The PoincaréDodecahedral Spaceand the Mystery of theMissing FluctuationsJeffrey Weeks

610 NOTICES OF THE AMS VOLUME 51, NUMBER 6

IntroductionBecause of the finite speed of light, we see theMoon as it was roughly a second ago, the Sun as itwas eight minutes ago, other nearby stars as theywere a few decades ago, the center of our Milky WayGalaxy as it was 30,000 years ago, nearby galaxiesas they were millions of years ago, and distantgalaxies as they were billions of years ago. If we lookstill deeper into space, we see all the way back tothe final stages of the big bang itself, when thewhole universe was filled with a blazing hot plasmasimilar to the outer layers of the modern Sun. Inprinciple we see this plasma in all directions; it fillsthe entire background of the sky. So why don’t wenotice it when we look up at the night sky? Thecatch is that the light from it—originally visible orinfrared—over the course of its 13.7 billion-yearvoyage from the plasma to us has gotten stretchedout as part of the overall expansion of the universe.Specifically, the universe has expanded by a factorof about 1100 from then till now, so what wasonce a warm reddish glow with a wavelengtharound 10,000 angstroms is now a bath of mi-crowaves with a wavelength of about a millimeter.So we cannot see the plasma with our eyes, but wecan see it with a microwave antenna.

If our eyes were sensitive to microwaves as wellas to visible light, a close-up view of the night skymight look something like Figure 1. In the fore-ground we see other galaxies as they were a fewbillion years ago. In the background we see the

omnipresent plasma as it was 380,000 years afterthe big bang, a mere three one-thousandths of onepercent of the universe’s present age of 13.7 bil-lion years. The plasma holds clues to the universe’sbirth, evolution, geometry, and topology. To har-vest these clues, NASA launched the Wilkinson Microwave Anisotropy Probe (WMAP) on 30 June2001. On 10 August 2001 WMAP reached its orbitabout the so-called second Lagrange point, wherethe combined gravity of the Sun and the Earth arejust right to keep the satellite orbiting the Sun insynchronization with the Earth, and WMAP beganits four years of observations (Figure 2). The ob-served radiation from the plasma, known as theCosmic Microwave Background (CMB) radiation, is extremely uniform across the sky. Nevertheless,it exhibits small temperature fluctuations on the order of 1 part in 105. These CMB temperaturefluctuations result from fluctuations in the density(not temperature!) of the primordial plasma: pho-tons arriving from denser regions do a little extrawork against gravity and arrive slightly cooler,while photons arriving from less dense regions doa little less work against gravity and arrive slightlywarmer. So in effect temperature fluctuations onthe microwave sky reveal density fluctuations in theearly universe.

Cosmologists’ current standard model positsan essentially infinite Euclidean space created byinflation and containing density fluctuations on all scales. On small scales WMAP observed thesefluctuations as predicted. However, on scales larger than about 60 degrees across the sky, the

Jeffrey Weeks is a MacArthur Fellow. He can be reachedat http://www.geometrygames.org/contact.html.

JUNE/JULY 2004 NOTICES OF THE AMS 611

fluctuations all but disappear,1 leaving cosmolo-gists with the Mystery of the Missing Fluctuations.

Finite UniverseOne possible explanation for the missing large-scale fluctuations is that the universe is simplynot big enough to support them. To take a simple1-dimensional analogy, an infinite line supportswaves of all sizes, while the circumference of a unitcircle supports no wavelength longer than 2π. Similarly, if the real universe is a closed 3-manifold,it can support no waves longer than its own “cir-cumference”.

What 3-manifolds shall we consider? Observa-tional evidence implies the observable universe ishomogeneous and isotropic to a precision of onepart in 104, so consider manifolds that locally looklike the 3-sphere S3, Euclidean space E3, or hyper-bolic space H3. To construct a finite universe, takethe quotient X/Γ of the simply connected spaceX = S3, E3, or H3 under the action of a discretefixed point free group Γ of isometries.

