6 θ = π 2π <θ 2π 5 ≤ 3 Figure 5.1: Five examples of organic causality.

△

In a euclidean plane, e2iπ = 1 is Euler’s identity. However we know that it may not hold in a non-euclidean plane (Chatelin 2011d). v 1 iθ iϕ(v) We set s = 1 0, s N N∗, z = e and Z = e . Thus vθ = π(v 2) = 2 − ≥ ∈ ∪ 2 − 2πs.

Lemma 5.1 z = Z iﬀ θ = ϕ(v) and Zv =( 1)v = 1 for v even, 1 for v odd . − { − } Proof. Since ϕ(v) is the angle in a euclidean v-polygon, we can apply Euler’s identity iﬀ θ = ϕ(v). The values for z = Z are the vth roots of 1 (resp. 1) for v even (resp. odd). −

20 5.3 The relation cos θ = 1 x at f =1/2 2 − 1 1 1 3 For f = 2 , cos θ = 2 x represents the distance from x [ 2 , 2 ] to the critical value 1 for which X =− α . ∈ − 2 | | | | We measure the complexity of the evolution by the number N of dimensions required by the real representation. If we limit the apparent complexity to N = 4 dimensions at most, we do not access to the real metric causality naturally associated π 1 with θ > 2 (v > 4). The study of the evolution of q = α + X for f = 2 gives us a computational path around the diﬃculty if we interpret cos θ as the distance 1 δ = 2 x given by the logistic iteration. This yields the following table valid for 1 − f = 2 :

x 1/2 0 1/2 1 3/2 − exact exact π π 2π θ 0 3 2 3 π v 2 3 46 ∞ N 1 1 1 2 2 24

We see that 1 x 1 N =1or2 2 v 6 and 1 6. We note that N = 1 occurs for v =3, 4 or 6. The⇐⇒ meta-rule for the complex dynamics at f =1/2 is z3 = 1. Hence we shall apply Lemma 5.1 with v 2. − ≥ Proposition 5.2 For f =1/2, we get the table

x 1/2 0 1/2 1 3/2 − exact exact v 2 3 46 ∞ s = v 1 0 1/2 1 2 2 − ∞ Zv 1 1 1 1 1 − {± } Proof. Clear.

We observe for future reference that the exact solutions for (3.1) yields Z3 = 1 (x = 0) and Z, Z =1 (x = 3 ) in Proposition 5.2. − { | | } 2 21 5.4 The relations cos2ψ =1 2x and x = sin2 ψ at f = 1 − − 1 2 The equivalence x = 2 (1 cos2ψ) = sin ψ suggests two possible causalities associated with ψ [0, π ] or 2ψ− [0, π]. This yields the table ∈ 2 ∈ x 0 1/4 1/2 3/4 1 exact exact

π π π π ψ 0 6 4 3 2

v1 = v(ψ) 2 34

s1 0 1/2 1 π π 2π 2ψ 0 3 2 3 π v = v(2ψ) 2 34 6 2 ∞ s 0 1/21 2 2 ∞

vj vj The analogue of Proposition 5.2 is obtained with sj = 2 1, j = 1, 2, Z = vj 3 2 − 3 ( 1) . The exact solution x = 0 (resp. 4 ) yields Z = 1 (resp. Z = 1 for j = 1, Z−6 = 1 for j = 2). −

5.5 The dynamics is real at f = 1 −2 The real dynamics is deﬁned by the relation ey = 1 2x > 0 for x =1/2 in ]0, 1[. | − | The meta-rule is now e−∞ = 0 in R. It diﬀers markedly from the algebraic rules z3 = 1 (f =1/2) and z6 =1(f = 1) in C. The− causality, which corresponds− to v = 2(ϕ(2) = 0), is associated with a segment on a line and not with v-polygons in a plane (v 3). The causality is ≥ linear; it corresponds to the only rationality currently acknowledged by scientists.

