Daniel Pecker

Poncelet's theorem and Billiard knots

Keywords: Poncelet's theorem, Jacobian elliptic functions, Billiard knots, Lissajous knots, Cylinder knots Mathematics Subject Classification 2000: 57M25

Let D be any elliptic right cylinder. We prove that every type of knot can be realized as the trajectory of a ball in D. This proves a conjecture of Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use Jacobi's proof of Poncelet's theorem by means of elliptic functions.

Introduction

The Poncelet closure theorem is one of the most beautiful theorems in geometry. It says that if there exists a closed polygon inscribed in a conic E and circumscribed about another conic, then there exist infinitely such polygons, one with a vertex at any given point of E.

When the conics are concentric circles the proof is very simple, each Poncelet polygon is obtained by rotating any one of them. What makes Poncelet's theorem great, is that it is impossible to generalize this simple proof.

Poncelet's proof ( [Po]) uses "pure" projective geometry, see [Be, DB, Sa] for modern proofs along these lines. Shortly after the publication of Poncelet's book, Jacobi ( [Ja] ) gave a proof by means of Jacobian elliptic functions. He discovered what is now called a uniformization of the problem by an elliptic curve. Most modern developments and generalizations follow Jacobi's proof, see [Lau, GH, BKOR, Sc, LT] .

A beautiful example of Poncelet polygonal lines is given by elliptic billiards. If a segment of a billiard trajectory in an ellipse E does not intersect the focal segment [F 1 F 2 ] of E, then there exists an ellipse C called a caustic, such that the trajectory is a Poncelet polygonal line inscribed in E and circumscribed about C, see [St, T, LT].

On the other hand, Jones and Przytycki defined billiard knots as periodic billiard trajectories without self-intersections in a three-dimensional billiard. They proved that billiard knots in a cube are very special knots, the Lissajous knots. They also conjectured that every knot is a billiard knot in some convex polyhedron. ( [JP], see also [START_REF] Lamm | There are infinitely many Lissajous knots[END_REF][START_REF] Cromwell | Knots and links[END_REF][START_REF] Bogle | Lissajous knots[END_REF][START_REF] Boocher | Sampling Lissajous and Fourier knots[END_REF][START_REF] Przytycki | Symmetric knots and billiard knots, Chapter 20 of the book Ideal Knots[END_REF]).

Lamm and Obermeyer [START_REF] Lamm | Zylinder-Knoten und symmetrische Vereinigungen[END_REF][START_REF] Lamm | Billiard knots in a cylinder[END_REF] proved that not all knots are billiard knots in a cylinder. Then Lamm conjectured that there exists an elliptic cylinder containing all knots as billiard knots [START_REF] Lamm | Zylinder-Knoten und symmetrische Vereinigungen[END_REF][START_REF] O'rourke | Tying knots with reflecting lightrays[END_REF]). It is easy to see that Lamm's conjecture implies the conjecture of Jones and Przytycki: if K is a billiard knot in a convex set, then it is also a billiard knot in the polyhedron delimited by the tangent planes. Dehornoy constructed in [D] (see also [O]) a billiard which contains all knots, but this billiard is not convex.

In this paper we will use Jacobi's method to study billiard trajectories in a right cylinder with an elliptic basis. We obtain a proof of Lamm's conjecture. Our result is more precise:

Theorem 20 Let E be an ellipse which is not a circle, and let D be the elliptic cylinder

D = E × [0, 1]. Every knot (or link) is a billiard knot (or link) in D.
Billiard trajectories in an ellipse are introduced in section 2. We show that an elementary theorem of Poncelet implies the existence of a caustic. We also show that Poncelet polygons in a pair of nested ellipses are projections of torus knots. Then, by a theorem of Manturov ([M]), we deduce that every knot has a projection which is a billiard trajectory in an ellipse.

In section 3, we recall the basic definitions and properties of the Jacobian elliptic functions sn(z) and cn(z). Then we give the Hermite-Laurent version of Jacobi's proof, which is based on Jacobi's uniformization lemma.

In section 4, we use Jacobi's lemma to compute the coordinates of the crossings and vertices of a billiard trajectory in E. We deduce that if the number of sides of a periodic billiard trajectory is odd, then it is generally completely irregular. This means that if we start at any vertex, 1 and the arc lengths of the other vertices and crossings are linearly independent over Q. We also see how our proofs generalize in the link case.

In section 5, we use Kronecker's density theorem to obtain our main result. The same strategy was used in [START_REF] Koseleff | Every knot is a billiard knot[END_REF] to give an elementary proof of the Jones-Przytycki conjecture. There is another application of Kronecker's theorem to the construction of knots in [START_REF] Koseleff | Chebyshev knots[END_REF].

Billiard trajectories in an ellipse

The study of billiard trajectories in an ellipse was introduced by Birkhoff in 1927 ( [Bi]), see also [T] for a modern exposition of the subject.

Some elementary facts

The following elementary theorem is due to Poncelet ([Po, Be]).

Theorem 1 (The second little Poncelet theorem)

Let E be an ellipse (or a hyperbola) with foci F 1 and F 2 . Let P M 1 and P M 2 be the tangents to E at the points M 1 and M 2 . Then the angles M 1 P M 2 and F 1 P F 2 have the same bisectors. Proof. Suppose that E is an ellipse (the case of a hyperbola is similar).

P F 1 F 2 M 1 M 2 F ′ 1 F ′ 2
Reflect

F 1 in P M 1 to F ′ 1 , and F 2 in P M 2 to F ′ 2 . As P M 1 is a bisector of F 1 M 1 F 2 , we see that F ′ 1 , M 1 , F 2 are collinear and F ′ 1 F 2 = M 1 F 1 + M 1 F 2 is the major axis of E. We deduce that F ′ 1 F 2 = F 1 F ′ 2 .
Consequently, the triangles F ′ 1 P F 2 and F 1 P F ′ 2 are congruent, because their sides are of equal length. This implies F ′ 1 P F 2 = F 1 P F ′ 2 , and then

F ′ 1 P F 1 = F ′ 1 P F 2 -F 1 P F 2 = F ′ 2 P F 1 -F 1 P F 2 = F ′ 2 P F 2 ,
which concludes the proof. 2

Theorem 2 Suppose that some segment of a billiard trajectory in an ellipse does not intersect the focal segment [F 1 F 2 ]. Then the billiard trajectory remains forever tangent to a fixed confocal ellipse called the caustic. Proof. Let P 0 P 1 be a segment of a billiard trajectory in an ellipse E, and suppose that

P 0 P 1 does not intersect [F 1 F 2 ].
Reflect F 1 in P 0 P 1 to F ′ 1 , and consider the ellipse

C = {M F 1 + M F 2 = F 2 F ′ 1 . We see that M 1 = F 2 F ′ 1 P 0 P 1 belongs to E, and since P 0 P 1 is a bisector of F 1 M 1 F 2 , it is the tangent to C at M 1 .
Draw P 1 M 2 the second tangent to C. By the second little Poncelet theorem, the angles M 1 P 1 M 2 and F 1 P 1 F 2 have the same bisectors. Hence P 1 M 2 is the second segment of our billiard trajectory, and is tangent to C.

2

Remark 3 When some segment contains only one focus, then every segment contains a focus, and there is no caustic. When some segment intersects the interior of the focal segment then there is a caustic, which is a hyperbola with foci F 1 and F 2 . But the tangency points need not be at a finite distance. This fact will be illustrated by the following example.

Example

Consider the points P 0 = (-1, -1), P 1 = (1, 1), P 2 = (1, -1), and P 3 = (-1, 1). The trajectory P 0 P 1 P 2 P 3 is a periodic billiard trajectory in the ellipse E = {x 2 + 2y 2 = 3}. The caustic is the hyperbola C = {x 2 -y 2 = 1}, and P 0 P 1 , P 2 P 3 are tangent to C at infinity. By Poncelet's theorem, if we apply the Poncelet construction from another point Q 0 ∈ E, then we obtain another periodic billiard trajectory

Q 0 Q 1 Q 2 Q 3 . Moreover, the diagonals Q 1 Q 3 and Q 0 Q 2 remain
forever parallel to the x-axis (this is a consequence of a theorem of Darboux that we will see later). Consequently, all these Poncelet quadrilaterals are symmetric with respect to the y-axis.

If we start from Q 1 = Q 3 = A = (0, 3/2), then the quadrilateral degenerates into a trajectory of the form A B A C A, where B = ( 5/3, -2/3), C = (-5/3, -2/3).

The preceding example shows that a billiard trajectory A 0 , . . . , A n such that A n = A 0 needs not be n-periodic. Moreover, a billiard trajectory in an ellipse is not necessarily a Poncelet polygonal line. Consider for example an ellipse E with foci F 1 , F 2 and a chord AB such that AB is the internal bisector of F 1 BF 2 . The polygonal line ABA is a billiard trajectory in E, but generally it is neither a Poncelet polygonal line, nor a periodic billiard trajectory.

Happily, these defects cannot occur for billiard trajectories that do not intersect the focal segment.

Corollary 4 Let P 0 , P 1 , . . . , P n-1 , P n = P 0 be a billiard trajectory in an ellipse E such that

P 0 P 1 does not intersect the focal segment [F 1 F 2 ].
Then it is a periodic billiard trajectory inscribed in E and circumscribed about a confocal ellipse C.

Proof. Since P n-1 P 0 does not intersect the focal segment, its reflection at P 0 cannot be P 0 P n-1 . Since it is another tangent to C through P 0 , it must be P 0 P 1 , and then P n+1 = P 1 .

2

Now, we shall prove the existence of billiard polygons with n sides and rotation number p. Following an idea due to Chasles and Birkhoff ( [Be, Bi]), we will obtain them as polygons of maximum perimeter among the n-polygons of rotation number p inscribed in E. We shall need the following classic lemma Lemma 5 Let A, B be (not necessarily distinct) points of an ellipse E. If the function

f (M ) = M A + M B, M ∈ E has a local maximum at C ∈ E, then CB is the reflection of CA in the normal to E at C.
Proof. See [LT].

