Marc Lachièze-Rey

Jeffrey Weeks

Orbifold construction of the modes of the Poincaré dodecahedral space

Keywords: Poincaré dodecahedral space, spherical 3-manifold, eigenmodes of the Laplace operator, Hopf fibration, multipole vectors, orbifold

We provide a new construction of the modes of the Poincaré dodecahedral space S 3 /I * . The construction uses the Hopf map, Maxwell's multipole vectors and orbifolds. In particular, the *235-orbifold serves as a parameter space for the modes of S 3 /I * , shedding new light on the geometrical significance of the dimension of each space of k-modes, as well as on the modes themselves.

Introduction

Cosmological motivations [START_REF] Lachièze-Rey | Cosmic Topology[END_REF] have inspired recent progress in understanding the eigenmodes of the spherical spaces S 3 /Γ * , i.e., the quotients of the three-sphere S 3 by a binary polyhedral group Γ * . Such modes may be seen as the Γ * -invariant solutions of the Helmoltz equation in the universal cover S 3 . Their numeration and degeneracy were given by Ikeda [START_REF] Ikeda | On the spectrum of homogeneous spherical space forms[END_REF]. Recent works [START_REF] Lachièze-Rey | Laplacan eigenmodes for spherical spaces[END_REF][START_REF]Predicting the CMB power spectrum for binary polyhedral spaces[END_REF][START_REF] Lachièze-Rey | Eigenmodes of Dodecahedral space[END_REF] have provided various means to calculate them.

Here we give a new point of view, using the Hopf map, multipole vectors and orbifolds to construct the modes of S 3 /Γ * and shed additional light on the geometrical significance of Ikeda's formula. Section 2 reviews the Hopf map and uses it to lift eigenmodes from S 2 to S 3 . Section 3 uses twist operators to extend the lifted modes to a full eigenbasis for S 3 . Section 4 generalizes the preceding results from the modes of S 3 to the modes of a spherical space S 3 /Γ * , showing that the latter all come from the lifts of the those eigenmodes of S 2 that are invariant under the corresponding (non-binary) polyhedral group Γ. We then turn to a detailed study of the Γ-invariant modes of S 2 . Section 5 recalls Maxwell's multipole vector approach and uses it to associate each mode of S 2 /Γ to a Γ-invariant set of multipole directions. Restricting attention to the case that Γ is the icosahedral group, Section 6 introduces the concept of an orbifold and re-interprets a Γinvariant set of multipole directions as a (much smaller) set of points in the *235-orbifold, which serves as the parameter space. Section 7 pulls together the results of the preceding sections to summarize the construction of the modes of the Poincaré dodecahedral space and state the dimension of the mode space for each k.

2 From S 2 to S 3 : lifting with the Hopf map

Spheres

We parameterize the circle S 1 as the set of points α ∈ C of unit norm α ᾱ = 1. The relationship between the complex coordinate α and the usual Cartesian coordinates (x, y) is the natural one: α = x + iy.

We parameterize the 2-sphere S 2 as the set of points (x, y, z) ∈ R 3 of unit norm x 2 + y 2 + z 2 = 1.

We parameterize the 3-sphere S 3 as the unit sphere in C 2 : the set of points (α, β) ∈ C 2 of unit norm α ᾱ+β β = 1. Hereafter, we will always assume that this normalization relation holds. The relationship between the complex coordinates (α, β) and the usual Cartesian coordinates (x, y, z, w) is the natural one: α = x + iy and β = z + iw.

The Hopf fibration

In S 3 , simultaneous rotation in the α-and β-planes defines the Hopf flow H t : S 3 → S 3 , H t (α, β) ≡ ( e it α, e it β ).

(

) 1
The Hopf flow is homogeneous in the sense that it looks the same at all points. An orbit

{ (e it α, e it β) | 0 ≤ t < 2π}. ( 2
)
is a great circle on S 3 called a Clifford parallel (Figure 1). Collectively the Clifford parallels comprise the Hopf fibration of S 3 . The fibers carry Clifford's name because William Kingdon Clifford (1845 -1879) discovered them before Heinz Hopf (1894 -1971) was born. However, while Clifford understood the fibration quite well, he did not, as far as we know, go on to consider the quotient map (Eqn. ( 3)).

As we walk along any given Clifford parallel (e it α, e it β), the ratio of its coordinates e it α e it β remains a constant α β , independent of t. The ratio α β labels uniquely each Clifford parallel, taking values in the extended complex numbers C ∪ {∞}, where ∞ represents the ratio α β = 1 0 . The extended complex numbers may be visualized as a Riemann sphere, proving that the Clifford parallels are in one-to-one correspondence with the points of a topological 2-sphere S. The Hopf map is defined as sending any point (α, β) of S 3 to the fiber its belong to, i.e., the point of S labelled by α β . Composing with a natural map from S to the unit 2-sphere S 2 gives an explicit formula for the Hopf map:

p : S 3 → S 2 (α, β) → p(α, β) = (x, y, z) = α β + ᾱβ, -i (α β -ᾱβ), β β -α ᾱ . (3)
It is easy to check that x 2 + y 2 + z 2 = 1, confirming that the Hopf map p sends S 3 to the the unit 2-sphere.

Lifts of functions

Any given function f on S 2 lifts to a function F on S 3 by composition with the Hopf map p from Equation (3),

F : S 3 p -→ S 2 f -→ R. (4)
In other words, F = p * f is the pull-back of f by p: explicitly,

F (α, β) ≡ f (p(α, β)) = f α β + ᾱβ, -i (α β -ᾱβ), β β -α ᾱ . (5)
For example, the quadratic polynomial

f (x, y, z) = x 2 -y 2 , (6)
lifts to the quartic polynomial

F (α, β) = (α β + ᾱβ) 2 -(-i (α β -ᾱβ)) 2 = 2(α 2 β2 + ᾱ2 β 2 ). ( 7
)
Definition 2.3.1. We call a function F : S 3 → R vertical if it is constant along every Clifford parallel (Formula (2)).

For every function f :

S 2 → R, the construction of the lift F (α, β) = f (p(α, β)) guarantees that F is vertical. Proposition 2.3.2.
The Hopf map lifts a polynomial f : S 2 → R of degree ℓ to a polynomial F : S 3 → R of degree 2ℓ.

Proof. The lifting formula (5) doubles the degree of any polynomial.

Eigenmodes

Basic definitions

Definition 3.1.1. An ℓ-eigenmode is an eigenmode f : S 2 → R of the Laplacian, with eigenvalue λ ℓ = ℓ(ℓ + 1).

An ℓ-eigenmode is a solution of the Helmholtz equation

∆ S 2 f = ℓ(ℓ + 1)f. ( 8
)
The index ℓ takes values in the set {0, 1, 2, . . .}. For each ℓ, the ℓ-eigenmodes (which are the usual spherical harmonics) form a vector space V ℓ of dimension 2ℓ + 1.

Definition 3.1.2. A k-eigenmode is an eigenmode F : S 3 → R of the Laplacian with eigenvalue λ k = k(k + 2).

A k-eigenmode is a solution of the Helmholtz equation

∆ S 3 F = k(k + 2) F. (9)
The index k takes values in the set {0, 1, 2, . . .}. For each k, the k-eigenmodes form a vector space V k of dimension (k + 1) 2 .

3.2 Eigenmodes of S 2 define eigenmodes of S 3 Proposition 3.2.1. An ℓ-eigenmode f on the unit 2-sphere lifts to a k-eigenmode F on the unit 3-sphere, with k = 2ℓ.

Proof. It is well-known that the ℓ-eigenmodes are precisely the homogenous harmonic polynomials of degree ℓ on R 3 , with domain restricted to the unit 2-sphere. Similarly the k-eigenmodes are the homogeneous harmonic polynomials of degree k on R 4 , with domain restricted to the unit 3-sphere. A harmonic function on IR 4 satisfies

∆ R 4 F ≡ 4 (∂ α ∂ ᾱ + ∂ β ∂ β )F = 0. ( 10
)
When F is the pull-back of f given by ( 5), direct calculations give

∆ R 4 F (α, β) = (∂ x ∂ x + ∂ y ∂ y + ∂ z ∂ z )f (x, y, z) = ∆ R 3 f (x, y, z). (11)
Thus, the pull-back of a harmonic function on IR 3 is a harmonic function on IR Let Y km0 = Y ℓm • p, with k = 2ℓ, denote the pullback of Y ℓm under the action of the Hopf map (3). In accordance with Proposition 2.3.2, its degree is k. For example,

Y 2,0 = 5 16π (x 2 + y 2 -2z 2 ) lifts to Y 4,0,0 = 5 4π (α 2 ᾱ2 -4α ᾱβ β + β 2 β2 ), of degree 4.
The Y km0 are simply the realization of the Y ℓm on the abstract 2-sphere S of Clifford parallels. As such, the linear independence of the Y ℓm immediately implies the linear independence of the Y km0 as well.