Observational data suggest the observable uni-verse either is flat or has a small curvature that ismore likely positive than negative. More precisely,on a scale where Ω < 1 indicates a hyperbolic uni-verse, Ω = 1 indicates a flat universe, and Ω > 1indicates a spherical universe, analysis of the WMAPdata yields Ω = 1.02± 0.02 at the 1σ level [2]. The

parameter Ω measures the average mass-energydensity of space, which by general relativity com-pletely determines the spatial curvature, with lowdensity yielding a hyperbolic universe and highdensity yielding a spherical universe. By definitionΩ is the ratio of the actual density to the criticaldensity that a flat universe would require.

The huge sphere on which we observe the pri-mordial plasma is called the horizon sphere, andits radius in the modern universe is our horizon radius. If we live in a finite universe X/Γ , then ourinjectivity radius is the radius of the smallest Earth-centered sphere that “reaches all the way aroundthe universe” and intersects itself. Equivalently,the injectivity radius is the radius of the largestEarth-centered sphere whose interior is embed-ded. Twice the injectivity radius is thus the mini-mal circumference of the universe, starting fromEarth. If our horizon radius exceeds our injectiv-ity radius, then we can trace two different lines ofsight to the same distant region of space, meaningthat in principle we see multiple images of thesame astronomical sources (galaxies, quasars,plasma,…) in different parts of the sky, making de-tection of the universe’s topology vastly easier.

A flat manifold E3/Γ allows no a priori rela-tionship between the horizon radius and the injectivity radius, because the latter is essentiallyarbitrary. Given any proposed Euclidean group Γ,we may easily stretch or shrink its translationalcomponents, via a similarity, to obtain any pre-scribed injectivity radius. We therefore have noreason to expect the injectivity radius to be com-parable to our horizon radius. Thus successfullydetecting a flat topology E3/Γ would require a huge amount of luck. In spite of the long a prioriodds against it, the possibility of a finite flat uni-verse E3/Γ continues to receive a fair amount of attention, because WMAP’s Ω = 1.02± 0.02 result

1Spergel et al. report the missing fluctuations in one of aseries of papers released along with the first-year WMAPdata [1]. Their Figure 16 compares predictions to obser-vations, showing, in their words, “the lack of any corre-lated signal on angular scales greater than 60 degrees.”More conservative observers point out that the signal isnot totally missing, but merely very weak. Spergel et al.estimate the probability that such a weak signal could ariseby chance to be either 0.0015 or 0.003, depending onwhich of their best-fit flat space models one compares to.

Figure 1. The primordial plasma provides a background for the whole sky. However, we see it not in the visiblespectrum but in microwaves. The left panel shows a portion of the Hubble Deep Field image as we see it with oureyes. The right panel simulates what we might see if our eyes were sensitive to microwaves as well as to visiblelight.

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Figure 2. A million miles fromEarth, the Wilkinson

Microwave Anisotropy Probeobserves deep space fromnear the second Lagrange

point, L2. In the (rotating) Sun-Earth coordinate system, the

Lagrange points are the criticalpoints of the effective

gravitational potential. Thesecond Lagrange point is a

saddle point, unstable in theradial direction but stable in

both the horizontal andvertical tangential directions.

The satellite traces a gentleorbit about L2 in the vertical

plane, with a slight nudgeevery few months to keep it

from drifting towards or awayfrom the Sun. The backward-

facing solar panels support a large protective disk blocking microwave interference from the Sun, Earth, andMoon, giving the outward-facing microwave receivers an unobstructed view of deep space.

612 NOTICES OF THE AMS VOLUME 51, NUMBER 6

comfortably includes it and because the Mysteryof the Missing Fluctuations still begs for an expla-nation.

Even though the 1σ estimate Ω = 1.02± 0.02leaves plenty of room for a hyperbolic universe, afinite hyperbolic topology H3/Γ would be difficultto detect. As the complexity of a hyperbolic groupΓ increases, the volume of its quotient H3/Γ in-creases as well, in contrast to spherical groups Γwhose quotient spaces S3/Γ get smaller as thegroup gets larger. Moreover, even in the smallesthyperbolic manifolds, the injectivity radius Rinj istypically larger than the horizon radius Rhor. As-trophysicists know our horizon radius to be about46 billion light-years.2 Geometers, however, wantto know our horizon radius not in light-years butin units of the curvature radius Rcurv. In otherwords, geometers want to know the dimension-less ratio Rhor/Rcurv , which tells the horizon radiusin radians or “the usual hyperbolic units”. Eventhough the horizon radius Rhor in light-years isknown, the curvature radius Rcurv depends stronglyon Ω. If Ω = 0.98, then the curvature radius Rcurvis about 98 billion light-years, the geometer’s hori-zon radius works out to Rhor/Rcurv = 0.47, andrandomly placed observers in the ten smallest