2

Proposition 6 Let P 0 be a point of an ellipse E. Let n and p be coprime integers such that n ≥ 2p + 1. There exists a billiard trajectory in E, of period n, winding number p and starting at P 0 .

Proof. We shall consider the following maximum problem. The domain of definition of the function to maximize is

A = {(α 1 , . . . , α n ) ∈ [0, π] n , α i = 2πp}
, it is a compact set. For α ∈ A and P 0 ∈ E, let us define the inscribed polygon P 0 , P 1 , . . . , P n = P 0 by the angular condition ( ----→

F P i-1 , --→ F P i ) = α i ,
where F denotes a focus of E. We want to maximize the perimeter of this polygon, which is f (α) = P 0 P 1 + P

1 P 2 + • • • + P n-1 P 0 .
Since A is compact and f continuous, this maximum exists. By lemma 5, it is a billiard trajectory at the points P 1 , . . . , P n-1 .

Let us show that no segment P i-1 P i intersects the open focal segment (F F ′ ) of E. If it was the case, then all segments of the trajectory would intersect (F F ′ ), hence the winding number of this trajectory about F would be zero, which is impossible by the definition of our polygon.

If some segment was the major axis, then all segments would be this axis, and all α i = π, which is impossible since

α i = 2pπ < nπ.
If some segment contains only one focus, then it is known that the billiard trajectory converges to the major axis and is not periodic ( see [St, Fr] ).

We deduce that P 0 P 1 does not intersect the focal segment [F F ′ ] of E. By lemma 5 and corollary 4, we conclude that P 0 , P 1 , . . . P n-1 is a periodic billiard trajectory in E.

The exact period d of our trajectory is a divisor of n, and we have n = d u. By the angular condition, u is a divisor of p. Since n and p are coprime u = 1 thus n is the exact period of our trajectory.

2

Remark 7 This does not not prove that the caustics C n do not depend on the initial point P 0 . This is true by Poncelet's theorem, which we shall prove later. Using a theorem of Graves [Be] , it can be shown that all the Poncelet polygons in two confocal ellipses have the same perimeter.

Poncelet polygons and toric braids

A toric braid is a braid corresponding to the closed braid obtained by projecting the standardly embedded torus knot into the xy-plane. A toric braid is a braid of the form

τ p,n = σ 1 σ 2 • • • σ p-1 n
, where σ 1 , . . . , σ p-1 are the standard generators of the full braid group B p .

Remark 8 Let E and C be nested ellipses such that there exists a Poncelet polygon inscribed in E and circumscribed about C. Every Poncelet polygon is the projection of a torus knot of type T (n, p), n ≥ 2p + 1. More precisely, if we cut the elliptic annulus delimited by E and C along a half-tangent, then we see that such a polygon is ambient isotopic to the projection of the closure of the toric braid τ p,n . Consequently, it is also ambient isotopic to the star polygon n p , see [START_REF] Koseleff | Every knot is a billiard knot[END_REF].

Figure 4: Some Poncelet polygons (or unions of Poncelet polygons) in nested ellipses. They are projections of the toric braids τ 3,10 , τ 2,10 and τ 3,9 , and are denoted 10 3 , 10 2 and 9 3 .

We shall need the following results on braids, due to Manturov [M]. A quasitoric braid of type B(p, n) is a braid obtained by changing some crossings in the toric braid τ p,n .

Manturov's theorem tells us that every knot (or link) is realized as the closure of a quasitoric braid ( [M]). More precisely, he proved that any µ-component link can be realized as the closure of a quasitoric braid of type B(pµ, nµ) where (p, n) = 1, p even and n odd.

The quasitoric braids form a subgroup of the full braid group, hence there exist trivial quasitoric braids of arbitrarily great length. Hence we can suppose n ≥ 2p+1 in Manturov's theorem. Using this theorem, proposition 6, and Poncelet's theorem, we obtain the main result of this section.

Theorem 9 Let E be an ellipse. Every µ-component link has a projection which is the union of µ billiard trajectories in E with the same odd period, and with the same caustic C.

Jacobi's proof of Poncelet's theorem

We shall only need the following properties of elliptic functions, see [WW] for proofs.

The Jacobian elliptic functions sn z, cn z and dn z.

They depend on the choice of a parameter k, 0 < k < 1, called the elliptic modulus.

The Jacobi amplitude ϕ = am(z) is defined by inverting the elliptic integral

z = ϕ 0 dt 1 -k 2 sin 2 t .
It verifies am(u + 2nK) = am(u) + nπ, where am(K) = π 2 , and n ∈ Z. The Jacobian elliptic functions are defined for z real by sn z = sin am(z) , cn z = cos am(z) , dn z = 1 -k 2 sn 2 z, and can be extended to meromorphic functions on C. When k = 0, these functions degenerate into the ordinary circular function sin z and cos z. But, contrarily to the circular functions, they are doubly periodic functions with periods 4K ∈ R, and 4iK ′ ∈ iR, and they have poles. For example, the poles of sn z are congruent to iK ′ (mod.2K, 2iK ′ ), its zeros are the points congruent to 0 (mod.2K, 2iK ′ ), and its exact periods are 4K, 2iK ′ . The zeros of cn z are the points congruent to K (mod.2K, 2iK ′ ). We have sn(z + 2K) = -sn z, and cn(z + 2K) = -cn z. We also have sn(K + iK ′ ) = k -1 , which implies that the zeros of dn z are the points congruent to K + iK ′ (mod.2K, 2iK ′ ).

We have the following addition formulas sn(x + y) = sn x cn y dn y + sn y cn x dn x 1 -k 2 sn 2 x sn 2 y , cn(x + y) = cn x cn y -sn x sn y dn x dn y 1 -k 2 sn 2 x sn 2 y When k = 0, these formulas degenerate into the usual addition formulas for the circular functions.

In the next section we will use the following formula due to Jacobi ([WW] p.529).

sin am

(u + v) + am(u -v) = 2 sn u cn u dn v 1 -k 2 sn 2 u sn 2 v

Jacobi's uniformisation

The next result is a variant of Jacobi's uniformization of the Poncelet problem. It is due to Hermite and Laurent ( [Lau]). Lemma 10 Let E and C be the ellipses defined by Using the addition formulas we obtain

P (ϕ + β) M (ϕ) P (ϕ -β)
E = x 2 a 2 + y 2 b 2 = 1 , a > b > 1, C = x 2 + y 2 = 1 .
S (1 -k 2 sn 2 ϕ sn 2 β) = cn 2 ϕ + sn 2 ϕ dn 2 β = cn 2 ϕ + sn 2 ϕ (1 -k 2 sn 2 β) = 1 -k 2 sn 2 ϕ sn 2 β.
Consequently, S = 1, and P (ϕ + β) belongs to the tangent to C at M (ϕ). Changing β to -β, we see that P (ϕ -β) also belongs to this tangent. 2

Remark 11 By affinity, Jacobi's lemma extends easily in the case of two nested ellipses with the same two axis, meeting transversally in P 2 (C). When this pair of ellipses is affinely equivalent to a pair of concentric circles, the elliptic parametrizations degenerate into the usual circular ones. We shall now present the Hermite-Laurent proof of Poncelet's theorem for a pair of confocal ellipses. Since any pair of conics meeting transversally in P 2 (C) is projectively equivalent to a pair of confocal ellipses ( [LT]), we obtain a proof of the generic case of Poncelet's theorem. For the nongeneric cases see [Be, Sa], and for the original proof of Jacobi see [BKOR]. Proof. Let P 0 , P 1 , . . . P n-1 P 0 be a Poncelet polygon inscribed in E and circumscribed about C. Let P j P j+1 be tangent to C at M j . We will use the Jacobi parametrizations of E and C. If M 0 = M (ϕ), then by Jacobi's lemma we can suppose P 1 = P (ϕ + β). Using Jacobi's lemma again, we have M 1 = M (ϕ + 2β), and by induction

Proof of

M j = M (ϕ + 2jβ).
Since the polygon closes after n steps, we have

M n = M 0 , or M (ϕ + 2nβ) = M (ϕ).
That means am(ϕ + 2nβ) = am ϕ + 2qπ = am(ϕ + 4qK) by the properties of the Jacobi amplitude. Consequently we obtain 2nβ = 4qK, or β = 2qK/n. Now, let us consider a Poncelet polygon starting from an arbitrary point

M ′ 0 = M (ϕ ′ ) of C. By Jacobi's lemma we have M ′ n = M (ϕ ′ + 2nβ) = M (ϕ ′ + 4qK) = M (ϕ ′ ) = M ′ 0 .
Consequently, we see that every Poncelet polygonal line closes after n steps.

2

Corollary 12 Let E and C be confocal ellipses, and let P be a Poncelet polygon with an even number of sides. Then P possesses a central symmetry.

Proof. We use the same notation as before. We have β = 2qK/n , where n = 2h is the number of sides of P. The numbers n and q are coprime, hence q is odd. For every ϕ, we have

M h (ϕ) = M (ϕ+2hβ) = M (ϕ+2qK) = M (ϕ+2K) = cn(ϕ+2K), sn(ϕ+2K) = -M 0 (ϕ).
2

By projective equivalence, this implies a remarkable theorem of Darboux ([Da]).

Theorem 13 Let P be a Poncelet polygon with an even number of sides inscribed in a conic E and circumscribed about another conic C. Then the diagonals of P pass through a point, which is the same for every Poncelet polygon.

Irregularity of Poncelet odd polygons

Most regularity properties of a polygon can be expressed by rational linear relations between some of its segments. Let us parameterize a (crossed) polygon by arc length, starting at a vertex P 0 . We shall say that this polygon is completely irregular if 1 and the arc lengths of its crossings and vertices (except P 0 ) are linearly independent over Q. The purpose of this section is to prove that if E and C is a pair of confocal ellipses possessing a Poncelet polygon with an odd number of sides, then there exists a completely irregular Poncelet polygon. We will give an analogous result for unions of finitely many Poncelet polygons.

Two lemmas on elliptic functions

We shall use elliptic functions to compute the arc lengths of the crossings and vertices of Poncelet polygons. We shall need the following two technical lemmas.