Twist

Each Y km0 is constant along Clifford parallels, but more general functions are not. As we take one trip around a Clifford parallel (e it α 0 , e it β 0 ), 0 ≤ t ≤ 2π, the value of the monomial α a ᾱb β c βd varies as e i(a-b+c-d) times the constant α a

0 ᾱ b 0 β c 0 β d 0 .
In other words, the value of a typical monomial α a ᾱb β c βd rotates counterclockwise (a -b + c -d) times in the complex plane as we take one trip around any Clifford parallel. The graph of the monomial is a helix sitting over the Clifford parallel, motivating the following definition. Definition 3.3.1. The twist of a monomial α a ᾱb β c βd is the power of the unbarred variables minus the power of the barred variables, i.e. a -b + c -d. The twist of a polynomial is the common twist of its terms, in cases where those twists all agree; otherwise it is undefined.

Proposition 3.3.2. The polynomials of well-defined twist (including all monomials) are precisely the eigenmodes of the operator

α∂ α -ᾱ∂ ᾱ + β∂ β -β∂ β , (12)
with the twist as eigenvalue.

Proof. Apply the operator to α a ᾱb β c βd and observe the result.

Geometrically, operator ( 12) is essentially the directional derivative operator in the direction of the the Clifford parallels, the only difference being that the directional derivative includes a factor of i that operator (12) does not, because the complex-valued derivative is 90 • out of phase with the value of the function itself.

Because we consider modes of even k only, the twist will always be even. Henceforth, for notational convenience, we shall take our twist-measuring operator to be

Z = 1 2 (α∂ α -ᾱ∂ ᾱ + β∂ β -β∂ β ). ( 13
)
The ad hoc factor of 1/2 transforms the range of eigenmodes from even integers to all integers.

Siblings and the twist operators

The twist operators

twist ≡ -β ∂ α + ᾱ ∂ β twist ≡ -β ∂ ᾱ + α ∂ β (14)
(defined in [START_REF] Weeks | Exact polynomial eigenmodes for homogeneous spherical 3-manifolds[END_REF]) increase and decrease a function's twist. That is, the twist operator converts an n-eigenmode of Z to an (n + 1)-eigenmode of Z, and inversely for twist. Here is the proof: It is easy to check that the commutator [Z, twist] = twist, so given ZF = λF it follows that

Z(twistF ) = (Ztwist)F = (twistZ + twist)F = (λ + 1)(twistF ).
Thus the operator twist increases by one unit the eigenvalue of an eigenfunction of Z, and similarly twist decreases it by one unit. Because ∆ S 3 and twist commute (see [START_REF] Weeks | Exact polynomial eigenmodes for homogeneous spherical 3-manifolds[END_REF]), the twist operator transforms each keigenmode into another k-eigenmode.

Being vertical, each Y km0 is an eigenmode of Z with eigenvalue 0. Repeatedly applying the operator twist gives eigenmodes of Z with eigenvalues 1, 2, . . . , k/2 (k is even), while repeatedly applying the operator twist gives modes with eigenvalues -1, -2, . . . , -k/2. Why do the sequences stop at n = ±k/2? The explanation is as follows. When written as a polynomial in the complex variables {α, ᾱ, β, β}, the original vertical mode Y km0 contains equal powers of the barred variables ᾱ and β and the unbarred variables α and β. The operator twist replaces a barred variable with an unbarred one, keeping the degree constant while increasing the difference #unbarred-#barred by two. After k 2 applications of the twist operator, the polynomial twist k/2 Y km0 contains unbarred variables alone: it has maximal positive twist and further application of the twist operator collapses it to zero. Analogously, the twist operator converts unbarred variables to barred ones, until twist k/2 Y km0 consists of barred variables alone, after which further applications of twist collapse it to zero. Let Y kmn be the resulting modes. That is, for n = 1, 2, . . . , k/2, define

Y k,m,+n = twist n Y km0 Y k,m,-n = twist n Y km0 (15)
Each Y k,m,n is simultaneously a k-eigenmode of the Laplacian and an n-eigenmode of Z.

The modes {Y k,m,n } n=-k/2...k/2 , being eigenmodes with different eigenvalues, are linearly independent [START_REF] Weeks | Exact polynomial eigenmodes for homogeneous spherical 3-manifolds[END_REF]. Conclusion: each Y ℓm generates, via the lift from S 2 to S 3 (Sections 2.3 and 3.2) and the twist operators, a (k + 1)-dimensional vector space V km• of k-modes, with basis {Y k,m,n } n=-k/2...k/2 (see Table 1). Thus the 2ℓ + 1 = k + 1 spherical harmonics Y ℓm generate the complete vector space of k-eigenmodes of S 3 ,

V k = m V km• , Y k,+ℓ,-k/2 ← • • • ← Y k,+ℓ,-1 ← Y k,+ℓ,0 → Y k,+ℓ,+1 → • • • → Y k,+ℓ,+k/2 ⇑ Y ℓ,+ℓ . . . . . . . . . . . . . . . Y k,+1,-k/2 ← • • • ← Y k,+1,-1 ← Y k,+1,0 → Y k,+1,+1 → • • • → Y k,+1,+k/2 ⇑ Y ℓ,+1 Y k,0,-k/2 ← • • • ← Y k,0,-1 ← Y k,0,0 → Y k,0,+1 → • • • → Y k,0,+k/2 ⇑ Y ℓ,0 Y k,-1,-k/2 ← • • • ← Y k,-1,-1 ← Y k,-1,0 → Y k,-1,+1 → • • • → Y k,-1,+k/2 ⇑ Y ℓ,-1 . . . . . . . . . . . . . . . Y k,-ℓ,-k/2 ← • • • ← Y k,-ℓ,-1 ← Y k,-ℓ,0 → Y k,-ℓ,+1 → • • • → Y k,-ℓ,+k/2 ⇑ Y ℓ,-ℓ
Table 1: Each ℓ-eigenmode Y ℓ,m of S 2 (i.e. each spherical harmonic; middle column, lower entries) lifts via the Hopf map (⇑) to a 0-twist k-eigenmode Y k,m,0 of S 3 (middle column, upper entries), with k = 2ℓ. The positive twist operator (→) then takes Y k,m,0 to its k 2 positively twisted siblings (right side) while the negative twist operator (←) takes Y k,m,0 to its k 2 negatively twisted siblings (left side), for a total of (k + 1) 2 linearly independent modes. with basis {Y k,m,n } m=-k/2...k/2, n=-k/2...k/2 , and thus of dimension (k + 1) 2 .

Proposition 3.4.1. Y k,m,n = Y k,-m,-n .
Proof. Each Y k,+m,0 is conjugate to the corresponding Y k,-m,0 because they are lifts of the standard 2-dimensional spherical harmonics Y ℓ,+m and Y ℓ,-m which have this symmetry. The twist operators (14) are complex conjugates of one another by construction. Therefore

when n ≥ 0, Y k,m,n = twist n Y k,m,0 = twist n Y k,-m,0 = Y k,-m,-n , (16)
and similarly when n ≤ 0.

Proposition 3.4.2. By choosing complex-conjugate coefficients c k,m,n = c k,-m,-n one may recover the real-valued modes of S 3 as

c kmn Y k,m,n + c kmn Y k,-m,-n (17)
In particular, whenever m and n are not both zero, the modes

Y k,m,n + Y k,-m,-n (18)
i Y k,m,n -i Y k,-m,-n (19)
are independent real-valued modes, analogous to cosine and sine, respectively.

Proof. The mode (17) is its own complex conjugate,

c kmn Y k,m,n + c kmn Y k,-m,-n = c kmn Y k,m,n + c kmn Y k,-m,-n (20)
and therefore real.

Convention 3.4.3. For the remainder of this article we will assume that all coefficients are chosen in complex-conjugate pairs c k,m,n = c k,-m,-n and therefore all modes are realvalued.

4 Eigenmodes of spherical spaces S 3 /Γ * A spherical space is a quotient manifold M = S 3 /G, with G a finite subgroup of SO( 4).

An eigenmode of M with eigenvalue k (k + 2) corresponds naturally to a k-eigenmode of S 3 that is G-invariant. The set of all such modes forms a subspace V k M of the vector space V k of all k-eigenmodes of S 3 . In the present article we focus on the case that G is a binary polyhedral group Γ * , because those spaces holds the greatest interest for cosmology as well as being technically easier.

4.1 Vertical modes of S 3 /Γ * generate all modes of S 3 /Γ * We will now show that when searching for Γ * -invariant eigenmodes, we may safely restrict our attention to the vertical ones. Proposition 4.1.1. Every Γ * -invariant mode of S 3 may be obtained from vertical Γ *invariant modes by applying the twist operators and taking a sum.