known hyperbolic topologies H3/Γ would haveroughly a 50-50 chance of living at a point wheretheir horizon radius exceeds the injectivity radius.However, if Ω = 0.99, then the curvature radiusRcurv increases to about 139 billion light-years, thedimensionless horizon radius drops toRhor/Rcurv = 0.33, and a randomly placed observerhas only about a 10 percent chance of living at apoint where the horizon radius exceeds the injec-tivity radius. As Ω approaches 1, the curvature ra-dius goes to infinity, and the chances of detectingthe nontrivial topology go to zero.

More promising from a purely topological pointof view is the possibility of a spherical universeS3/Γ. The curvature radius Rcurv, which here is sim-ply the radius in meters of the 3-sphere S3 fromwhich the universe S3/Γ is constructed, comes outto 98 billion light-years when Ω takes the observedvalue of 1.02. Thus the astrophysicist’s horizon radius Rhor = 46 billion light-years translates tothe geometer’s horizon radius Rhor/Rcurv = 0.47 radians. In other words, at the nominal value of Ω = 1.02, our horizon sphere’s radius on the 3-sphere is 0.47 radians, meaning that we are see-ing a modest yet nontrivial portion of the 3-sphere(Figure 3). Luckily, a horizon radius of 0.47 sufficesto see the topology S3/Γ for many of the simplestand most natural groups Γ, to be discussed below.Moreover, for more complicated groups Γ, the quo-tient S3/Γ gets smaller, making the topology eveneasier to detect. This potential detectability, alongwith WMAP’s observation of Ω ≈ 1.02, has fuelledconsiderable interest in the possibility of a spher-ical universe S3/Γ.

Whether we consider hyperbolic, flat, or spher-ical manifolds, the question remains: which spacesbest account for the Mystery of the Missing

2Readers may wonder why the horizon has a 46 billionlight-year radius when the universe is only 13.7 billion yearsold. The expanding universe provides the answer. Thephotons now reaching us from our horizon began theirjourney when the universe was 1100 times smaller thanit is today. Thus the first light-year of space that a givenphoton traversed has since expanded to roughly 1100light-years of space in the modern universe. In otherwords, the present day horizon radius is 46 billion light-years, but that same volume of space was much smaller,and more easily traversable, in the distant past.

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Fluctuations? Surprisingly, not all small-volumeuniverses suppress the large-scale fluctuations. Aswe will see below, some small-volume universeseven elevate them!

ModesA sustained musical tone may be expressed as thesum of a fundamental, a second harmonic, a thirdharmonic, and so on—in effect, its Fourier de-composition. This Fourier approach provides greatinsight into musical tones; for example, weak sec-ond and fourth harmonics characterize the soundof a clarinet. Similarly, any continuously definedfield in the physical universe—for example, thedensity distribution of the primordial plasma—

may be expressed as a sum of harmonics of 3-dimensional space. Technically these harmonicsare the eigenmodes of the Laplace operator; intu-itively they are the vibrational modes of the space, analogous to the vibrational modes of a 2-dimensional drumhead, so henceforth we willsimply call them the modes of the space.

Just as the relative strengths of a clarinet’s har-monics, its spectrum, characterize its sound, therelative strengths of the universe’s modes charac-terize its physics. That is, just as the pressure anddensity fluctuations within the clarinet must con-form to the clarinet’s size and shape, the pressureand density fluctuations in the primordial plasmamust conform to the size and shape of the universe.When we look out into space at our horizon, we seethese density fluctuations. Of course we do not seethe full 3-dimensional modes, but only their in-tersection with the 2-dimensional horizon sphere.Nevertheless, it is straightforward to calculate howa 3-manifold’s modes restrict to 2-dimensionalmodes of the horizon sphere, ultimately allowingdirect comparison to observations. Various sourcesof noise and other physical effects complicate theprocess but seem not to obscure the underlyingtopological and geometrical signatures. Thestrengths of the modes we observe on our 2-dimensional horizon sphere are called the CMBpower spectrum (Figure 4). The CMB power spec-trum’s weak low-order terms conveniently quantifythe Mystery of the Missing Large-Scale Fluctua-tions, just as the lack of low tones in a piccolo’sspectrum reflect its small size.