Lemma 14 Let n and p be coprime integers, with n odd. For every integer j, let us define the function f j (z) = sn 2 (z + jθ) + r 2 , where r 2 > 0 and θ = 4pK/n. Then, if h ≡ j (mod n), the functions f j (z) and f h (z) do not possess any common zero.

Proof. First, let us study the zeros of the elliptic function g(z) = sn z + ir, r > 0. By considering its restriction to the y-axis, we see that there exists a pure imaginary α, such that g(α) = 0. Since we have sn(2K -α) = sn α, we see that 2K -α is another zero of g(z). As g(z) is an elliptic function of order two, its zeros are the points congruent to α or 2K -α (mod. 4K, 2iK ′ ). By parity, we deduce that the zeros of f j (z) are the numbers which are congruent to ±α -jθ, or 2K ± α -jθ, (mod. 4K, 2iK ′ ).

If we had α-jθ ≡ α-hθ, or α-jθ ≡ 2K +α-hθ (mod.4K, 2iK ′ ), then we would deduce (h -j)θ ≡ 0 (mod 2K). This implies that 2(h -j)p/n is an integer, which is impossible since n is odd, (n, p) = 1, and h ≡ j (mod n).

If we had α -jθ ≡ -α -hθ or α -jθ ≡ 2K -α -hθ, (mod.4K, 2iK ′ ), then we would have 2α ≡ ((j -h)θ (mod. 2K, 2iK ′ ). Taking the real parts, we would obtain (j -h)θ ≡ 0 (mod 2K) which is impossible.

Consequently, α -jθ cannot be a zero of f h (z). The proof that the other zeros of f j (z) cannot be zeros of f h (z) is entirely similar.

2

Remark 15 As the proof shows it, the condition n odd is necessary in lemma 14.

Lemma 16 Let n and p be coprime integers. For j ≡ 0 (mod n), let us define the functions D j (z) and F j (z) by D j (z) = sn(z + jθ) cn z -cn(z + jθ) sn z, F j (z) = sn(z + jθ) -sn z D j (z)

, where θ = 4pK n .

Then, for every integer j there exists a complex number α j such that F j (α j ) = ∞, and if h ≡ j (mod n), then F h (α j ) = ∞.

Proof. We have D j (z) = sin am(z + jθ) -am z = sin am(z + jθ) + am(-z)

Now, using the Jacobi formula for sin am(u + v) + am(u -v) , we obtain

D j (z) = 2 sn(jβ) cn(jβ) dn(z + jβ) 1 -k 2 sn 2 (jβ) sn 2 (z + jβ) , where β = θ 2
Let α j = -jβ + K + iK ′ . We have dn(α j + jβ) = dn(K + iK ′ ) = 0. Since dn 2 z + k 2 sn 2 z = 1, we obtain sn 2 (α j + jβ) = 1/k 2 , and then D j (α j ) = 0. The numerator of F j (α j ) is

N (α j ) = sn(α j + jθ) -sn(α j ) = sn(K + iK ′ + jβ) -sn(K + iK ′ -jβ).
Using the addition formula for the function sn z, we obtain

N (α j ) = 2 sn(K + iK ′ ) cn(jβ) dn(jβ) 1 -k 2 sn 2 (K + iK ′ ) sn 2 (jβ) .
Since sn(K + iK ′ ) = k -1 , we obtain N (α j ) = 2k -1 dn(jβ) cn(jβ) = 0, and then F j (α j ) = ∞.

On the other hand, if h ≡ j (mod n), we have

α j + hβ = K + iK ′ + 2(h -j)pK/n.
First, we see that α j +hβ ≡ K+iK ′ (mod.2K, 2iK ′ ), which implies that dn(α j +hβ) = 0. We also see that α j + hβ ≡ iK ′ (mod.2K, 2iK ′ ), which implies that sn(α

j + hβ) = ∞. We conclude that D h (α j ) = 0. Let us show that if sn z = ∞, then F h (z) = ∞.
Since the functions sn z and sn(z + hθ) do not have common poles, sn(z + hθ) = ∞. On the other hand, as sn 2 z + cn 2 z = 1, we obtain cn 2 z sn 2 z = -1, and then

F h (z) = -1 sn(z + hθ) cn z sn z -cn(z + hθ) If we had F h (z) = ∞, then sn(z + hθ) cn z sn z = cn(z + hθ),
whence sn 2 (z + hθ) = -cn 2 (z + hθ) = ∞, and sn 2 (z + hθ) + cn 2 (z + hθ) = 0, which is impossible.

Similarly, we see that if sn(z + hθ) = ∞, then F h (z) = ∞. Now, let us prove that F h (α j ) = ∞. We have D h (α j ) = 0, and we have proved that we can suppose sn(α j ) = ∞ and sn(α j + hθ) = ∞, then

F h (α j ) = sn(α j + hθ) -sn α j D h (α j ) = ∞. 2

Irregular Poncelet polygons with an odd number of sides

Proposition 17 Let E and C be confocal ellipses such that there exists a Poncelet polygon P inscribed in E and circumscribed about C. We suppose that the number of sides of P is odd. Then there exists a Poncelet polygon satisfying the following condition. If the arc lengths t i of the vertices and crossings are measured from a vertex P 0 , then the numbers 1 and t i , t i = 0 are linearly independent over Q.

Proof

. Let Q h,j = ℓ h ℓ h+j , j ≡ 0 (mod n). The abcissa x h,j of Q h,j is

Let E = { x 2 a 2 + y 2 b 2 = 1},
x h,j = -sn ϕ h + sn(ϕ h + jθ) sn(ϕ h + jθ) cn ϕ h -cn(ϕ h + jθ)sn ϕ h = F j (ϕ h ) where F j is the function defined in lemma 16. The abcissa of P h = Q h,-1 = Q h-1,1 will also be denoted by x h = x h,-1 . The distance P h Q h,j is |d h,j | where

d h,j = d h,j (ϕ) = √ 1 -c 2 sn ϕ h sn 2 ϕ h + c 2 1 -c 2 x h -x h,j
Since c 2 /(1 -c 2 ) > 1, the function d h,j (ϕ) is meromorphic in a neighborhood of the real axis.

Our first step is to prove that the functions 1 and d h,j (ϕ), j ≡ -1 (mod n) are linearly independent over C.

Let λ h,j and λ be complex numbers such that n h=1 n-2 j=1 λ h,j d h,j = λ, or

n h=1 √ 1 -c 2 sn ϕ h sn 2 ϕ h + c 2 1 -c 2 n-2 j=1 λ h,j (x h -x h,j ) = λ.
Since c 2 /(1-c 2 ) > 0, we see by lemma 14 that the functions f h (ϕ) = sn 2 ϕ h + c 2 /(1 -c 2 ) do not possess any common zero. Hence, in the neighborhood of a zero of f h (ϕ) this function is not meromorphic, while the other functions are.

This implies that for every h = 1 . . . n we have n-2 j=1 λ h,j x h -x h,j = 0, and then λ = 0.

Using our expressions of the abcissas x h,j , we obtain the following relation between meromorphic functions n-2 j=1 λ h,j F -1 (z) -F j (z) = 0.

By lemma 16, for every integer j = 0 there exists a number α j such that F j (α j ) = ∞, and F h (α j ) = ∞ if h ≡ j (mod n). Letting z = α j , we obtain λ h,j = 0, which concludes the proof of the linear independence of our functions. Now, we shall prove that for most ϕ ∈ R, the numbers d h,j (ϕ) and 1 are linearly independent over Q.

Figure 1 :

 1 Figure 1: The second little Poncelet theorem.

Figure 2 :

 2 Figure 2: Existence of an elliptic caustic.

  Figure 3: A billiard quadrilateral in an ellipse. The caustic is a hyperbola.

Figure 5 :

 5 Figure 5: Jacobi's lemma.

  Let us parameterize E by P (ψ) = (a cn ψ, b sn ψ), and C by M (ϕ) = (cn ϕ, sn ϕ), where the elliptic modulus k is defined by k 2 (a 2 -1) = (a 2 -b 2 ). Let β be a real number such that cn β = 1/a.Then the tangent to C at M (ϕ) intersects E at P (ϕ -β) and P (ϕ + β).Proof. We have dn2 β = 1 -k 2 sn 2 β = b 2 /a 2 , hence dn β = b/a.Let us show that P (ϕ + β) belongs to the tangent to C at M (ϕ). The equation of this tangent is x cn ϕ + y sn ϕ = 1. Let us compute S = a cn(ϕ + β) cn ϕ + b sn(ϕ + β) sn ϕ.

Figure 6 :

 6 Figure 6: Proof of Poncelet's closure theorem.

  a > b > 1, and C = {x 2 + y 2 c 2 = 1}, c < 1 be our ellipses. The condition on the eccentricity of C means that 2c 2 > 1.Let us consider the Jacobi parametrizations of E and C by means of elliptic functions, and let θ = 4pK/n. To each real number ϕ corresponds a Poncelet polygon P ϕ through M (ϕ) = cn ϕ, c sn ϕ . Let us denote ϕ j = ϕ + jθ, M j = M (ϕ j ), and let ℓ j be the tangent to C at M j . The equation of ℓ j isx cn ϕ j + y c sn ϕ j = 1.

For every nonzero collection of rational numbers Λ = (λ, λ h,j ), let us define the function F Λ by F Λ (ϕ) = λh,j λ h,j d h,j (ϕ). By our first step, this function is not identically zero, and it is meromorphic in a neighborhood of R. Therefore, the set of its real zeros is countable. Consequently, the set of all real numbers ϕ such that 1 and the numbers d h,j (ϕ) are linearly dependent over Q is countable. By cardinality, we deduce that the complementary set is not countable, hence nonempty. Consequently, there exists a real ϕ such that 1 and the numbers |d h,j (ϕ)| are linearly independent over Q.

Now, let us parameterize our Poncelet polygon by arc length, starting from

The result follows from the independence of the numbers 1 and |d h,j |.

2

We shall also need an analogous result for links.

Proposition 18 Let E and C be confocal ellipses such that there exists a polygon of an odd number of sides inscribed in E and circumscribed about C.