Proof. Let F be an arbitrary Γ * -invariant mode of S 3 (not necessarily vertical). Express F relative to the basis Y kmn (Table 1) as

F = kmn c kmn Y kmn = kn m c kmn Y kmn = kn F kn , (21)
where F kn ≡ m c kmn Y kmn is the component of F that is simultaneously a k-eigenvalue of the Laplace operator ∆ S 3 and an n-eigenvalue of the twist-measuring operator Z (Equation ( 13)). By assumption each element γ ∈ Γ * preserves F . Because γ commutes with both ∆ S 3 and Z, it must preserve each F kn individually. (Unlike an arbitrary element of SO(4), the isometry γ commutes with Z because γ takes Clifford parallels to Clifford parallels.) Thus each F kn is Γ * -invariant.

Because F kn has constant twist, it is easily obtained by applying the twist operator to a vertical function,

F kn = m c kmn Y kmn = m c kmn twist n Y km0 = twist n m c kmn Y km0 , (22)
where for negative n, twist n means twist |n| . Because the twist operators twist and twist commute with each γ, each vertical function m c kmn Y km0 is Γ * -invariant, thus completing the proof.

Like for S 3 , the search for the eigenmodes of S 3 /Γ * reduces to a search for the vertical ones, since each vertical Γ * -invariant k-eigenmode generates, through the action of the twist operators, a (k + 1)-dimensional vector space of generic Γ * -invariant k-eigenmodes.

4.2 Modes of S 2 /Γ generate all vertical modes of S 3 /Γ * Section 3.2 showed that the vertical modes of S 3 are the pullbacks of the modes of S. Thus in a direct geometrical sense, the modes of S 2 are the vertical modes of S 3 , and Γ * -invariance on S 3 corresponds directly to Γ-invariance on S 2 . 5 Γ-invariant eigenmodes of S 2

Multipole vectors

Consider V ℓ , the vector space of ℓ-eigenmodes. According to Maxwell's multipole vector decomposition of modes [START_REF] Clerk | A Treatise on Electricity and Magnetism[END_REF][START_REF] Dennis | [END_REF]9,10,11,12,13], we may write each eigenmode f ℓ ∈ V ℓ as

f ℓ (x, y, z) = c r 2ℓ+1 ∇ v ℓ • • • ∇ v 2 ∇ v 1 1 r , (23)
where r = x 2 + y 2 + z 2 and the decomposition is well defined up to flipping the signs of the direction vectors {v 1 , . . . , v ℓ } and the scale factor, two at a time. The ordering of the direction vectors is irrelevant. Define an equivalence relation on V ℓ setting two functions f and f ′ to be equivalent whenever they are nonzero real multiples of each other:

f ≃ f ′ ⇔ f = cf ′ , c ∈ R -{0}.
All the elements of each equivalence class [f ] share the same decomposition (23) up to the choice of signs for the direction vectors {v 1 , . . . , v ℓ } and the leading constant c. Therefore each equivalence class [f ] is uniquely represented by a set of directions {d 1 , ..., d ℓ }, where each direction d i represents a line ±v i , with no concern for the sign. The set of all possible directions forms a real projective plane IRP 2 = S 2 /±Id.

Invariant sets of directions

A class [f ] of modes is Γ-invariant iff the associated set {d 1 , ..., d ℓ } is Γ-invariant. Note that although each symmetry γ ∈ Γ is nominally a map γ : S 2 → S 2 , its action on IRP 2 is well defined. To understand the possible classes [f ] of Γ-invariant modes, we need to understand the possible Γ-invariant sets {d 1 , ..., d ℓ } of directions.

6 Eigenmodes of the Poincaré dodecahedral space S 3 /I * Let us now further restrict our attention to the Poincaré dodecahedral space, because of the interest it holds in cosmology as well as its greater technical ease. In other words, let Γ be the icosahedral group I comprising the 60 orientation-preserving symmetries of a regular icosahedron. We will consider sets of directions {d 1 , ..., d ℓ } that are invariant under I. Because each direction d i is automatically invariant under the antipodal map, the set {d 1 , ..., d ℓ } will be invariant under the full group I h of 120 symmetries of a regular icosahedron, reflections included.

The orbifold

The quotient S 2 /I h is an orbifold consisting of a spherical triangle with mirror boundaries and corner reflectors with angles π/2, π/3 and π/5 (see Figure 2). In Conway's notation this orbifold is denoted *235.

• Each point in the interior of the triangle lifts to an invariant set of 120 points on S 2 , which in turn defines an invariant set of 60 directions.

• However, each point on a mirror boundary lifts to only 60 points on S 2 , defining only 30 directions. For this reason it's convenient to think of a point on the mirror boundary as a "half-point".

• A point at the corner reflector of angle π/2 lifts to 30 points on S 2 or 15 directions, so it's convenient to think of it as a "quarter point".

• Similarly, the points at the corner reflectors of angle π/3 and π/5 may be considered a 1/6 point and a 1/10 point, respectively.

In all cases, a 1 F fractional (F = 1, 2, 4, 6 or 10) point of S 2 /I h represents an invariant set of ℓ = 60 F directions. Another way to think about it is that a half-point on the mirror boundary lifts to 120 half-points on S 2 , and then each pair of identically positioned half-points combines to form a single full point, and similarly for the other fractional points.

+ 10C 1 6 + 15C 1 4 + 30C 1 2 + 60C 1 . ( 24
)
Some care is required here: knowing that an equivalence class [f ] of modes is I-invariant does not immediately imply that each representative f of that class is I-invariant. It is a priori possible that some symmetry γ ∈ I could send f to -f . The next proposition shows that this does not happen.

Proposition 6.1.3. If an equivalence class [f ] of modes of S 2 is invariant under the icosahedral group I, then each representative f is also invariant under I.

Proof. Let {d 1 , ..., d ℓ } be the set of I-invariant directions defining the class [f ] (Section 5.2), and let γ ∈ I be a symmetry of the icosahedron. By the assumed I-invariance of [f ], we know that γ sends each d i to ±d j (for some j). To prove that f itself is invariant, it suffices to prove that γ sends d i to -d j (rather than to +d j ) for an even number of the d i .

First consider the case that a given d i lies in the "interior" of the *235-orbifold (Figure 2c). This implies that 59 other d i (for different values of i) lie in the interiors of other copies of the fundamental triangle (Figure 2b), arranged symmetrically . Each right-handed copy of the fundamental triangle lies antipodally opposite a left-handed copy (Figure 2b). If we make the convention to orient each of the 60 d i in question so that it points toward a right-handed copy of the triangle and away from a left-handed copy, then every γ ∈ I will preserve those d i exactly, always sending a d i to a +d j , never to a -d j .

Next consider the case that some d i lies on the perimeter (the mirror boundary) of the *235-orbifold's fundamental triangle. In this case it has only 30 translates under the group (including itself). The icosahedral group I consists entirely of rotations, each about some vertex of the tiling (Figure 2b). Let γ be some such rotation. In the generic case that none of the 30 d i lies exactly 90 • from the rotation axis of γ, we may orient all 30 d i to point towards the "northern hemisphere" (relative to γ's rotation axis) and away from the "southern hemisphere". In this case γ sends each d i to a +d j , never to a -d j . In the non-generic case that some of the d i lie exactly on the "equator" relative to γ's rotation axis, consider the three sub-cases that the rotation γ has order 2, 3 or 5. When γ is a rotation of order 3 or 5, easy ad hoc conventions serve to orient the equatorial d i so that γ respects their orientations. When γ is a rotation of order 2, it perforce takes each d i to -d i , but there are exactly two such d i , so the net effect is still that γ maps the mode f to +f , not -f . Finally, consider the case that some d i lies isolated at one of the fundamental triangle's vertices. According to whether the vertex is a corner reflector of order 2, 3 or 5, d i will have 15, 10 or 6 translates (including itself), respectively. Imitating the method of the preceding paragraph, we consider a rotation γ ∈ I, and wherever possible orient the d i to point towards the northern hemisphere and away from the southern hemisphere, thus ensuring that γ permutes such d i respecting orientation. It remains to consider only the d i that lie on the equator relative to γ's rotation axis. When γ has order 3 or 5, its equator contains corner reflectors of order 2 only, and an ad hoc convention serves to orient them consistently. When γ has order 2, it maps each equatorial d i to -d i , but the equator contains exactly four corner reflectors of order 2, four corner reflectors of order 3 and four corner reflectors of order 5, so in each sub-case the equator contains exactly two of the directions d i (from among the complete set of 15, 10 or 6 directions under consideration), and because exactly two directions get flipped, we conclude that γ maps the mode f to +f , not -f . Corollary 6.1.4. Any value of ℓ not expressible in the form (24), for example ℓ = 14, cannot be the degree of an eigenmode of S 2 /I. Corollary 6.1.5. The nontrivial I-invariant mode of S 2 of least degree has degree l = 6.