5.6 Some epistemological consequences The computational interpretation by means of (3.2) at f = 1/2 indicates that 2 π π≤ v 4 for x 1/2 0 θ 2 . Moreover, v 3 for x 0 0 θ 3 . The≤ reference| | ≤ to (3.2)⇐⇒prevents≤ the≤ quantisation artifact≤ described≤ ⇐⇒ in Example≤ ≤ 8.1 π (Chatelin 2011b). Without such a reference, it is possible for θ, 0 < θ 3 to be analysed with v = 4 (instead of 3), then 1 ρ (θ) 1: this creates≤ a spurious √2 ≤ 4 ≤ lower bound l(4) = 1 resulting from the choice v = 4 which is too large for the √2 optimal value v = 3.

22 This indicates the emergence of the discrete from the continuous, a possibility that we study further in (Chatelin 2011c).

The reference to (3.2) gives a meaning to cos θ < 0, a fact that is impossible in 2 1 P4-causality which interprets cos θ as the square (ρ4(θ)) . The situation x 2 can be interpreted by triangles and squares in a hyperbolic plane without an≤ explicit π π reference to (3.2). On the other hand, when x> 1/2, hence θ> 2 , hence θ> 2 , the 1 interpretation cos θ = 2 x makes an explicit use of (3.2). The same remark holds − π for cos2ψ = 1 2x: v2 = v(2ψ) > 4 for 2 < 2ψ < π. When causality is derived − 1 from θ are 2ψ, 2 v 4 iff x< 2 . Whereas when≤ it is≤ derived from ψ at f = 1, 2 v 4 is the rule for x [0, 1]. We conclude that the empirical observation of− movement≤ ≤ based on 4 dimensions∈ (3 for space +1 for time) may reveal only half of the complete picture, the half defined by the positive cosine of θ (f = 1 )or2ψ (f = 1); in other words when the distance, 2 − either δ = 1/2 x or d = 2δ = 1 2x, is positive. The limitation disappears at − − f = 1 when one considers the exponential representation for g based on 1X and ± ψ, with− x = sin2 ψ = α2 [0, 1]. − ∈ When v 2 is finite, the organic causality means that v 1 1 causes can be equally attributed≥ to the phenomenon (here the intersection− of two≥ geodesics in a hyperbolic plane). The value v = 2 leads to a single cause as is expected from linear causality. Such a causality underlies the real dynamics at f = 1/2 and the complex one at the two points R and S, see Figure 4.3. When v 3, there− are more ≥ than one possible cause, leading to a complex causality. Generically, the complex dynamics of (3.2) implies an algebraic causality (3 v < ) with v 1 causes. The causality is transcendent when v = : the v polygon≤ becomes∞ a straight− line ∞ − in the limit v . There are uncountably many possible causes: this is the case at P . At Q→(x ∞= 1), a choice is offered between v = (associated with the ∞ distance 1 2x = cos2ψ = 1) hence transcendence, or v = 4 (associated with √x = sin ψ−= 1) hence 3 causes.− The reader is referred to Chapters 3 and 12 for a detailed analysis of the linear and complex causalities in relation with R and C thinking. − − It is quite remarkable that the point Q lends itself to a twofold causal analysis corresponding to either 3 or an uncountable infinity of organic causes. The two views underlie respectively the physical and the metaphysical approaches to life. Computation provides a means to overcome the western matter-based bias to- ward the first view exclusively, de facto rejecting the latter. In spite of the current scientific dogma, computation tells us that the two views are equally valid by expos-

23 ing the implicit presuppositions in each view. The complete picture at Q consists of two possible views. Moreover “metaphysics” is mandatory at the point P which is therefore a blind spot for scientiﬁc rationality. All organisms in a state corresponding to P are beyond the reach of any explanation with a ﬁnite number of organic causes.

5.7 Einstein vs. Poincaré 1 The relations cos θ = 2 x and cos2ψ = 1 2x reveal that a deeper connection exists between the explanations− of Special Relativity− presented in 1905 by Einstein (+ ) and Poincaré( ). These two different views are traditionally opposed by physicists◦ and philosophers× (Chatelin 2011b). Such an opposition is nothing more than an artificial creation of the intellect lacking any solid basis in computation. No vision is “truer” than the other: they both belong to a deeper computational reality stemming from (3.2).