For any integer µ, there exist µ Poncelet polygons P (0) , P (1) , . . . , P (µ-1) satisfying the following condition:

for each such polygon, if t i are the arc lengths corresponding to its vertices , its crossings, an its intersections with the other polygons, then the numbers 1 and t i , i = 0 are linearly independent over Q.

Proof. Let τ = θ µ , and let us denote M h = M (ϕ + hτ ) ∈ C, and ℓ h the tangent to C at M h . Let us consider the Poncelet polygons P = P (0) , P (1) , . . . , P (µ-1) through the points

The vertices and crossings of P are the points Q h,j = ℓ h ℓ h+j , where h ≡ 0 (mod µ).

Just as before, it can be proved that the distances 1 and |d h,j (ϕ)|, h ≡ 0, j ≡ 0, j ≡ -1 (mod µ) are linearly independent over Q, except for a countable set of numbers ϕ.

Consequently, the number 1 and the arc lengths t i , i = 0 of the crossings and vertices of P are linearly independent over Q except on a countable set of values of ϕ.

By cardinality, we can suppose that the same property is true for each polygon P (j) , j = 0, . . . , µ -1, which proves our result. 2

Proof of the theorem

We will use Kronecker's theorem ( see [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF]Theorem 443]):

Theorem 19 If θ 1 , θ 2 , . . . , θ k , 1 are linearly independent over Q, then the set of points (nθ 1 ), . . . , (nθ k ) is dense in the unit cube. Here (x) denotes the fractional part of x.

Now, we can prove our main theorem.

Theorem 20 Let E be an ellipse which is not a circle, and let D be the elliptic cylinder

Proof. First, we consider knots. By theorem 9 there exists a knot isotopic to K, whose projection on the xy-plane is a billiard trajectory of odd period in the ellipse E. If t 0 , t 1 , . . . , t k are the arc lengths corresponding to the vertices and crossings, we can suppose by proposition 17 that the numbers t 1 , . . . , t k , and 1 are linearly independent over Q. Using a dilatation, we can suppose that the total length of the trajectory is 1.

Let us consider the polygonal curve defined by (x(t), y(t), z(t)), where z(t) is the sawtooth function z(t) = 2|(nt+ϕ)-1/2| depending on the integer n and on the real number ϕ. If the heights z(P j ) of the vertices are such that z(P j ) = 0, z(P j ) = 1, then it is a periodic billiard trajectory in the elliptic cylinder D = E × [0, 1] (see [START_REF] Jones | Lissajous knots and billiard knots[END_REF][START_REF] Lamm | There are infinitely many Lissajous knots[END_REF][START_REF] Lamm | Billiard knots in a cylinder[END_REF][START_REF] Przytycki | Symmetric knots and billiard knots, Chapter 20 of the book Ideal Knots[END_REF][START_REF] Koseleff | Chebyshev knots[END_REF]). If we set ϕ = 1/2 + z 0 /2, z 0 ∈ (0, 1), we have z(0) = z 0 . Now, using Kronecker's theorem, there exists an integer n such that the numbers z(t i ) are arbitrarily close to any chosen collection of heights, which completes our proof.

The case of µ-component links is similar. First, by theorem 9, we find a diagram that is the union of µ Poncelet polygons with the same odd number of sides. Then, by proposition 18 and Kronecker's theorem, we parameterize each component so that the heights of the vertices and crossings are close to any chosen numbers.

2


Janice Hau

Andrei Gorea May

The influence of task difficulty, success and confidence on preceived brief durations

Keywords: perception of duration, millisecond timing, task difficulty, attention, success, confidence, psychophysics

Time is constant but it has been repeatedly shown that our perception of time is impacted by many factors. This is evidenced by the growing literature on temporal distortions.

The interference effect, the finding that we tend to underestimate temporal durations when temporal processing is disrupted by a secondary (non-temporal) task, has been widely observed in the time literature. However, within the sub-second range the influence of a secondary task on temporal processing has been less-studied and the subject of much debate. In this study we looked at the effect of task difficulty on perceived duration under a dual-task paradigm.

Additionally, we investigated two potential factors that may play a role in perceived duration: success and confidence. While no significant effects were found for our three main factors, we propose that our main contribution is our original dual-task visual search paradigm. Our negative results are discussed further with regard to the distinct timing hypothesis, which proposes distinct mechanisms for sub-second and supra-second timing.

Introduction

Our perception of time is variable from one situation to another. The expressions "a watched pot never boils" and "time flies when you're having fun" are illustrations of the common variability of our duration perception. The first to write about a time sense was [START_REF] Czermak | Ueber den schallerzeugenden apparat von crotalus[END_REF] in the 19th century, the idea that time is a perceived quality processed internally. In addition to physical duration itself time perception is actually influenced by many factors such as emotional context [START_REF] Langer | The effect of danger upon the experience of time[END_REF], personality traits [START_REF] Orme | Time, experience and behaviour[END_REF], age [START_REF] Wallach | On age and the subjective speed of time[END_REF], drugs use [START_REF] Frankenhaeuser | Estimation of time: An experimental study[END_REF], saccadic eye movements [START_REF] Morrone | Saccadic eye movements cause compression of time as well as space[END_REF], visual motion [START_REF] Kanai | Visual motion dilates the time[END_REF], and visual space [START_REF] Casasanto | Time in the mind: Using space to think about time[END_REF]. The implications for studying perceived time, particularly of brief durations holds significant importance for motor, sensory, and cognitive processes. Timing in the milliseconds is required for immediate perception (i.e., seeing, hearing) and also for the actions we produce in relation to these events (i.e., motor coordination is just one example). Without our ability to process time (relatively) precisely, our understanding of speech and music would be impaired, we would have difficulty catching a ball, driving a car, and doing simple tasks such as crossing the street -judging the moment when a car would pass by. In short, our daily functioning would suffer extensively.

The interference effect is a classical phenomenon involved in time perception that has been consistently demonstrated in the literature [START_REF] Brown | Attentional resources in timing: Interference effects in concurrent temporal and nontemporal working memory tasks[END_REF]. The interference effect is the tendency to underestimate temporal durations when time processing is disrupted by a concurrent task. It also translates into less accurate (i.e., more variable) duration judgements [START_REF] Brown | Multiple timing and the allocation of attention[END_REF][START_REF] West | Attention, multiple timing, and psychophysical scaling of temporal judgments. Time, action, and cognition: Towards bridging the gap[END_REF], see review by [START_REF] Brown | Time and attention: Review of the literature[END_REF]. It is usually assessed by means of a dual-task paradigm, where a primary (duration estimation) task and an arbitrary (non-temporal) secondary task are performed simultaneously. These duration estimates are then compared to estimates obtained in a single (timing-only) task condition. Gradual effects when changing the difficulty of the secondary (non-temporal) task can also be observed. The more difficult the non-temporal task, the shorter the judged duration. This has held for a variety of secondary tasks involving perceptual [START_REF] Brown | Time perception and attention: The effects of prospective versus retrospective paradigms and task demands on perceived duration[END_REF][START_REF] Coull | Functional anatomy of the attentional modulation of time estimation[END_REF][START_REF] Field | Temporal interval production and short-term memory[END_REF][START_REF] Wundt | Relative and absolute duration judgements under prospective and retrospective paradigms[END_REF], verbal [START_REF] Fortin | Order information in short-term memory and time estimation[END_REF][START_REF] Fortin | Short-term memory and time estimation: Beyond the 2second" critical" value[END_REF][START_REF] Rammsayer | No evidence for qualitative differences in the processing of short and long temporal intervals[END_REF][START_REF] Miller | Effects of concurrent verbal rehearsal and temporal set upon judgements of temporal duration* 1[END_REF], and memory [START_REF] Fraisse | The psychology of time[END_REF][START_REF] Brown | Cognitive effort and temporal information processing[END_REF][START_REF] Zakay | The influence of task difficulty and external tempo on subjective time estimation[END_REF] processes. The lower duration estimation performance in the presence of a secondary task is usually accounted for in terms of a limited attentional capacity [START_REF] Kahneman | Attention and effort[END_REF] that is taxed by the secondary task in proportion to its cognitive load [START_REF] Hicks | Prospective and retrospective judgments of time as a function of amount of information processed[END_REF][START_REF] Thomas | Cognitive processing and time perception[END_REF]. When a secondary task is highly-demanding, less attention can be allocated to "keeping track" of time hence entailing degraded performance [START_REF] Brown | Time perception and attention: The effects of prospective versus retrospective paradigms and task demands on perceived duration[END_REF]. Inasmuch as less attention entails a lesser number of processed events by the count of which perceived duration is putatively calibrated [START_REF] Thomas | Cognitive processing and time perception[END_REF], less attention will also entail a duration underestimation [START_REF] Treisman | Temporal discrimination and the indifference interval. Implications for a model of the" internal clock[END_REF].