The modes of a multiply connected space X/Γ ,with X = S3 , E3, or H3 as before, lift in the obviousway to Γ-periodic modes of the simply connectedspace X. Conversely, each Γ-periodic mode of X

WMAP dataSmoothed WMAP dataEuclidean best fit

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Figure 3. At the nominal observed densityΩ = 1.02 the universe is positively curvedwith radius 98 billion light-years. Ourhorizon sphere has radius 46 billion light-years, which works out to 46/98 = 0.47radians on the 3-sphere. Thus we see amodest but nontrivial portion of the3-sphere.

Figure 4. Just as a musical tone splits into a sum of ordinary harmonics, the temperaturefluctuations on our horizon sphere split into a sum of spherical harmonics. The resulting CMBpower spectrum, shown here, tells much about the birth, evolution, geometry, and topology of theuniverse. The peaks in the spectrum fall more or less where expected, confirming physicists’theoretical understanding of the primordial plasma. The surprise lay in the weak lowest-orderterms, which hint at a finite space.

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614 NOTICES OF THE AMS VOLUME 51, NUMBER 6

Figure 5. If the speed of light were infinite,inhabitants of the binary tetrahedral space

S3/T∗ would see 24 images of everycosmological object (upper left); like atoms ina crystal the images repeat along a tiling of S3

by 24 copies a fundamental octahedral cell. Inthe binary octahedral space S3/O∗ the images

repeat along a tiling by 48 truncated cubes(upper right), and in the binary icosahedralspace S3/I∗, better known as the Poincaré

dodecahedral space, the images repeat alonga tiling by 120 octahedra (lower right).

Because these still images provide only aweak understanding of the tiling, the reader isencouraged to fly around in them in real time

using the free simulator available athttp://www.geometrygames.org/

CurvedSpaces.

Figure 6. If our horizon radius exceeds our injectivity radius, the horizon sphere wraps all the wayaround the universe and intersects itself. Viewed in the universal cover, repeating images of the hori-

zon sphere intersect. Observationally, we see the same circle of intersection on oppositesides of the sky.

JUNE/JULY 2004 NOTICES OF THE AMS 615

projects down to a mode of the quotient X/Γ . Thus,even though we think of the modes conceptuallyas modes of X/Γ , in practice we invariably repre-sent them as Γ-periodic modes of X.

In contrast to the irregular, hard-to-understandmodes of hyperbolic space H3, the modes of spherical space S3 are regular and predictable. As a simple point of departure, consider the modes,or harmonics, of a circle S1. Traditionally one writesthem as cosθ, sinθ, cos 2θ, sin 2θ, . . . . Yetif we embed the circle as the set x2 + y2 = 1 in thexy-plane, we see that the transcendental functionscosθ and sinθ are completely equivalent to the linear functions x and y . Similarly, cos 2θ andsin 2θ become the quadratic polynomials x2 − y2

and 2xy , and so on. In general, cosmθ and sinmθbecome mth -degree harmonic polynomials in xand y. By definition a polynomial p(x1, . . . , xn) isharmonic if and only if it satisfies Laplace’s equation ∇2p ≡ ∂2p

∂x21+ · · · + ∂2p

∂x2n= 0.

In perfect analogy to the modes of the circle S1,the modes of the 2-sphere S2 are precisely the homogeneous harmonic polynomials in x, y, andz , and the modes of the 3-sphere S3 are preciselythe homogeneous harmonic polynomials in x, y, z ,and w . The only difference lies in the number ofmodes. On the circle the space of mth -degree harmonic polynomials always has dimension 2, independent of m. On the 2-sphere the space of "th-degree harmonic polynomials has dimension2"+ 1, and on the 3-sphere the space of kth-degreeharmonic polynomials has dimension (k+ 1)2.