The connection can be rooted in the behaviour of the logistic iteration at f =1/2 1 H C R by cos θ = 2 x. Then the evolution context is a ﬁeld ( , or ). Or it can be rooted at f =− 1 by either cos 2ψ =1 2x or sin2 ψ = x in a nonassociative context for 0

24 In general, the cloth W is derived from V and G = Bλ V or V itself . The Chen-Ungar formalism generalises the local isomorphism{ between⊂ SL(2, C) and} SO(3, 1), the (3+1)-dimensional Lorentz group. The formalism involves two maps: + + φ : V R , v φ(v)= φv, φ(0) = 0, is a continuous map from Bλ into R . • → → v v m : R R is a continuous bijective map such that, with m(φ ) = = , • → | v | λ λ the condition sinh φv = 0 holds for all 0 = v B . m(φv) λ | | ∈ v We set vλ = λ . The choice for m(r) specifies the scalar multiplication which 1 1 ◦× satisfies for 0 = v G, r R, r v = (r)v with (r)= m(rm− ( vλ )). ∈ ∈ ◦× vλ The formula for (r) evokes irresistibly the right hand side in the formula listed for r = 2 in Table 4.1. Indeed if we set (2) = 1 2 v = v, we get the fixed-point relation for v > 0: ⇐⇒ ◦× λ 1 v = m(2m− ( v )). (5.1) λ λ which corresponds in Table 4.1, r = 2, to

1 d = h(2h− ( d ) (5.2) | | | | y where h is the real exponential y R− e = d < 1. ∈ → | | The solution of (5.2) y =2y < 0 is y = e−∞ =0 d =0 ⇐⇒ 1 1 −∞ ⇐⇒ 1 ⇐⇒ ⇐⇒ x = 1/2. Next, (5.1) yields m− ( vλ )=2m− ( vλ ) m− ( vλ ) unbounded v = m( ) . Since m(φ) = v , the condition⇐⇒ | requires | that φ = ⇐⇒ λ | ±∞ | | v | λ v φ(v) = for 0 = v Bλ. The equality (5.1) is at odds with 2 v = v v = 0 for all v∞ B . ∈ ◦× ⇐⇒ ∈ λ

When the set of relativistically admissible vectors is Bλ V , vλ < 1 for v B . The contradiction in (5.1) cannot be resolved. When +⊂ is not restricted to B∈ λ ◦ λ (as in Example 2.3 of Chatelin 2011b) then 1 vλ is possible for v V Bλ. The contradiction can be metrically bypassed≤ if v ≤∞= and m( ) =∈ .\ λ ∞ | ±∞ | ∞ Example 5.2 We illustrate the discussion with the three basic examples of Special Relativity, see Section 2.6 in (Chatelin 2011b). To each example is associated a classical model of hyperbolic geometry, respectively the Poincar´eball, the Beltrami-Klein ball, the hyperboloid model, see Section 8.1 in TR/PA/11/27. The Poincar´eaddition + P (Example 2.1 generalised to n 2, 0 <λ< ) corresponds to ◦ sinh φv ≥ ∞ m(r) = tanh r such that m( ) = 1, = cosh φv 1. The Einstein addition + E (Example | ±∞ | m(φv) ≥ ◦ | | r sinh φv φv 2.2, λ = c in physics) corresponds to m(r) = tanh 2 such that m(φv) = 2cosh 2 2. | | ≥ They both induce to the same scalar multiplication (Example 3.1). Since W is based on Bλ, the contradiction in (5.1) cannot be resolved.

25 The addition + in Example 2.3 corresponds to m(r) = sinh r such that sinh φv = 1. The ◦ m(φv) contradiction in (5.1) is metrically resolved with v = . | | One ﬁnal remark: the underlying identity µ(2) = 1 ∞ 2 v = v which leads to a contradiction in (5.1) sheds light on the key role played by the isomorphism⇐⇒ ◦× between the ball-models realised 1 through a homothety. Indeed vP = 2 vE vE = 2 vP : the pair Poincar´e-Einstein is the cloth geometric answer to the impossible◦× identity⇐⇒ v =2 ◦×v when 0 < v{ < . } ◦× ∞ △