Dimension of the space of modes

Proposition 6.2.1. The I-invariant mode of degree l = 6 is unique up to a constant multiple. Thus dim(V 6 ) = 1.

Proof. To construct this mode, take the *235 orbifold and place a single 1/10 point at the corner reflector of angle π/5. This 1/10 point lifts to 12 points of S 2 which in turn define 6 directions. According to Maxwell's formula (23), those 6 directions define an Iinvariant class of modes [f ] of degree 6. By Proposition 6.1.3, each representative f of [f ] is I-invariant. Assuming a fixed realization of the icosahedral group I, the 6 directions are well defined -they align with the vertices of an icosahedron or the face centers of a dodecahedron. Therefore the class [f ] is also well defined, and the only degree of freedom for the mode f is the scale factor inherent in the equivalence class [f ]. Thus V 6 is of dimension 1.

The method of the preceding proposition lets us construct I-invariant modes of degree 10 (place a 1/6 point at the corner reflector of angle π/3) and degree 15 (place a 1/4 point at the corner reflector of angle π/2), while proving that I-invariant modes of most other low degrees cannot exist. V 10 and V 15 have dimension 1.

The case of degree 30, realized by placing a half-point on the *235 orbifold's mirror boundary, is more interesting because we have an extra degree of freedom corresponding to where we choose to place the half-point. Allowing for the scale factor inherent in the equivalence class [f ] gives a total of two real degrees of freedom: V 30 has dimension 2.

The case of degree 60, corresponding to one full point in the *235 orbifold, is more interesting still, because now we have a choice as to how we realize that one full point:

• Case 1. We may place a single full point anywhere in the orbifold.

• Case 2. We may place two half-points on the orbifold's mirror boundary. In the special case that the two half-points coincide, we get a single full point as in Case 1.

• Case 3. We may place any combination of fractional points at the orbifold's corner reflectors, just so the fractions sum to one. However it turns out that the only ways to do this are to place a full point at a single corner (for example realized as ten 1/10 points at the corner of angle π/5) or to place a half-point at each of two corners (for example realized as five 1/10 points at the corner of angle π/5 plus three 1/6 points at the corner of angle π/3). The full point corresponds to Case 1 while the two half-points correspond to Case 2, so nothing new arises here and we will henceforth ignore this Case 3.

• Case 4. We may place a half-point on the mirror boundary and a half-point's worth of fractional points at the corner reflectors, but as in Case 3 nothing new arises here so we may ignore this possibility.

by

dim(V ℓ ) = 1 + C 1 2 + 2C 1 . (25)
Note that no matter how many half points may or may not combine into whole points, the half and whole points together contribute C 1 2 + 2C 1 degrees of freedom.

Improved dimension formula

The dimension formula (25) is nice, but we would much rather have a formula in terms of ℓ, to save us the trouble of manually decomposing ℓ into a linear combination of the C i .

Here is the improved formula, Proposition 6.3.1. The dimension of the space of I-invariant ℓ-eigenmodes of S 2 is given by But the number of isolated 1 10 points may only be 0, 1, 2, 3 or 4, because if we had 5 or more 1 10 points they would combine to form half points and acquire an additional degree of freedom. So the number of isolated This agrees with Ikeda's formula [START_REF] Ikeda | On the spectrum of homogeneous spherical space forms[END_REF], while at the same time providing a concrete construction of the modes and shedding additional light on the formula's geometrical origins, as degrees of freedom in an orbifold.

dim(V ℓ ) = 1 + ⌊ ℓ 2 ⌋ + ⌊ ℓ 3 ⌋ + ⌊ ℓ 5 ⌋ -ℓ. (26

Conclusion

Returning to the 3-dimensional Poincaré dodecahedral space S 3 /I * , the results of the preceding sections may be summarized as follows. Keep in mind that S 3 /I * admits k-modes for even k only; odd k-modes cannot exist because I * contains the antipodal map. • Each mode of S 3 /I * corresponds to an I * -invariant mode of S 3 (elementary).

• Each I * -invariant mode of S 3 is a sum of twists of I * -invariant vertical modes of S 3 (Proposition 4.1.1).

• Each I * -invariant vertical k-mode of S 3 is the pull-back, under the Hopf map, of an I-invariant ℓ-mode of S 2 , with k = 2ℓ (Proposition 3.2.1).

• The I-invariant ℓ-modes of S 2 are parameterized by ℓ/60 points on the *235-orbifold, possibly including fractional points (Section 6.1).

Theorem 7.2 The space of k-modes of the Poincaré dodecahedral space S 3 /I * has dimension

(k + 1)(1 + ⌊ k/2 2 ⌋ + ⌊ k/2 3 ⌋ + ⌊ k/2 5 ⌋ - k 2 ). (29)
Proof. The space of I-invariant k/2-modes of the 2-sphere has dimension 1 + ⌊ k/2 2 ⌋ + ⌊ k/2 3 ⌋ + ⌊ k/2 5 ⌋ -k 2 (Proposition 6.3.1) and thus the space of vertical I * -invariant k-modes of the 3-sphere has this same dimension (Theorem 7.1). The twist operators then take each vertical mode to a (k + 1)-dimensional space of generic I-invariant modes (Table 1 and Proposition 4.1.1,), completing the proof.

Figure 1 :

 1 Figure 1: A computer generated view of the Clifford parallels in S 3

Conclusion 4 . 2 . 1 .

 421 The search for Γ * -invariant eigenmodes of S 3 reduces to the search for Γ-invariant eigenmodes of S 2 .

Figure 2 :

 2 Figure 2: How to construct the *235 orbifold. (a) Begin with an icosahedrally symmetric pattern on the 2-sphere. (b) Locate all lines of mirror symmetry. Each is a great circle, and together they divide the sphere into 120 congruent triangles. (c) Fold the sphere along all mirror lines simultaneously, so that the whole sphere maps 120-to-1 onto a single triangle. The resulting quotient is the *235 orbifold. The Conway notation *235 may be understood as follows: the '*' denotes the mirror-symmetric origin of the triangle's sides, while the 2, the 3 and the 5 denote the fact that 2, 3 and 5 mirror lines met at each corner, respectively.

denote the number of 1 6 1 4 1 2

 611 points at the vertex of angle π/3, C denote the number of 1 4 points at the vertex of angle π/2, C denote the number of half points on the triangle's perimeter, and C 1 denote the number of whole points in the triangle's interior. The preceding discussion has shown that Proposition 6.1.2. Each I-invariant equivalence class [f ] of modes of S 2 corresponds to a unique choice of ℓ I-invariant multipole vectors. The degree of a representative mode f is ℓ = 6C 1 10

10 , 10 (

 1010 the C i depend on ℓ. First consider C 1 the number of1 10 points. Taking Equation (27) modulo 5 we get ℓ ≡ C 1 mod 5).

1 10 points must be C 1 10 = 6 = ℓ -3⌊ ℓ 3 ⌋ and C 1 4 = 1 2+

 106341 ℓ-5⌊ ℓ 5 ⌋. The same argument, repeated mod 3 and mod 2, gives C 1 ℓ -2⌊ ℓ 2 ⌋, respectively. Rearranging Equation (27) now gives C 28) into Equation (25) gives the final result (26) as stated above.

Theorem 7 . 1

 71 To construct the modes of the Poincaré dodecahedral space S 3 /I * ,

Proposition 6.2.2. The I-invariant classes [f ] of modes of S 2 of degree l = 60 are parameterized by a real projective plane.

Proof. Each class [f ] of degree 60 corresponds to 60 directions {d 1 , ..., d 60 } that are invariant under the icosahedral group I, which in turn correspond either to a single point in the *235 orbifold (Case 1 above) or to a pair of half-points on the mirror boundary (Case 2 above).

The possible locations for a whole point are obviously parameterized by the points of the orbifold itself, which is topologically a disk.

The possible locations for a pair of points on the orbifold's mirror boundary are parameterized by a Möbius strip. To see why, first note that the mirror boundary is topologically a circle S 1 . Parameterize this circle in some arbitrary but fixed way, with the parameter angle defined modulo 2π, and then for any pair of points define θ = the position of the two points' "center of mass" (θ ∈ S 1 = R/2π) φ = the separation between the two points (φ ∈ [0, π]) At first glance this gives a cylinder parameterized by (θ, φ). But (θ, π) and (θ + π, π) define the same pair of points, so we must identify opposite points on the cylinder's upper boundary circle (θ, π) ∼ (θ + π, π), which transforms the cylinder into a Möbius strip. The cylinder's lower boundary circle (θ, 0) becomes the Möbius strip's edge.