Cosmologists model the physics of a multicon-nected spherical universe S3/Γ using the Γ-periodicmodes of S3. For each degree k, the Γ-periodic modesform a subspace of the full (k+ 1)2-dimensionalmode space of S3. Finding an orthonormal basis forthat subspace requires, in principle, nothing morethan a simple exercise in sophomore-level linear algebra. In practice the linear algebra works greatfor single-digit values of k but quickly bogs downas the size of the (k+ 1)2 -dimensional function space grows. Even simple cubic-time numerical matrix operations require O(k6) time on (k+ 1)2-by-(k+ 1)2 matrices, and efforts to work directly withthe polynomials slow down even more dramatically.Adding insult to injury, accumulating round-off errors in floating point computations often renderunusable the results of those computations that canbe carried out within a reasonable time.

Thus for the past two years the main bottleneckfor understanding and simulating the physics ofmulticonnected spaces X/Γ has been the efficientand accurate computation of the modes, the underlying local physics being already well understood. News Flash: Jesper Gundermann of the Danish Environmental Protection Agency hasovercome the bottleneck and extended the CMB

power spectra for various spherical spaces S3/Γfrom "max = 4 out to "max = 15. As this article goesto press, he is completing a rigorous statisticalanalysis, with results expected soon. His tentativeresults show an excellent fit.

Spherical SpacesTo recognize a spherical universe S3/Γ by its im-print on the CMB power spectrum, we must firstknow which such spaces are possible. Fortunately3-dimensional spherical spaces were classified by1932 [3]. The possible groups Γ turn out to bear aclose relationship to the symmetry groups of an ordinary 2-sphere! The easiest way to see the tightrelationship between the symmetries of a 2-sphereand the symmetries of a 3-sphere is via the quater-nions. Recall that the quaternions provide a non-commutative algebraic structure on R4 analogousto the commutative algebraic structure that thecomplex numbers provide on R2. Specifically thequaternions are spanned by 1, i, j,k subject to therules i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = iand ki = −ik = j . The length of a generic quater-nion a1+ bi + cj+ dk is just its obvious Euclidean length

√a2 + b2 + c2 + d2 . Just as the complex

numbers restrict to a multiplicative group on theunit circle S1, the quaternions restrict to a multi-plicative group on the unit 3-sphere S3.

Visualize the 3-sphere S3 as the multiplicativegroup of unit length quaternions. A given quater-nion q ∈ S3 can act on S3 in two ways: by multi-plication and by conjugation. When q acts by multiplication, taking each point x ∈ S3 to qx , the result is a fixed-point free rotation of S3. Anyfinite group Γ of such fixed-point free rotations defines a spherical space S3/Γ.

When q acts by conjugation, taking each pointx ∈ S3 to qxq−1, the result is a rotation with fixedpoints. Indeed, the point 1 ∈ S3 is fixed by all suchrotations, because q1q−1 = 1 for all q, so in effectconjugation by q defines a rotation of the equato-rial 2-sphere, which is the intersection of S3 withthe 3-dimensional subspace of purely imaginaryquaternions bi + cj+ dk .

We now have a way to transfer symmetries fromthe 2-sphere to the 3-sphere. Start with a finite groupof symmetries of S2, for example, the tetrahedralgroup T consisting of the twelve orientation-preserving symmetries of a regular tetrahedron. Represent T as a set G of quaternions acting by conjugation. Now let the same set G act on S3 by mul-tiplication. Voilá! There is our group Γ of fixed-pointfree symmetries of the 3-sphere. The only catch isthat each of the original symmetries of S2 is realizedby two different quaternions, q and −q, so the groupG has twice as many elements as the original group.In the present example, with the original group being the tetrahedral group T, the final group Γ is the binary tetrahedral groupT∗, of order 24.

616 NOTICES OF THE AMS VOLUME 51, NUMBER 6

The finite symmetry groups of S2 are well known:• The cyclic groups Zn of order n, generated by a

rotation through an angle 2π/n about someaxis.

• The dihedral groupsDm of order 2m , generatedby a rotation through an angle 2π/m aboutsome axis as well as a half turn about some perpendicular axis.

• The tetrahedral group T of order 12 consistingof all orientation-preserving symmetries of a regular tetrahedron.