The identity “1 = 2” is impossible in classical (absolute) arithmetic. By pulling the thread of hypercomputation plus quadratic iteration, we discover that cloth geometry replaces the impossible synthesis 1 = 2 by the exact product 2 1 = 1, × 2 which is realised in hyperbolic geometry by the two models associated with the y “twin” additions + E and + P . In the process h(y) = e , y R−, h(0) = 1, ◦ ◦ r ∈ R h( ) = 0 is replaced by m(r) = tanh r (for + P ) and tanh 2 (for + E) for r , m(0)−∞ = 0, m(+ ) = m( ) = 1: the additive◦ neutral 0 and the◦ multiplicative∈ unit 1 are exchanged∞ at− 0 and−∞ . ∞ 6 Poincaréintegrates Fourier at f = 1/2 − 6.1 Trigonometry with the Fourier scalar product The reader is referred to Chapter 10, Sections 10.8 and 10.9 for the background material and notation about the Fourier transform. In what follows we shall consider the Fourier scalar product σ(s)=< ∂ts,̟ts>= 2 when s : t f(t) is a smooth signal (s′ L ) and lim t t s(t)=0. → R R ∈ | |→∞ | | 1 2 We assume also that s i so that σ(s) is real. We recall that σ(ps)= 2 s 1 2 ∈ R ∪ R − (resp. + 2 s ) for s ∗ (resp. i ∗). It follows that σ(s) is minimum (resp. ∈ t2/4 maximum) iff s(t)= e− with R (resp. iR) (Proposition 10.9.1). C C ∈ C ∈ It was observed in Section 10.9.3 that ∂ and ̟ do not commute when applied to an arbitrary s: ∂(̟s) = s + ts′ whereas ̟(∂s) = ts′, hence [∂,̟]s = s. The commutator is computed as the identity map I : s s. This result holds for t2/4 → π the real exponential η(t) = e− such that the scalar product σ(η) = 2 is 1 1 − minimum: η satisfies η′ = 2 tη, that is ∂η = 2 ̟η. p We observe that η( −) = 0 implies that− at t = , [∂,ω]η( ) = 0. The property [∂,ω] = I becomes±∞ indistinguishable from commutativity±∞ ±∞ for η( )= 0. ±∞ The following result is a reciprocal of proposition 10.9.1. s′ We suppose that s , s′ ]0, [, and we set S(s)= , 0

26 Proposition 6.1 Trigonometry on the negative Fourier scalar product σ(s) char- t2/4 acterises the exponential function s = Ce− , C R. ∈ R 1 2 ∡ Proof. For t , = s′ ts cos θ(t)= 2 s with θ(t)= (s′(t), ts(t)). ′ ′ ′ ∈ ts s s− t s 2t 2 ′ By trigonometry ts = s cos θ(t). Therefore s = cos θ(t) ts = s 2 s′ = 2 2 − 2 St 2S t. That is s(t)= Ce− , C R. The proof of Proposition 10.9.1 shows that −necessarily S = 1 , that is θ(t) 0∈ (mod π). 2 ≡ Proposition 6.1 tells us that all signals s such that θ(t) 0 ( mod π) are ﬁl- tered out by trigonometry, making the Fourier scalar produc≡t appear extremum: either σ( η) = π or σ( iη) = π . When used in conjunction with trigono- ± − 2 ± 2 metric measurement,p the Fourier scalarp product favours the emergence of the family Cη, Cη R, iR of exponential signals: this entails a signiﬁcant epistemological ∈ { } 2 t2/2 reducion. Observe that φ0(t)= η = e− is an eigenfunction for the Fourier transform TF associated with the eigenvalue 1 (Theorem 10.8.1). We recall that other reductionist aspects of trigonometry related to exponentiation in Dickson algebras were already discussed (Sections 2.9 and 2.11 in Chapter 2).