The Möbius strip's edge, parameterized by (θ, 0), corresponds to the case that the two half-points fuse together to form a single whole point on the triangle's perimeter. This corresponds exactly to the boundary of the disk in the whole point parameter space. In other words, the total parameter space is the union of a disk and a Möbius strip glued together along their boundary circles, which yields a real projective plane.

It's no surprise that the parameter space is a real projective plane. The space of Γinvariant harmonic functions f on S 2 of any fixed degree ℓ is a vector space of some finite dimension n. When we pass from functions f to equivalence classes [f ] we identify each line through the origin to a single point, giving in all cases a real projective space RP n-1 . In the case just considered, with degree ℓ = 60, we found the projective space to be RP 2 meaning the total function space, including the scale factor, is R 3 .

To construct a generic I-invariant mode, we may place any combination of whole points (anywhere in the orbifold), half points (on the orbifold's mirror boundary), and other fractional points (isolated at the orbifold's corner reflectors). Each whole point contributes two degrees of freedom to the space of modes (corresponding to the point's location in the 2-dimensional triangle), each half point contributes one degree of freedom (corresponding to its location along the triangle's 1-dimensional perimeter), and each isolated fractional point contributes nothing. The overall scaling factor contributes one more degree of freedom for any nontrivial mode. In summary, Proposition 6.2.3. The dimension of the space of I-invariant ℓ-eigenmodes of S 2 is given


Federica Appendino
email: federica.appendino@u-pec.fr

Charlotte Roux
email: charlotte.roux@eivp-paris.fr

Myriam Saadé

Bruno Peuportier
email: bruno.peuportier@mines-paristech.fr

Track 10 -Urban metabolism and circular economy Circular economy in urban projects: a case studies analysis of current practices and tools

Keywords: Circular Economy, Life Cycle Assessment, Urban Project, Circular neighbourhood

published or not. The documents may come

Circular economy in urban projects: a case studies analysis of current practices and tools

Federica Appendino, Charlotte Roux, Myriam Saadé, Bruno

Introduction

Over the last 10 years especially, the concept of a circular economy (CE) has been gaining momentum in politics, business and academia [START_REF] Reike | The Circular Economy: New or Refurbished as CE 3.0? -Exploring Controversies in the Conceptualization of the Circular Economy through a Focus on History and Resource Value Retention Options[END_REF][START_REF] Kampelmann | Mesurer l'économie Circulaire à l'échelle Territoriale[END_REF] in the pursuit of overcoming the contradictions between economic and environmental prosperity [START_REF] Geissdoerfer | The Circular Economy -A New Sustainability Paradigm?[END_REF]. Indeed, the current economic model, which can be characterised as 'linear' based on a 'take-make-consume-throw away' approach of resources, is presently reaching its limits. Conversely, the CE forms an "economic system of trade and production which, at all stages of the product lifecycle, aims to increase the efficiency of resource use and reduce the impact on the environment, while developing the well-being of individuals" (ADEME 2014). For these reasons, CE already represents the core theme of major European plans and regulations [START_REF] Petit-Boix | Circular Economy in Cities: Reviewing How Environmental Research Aligns with Local Practices[END_REF], such as the 'Circular Economy Package' adopted in 2015.

Today, CE is studied by several disciplines ranging from economics to urban planning, though it remains closely linked to sustainable development issues [START_REF] Kirchherr | Conceptualizing the Circular Economy: An Analysis of 114 Definitions[END_REF]. However, to date, no univocal and shared definition of CE has yet been developed, despite a wide dissemination of the concept (Prieto-Sandoval, Jaca, and Ormazabal 2017). Thus, CE constitutes an evolving notion [START_REF] Merli | How Do Scholars Approach the Circular Economy? A Systematic Literature Review[END_REF], rather ambiguous and vague [START_REF] Korhonen | Circular Economy as an Essentially Contested Concept[END_REF].

The built environment, given its important contribution to several environmental issues, is supposed to comprise one of the main targets of CE strategies. However, current scientific literature on this subject remains limited, and concrete application of the principle is slow to be implemented (Densley [START_REF] Tingley | Applying Circular Economy Principles To Reduce Embodied Carbon[END_REF][START_REF] Adams | Circular Economy in Construction: Current Awareness, Challenges and Enablers[END_REF][START_REF] Pomponi | Circular Economy for the Built Environment : A Research Framework[END_REF]. Furthermore, little general consensus exists concerning how to best approach and deal with this concept, and the knowledge and tools required to enact it remain to be developed [START_REF] Leising | Circular Economy in the Building Sector: Three Cases and a Collaboration Tool[END_REF]. This is especially true for the building sector, to which CE remains a relatively new topic [START_REF] Adams | Circular Economy in Construction: Current Awareness, Challenges and Enablers[END_REF][START_REF] Bocken | Taking the Circularity to the Next Level: A Special Issue on the Circular Economy[END_REF]. Furthermore, there is still a need to demonstrate and assess its positive environmental impacts. Indeed, implementing CE initiatives not only generates potential benefits, but also a number of environmental risks. 'Closing the loop' does not always positively affect the environment, and therefore, 'circularity' should be analysed against relevant indicators [START_REF] Petit-Boix | Circular Economy in Cities: Reviewing How Environmental Research Aligns with Local Practices[END_REF][START_REF] Kampelmann | Mesurer l'économie Circulaire à l'échelle Territoriale[END_REF]. In addition, it is also necessary to ensure that the most environmentally relevant initiatives are realised. For this reason, the need for systemic methods and tools to validate the environmental relevance of the CE applied to the built environment is now evident [START_REF] Haupt | How Can LCA Support the Circular Economy?-63rd Discussion Forum on Life Cycle Assessment[END_REF][START_REF] Haupt | Do We Have the Right Performance Indicators for the Circular Economy?: Insight into the Swiss Waste Management System[END_REF].

Accordingly, this paper aims to contribute to this growing area of research, which currently remains in its infancy. The results indicate a diverse representation of the CE paradigm and confirm that assessment tools could bring a useful contribution because if CE remains a vague trendy expression, there is a high risk of greenwashing. Life cycle assessment (LCA) appeared in the early 1990's and even if the expression CE was not used at that time, most ideas corresponding to CE were already integrated. For instance, recycling was one of the aspects studied in order to reduce environmental impacts of the built environment. A method has been elaborated that accounted, among other aspects corresponding to the CE approach, for environmental benefits of recycling at the fabrication stage but also after deconstruction [START_REF] Polster | Evaluation of the Environmental Quality of Buildings towards a More Environmentally Conscious Design[END_REF].

CE in the built environment: from the 'circular city' to the 'circular neighbourhood'

As mentioned, the CE approach is gaining momentum in the field of urban sustainability. Several studies, as well as some international meetings, have investigated the role that the CE can play in ensuring a more sustainable city, and the literature on this subject continues to grow (Cities Foundation 2017;Ellen MacArthur Foundation 2017;[START_REF] Prendeville | Circular Cities: Mapping Six Cities in Transition[END_REF].

According to [START_REF] Pomponi | Circular Economy for the Built Environment : A Research Framework[END_REF], three scales of CE deployment can be identified from the scientific literature: the 'macro scale' of the city, the 'meso scale' of the buildings, and finally, the 'micro scale' of the construction elements. However, while academic research is more consistent concerning the macro scale (particularly in urban metabolism and eco-parks), as well as the micro scale (particularly in materials and building components), the meso scale remains poorly explored [START_REF] Pomponi | Circular Economy for the Built Environment : A Research Framework[END_REF].

Considering their pressures on the environment, to date, the scientific literature on CE has focused more attention on the 'circular city' topic.

Several cities, such as Berlin, Rotterdam, Paris, London, Milan and Amsterdam, have recently adopted strategic plans and are launching specific actions and projects to make their economy more circular. In 2014, for instance, the city of Amsterdam adopted 'The Circular Metropolis Amsterdam 2014-2018' 1 , a strategic document aimed at transforming the city into a competitive and sustainable European metropolis. This document, which comprises part of the Amsterdam Smart City initiative, is based on the 'The City Circle Scan' approach, enabling areas to be identified where major CE progress can be made. Thanks to this tool, Amsterdam has decided to focus on the construction sector as well as the organic production and biomass sector. In addition, in 2016, Amsterdam became a Fab-City, an international initiative bringing approximately 20 cities together with the goal of becoming self-sufficient 2 . Similarly, the city of Rotterdam also linked the EC to the smart city initiative, adopting the 'Roadmap Circular Economy Rotterdam' 3 in 2016. The proposed actions to ensure the city's sustainable and circular development by 2030 are based on the results of the 'Rotterdam Metabolism' study, which provided a comprehensive picture of urban flows. Rotterdam's EC strategy focuses primarily on the city's port area for implementing biosourced projects [START_REF] Prendeville | Circular Cities: Mapping Six Cities in Transition[END_REF].