• The octahedral group O of order 24 consistingof all orientation-preserving symmetries of a regular octahedron.

• The icosahedral group I of order 60 consistingof all orientation-preserving symmetries of a regular icosahedron.

Transferring those groups from S2 to S3, as ex-plained above, yields the single action symmetrygroups of S3:• The cyclic groups Zn of order n.• The binary dihedral groups D∗

m of order 4m ,m ≥ 2.

• The binary tetrahedral group T∗ of order 24.• The binary octahedral group O∗ of order 48.• The binary icosahedral group I∗ of order 120.

The corresponding quotients S3/Γ are the single action spaces (Figure 5). To fully understand thesespaces, the reader is encouraged to fly around inthem using the free simulator available atwww.geometrygames.org/CurvedSpaces.

The full classification of finite fixed-point freesymmetry groups of S3 is more complicated, butonly slightly. Geometrically, the mapping x→ qxacts as a right-handed corkscrew motion. If weswitch to right multiplication x→ xq , we get left-handed corkscrew motions instead of right-handedones, and the quotient space S3/Γ becomes themirror image of what it had been. (To understandwhy changing qx to xq reverses the chirality of thecorkscrew motion, multiply out the two products

(cosθ1+ sinθi)(a1+ bi + cj+ dk)

and

(a1+ bi + cj+ dk)(cosθ1+ sinθi)

and observe the results.) Two groups, Γ and Γ ′,may act on S3 simultaneously, one by left multi-plication and the other by right multiplication. Theresulting quotient space is called a double actionspace (for details see [4]). So far cosmologists havelargely neglected double action spaces and the related linked action spaces. One reason for theneglect is that they are more complicated and oftenrequire unrealistically large groups Γ. A more fundamental difficulty, however, is that a doubleaction space is globally inhomogeneous, meaningits geometry and therefore the expected CMB powerspectrum look different to observers sitting at

different locations within the same space, while asingle action space is globally homogeneous, mean-ing its geometry and the expected CMB fluctuationslook the same to all observers. Obviously the single action spaces are far easier to simulate, because a single simulation suffices for all observers.

Encouraging News

Resolving the Mystery of the Missing Fluctuationsturned out to be harder than expected. The prin-ciple that a small universe cannot support broadfluctuations was clear enough; the challenge lay inaccounting for the particular CMB power spectrumthat WMAP observed. In reality the " = 2 term,corresponding to a quadratic polynomial and gen-erally called the quadrupole, was anomalously low(at the 1-in-100 level), while the " = 3 term, corre-sponding to a cubic polynomial and generally called the octopole, was somewhat low as well (at the 1-in-5 level). These 1-in-500 odds of weak low-" fluc-tuations beg for an explanation, even though sim-ple random chance cannot be excluded.

The reader may wonder what became of the lower-" terms. The " = 0 constant term is simply the aver-age CMB temperature. The " = 1 linear term, the di-pole, is swamped by the far stronger dipole inducedby the solar system’s 300 km/sec motion relative tothe CMB and is therefore unavailable.

Returning to the mysteriously low quadrupoleand octopole, the simplest possible explanation, aflat 3-torus universe made by identifying oppositefaces of a cubical block of space, failed to accountfor the observed power spectrum. A sufficientlysmall cubic 3-torus can of course suppress thequadrupole (" = 2) as strongly as desired, but notwithout suppressing other low-order modes(" = 3,4,5) along with it, contrary to observations.Making the 3-torus from a more general paral-lelepiped offers greater flexibility. The full 6-parameter space of parallelepipeds has yet to be fully explored, but initial investigations show that while noncubic rectangular 3-tori suppressthe low-" portion of the spectrum, they suppressthe high-" portion even more, leaving the low-" portion relatively elevated [5], contrary to observation.

Among the spherical topologies, the lens spaceswere considered first. A lens space L(p, q) is thequotient S3/Zp of S3 under the action of the cyclicgroup generated by

JUNE/JULY 2004 NOTICES OF THE AMS 617

cos 2πp − sin 2π

p 0 0

sin 2πp cos 2π

p 0 0

0 0 cos 2πqp − sin 2πq

p

0 0 sin 2πqp cos 2πq

p

.