The study of time perception is complicated by the putative existence of different time scales. [START_REF] Mauk | The neural basis of temporal processing[END_REF] have categorized the limitless span of time into four time ranges: microseconds, milliseconds, seconds and circadian rhythms. It is remarkable that the central nervous system processes temporal information across all of these ranges throughout life. The way circadian rhythms function in the body is relatively well understood [START_REF] Perreau-Lenz | The biological clock: the bodyguard of temporal homeostasis[END_REF], however surprisingly little is known about how we process time in the intervals spanning milliseconds, seconds, and minutes. From the beginning of psychological time research scientists have raised the question of whether we process time differently in the brain within each of these time scales [START_REF]Beiträge zur experimentellen Psychologie[END_REF][START_REF] Wundt | Relative and absolute duration judgements under prospective and retrospective paradigms[END_REF]. More recently, researchers have suggested distinct mechanisms specifically for sub-second and supra-second timing (Lewis & Miall, 2003a;[START_REF] Ivry | Dedicated and intrinsic models of time perception[END_REF][START_REF] Buonomano | How do we tell time[END_REF][START_REF] Buhusi | Relativity theory and time perception: Single or multiple clocks[END_REF][START_REF] Michon | The compleat time experiencer[END_REF]. [START_REF] Gibbon | Toward a neurobiology of temporal cognition: advances and challenges[END_REF] identified time ranges that showed distinct psychophysical characteristics. He noted differences in the coefficients of variation (standard deviation of interval estimates divided by the estimated mean) in the time ranges below 1-2 s and the ones above, and proposed the existence of different neurobiological mechanisms1 . But recent evidence from [START_REF] Lewis | The precision of temporal judgement: milliseconds, many minutes, and beyond[END_REF] revealed no such discrete differences, instead showing a continuous logarithmic decrease for durations ranging from 68 ms to 17.7 min. Pharmacological [START_REF] Rammsayer | On dopaminergic modulation of temporal information processing[END_REF][START_REF] Rammsayer | Neuropharmacological evidence for different timing mechanisms in humans[END_REF][START_REF] Rammsayer | Pharmacologic properties of the internal clock underlying time perception in humans[END_REF][START_REF] Mitrani | Identification of short time intervals under lsd25 and mescaline[END_REF], lesion [START_REF] Breukelaar | Effects of lesions to the cerebellar vermis and hemispheres on timing and counting in rats[END_REF][START_REF] Clarke | Exploring the domain of the cerebellar timing system[END_REF] and neuroimaging (Lewis & Miall, 2003a) studies also provide support for separate mechanisms. This so-called distinct timing hypothesis proposes that temporal discrimination of intervals in the range of seconds is cognitively mediated, whereas brief intervals in the sub-second range are processed automatically and therefore beyond cognitive control [START_REF] Lewis | Distinct systems for automatic and cognitively controlled time measurement: evidence from neuroimaging[END_REF][START_REF] Michon | The compleat time experiencer[END_REF][START_REF] Johnston | Spatially localized distortions of event time[END_REF][START_REF] Ivry | The neural representation of time[END_REF] 2 . According to the distinct timing hypothesis then, temporal processing in the range of milliseconds should be unaffected by cognitive tasks and attention. But, whereas perceptual factors have been shown to influence duration judgments in this interval range [START_REF] Rammsayer | Duration discrimination of filled and empty auditory intervals: Cognitive and perceptual factors[END_REF][START_REF] Tse | Attention and the subjective expansion of time[END_REF][START_REF] Rammsayer | No evidence for qualitative differences in the processing of short and long temporal intervals[END_REF][START_REF] Kanai | Time dilation in dynamic visual display[END_REF], the role of higher cognitive processing in this range is less clear. Only a few studies have looked into this issue. Of particular interest are the studies that involved dual-tasks, which we will consider in more detail.

Mattes & Ulrich (1998) manipulated attention by using a valid vs. invalid pre-cue to indicate the modality (Exp. 1-3) or spatial location (Exp. 4-6) of a stimulus whose duration was to be estimated. In the valid pre-cue trials subjects were correctly oriented to the stimulus (more attention) and in the invalid pre-cue trials subjects were misdirected away from the stimulus (less attention). They showed that directed attention (via the pre-cue) to a stimulus prolongs its perceived duration, a finding that contradicts the distinct timing hypothesis, given that milliseconds timing was mediated by attention. In dual-task studies, if the distinct timing hypothesis held, then differential impairment should be observed between temporal judgements for sub-second intervals and supra-second intervals, indicating different underlying mechanisms. Under a dual-task paradigm, [START_REF] Rammsayer | No evidence for qualitative differences in the processing of short and long temporal intervals[END_REF] found no differential interference between subjects' duration discrimination centred around 100 ms and 1000 ms. Both intervals were systematically influenced in the same way by a secondary mental arithmetic task and unaffected by secondary memory search and visuo-spatial memory tasks. In contrast, [START_REF] Rammsayer | Duration discrimination of filled and empty auditory intervals: Cognitive and perceptual factors[END_REF] (see Exp.'s 2-3) showed differential impairment in duration discrimination in dual-task conditions for intervals in the sub-second and supra-second ranges. The secondary task was a word learning task and involved subjects having to learn a visually presented word on each trial and recall the words at the end of the experiment. Performance on the timing task in the milliseconds (centred around 50 ms) showed no difference in the presence of a secondary task versus in the single task (timing-only) condition. But for the intervals centred around 1-second, performance on the timing task was affected by the secondary task. [START_REF] Rammsayer | Elaborative rehearsal of nontemporal information interferes with temporal processing of durations in the range of seconds but not milliseconds[END_REF] showed the same pattern of differential impairment on temporal judgements between sub-second and supra-second intervals using the same word learning task, referred to as elaborative rehearsal, essentially replicating the results of [START_REF] Rammsayer | Duration discrimination of filled and empty auditory intervals: Cognitive and perceptual factors[END_REF] but also showing that it was not the case for a different word learning task referred to as maintenance rehearsal. In maintenance rehearsal subjects only needed to recall the last word beginning with a designated letter such as the letter "B". According to these authors, the elaborative rehearsal task required continuous updating of information and transfer to long-term memory, producing interference in sub-second temporal processing. Instead, maintenance rehearsal presumably only requires passive information storage in working memory and therefore should not affect timing processes. This result is consistent other studies showing no interference effects for passive storing of information in working memory [START_REF] Brown | Processing resources in timing and sequencing tasks[END_REF][START_REF] Fortin | Temporal interval production and processing in working memory[END_REF].

In short, factors such as the type of the secondary task and the experimental design need to be considered further for a better appraisal of the interference effect in duration estimation (for discussion, see [START_REF] Bangert | Dissecting the clock: Understanding the mechanisms of timing across tasks and temporal intervals[END_REF]).

Here we address the issue of the impact of the difficulty of a secondary, perceptual task on sub-second duration estimations. In accord with a wide literature [START_REF] Carrasco | Visual attention[END_REF] we posit that attention intervenes at all processing levels and at all psychological time scales and therefore hypothesize that a more difficult visual task requires greater attention, hence less resources devoted to keeping track of time, and therefore a lower time units count leading to a duration underestimation.

In addition to manipulating the task difficulty, the present study tackles two duration estimation issues never addressed before, namely subject's success (vs. failure) and level of confidence in accomplishing the visual task. Intuitively, there is reason to believe the duration of a successfully accomplished task should be overestimated by comparison with the same duration of an unsuccessfully accomplished task. Retrospectively, success might entail the impression that sufficient time had been available to accomplish the task and if so failure should have the opposite consequence. Alternatively, it can be argued that the difficulty of the task may lie in the trial by trial strength of the evoked brain response (conditional on internal noise) and not on the overall task difficulty as assessed by a global performance/sensitivity index such as d'. Accordingly, successful secondary task trials should be associated with shorter and more accurate duration estimates than unsuccessful trials. Inasmuch as correct responses generally correlate with the level of confidence, (e.g., [START_REF] Henmon | The relation of the time of a judgment to its accuracy[END_REF]; [START_REF] Harvey | Confidence in judgment[END_REF]; [START_REF] Barthelmé | Evaluation of objective uncertainty in the visual system[END_REF]) the same should be true for duration estimates associated with 'confident' and 'not-confident' secondary task trials, respectively. As the probability of succeeding is inversely proportional with task difficulty, our experimental design should disentangle these factors. To our best knowledge, the dependence of duration perception on the correctness of the responses in a secondary task and on the confidence subjects have in these responses has never been studied.

The present study

The large majority of studies involving the effects of task difficulty in time perception have focussed on durations in the supra-second range (2 s to several minutes). Several neuroimaging studies (fMRI) have manipulated duration discrimination difficulty itself in the milliseconds range either by means of varying the difference between the two durations to be discriminated [START_REF] Tregellas | Effect of task difficulty on the functional anatomy of temporal processing[END_REF] or the time interval between two events whose temporal order was to be discriminated [START_REF] Lewandowska | Changes in fmri bold response to increasing and decreasing task difficulty during auditory perception of temporal order[END_REF]. Task difficulty manipulations have also been used to study sensitivity in timing [START_REF] Ferrara | Changing sensitivity to duration in human scalar timing: An experiment, a review, and some possible explanations[END_REF], and the perception of one's own response times [START_REF] Petrusic | Probability assessment with response times and confidence in perception and knowledge[END_REF].

There are a number of methodological issues that are associated with a task difficulty manipulation. One is to design a task whose difficulty is equally maintained for the entire duration to be estimated and which does not differ with time. [START_REF] Petrusic | Probability assessment with response times and confidence in perception and knowledge[END_REF] investigated the effects of task difficulty on the perceived duration of subjects' own response times which, of course, correlated with task difficulty. Hence, not only was the duration to be estimated varying from trial to trial but subjects' estimates could simply be based on their estimate of the task difficulty, on their response time or on both. In our paradigm task difficulty is manipulated within a fixed physical duration given to subjects to accomplish their perceptual (secondary) task with no speeded response. Hence duration estimation can be achieved based on time processing only. [START_REF] Fortin | Time estimation and concurrent nontemporal processing: Specific interference from short-term-memory demands[END_REF] used a visual search task in which subjects were asked to produce a fixed 1.1 The present study 3 s temporal interval while simultaneously finding a target in a visual search array. Regardless of the task difficulty, the time interval needed to accomplish the search task was significantly shorter (by at least 1 s) than the interval to be produced. Hence the secondary task only partly overlapped with the required time processing that remained partly undisturbed.

In our study, we use a visual search task within a modified version of the Rapid Serial Visual Presentation (RSVP) [START_REF] Forster | Visual perception of rapidly presented word sequences of varying complexity[END_REF], of a constant total duration and tailored so that subjects needed to sustain their attention until the very end of the given interval. Another difference between our study and previous studies concerns the fact that duration estimation and difficulty were related to disjoint, independent events. As such the secondary task added a cognitive load difficulty to this task's difficulty per se. In the present experiment, subjects were asked to estimate the duration of the very same time interval that they were given to perform the visual search task knowing that this time interval is kept constant for all difficulty levels. Also in contrast with other studies that have manipulated difficulty within a poorly defined dimension space, i.e., using different qualitative tasks [START_REF] Axel | Estimation of time[END_REF][START_REF] Gulliksen | The influence of occupation upon the perception of time[END_REF], the present study varies task difficulty along a single dimension, i.e. orientation.