More recently, London and Paris also presented guidance documents in 2017. Following the 2015 General Assembly of the Circular Economy, Paris adopted its first 'Circular Economy Plan 2017-2020' 4 and its operational roadmap. London similarly published a 'Circular Economy Route Map' 5 , containing actions involving the construction, food, textile, plastic and electrical industries. This document is accompanied by an economic analysis, which estimates the benefits to the economy in terms of wealth creation, activities and employment at £2.8bn.

Initiatives and actions are multiplying and global networks are being created, bringing together several cities. The Circular Europe Network (CEN) 6 , for example, brings together dozens of European cities to promote the exchange of best practices. At the international level, the Open Source Circular Economy (OSCE) 7 collects innovative solutions linking EC and open data.

Otherwise, current research on CE has dedicated little attention to the meso scale, whereas some authors have stressed the importance of orienting CE research towards the built environment [START_REF] Leising | Circular Economy in the Building Sector: Three Cases and a Collaboration Tool[END_REF][START_REF] Pomponi | Circular Economy for the Built Environment : A Research Framework[END_REF][START_REF] Glass | Editorial: Circular Economy in the Built Environment[END_REF]. Indeed, in Europe, the built base represents almost half of the total energy consumption, and more than 50% of all extracted materials (European Commission 2011). In France, nearly 40% of energy consumption and 60% of electricity are attributed to this, which is also responsible for the emission of approximately a quarter of national greenhouse gases (ADEME 2012). In addition, the construction sector generates nearly three-quarters of the national waste 8 volume and consumes approximately 600 km 2 of natural areas per year 9 .

In this context, the built environment could truly represent an essential cornerstone for implementing effective CE strategies in the city. Concerning the meso scale, several authors have pointed out that the 'neighbourhood scale', the link between the city and the building, is the most relevant for addressing numerous environmental problems [START_REF] Lotteau | Integration of Morphological Analysis in Early-Stage LCA of the Built Environment at the Neighborhood Scale[END_REF]. In Europe, the attention paid to the neighbourhood scale has even become central to the sustainable city discourse [START_REF] Souami | Écoquartiers, Secrets de Fabrication[END_REF]. By contrast, however, limited research has been conducted concerning the application of CE principles in neighbourhoods, and there is a lack of comprehensive studies summarising recent development. Nevertheless, a number of pathfinder projects are beginning to emerge in the practice. Assessment tools have been developed [START_REF] Popovici | Using life cycle assessment as decision support in the design of settlements[END_REF][START_REF] Herfray | Life Cycle Assessment applied to urban settlements[END_REF][START_REF] Roux | Dynamic LCA applied to buildings and urban districts[END_REF] and applied to the design at neighbourhood scale in Lyon, Champs sur Marne and Paris [START_REF] Peuportier | Towards sustainable neighbourhoods, the eco-housing project[END_REF][START_REF] Peuportier | LCA application in urban design[END_REF][START_REF] Merli | How Do Scholars Approach the Circular Economy? A Systematic Literature Review[END_REF].

On these bases, the present paper aims to address the two following questions: Q1) How can the CE be implemented in a neighbourhood? Q2) What assessment tools are used?

Methodology

The study proposes analysing and comparing four case studies concerning a 'circular neighbourhood'. The case study method was selected because this enables integrating theory and practice, aptly suiting the exploratory nature of this research [START_REF] Leising | Circular Economy in the Building Sector: Three Cases and a Collaboration Tool[END_REF].

To begin, a literature review was undertaken to identify relevant 'circular neighbourhood' cases. Attention was dedicated not only to scientific papers, but also to other types of documents, such as reports and urban planning documents. Indeed, scientific literature was mostly limited to theoretical discourse and only little attention was given to the neighbourhood scale. The research was conducted in English, French and Italian. A total of only four case studies were found and selected: the first case concerns the neighbourhood of Buiksloterham (Amsterdam), which will develop into a sustainable district based on circular principles; the second neighbourhood is Kera (Espoo), an industrial area destined to become a 'liveable circular economy neighbourhood'; the last two cases are based in Paris, namely, the Groues and Saint-Vincent-de-Paul eco neighbourhoods, both considered an 'EC living lab'. All these cases were selected on the basis of two fundamental criteria: the willingness to implement the principles of the CE at the neighbourhood scale and the presence of a comprehensive CE strategy at the city level, within which the project fits in.

Following the case selection, CE initiatives and actions were identified and classified using document analysis and integrating data collected through semi-structured interviews with local stakeholders involved in the projects.

Finally, a conceptual framework analysis was defined based on three main criteria: CE practices, strategic city scale integration, and tools employed. This analytical framework has been applied to the four case studies in order to compare them.

Case studies analyses

Four case studies were selected based on their innovative character and central relevance of the CE. Indeed, in all these cases, CE is put forward as one of the key pillars for the urban projects. As illustrated in Table 1, despite the differences in size and location, the analysed neighbourhoods present some common features. For instance, all four cases constitute urban regeneration projects, and at the same time, they are also experimental and function as showcases to test the CE principles. Moreover, it is important to underline that all the projects are recent and have not yet been carried out or completed. For this reason, this analysis can only be based on the design phase.

Buiksloterham, Amsterdam

Amsterdam represents one of Europe's pioneering cities in terms of CE. In fact, CE comprises one of the main pillars of the Sustainable Amsterdam Agenda (2015), which aims to, by 2020, reduce energy consumption by 20% and increase renewable production by 20% compared to 2013 (van der Hoek, Struker, and de Danschutter 2017). In this strategic document, the Buiksloterham neighbourhood is considered "an engine for the broader transition of Amsterdam" (Metabolic 2015) towards a circular city.

Part of a larger redevelopment plan of the northern banks of the river, Buiksloterham is characterised by abandoned factories, wastelands and docks. In the city's vision, however, this neighbourhood, once the site of Amsterdam's most polluting industries, can become "a key innovation zone for circular urban development" (Metabolic 2016). Furthermore, the municipality proposed a bottom-up approach for the area's redevelopment in order to develop a more comprehensive sustainability strategy. For this reason, in 2015, the 'Circular Buiksloterham Manifesto' was signed by approximately 20 stakeholders, including local actors, organisations, associations and companies. This innovative manifesto includes the shared guiding principles for redeveloping Buikloterham, such as the zerowaste objective, the implementation of clean technologies or the use of biosurced materials.

Recognising the urgency for a clear operational strategy, all the stakeholders involved commissioned an Urban Metabolism Scan in order to understand the neighbourhood's complete workings from a systemic perspective. The analysis, carried out by the Metabolic and published in 2015, was divided into three stages (context analysis, stakeholder analysis and metabolism analysis). This 'Urban Metabolism Scan' focused on material and energy flows, biodiversity, environmental conditions, socio-economic factors, local actors, urban planning documents and plans, health, and living environment.

Following this analysis, a study was conducted concerning the neighbourhood's CE potential. On these bases, the priority objectives for redeveloping Buikloterham as a 'living lab for CE' by 2034 were divided into eight priority issues (Table 2).

OBJECTIVES Energy

Buiksloterham is energy self-sufficient with a fully renewable energy supply

Materials & products

Buiksloterham is a zero waste neighbourhood that with a near 100% circular material flow Water

Buiksloterham is rainproof and has near 100% resource recovery from waste water

Ecosystems and biodiversity

Buiksloterham's ecosystems are regenerated and its base of natural capital is self-renewing

Infrastructure & mobility

Buiksloterham's Infrastructure is maximally-used and local mobility has zero emissions

Socio-cultural

Buiksloterham has a diverse and inclusive culture, and a high quality, livable environment Economy Buiksloterham has a strong local economy that stimulates entrepreneurship and encourages the creation and exchange of multiple kinds of value (social, environmental, cultural)

Health & wellbeing

Buiksloterham is a healthy, safe and attractive environment with recreational activity space for all residents.

Table 2. Buikloterham's objectives.

In particular, with regard to the built environment, it is interesting to note that a 'Circular Building Standard' will be introduced for all renovations or new constructions. This innovative assessment tool, which is still in the development phase, would allow tax credits once the standard has been reached. Among the key recommendations, all building roofs are equipped for clean energy production and rainwater collection, and all materials are registered in a digital passport to facilitate their identification. In addition, prefabricated building elements are preferred, facilitating deconstruction and reuse.

To ensure these objectives, a first action plan was developed. The proposed actions consist of two types: systemic actions, aimed at ensuring the district's long-term transition, and technical actions, concerning more specific issues. For the definition of the actions, prioritisation work was carried out.

In particular, the actions considered most urgent consist of those related to new construction and infrastructure. Consequently, the priority actions concern the energy efficiency of the built stock, the flexibility of new infrastructures, the development of fresh mobility, water recovery and management.