6.2 The Fourier interpretation of (3.2) at f = 1/2 − At f = 1/2, one can write 1 2x = ey, y < 0 by means of y = ( t )2 = ln 1 2x . − | − | − 2 | − | Therefore η(t)= 1 2x . | − | t2/4 Lemma 6.2 The Fourier scalar product is minimum for the function η(t)= e− = 1 2x , a quantity which measures the convergence of (3.2) at f = 1/2, 0 x 1. | − | − ≤ ≤ Proof. Clear. t2/4 The exponential η : t R e− ]0, 1] is replaced by the piecewise linear function: x ]0, 1[ 1 ∈2x → [0, 1[ through∈ the parabolic change of variable 1 2 ∈ → | − | ∈ y = t R−. − 4 ∈ Remark 6.2.1 Spectral methods are efficient tools to solve numerically time-dependent problems such as the (parabolic) heat equation (Fourier 1822) and the (hyperbolic) wave equation. They use the Fourier transform and its fast version in the discrete case. The Fast Fourier Transform is part of the revolution in digital image processing which started in the 1970s. It is perhaps less known that the works of Fourier are also deeply connected with the Pascal triangle and the Sierpiński triangle, itself closely connected to the logistic (3.2) (Chapter 8). The interested reader is referred to (Robert 1992) which presents the Sierpiński triangle as a discrete wave obeying discrete versions mod 2 of parabolic, hyperbolic and elliptic differential equations.

We turn in the next Section to the hypercomplex context at f = 1/2. − 27 6.3 The Fourier-Einstein-Poincar´econnection between G2 and R at f = 1/2 − Lemma 6.1 provides a direct connection in R between the real dynamics for (3.2) at f = 1/2 and the minimisation of the Fourier scalar product min (σ(s) = − s < s′,ts >) realised by the exponential signal η(t), t R and leading to the usual linear causality that is of common experience in the physica∈ l world. This empirical causality is founded on an implicit reduction of the relative evolution framework from G to R. Notwithstanding experience, this reduction is a computational loss of information: we indicated in Section 4.5 that the octonionic context G is not reducible at f = 1/2 (Figure 4.3). Indeed q = α + X yields g = X α 1˜ ± with X = α =−1 , g = 1 , and the unit sedenions (g ,g ) and (g ,g±) are 2 √2 + + ± G − − H zerodivisors in Zer (A4) ∼= Aut ( ). These two sedenions stemming from q in can be interpreted as two automorphisms G G with norm= 1. The loss of information resulting from→ the mechanical use of linear causality results in the physical phenomenon known as entropy in thermodynamics. In me- chanics, energy is dissipated as heat and no perpetual machine can exist in the physical world.

Fourier analysis and Special Relativity play an essential role in the development of our information-based technology, from DVDs to GPS and cell-phones. It is quite a computational revelation to discover that both theories reside in a theoretical conundrum: in Fourier analysis a trigonometric interpretation leads to a loss of information, and in SR a theoretical impossibility is realised by computation! These feats are but distant echoes of the founding contradiction for computation: the paradox of zero (turned into a number in India) lies of at heart of the whole mathematical enterprise (Chatelin 2011e). We add that the heat equation is viewed in physics theory as a violation of special relativity because its solutions involve instantaneous propagation of a disturbance. The evolution that we witness on earth is but a reﬂection of the amazing trans- formative capabilities of mathematical computation. Because it based on repeated addition and multiplication, exponentiation plays a central role to connect + and × in the form eX eY = eX+Y × when the commuting pair X,Y consists of: (i) either square matrices deﬁned on { } complex or algebraic numbers, (ii) or doubly pure vectors in Ak, k 4 such that Y = tX + Z, t R, 0 = Z Zer (X) and X Y = is≥ real (Chapter ∈ ∈ × −

28 2, Section 2.9.7). The interested reader is referred to (Chatelin 2011c) for more on the topic of computational experimentation. Computation shapes our mental image of the world. To live up to the challenges ofa physically limited planet it is vital that science embraces the larger picture oﬀered by hypercomputation. All it takes is to view the essence of information as immate- rial, rather than having a purely physical origin (Gleick 2011, Chatelin 2011a). For Shannon’s heirs in computer science and biology, information is mechanical entropy. True but reductionist: paradoxical mathematics tells us that information is, above all, eidetic intelligence ruled by an organic logic.

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