The modified RSVP task consisted in visual stimuli (vertical Gabors) briefly displayed at non-overlapping (hence lateral-masking free) iso-eccentric locations with a deviant orientation target presented at the end of the sequence. Such presentation was meant to entail a high and constant attentional level throughout a given RSVP sequence while dissociating the difficulty of the task (the difference between target and distracters) from the duration of the RSVP sequence to be estimated. A standard visual search task and Multiple Object Tracking (MOT) task were also considered and rejected as they entail such confounds and were not feasible for sub-second durations.

In sum, our study contributes in several ways: it measures difficulty effects on sub-second duration estimation with an original visual search task that circumvents confounds identified in the literature, and addresses two unstudied putatively interfering factors in duration estimation namely success and confidence.

Method

Participants

Six subjects took part in the study (5 female). Five subjects had ages that ranged from 22 to 26 years (mean = 23.6 yrs) and one subject was 59 years of age. All subjects had normal or correctedto-normal vision and gave their informed consent to participate. All subjects were compensated 30 euros for their participation in the study.

Apparatus and stimuli

The stimuli were displayed on a 19" E96f + SB ViewSonic screen (1024 × 768 pixels, 100 Hz refresh rate) 60 cm away from observers' eyes. Lighting conditions in the room were dim. The presentation of stimuli and response recording were controlled using the Psychophysics toolbox [START_REF] Brainard | The psychophysics toolbox[END_REF][START_REF] Pelli | The videotoolbox software for visual psychophysics: Transforming numbers into movies[END_REF] in Matlab. The target and distracter stimuli were Gabor patches with a sigma of 0.5 deg, a spatial frequency of 4 cpd, and a 99% contrast. They were presented on a 35 cd/m 2 grey background. The stimuli were presented in two successive modified rapid serial visual presentation (RSVP) sequences. These displays differed from the standard RSVP mode in that the stimuli appeared at pseudo-random locations on a 6.5 deg radius virtual circle around fixation (see Figure 1). The pseudo-randomization was such that the locations of items i+1 to i+n in the RSVP sequence were at least 45 deg of visual angle away from the location of item i within a sliding temporal window of 100 ms. This presentation scheme was meant to prevent lateral interactions between items. Each item i was displayed for 20 ms with inter-stimulus intervals randomized in-between 10, 20 and 30 ms so that the number of items within the "non-interaction" window varied from trial to trial. The randomization of the inter-stimulus interval was meant to prevent subjects from potentially basing their duration judgments on the number of displayed items 3 . The number of distracters (vertical Gabors) displayed before the target was varied so as to yield one of three total RSVP durations (300, 400 and 500 ms), the 'standards'. The target (a tilted Gabor) was always followed by one distracter that terminated a given RSVP sequence. Fixating the rank order of the target one item before the end of a random (one out of three) RSVP durations insured that observers had to sustain their attention until the end of that duration. A target 2.3 Procedure item concluding the RSVP sequence would have allowed observers to passively wait until the end of the sequence and base their orientation discrimination on this last item/target orientation stored in iconic memory. The target item was always presented in the second RSVP sequence, the standard. The duration of the first RSVP sequence (the 'probe' duration) was randomly chosen out of 5 durations centred logarithmically on the duration of one of the three standards. Target orientation was randomly rotated clockwise or counter-clockwise by a predefined amount to yield two orientation discrimination difficulty levels (67 and 96 percent correct as assessed in a twoalternative forced-choice (2AFC) preliminary experiment; see Procedure). (k-1). Location constraints are placed on the target/distracters so that items never overlap within a 100 ms sliding window. The illustration is for the 'standard' RSVP sequence that could be of 300, 400 or 500 ms. The 'probe' sequence, always presented first, included no target. Its duration was randomly chosen out of 5 durations centred logarithmically on the duration of the standard. procedure used in the Main experiment was applied here. Observers were required to specify the clockwise/counter-clockwise orientation of the target (2AFC method with constant stimuli) by pressing one of four keys on a standard keypad. The four keys were aligned horizontally with the two extreme keys designating a high level of confidence that the target was tilted counter-or clockwise and with the middle keys designating a low level of confidence. On each trial observers were also required to perform the duration discrimination task (see Main experiment). The duration of the standard for these trials was fixed and set at 450 ms. Four target orientations (+/-2, 4, 8 and 15 deg away from the vertical) were presented 50 times each in a randomized order once the first 24 -48 (training) trials were discarded for each observer. A cumulative Gaussian was used to fit the four performance levels and the target orientations yielding theoretical correct responses of 67% (d' = .62) and 96% (d' = 2.48) were used in the main experiment. The preliminary experiment together with the training trials was run in 15 to 25 min.

Procedure

Main experiment. The main task focused on observers' estimation of the duration of the RSVP sequences while also performing the target orientation discrimination task described above. The probe sequence appeared first and was followed 300 ms later by the onset of the standard sequence. At its end observers first provided a 2AFC duration judgment (standard duration longer or shorter than the probe) and then the 2AFC orientation discrimination response. Probe durations varied in proportion to the standard durations; specifically, they were either equal to the standard or 17 and 34% shorter or longer than the standard). These values were chosen based on preliminary trials meant to bracket the duration discrimination psychometric function. As noted above, subjects also qualified their target orientation choice on a scale of 1 to 2 by means of pressing one of four keys. The following trial started 500 ms after the orientation response. The two levels of difficulty (as determined for each subject in the preliminary experiment) were run in separate blocks. Each block consisted of 150 trials, i.e., 10 trials per each of the 3 standards and per each of the 5 probe durations associated with each standard. Within each block both standard and probe durations were drawn randomly across trials. Each block was repeated 5 times in a random order for each observer to yield a total 250 trials per orientation discrimination difficulty level and per standard duration (i.e., 50 trials per each probe duration). The estimated Points of Subjective Equality of each standard duration and for each orientation discrimination difficulty level were taken to be the probe durations entailing 50% "shorter standard" responses as derived from the fit cumulative Gaussians to observers' percentages of "shorter standard" responses. The accuracy of the duration judgments was the fit standard deviation of these cumulative Gaussians.

Trials were classified into two confidence levels depending on subjects' responses for the orientation discrimination task (whether they pressed the extreme or middle keys). The 1500 trials total per observer (750 trials per difficulty level x 2 difficulty levels) were run in about 2h30 not including short brakes after no more than 150 trials. Participants were instructed to concentrate on both duration and orientation tasks equally, without giving priority to one or the other. The experiment was run in three hour-long sessions. Figure 2: Fitted Weibull and cumulative Gaussian of a typical subject's orientation discrimination data (a) and duration discrimination data (b) in the preliminary experiment. (a) shows the percentage of correct orientation responses for orientation differences from the vertical of ±2, 4, 8, and 15 • , for a fixed 450 ms standard; (b) shows the percentage the standard was judged shorter than the probe. In the insets, the error of the fits are given between parentheses and the parameters µ and σ of the respective fits are given by 'm' and 's', respectively.

Results

Preliminary experiment

Table 1: Points of Subjective Equality (µ) and sensitivity indices (σ) derived from the duration estimates of a 450 ms standard by the 5 participants for each of the 4 orientation discrimination difficulty levels (i.e. angles relative to the vertical) in the preliminary experiment.

!""# $% #! &# '! (
σ are free parameters corresponding to its mean (the detection threshold) and standard deviation (an index of sensitivity) 4 . Subjects' thresholds (µ) ranged from 4.42 to 7.08 deg with a mean of 5.36 deg. Their fit standard deviations (σ) ranged from 0.89 to 2.83 deg with a mean of 1.68 deg.

The orientations yielding performance levels of 67% and 96% (corresponding to d' values of .62 and 2.48) for the six participants were ±2.6

• , ±2.6 • , ±2.9 • , ±4.8 • , ±2.5 • , ±5.4 • (67% performance) and ±8.4 • , ±8.4 • , ±8.5 • , ±9.1 • , ±8.1 • , ±25.6 • (96% performance
). These values were used in the main experiment and specified the "easy" and "difficult" detection conditions run concurrently with the duration estimation task. Figure 2.b shows percentages of the standard 450 ms duration being judged "shorter" than the probe (abscissa) for the same observer together with a rescaled cumulative Gaussian function of the form P (x) = γ + (1 -γ)p(x) where p(x) is the cumulative Gaussian function Φ(µ, σ 2 ) where µ is this time the observer's Point of Subjective Equality (PSE) and σ is the sensitivity index 5 . Table 1 shows these fit values for each of the five observers and for each of the four orientations (difficulty levels) of the target stimulus. Observer NP could not do the task and her fit values are not included in the overall averages. The PSE of the remaining 4 observers bracket more or less the duration of the standard (450 ms) with a grand mean (over the 4 orientations and 5 observers) of 442 ms. Notice however the very large standard deviations (σ) of the fits (grand mean of 499 ms, σ/µ > 100%) that indicate that the task was particularly difficult. This is partly due to observers not being trained enough at this stage of the experiment. 4 Percentages correct were also fit with cumulative Gaussian [p(x) = (1 + erf (-(x -µ)/σ)/2] and logistic [p(x) = 1/(1 + e(-(x -µ)/σ))] functions that yielded comparable error functions.

5 Percentages shorter were also fit with Weibull [p(x) = 1 -e(-(x/µ)σ] and logistic [p(x) = 1/(1 + e(-(x -µ)/σ))] functions that yielded comparable error functions. As shown below, all observers improved their duration discrimination in the main experiment.

Main experiment

Main experiment

Performance on the orientation discrimination task was between 81 -98% (mean = 92.8%) for the easy condition and 67 -83% (mean = 74.6%) for the difficult condition across observers. The large standard deviations observed for the duration task in the preliminary.

Duration discrimination performances and their cumulative Gaussian fits in this experiment 2 presents the fit parameters for each observer and for each of the three data partitions. One subject (np) was not included in the analysis due to deviant performance on the duration discrimination task and the low number of high confidence responses reported (see Supplementary information in Appendix C).