Kera, Espoo

The city of Espoo, Finland is one of the pioneering cities in terms of sustainable development, as demonstrated in a comparative assessment study of 15 European cities carried out in 2017 by the University of Tilburg10 . Indeed, initiatives led by the municipality within the framework of the Helsinki Metropolitan Plan, as well as the Helsinki Metropolitan Area Smart and Clean Cooperation project, are multiplying, with the objective of becoming carbon neutral by 205011 . In most cases, CE constitutes a central issue in these initiatives.

In this context, the Kera neighbourhood, located within the eastern part of Espoo and close to the rain station, represents a unique opportunity for the municipality to experiment in innovative CE solutions. Previously an industrial area, and the headquarters of the Finland's largest distribution group, Kera will thus be transformed into a mixed-use and dense neighbourhood of 14.000 residents. In addition to commercial services and offices, the project includes day-care centres, schools, sports and recreation services 12 .

The municipality's objective is to transform this industrial park into a liveable neighbourhood with a strong CE focus by 2035. The goal of this project is to make Kera "a showcase district for circular economy" 13 , as well as "an international example of circular economy" 14 . It is with this perspective that the case of Kera was presented as an example of a 'circular neighbourhood' at the World Circular Economy Forum of 2017 15 .

The ongoing project was the winner of the Kera Challenge 16 , launched in 2015 with the aim of identifying a vision and project for Kera's future, based on the principles of sustainable urban planning and EC. In the winning project, Co-op City, CE is supposed to be achieved through a "large range of different measures, from boosting resources efficiency and creating closed loop systems to involving the local residents" 17 .

The main solutions to support circular economy put forward in the project were analysed and summarised in Table 3: In particular, with respect to the built environment, the EC practices to be implemented are contained within the 'Kera Design Manual'. In the manual, all constructions are required to be biodegradable or fully recyclable in order to gradually phase out construction waste. Indeed, the flexibility of the constructed buildings represents one of the document's first fundamental points. In fact, this flexibility provides the basis for the possibility of a future 'circular regeneration' of the built stock. In this perspective, LCA will be mandatory. In addition, for new constructions, all materials used are required to be fully biodegradable or recyclable in order to reduce construction waste.

Concerning reuse and recycling, attention is paid to both existing materials, such as asphalt, which must be recovered, and to the construction elements of existing halls, such as beams, slabs and columns. In addition, the temporary use of some existing buildings is highlighted as an EC practice. For example, during the building construction, the ground floor of the halls will be used as for temporary storage for materials and elements to be reused or recycled. Furthermore, the halls' structure will be reused in constructing new buildings or outdoor spaces.

In addition, it is interesting to note that 100% of the primary energy demand will be produced from renewable sources, some of which will be produced on site. Solar, geothermal and wind energy are planned and will feed into an intelligent energy grid. For new constructions, passive solutions are preferred.

Les Groues, Nanterre

The urban redevelopment project of the Groues in Nanterre, led by the Etablissement Public d'Aménagement de la Défense Seine-Arche (EPADESA), aims to create a mixed district, offering housing, office space, shops, services and equipment and accommodating nearly 12.000 inhabitants and new jobs18 . Near the business district of La Défense and served by a future line of the Grand Paris metro, the Groues neighbourhood, covering approximately 65 hectares, is characterised by numerous wastelands and distressed buildings. The project is recent, as is its realisation. In 2015, a Strategic Operational Project was approved by EPADESA, and in December 2016, the Concerted Development Areas (ZAC) were created. The first lots are currently being awarded, and the operation is expected to continue until 2030 (ADEME 2017).

The goal of the Groues development project is to become a "laboratory for a dynamic, green and inclusive neighbourhood"19 , and more generally, to become "an experimental laboratory for the sustainable city of tomorrow" 20 . In particular, the project aims to be exemplary in environmental matters and to obtain the EcoQuartier Label, becoming a positive energy territory 21 .

The project is based on five strategic axes: As presented in the figure above, the actions planned for CE are multiple. In particular, concerning the built environment, attention is paid to the local management of construction site waste (choice of materials, grey energy, local management of backfill/burial) (ADEME 2017). In addition, the ZAC project plans to place the built environment at the centre of EC's approach, as well as the project's overall energy efficiency ambition. To this end, 'lifecycle thinking' is encouraged here: "The building must be understood in all its spatial and temporal integrity by real estate operators, who must understand the lifecycle of their building: its manufacturing processes and materials, its duration over time and its capacity to adapt and evolve up to its deconstruction"22 . In addition, the concepts of 'grey energy' and 'transformation capacity' are central. Other EC practices are highlighted as well, such as rainwater harvesting, building flexibility and modularity, neighbourhood waste harvesting and reuse of existing buildings.

In addition, in 2016, EPADESA launched two calls for projects, aimed at inspiring innovative reflections and experiments on the CE theme. The first one concerns temporary urban planning approaches, allowing the lifespan of existing buildings to be extended. The ephemeral initiatives presented were highly diverse, ranging from soil remediation to the reuse of building materials, or even innovative start-up incubators. The second one directly concerns new constructions and aims to develop innovative CE solutions in the construction sector. LCA has been applied to the design of 5 office buildings, and the environmental benefit of recycling has been studied.

Saint-Vincent-de-Paul, Paris

Located in the 14 th arrondissement of Paris, the Saint-Vincent-de-Paul ancient hospital remained under decommissioning for approximately 10 years. In 2014, however, it was acquired by the municipality of Paris with the aims of transforming it into an innovative eco-neighbourhood (City of Paris 2017a). Covering an area of 3.4 hectares, the redevelopment project of the former Saint-Vincent-de-Paul Hospital represents a rare opportunity for urban transformation in the heart of Paris' particularly dense urban fabric.

In particular, the objectives pursued by Paris are as follows:

Objectives

Create a predominantly residential area, promoting social diversity Lead an exemplary environmental approach, making Saint-Vincent-de-Paul an innovative and emblematic eco-neighbourhood for the city Think of public and open spaces as green spaces, whether on roofs, floors or facades Enhance the heritage and history of the site Table 5. Saint-Vincent-de-Paul's objectives.

In December 2016, the ZAC was created and granted to the developer Paris Batignolles Aménagement. The construction work began in 2018, and the planned programme provides for approximately 60,000 m 2 of total floor area, broken down in housing (including 50% social housing), facilities, equipment (including a school and a gymnasium), shops and a public garden23 .

Concerning the built environment, the future district aims to become an exemplary showcase for the entire city thanks to an ambitious environmental approach. Specifically, as stated in the Resilience Strategy adopted in 2017, Saint-Vincent-de-Paul aims to be the city's first resilient and carbon-neutral neighbourhood (City of Paris 2017b). In compliance with the city's framework documents, the project thus aims to create a "pilot district for sustainable development" (City of Paris 2017a), reducing its impact on the environment and promoting innovative technologies while also involving the inhabitants. In this perspective, the new project provides for reversible buildings, pooling resources, converting 60% of existing buildings, developing renewable energies, certifying new constructions, optimising energy systems and recovering waste.

In addition, ZAC Saint-Vincent-de-Paul aims to be "a privileged space to develop the principle of CE"24 . Several different actions are being put forward in the field of CE, and the project must respect the orientations of the Parisian CE plan. With regard to the built environment, attention is being paid to reusing certain buildings, which would make it possible to limit demolitions, short circuits and temporary occupation of existing buildings, as well as to ensure continuity with the ephemeral urban planning experimentation of the Grands Voisins, including the smart grids, urban agriculture and biowaste.

More specifically, regarding waste, more attention is paid to the construction phase, and the following objectives in particular:

 Source reduction of the quantity of waste;

 Material recovery by reuse in place or elsewhere (in particular, building elements such as doors, windows, etc., which can easily find a new use, or concrete waste and bricks, which can be used as aggregates);

 Energy recovery from waste;

 Storage in a technical centre, reserved for final waste.

To this end, a process of recovering dismountable elements likely to be reused is planned for all site operations, thanks to an inventory distributed to various local, potentially interested structures. In this respect, a specific project management assistance service for sustainable development is planned from the project's amelioration. Applying LCA to the design of a project including deconstruction, renovation and new construction is planned in the frame of the PULSE-PARIS project funded by ADEME25 . First of all, the four projects are integrated into the strategic documents regarding the sustainable development and EC of each city, such as the Smart City Agenda, Climate Change Plan, and so on. Furthermore, in these documents, the projects are always presented as 'experimental demonstrators' of CE in urban projects. However, it is important to note that, in all the cases reviewed, CE is often seen as one of the pillars of sustainable development, and sometimes, no distinction is made among the proposed actions between those related to the CE or to sustainable development.