As a preliminary observation, note that the precision of observers' duration estimate (i.e., σ)

improved with respect to the preliminary experiment (grand mean of 327 ms, σ/µ = 78%) but remained rather high. Also note that the coefficients of variation (σ/µ) were not constant over the three standard durations tested which is evidence against Weber's law (or scalar property; see Figure 4). With one exception (incorrect responses; Figure 4.) out of 6 cases they decreased with the standard duration which is to say that σ increased less than required by the scalar property within this 400 -600 ms range (see Figure 3). Departures from Weber's law within this duration range have been frequently reported [START_REF] Allan | The perception of time[END_REF][START_REF] Gorea | Introspective duration estimation of reactive and proactive motor responses[END_REF][START_REF] Kristofferson | A quantal step function in duration discrimination[END_REF][START_REF] Kristofferson | Quantal and deterministic timing in human duration discrimination[END_REF][START_REF] Mauk | The neural basis of temporal processing[END_REF][START_REF] Wearden | Decision processes in models of timing[END_REF][START_REF] Wright | Learning and generalization of auditory temporal-interval discrimination in humans[END_REF].

This being said, inspection of Figure 3 already tells us that none of the three factors of interest (difficulty of the concurrent orientation discrimination task, success and confidence) appears to play a systematic role in observers' duration estimation. This is the case both for observers' PSEs (µ; Figure 3 top row) and for their accuracies (σ; Figure 3 botton row).

In order to add statistical validity to these observations, six 2-way ANOVAs with repeated measures were run separately for the fit PSEs and σ-s and for each partition (difficulty, success, and confidence). As expected, for PSEs (µ) the standard duration factor (3 modalities, 300, 400 and 500 ms) was always significant (p < 10 -9 ). The three main factors (difficulty, success, and confidence) failed to reach significance (0.10 < p < 0.30). For the standard deviations (σ) the difficulty and success factors were close to but did not reach significance (difficulty: p = 0.09, success: p = 0.08, confidence: p = 0.27).

The only clear and consistent result is the slope of the PSE function of the standard duration which is always larger than 1. This may be explained by an effect of regression to the mean within the duration discrimination task (see Discussion).

Discussion

The aim of this study was to better understand the influence of task difficulty on perceived duration in the sub-second range. Additionally, two factors that have never before been considered in the time perception literature -the effects of success and of confidence, were investigated. We predicted that given a more difficult secondary task, more attention will be allocated to accomplishing the secondary task and less attention for the primary task, leading to fewer time units processed and time shrink [START_REF] Thomas | Cognitive processing and time perception[END_REF]. This has been widely found in the literature [START_REF] Brown | Time perception and attention: The effects of prospective versus retrospective paradigms and task demands on perceived duration[END_REF][START_REF] Zakay | The influence of task difficulty and external tempo on subjective time estimation[END_REF]; ?; [START_REF] Hicks | Prospective and retrospective judgments of time as a function of amount of information processed[END_REF]. In addition, as a result of less attention the accuracy of estimates would also vary with difficulty (the more difficult task being less accurate) [START_REF] Brown | Multiple timing and the allocation of attention[END_REF][START_REF] West | Attention, multiple timing, and psychophysical scaling of temporal judgments. Time, action, and cognition: Towards bridging the gap[END_REF]. For success, we predicted an overestimation of time in the "correct" trials as a result of subjects retrospectively assessing the time duration. They would perceive that they had adequate time for successful trials and hence overestimate the duration relative to failed trials. However, without feedback in this Table 2: Points of Subjective Equality (µ) and sensitivity indices (σ) of duration estimates by the 5 participants grouped according to the difficulty, correct/incorrect responses and confidence levels in the concurrent orientation discrimination task.

! "#

experiment subjects had no indication of success or failure meaning that any hypothetical effect would have arisen only from a highly cognitive nature below one's awareness [START_REF] Kunimoto | Confidence and accuracy of near-threshold discrimination responses[END_REF]. We predicted the same for confidence, that their estimates would be based on retrospective assessment and therefore perceive trials in which they responded "less confident" as shorter in duration compared with high confidence trials. We proposed a second hypothesis: that success/failed trials and confidence responses would be an indication of the strength of the evoked signal in the brain and therefore reflect the influence of the difficulty of the orientation discrimination task (easy condition = stronger signal). Confidence has been linked to success and show sensitivities to variations in the difficulty of a task [START_REF] Petrusic | Probability assessment with response times and confidence in perception and knowledge[END_REF]. Under this hypothesis, successful and high confidence trials should be associated with shorter and more accurate duration estimates.

Lack of rigorous methods in past and current task difficulty manipulations have led to some weaknesses in the results obtained thus far. These include: 1) perceived duration-difficulty confounds (i.e., if the estimated durations were based on subjects' own response times, then response time would covary with difficulty [START_REF] Petrusic | Probability assessment with response times and confidence in perception and knowledge[END_REF], 2) variable attention over the duration to be estimated (i.e., if the overlap in time between the primary and secondary tasks was variable from trial to trial, for example if the secondary task was finished before the primary timing task so that the subject is left to wait passively until the end of the duration, then the attention paid to the duration would be variable depending on when they finished the secondary task [START_REF] Fortin | Time estimation and concurrent nontemporal processing: Specific interference from short-term-memory demands[END_REF][START_REF] Rammsayer | No evidence for qualitative differences in the processing of short and long temporal intervals[END_REF], 3) the secondary (non-temporal) task and the primary task being disjoint events (such that the subject must be simultaneously aware of two separate sets of stimuli for a single duration (i.e., two different modalities), and their onsets may occur at different times within the trial,) may pose too high demands on subjects [START_REF] Fortin | Time estimation and concurrent nontemporal processing: Specific interference from short-term-memory demands[END_REF][START_REF] Rammsayer | No evidence for qualitative differences in the processing of short and long temporal intervals[END_REF][START_REF] Rammsayer | Duration discrimination of filled and empty auditory intervals: Cognitive and perceptual factors[END_REF][START_REF] Rammsayer | Elaborative rehearsal of nontemporal information interferes with temporal processing of durations in the range of seconds but not milliseconds[END_REF], and 4) poorly defined difficulty modulation involving discrete levels and/or levels differing on not one but multiple dimensions [START_REF] Axel | Estimation of time[END_REF][START_REF] Gulliksen | The influence of occupation upon the perception of time[END_REF][START_REF] Brown | Time perception and attention: The effects of prospective versus retrospective paradigms and task demands on perceived duration[END_REF]. Based on the weaknesses identified here, we implemented an original method for systematic and controlled study of task difficulty in the sub-second duration range. Our method has several key features: 1) it has controlled fixed standard durations (from which we are able to fit psychometric functions to approximate PSEs), 2) it requires a high constant attentional level (the targets appear at the end of sequences of variable durations), 3) the secondary (non-temporal) task is embedded in the actual duration event to be estimated (therefore both tasks are presented in a single coherent stimulus, from which duration is to be judged; hence biases in the duration estimates must be linked to the characteristics of the secondary task), and 4) task difficulty varies along a single continuous dimension (allowing for the systematic study of task difficulty and avoiding potential confounds, such as a third variable). In previous studies, it could be argued that subjects' attentional level may have weaned for the start of each duration, knowing that the target always appeared at the end of a sequence, and waited till the end of the sequence. However, given the brief durations studied here (the lowest being 300 ms) and the unpredictability of the duration to be estimated on each trial, it is unlikely that subjects could have selectively attended to the end of the sequence only. Concerning the manipulation of the difficulty of the secondary task (orientation discrimination), it became apparent during the course of the experiment that it was drowned out by the overall difficulty of simultaneously monitoring the two concurrent (primary and secondary) tasks. While this drawback is probably shared by all dual task paradigms involving duration estimations (as primary tasks; see [START_REF] Brown | Time and attention: Review of the literature[END_REF], it definitely hampered in the present experimental design the successful manipulation of the orientation discrimination difficulty effects on duration estimation (discussed below).

As expected, the results show a significant main effect of the physical standard duration on its perceived duration (Points of Subjective Equality, PSEs). Instead, statistical analyses revealed no significant PSE effects of any of the three main factors, namely task difficulty, success, and confidence. No such effects were obtained either for the standard deviations of the estimated durations.

Difficulty in our experiment referred to the orientation discriminability (d') measured in the secondary (non-temporal) task. Unfortunately, both subjects' reports and the large standard deviations of the perceived duration judgments (the primary task in both the preliminary and the main experiments), indicated that the manipulation of the orientation discrimination difficulty was negligible in comparison with the overall difficulty of simultaneously monitoring these two tasks. Hence, the absence of a difficulty effect is most likely due to this "drowning out" effect.

While one might hope that rendering the duration estimation/discrimination task easier would render the difficulty manipulation of the secondary task more effective, such means would also hamper the assessment of the duration estimation psychometric function. A better alternative would be to replace the duration discrimination method with a duration reproduction one.

Nevertheless, several other studies have also failed to obtain significant effects on task difficulty in the sub-second range [START_REF] Rammsayer | Duration discrimination of filled and empty auditory intervals: Cognitive and perceptual factors[END_REF][START_REF] Rammsayer | Elaborative rehearsal of nontemporal information interferes with temporal processing of durations in the range of seconds but not milliseconds[END_REF]. Moreover, some researchers have suggested that temporal processing mechanisms in the milliseconds range are automatic and therefore unimpaired by additional cognitive processing. Our study fits into this distinct timing hypothesis. Interestingly, there has been no account of why [START_REF] Rammsayer | No evidence for qualitative differences in the processing of short and long temporal intervals[END_REF] produced interference in the temporal judgments of supra-second intervals but

Rammsayer & Lima (1991) study did not. One critical difference between the two studies is that the secondary task was not done simultaneously for the latter study, rather they induced cognitive load (so that rehearsal occurred during the temporal discrimination task but the word to be learned was shown before the onset of this task). It is also important to note that the three dual-task studies discussed above all used an adaptive time comparison method and in our study we used the classic time comparison method. These contradictory results and the lack of a theoretical account indicate that very little is known about how and when a concurrent task influences temporal judgements (more specifically, which types of tasks cause interference on temporal judgements).

At first sight the observation that subjects underestimated the shortest standard duration (300 ms) and overestimated the longest one (500 ms; figure 3 is contrary to Vierordt's law [START_REF] Vierordt | Der Zeitsinn: Nach Versuchen von Karl Vierordt[END_REF] (regression to the mean) observed in many duration reproduction studies (e.g., [START_REF] Kanai | Time dilation in dynamic visual display[END_REF]; [START_REF] Tse | Attention and the subjective expansion of time[END_REF]; [START_REF] Brown | Time perception and attention: The effects of prospective versus retrospective paradigms and task demands on perceived duration[END_REF]; [START_REF] Schiffman | Effects of stimulus complexity on the perception of brief temporal intervals[END_REF]; Jazayeri & Shadlen (2010)).

However, when taking into account that the duration judgments consisted in a comparison between two durations (probe and standard), these findings can be taken as essentially Vierordt's law. Perceiving the probe duration as longer for short physical durations and shorter for long durations (Vierordt's law) implies that the standard was judged respectively shorter and longer than the physical duration. However, this interpretation raises the question of why the regression to the mean applies only (or possibly more) to the probe stimuli (than to the standards). An interesting observation is that the standards appear five times more frequently than the probe durations. The conditions for regression to the mean and its relative effects require further investigation.

The coefficients of variation revealed that Weber's law did not hold in this experiment.

According to the Scalar Expectancy Theory (SET) [START_REF] Gibbon | Scalar expectancy theory and Weber's law in animal timing[END_REF][START_REF] Gibbon | Scalar timing in memory[END_REF], standard deviations should be proportional to the estimated duration (Weber's law). Figure 4 shows the coefficients of variation for the three factors studied (task difficulty, success, and confidence) and clearly shows non-monotonic increases/decreases with the standard duration. Deviations from Weber's law have been frequently observed, particularly in the sub-second range [START_REF] Allan | The perception of time[END_REF][START_REF] Kristofferson | A quantal step function in duration discrimination[END_REF][START_REF] Mauk | The neural basis of temporal processing[END_REF][START_REF] Wearden | Applying the scalar timing model to human time psychology: Progress and challenges[END_REF][START_REF] Wright | Learning and generalization of auditory temporal-interval discrimination in humans[END_REF]. It is possible that failure to reveal a constant Weber fraction is due to the restricted duration range used here, which happens to be within [START_REF] Kristofferson | A quantal step function in duration discrimination[END_REF] slow rise range.

Conclusion

In this study we investigated the effects of task difficulty, success and confidence on perceived temporal durations. We found no significant effects for any of the three factors and discussed that while it may be evidence to support the distinct timing hypothesis, the absence of any effect could also have been a result of potential ceiling effects due to the difficulty of simultaneously performing both tasks. This was indicated by the large standard deviations observed in both the preliminary and main experiments. Our main contribution here is our proposed (dual-task)

RSVP visual search paradigm to study the effect of task difficulty, and our exploration of two potential factors (success and confidence) that may play a role in perceived time.

Figure 5: Illustration of the random location of gabor positions θ 1 , θ 2 and θ 3 on a virtual circle centered on a fixed point F with a radius R and the free-zone A. The free-zone A is the set of points ouside intervals located at ±ǫ from the gabor positions θ.

Eq. ( 2). With this method, we obtain random positions uniformly distributed on [0, 2π[. We want gabors not to overlap during a given period of time T . Each time a position is sampled, an area where the next positions can only be sampled, a free zone area, is updated. The method can be described as follows:

•

B Estimation of the parameters of psychometric functions

In our problem, we have N observations (x i , y i ), for i = 1, . . . , N . The variable x i represents a parameter of our experiment (for instance, the angle orientation difference) and y i probabilities for a given response (for instance, the probability of answering correctly on the orientation task). Our purpose is to fit a function to the observations, i.e., to find a function that explains the observations but is also able to predict probabilities for different values of the experimental parameters. In our problem, we consider three parametric functions (namely, the erf, logistic and Weibull) defined by two parameters µ and σ: µ is a shift parameter that represents the mean of the psychometric function and σ is a scale parameter that is taken as the standard deviation of the psychometric function. The three psychometric functions are respectively defined by:

f (erf ) µ,σ (x) = γ + (1 -γ) 1 2 erf x -µ σ (3) f (logistic) µ,σ (x) = γ + (1 -γ) 1 1 + exp -x-µ σ (4) f (W eibull) µ,σ (x) = γ + (1 -γ)(1 -exp - x µ s . (5)
where 0 ≤ γ < 1. All these functions have values between γ and 1. The first two functions are defined for all real values x while the Weibull function is only defined for positive real values. Figure 6 gives an illustration of such a fitting function and with its parameters µ, σ and γ = 0.5 for N = 5 observations.

Mathematically, the optimal parameters µ and σ can be obtained by solving the least squares problem, i.e., by minimizing the sum squared error between f µ,σ (x i ) and y i defined by:

N i=1 (f µ,σ (x i ) -y i ) 2 . ( 6
)
The errors between the observed probability y i and the prediction given by f µ,σ at positions x i are represented on Figure 6. The solution (m, s) of such an optimization problem is the point at which the differentials of the sum squared error equal to zero (i.e., the solution of the following system):

   ∂ ∂µ N i=1 (f µ,σ (x i ) -y i ) 2 = 0 ∂ ∂σ N i=1 (f µ,σ (x i ) -y i ) 2 = 0 . ( 7
)
For our psychometric functions, such a system cannot be solved analytically. Instead, we choose to map the problem to a linear regression problem, for which an analytic solution can be obtained (see next sections).

This solution does not minimize Eq. ( 6) but another least square problem. However, we assume it is close

Figure 1 :

 1 Figure1: Spatio-temporal sequence of stimuli presented in the RSVP visual search task. The target (an oblique Gabor) is always presented in the first to last frame (k-1). Location constraints are placed on the target/distracters so that items never overlap within a 100 ms sliding window. The illustration is for the 'standard' RSVP sequence that could be of 300, 400 or 500 ms. The 'probe' sequence, always presented first, included no target. Its duration was randomly chosen out of 5 durations centred logarithmically on the duration of the standard.

Figure 2 .

 2 Figure 2.a shows the psychometric function of a typical subject's orientation discrimination data fitted with a rescaled Weibull function of the form P (x) = γ + (1 -γ)p(x), where P (x) is the probability of a correct response in the 2AFC paradigm as a function of the orientation x of the target, γ is the guess factor (0.5 in 2AFC) and p(x) is the Weibull function W (µ, σ 2 ) where µ and

Figure 3 :

 3 Figure 3: PSEs (top row) and standard deviations (bottom row) of the fit cumulative Gaussians in the duration estimation task (averaged over all subjects) as a function of the standard duration for easy and difficult conditions (a), correct and incorrect orientation discrimination (b), and high ("sure") and low ("not sure") confidence levels (c). Vertical bars are ±1 SE. The dotted line represents accurate perception of physical duration (slope = 1).

Figure 4 :

 4 Figure 4: Coefficients of variation for easy and difficult conditions (circles), correct and incorrect orientation discrimination (squares), and high and low confidence levels (triangles).

-----

  Initiate the free zone area: A = [0, 2π[ • Initiate the history of gabor positions: P = Ø • Initiate current time: t curr = 0 • Repeat Sample a new position θ curr ∈ A Add the new position to the history of gabors positions: P = P ∪ {(t curr , θ curr )} Delete old positions in the history of gabors positions: P = {(t pre , θ pre ) ∈ P such that (t currt pre ) < T } Update the free zone area: A = [0, 2π[ (t,θ)∈P [θ -ǫ, θ + ǫ[ Update the current time: t curr = t curr + dt

Equally interesting is the fact that similar Weber fraction values apply across a variety of tasks and species, which may indicate that the mechanisms that process time may be more similar than different, an observation that may favour a unified "clock".[START_REF] Matell | Neuropsychological mechanisms of interval timing behavior[END_REF] approach this issue from an evolutionary perspective, favouring coincidence-detectors as a fundamental singular timing mechanism. For timing across different species, see(Lejeune & Wearden,

2006;[START_REF] Penney | Categorical scaling of duration bisection in pigeons (columba livia), mice (mus musculus), and humans (homo sapiens)[END_REF].

It is important to note that this boundary is not clear cut, 250 ms:[START_REF] Spencer | Evaluating dedicated and intrinsic models of temporal encoding by varying context[END_REF]; 500 ms:[START_REF] Michon | The compleat time experiencer[END_REF].

An unlikely event given the fast presentation pace.

& & & & ( &

Appendices

A Locations of Gabor patches

Gabor patches are used as our stimuli in our experiment. Gabors are displayed on a virtual circle of radius R around the fixation point of coordinates (x F , y F ) whose equation is given by:

(1)

The solutions of Eq. 1 are given by:

To obtain random locations on the virtual circle, we uniformly sample values in the interval [0, 2π[ using the Matlab command 2 * pi * rand. This provides one angle θ which then gives us one position (x, y) using The mean µ and the standard deviation σ are indicated as well as the errors of prediction on the observations:

enough to the optimal solution, and then can be used to initialize an iterative gradient descent to solve the orginal optimization problem:

where t is the current time index during the gradient descent and ρ > 0 is the descent step.

B.1 The case of the erf psychometric function

We are looking for the values of σ and µ such that for all observations i:

We can show that solving the problem in ( 9) is equivalent to solving the following:

Estimates of σ and µ are then obtained by minimizing a linear least squares problem in the set of observations (ŷ i , x i ) which can be performed easily with the Matlab function polyfit.

B.2 The case of the logistic psychometric function

We are looking for the values of σ and µ such that for all observations i:

We can show that solving the problem in ( 12) is equivalent to solving the following:

where ŷi = log y -γ 1 -y (14)

Estimates of σ and µ are then obtained by minimizing a linear least squares problem in the set of observations (ŷ i , x i ) which can be performed easily with the Matlab function polyfit.

B.3 The case of the Weibull psychometric function

We are again looking for the values of σ and µ such that for all observations i:

We can show that solving the problem in ( 15) is equivalent to solving the following:

where ŷi = log -log 1 -

Estimates of σ and µ are then obtained by minimizing a linear least squares problem in the set of observation (ŷ i , x i ) which can be performed easily with the Matlab function polyfit.

C Supplementary information