Second, the analogies are also evident with respect to CE practices identified in the projects. Following the case studies analysis, it is possible to classify four categories of recurrent practices: energy, water, waste and other. In particular, all cases insist on flexibility and temporary occupancy of buildings, the reuse of both building materials and existing buildings, and eco-construction. On this subject, an important focus is placed on the energy aspects of new buildings, with precise standards to be achieved. Other practices highlighted by at least two cases relate to waste management, particularly construction site waste, as well as water management and urban agriculture.

More generally, the comparative table indicates that the vast majority of the CE practices highlighted by the cases focused more on environmental issues. In fact, the other two pillars, economic and social, would not appear to be central.

However, no consensus is present concerning the tools employed. The Dutch and Finnish cases, for instance, are based on ad hoc assessment methodologies still under development. Furthermore, only in the case of Buiksloterham is a material flow analysis (MFA) conducted. This well-identified assessment tool is often coupled with CE, but according to [START_REF] Elia | Measuring Circular Economy Strategies through Index Methods: A Critical Analysis[END_REF], it is not sufficient for validating the pertinence of CE practices, because it does not enable explicitly accounting for all of the environmental impacts. Indeed, MFA is an important territorial knowledge tool, but it does not allow to prioritise and make decisions among the different CE actions. Other tools, such as the LCA, could conversely support such decisions.

In this regard, it is interesting to note that the 'lifecycle perspective' seems to be central in all cases, but it is not necessarily associated with lifecycle assessment (LCA) tools. The LCA method seems to be the most complete methodology for assessing environmental impacts and CE requirements [START_REF] Elia | Measuring Circular Economy Strategies through Index Methods: A Critical Analysis[END_REF], but nevertheless, in the Kera cases, LCA is planned only at the building scale and for new constructions. The assessment of CE practices is not directly mentioned except in Saint Vincent de Paul. This can lead to contradictions, because as demonstrated during the 63 rd discussion forum on LCA 26 , 'circularity' does not always positively affect the environment.

Finally, it is important to note that all these projects are currently underway or were recently started, probably contributing to a lack of precision regarding the performance that will actually be achieved upon delivery.

Conclusions

The literature review reveals a lack of research concerning the application of CE principles to urban projects and their assessment. Accordingly, this paper aims to provide a comparative analysis of four 'circular neighbourhood' case studies in order to discuss the CE practices implemented and the assessment tools utilised.

The results of this analysis indicate a large panel of CE practices and a limited use of indicators and assessment tools. Further steps in this research field are required concerning the study and analysis of other cases and tools. Moreover, several authors have highlighted the interest of LCA in evaluating CE [START_REF] Ghisellini | Exploring Environmental and Economic Costs and Benefits of a Circular Economy Approach to the Construction and Demolition Sector. A Literature Review[END_REF][START_REF] Zanni | Life Cycle Assessment Applied to Circular Designed Construction Materials[END_REF][START_REF] Elia | Measuring Circular Economy Strategies through Index Methods: A Critical Analysis[END_REF][START_REF] Giorgi | Circular Economy, Waste Management and Life Cycle Thinking[END_REF][START_REF] Haupt | How Can LCA Support the Circular Economy?-63rd Discussion Forum on Life Cycle Assessment[END_REF][START_REF] Fregonara | Economic-Environmental Indicators to Support Investment Decisions: A Focus on the Buildings' Endof-Life Stage[END_REF]. Two key characteristics of LCA are highlighted by these authors: the 'life-cycle thinking', which should also be the basis for EC, and the environmental impact assessment. Thus, the relevance of LCA is also becoming increasingly legitimate to experts, who are beginning to emphasise the importance of LCA to implement EC strategies. In this context, one of the objectives of the PULSE-PARIS project

Table 1 .

 1 Case studies.

	Case	Buiksloterham	Kera	Les Groues	Saint-Vincent-de-Paul
	City and Country	Amsterdam, Netherlands	Espoo, Finland	Nanterre, France	Paris, France
	Size	1000 hectares	22 hectares	65 hectares	4 hectares
	Site	Requalification industrial areas	Requalification industrial areas	Requalification industrial areas	Requalification hospital complex
	Main Objectif	"key innovation zone for circular development"	"a showcase district for circular economy"	"circular economy living-lab"	"a privileged space to develop and test circular economy"
	Starting date	Around 2015	Around 2018	Around 2018	Around 2018

Table 3 .

 3 Kera's EC actions.

	Green Infrastructure A network of multifunctional public places	Urban Fabric Sustainable Planning and Construction process	Function and Services Mobility Services in a 20 minute city Sustainable energy solutions	Social Environment Livability through co-creation
			Mobile platforms for emerging
			services models
			Autonomous vehicle traffic
	Ecosystem services Implement green and blue tools Design for multifunctional Evaluate generated values	Cyclical flexibility in buildings Material cycles Temporary use of the logistics Reuse and recycling of asphalt Recycling of the logistic halls	Logistics Lab Access and smarter use of the spaces Smart parking and shared cars 100% renewable energy	Participatory processes Common spaces Social engineers Affordable housing
			On site energy
			Smart energy grid
			Passive energy optimization

Table 4 .

 4 Les Groues' objectives.As such, CE represents one of the main pillars of the project's sustainable development strategy. Winner of the Call for Expression of Interest 'Circular Economy and Urban Planning' launched by the Ademe in 2015, the Groues project offers a place to experiment with a CE at the neighbourhood and territorial scale.

	Objectives

Table 6 .

 6 Cross-case comparison

			Sustainable Agenda Smart City Initiative Circular Amsterdam	Lien stratégie durabilité et smart city	EcoQuartier Label	Resilience Plan Territorial Climate Plan CE Plan
	Tools employed		Circular Building Standard Materials digital Passport MFA	Kera design manual LCA Buildings	(Perspective cycle de vie)	LCA Building Carbon Footprint Municipality's assessment tool
			100% renewable energy PassivHaus Label Local energy production	100% renewable energy PassivHaus Label Local energy production	70% renewable energy	40% electricity by photovoltaic panels PassivHaus Label
		Energy	100 % energy recovery from wastewater	(Géothermal, Eolic) Smart Grid	Energy recovery from wastewater	Smart Grid
					Reuse of materials and	Reuse of materials and
			Reuse materials		existing buildings	existing buildings
			100% « circular
	CE practices	materials and Waste	material flow » Deconstruction Zero waste objectif	Reuse materials and construction elements Biodegradable or recyclable materials	Local management of construction and demolition waste Waste recovery and valorisation	Local management of construction and demolition waste Waste recovery and valorisation
		Water	Rainwater collection	« Green and Blue Tools »	Rainwater collection
			Temporary occupancy			Temporary occupancy of
			of buildings Prefabricated constructions and Buildings flexibility	buildings Temporary occupancy of	Buildings flexibility buildings Temporary occupancy of	buildings
			structures	Buildings flexibility		Urban agriculture
		Other	Auto-construction Urban agriculture and local food production		Urban agriculture	Short circuits CE stakeholders point of reference

https://www.espooinnovationgarden.fi/en/espoo-innovation-garden/media/news/espoo-remains-the-mostsustainable-city-in-europe/

11 https://www.espoo.fi/en-US/Housing_and_environment/Sustainable_development

https://fr.calameo.com/read/00398144113f3e65fce93

EPADESA.

Les Groues. Plan Guide. 20 https://fr.calameo.com/read/00398144113f3e65fce93 21 http://fr.calameo.com/read/0039814413c1330021bdd

EPADESA. 2016. Les Groues. Plan Guide.

https://www.paris.fr/services-et-infos-pratiques/urbanisme-et-architecture/projets-urbains-etarchitecturaux/saint-vincent-de-paul-14e-2373

http://www.paris-batignolles-amenagement.fr/pba/sites/default/files/publications/dp_svp_1506_deflite_bd.pdf

The PULSE-PARIS research project aims to improve the relevance and operationality of eco-design approaches for urban projects in line with the CE strategic plans of the City of Paris. In particular, the project focuses on life cycle assessment (LCA) tools at the neighborhood scale, which are still in their infancy. The project will synthesise strategic approaches at the city scale and eco-design approaches on the urban project, in order to verify the coherence and articulation between these decision-making levels. The evaluation of CE practices on this scale using LCA is innovative and would make it possible to better understand the environmental benefits of these practices. (The research project is led by EIVP and MinesParisTech).

Acknowledgement

This article further presents some of the first results of the PULSE-PARIS research project, currently still in progress and financed by Ademe.

Cross-case comparison and discussion

The results of the analysis are summarised in Table 6 and are compared by applying the analytical framework based on the following criteria: CE practices, strategic city scale integration, and tools employed. Following this three-step analysis, some significant similarities have been pointed out.

Buiksloterham

Kera

Les Groues Saint-Vincent-de-Paul Strategic city scale integration involves the concrete application of the LCA method for evaluating the identified CE practices in order to study its relevance and propose possible improvements.