Because the group acts in only one direction, thequotient behaves roughly like a rectangular 3-torusthat is narrow in one direction but wide in two others. Like a noncubic 3-torus, a lens space L(p, q)suppresses the high-" portion of the spectrummore heavily than the low-" portion, in effect elevating the quadrupole and contradicting observations for all but the smallest choices of p.

The failures of both the rectangular 3-tori and thelens spaces to account for the low quadrupole com-bined to teach a useful lesson: a low quadrupolerequires a well-proportioned space, with all dimensions of similar magnitude. Only in a well-proportioned space is the quadrupole suppressedmore heavily than the rest of the power spectrum.This insight, combined with WMAP’s hints of slightpositive curvature, led researchers to consider thebinary polyhedral spaces S3/T∗, S3/O∗, and S3/I∗,all of which have both positive curvature and well-proportioned fundamental domains, namely a regular octahedron, a truncated cube, and a regu-lar dodecahedron, respectively, as shown in Figure 5.

Of the binary polyhedral spaces, S3/I∗ seemedthe most promising candidate, for the simple geo-metrical reason that its fundamental domain’s inradius of π/10 ≈ 0.31 fits easily within the horizon radius Rhor/Rcurv = 0.47 correspondingto Ω = 1.02. Topologists know S3/I∗ as the Poincaré dodecahedral space. Curiously, Poincaréhimself never knew his namesake manifold couldbe constructed from a dodecahedron. Rather, hediscovered the manifold in a purely topologicalcontext as the first example of a multiply connectedhomology sphere. A quarter century later Weberand Seifert glued opposite faces of a dodecahedronand proved that the resulting manifold was home-omorphic to Poincaré’s homology sphere.

Lacking explicit formulas for the modes of the bi-nary polyhedral spaces, the author and his colleagues computed the modes numerically. Unfortunately, in the case of the Poincaré dodeca-hedral space S3/I∗, accumulating numerical errorslimited the computation to the modes k ≤ 24 of the3-dimensional space, in turn limiting the reliableportion of the predicted CMB power spectrum to " = 2,3,4 . Nevertheless, the results were delightful: the predicted quadrupole (" = 2) and octopole (" = 3) matched observations [6]! (The " = 4term was used to set the overall normalization.)

Moreover, the best fit occurred in the range1.01 < Ω < 1.02, comfortably within WMAP’s observation of Ω = 1.02± 0.02.

Several factors made this result especially ele-gant. First and foremost was the near total lack offree parameters. Unlike the 3-torus, which can bemade from an arbitrary parallelepiped (six degreesof freedom), the dodecahedral space can be madeonly from a perfectly regular dodecahedron (zerodegrees of freedom). Second, the dodecahedralspace is globally homogeneous. Unlike in a typical3-manifold where the observer’s position affects theexpected CMB power spectrum (three degrees offreedom), the dodecahedral space looks the sameto all observers (zero degrees of freedom). Theonly free parameter in our simple initial study was the density parameter Ω. Amazingly, with only one parameter to vary, the model correctly accounted for three independent observations: the quadrupole (" = 2 ), the octopole (" = 3 ), and the observed density itself. While far from a proof, such results were most encouraging.

Discouraging NewsThe dodecahedral model makes three testable predictions:1. the weak large-scale CMB fluctuations,2. matching circles in the sky (to be discussed

momentarily), and3. a slight curvature of space.

The WMAP satellite had already observed theweak large-scale CMB fluctuations, so there is noproblem with the first prediction.

As for the third prediction, current measure-ments (Ω = 1.02± 0.02) fail to distinguish flatspace (Ω = 1) from the slight curvature that the dodecahedral model requires (Ω ≈ 1.02). Fortu-nately, upcoming data may suffice, either within ayear or two by combining the WMAP results withother data sets to narrow the error bars on Ω, orby the end of the decade if we wait for more pre-cise CMB measurements from the European SpaceAgency’s Planck satellite.

This leaves the second prediction, currently themost controversial one. The basic insight is as fol-lows. If the fundamental dodecahedron is smallerthan our horizon sphere, then the horizon spherewill “wrap around the universe” and intersect itself.This is most conveniently visualized in the universalcovering space (Figure 6) where repeating imagesof the horizon sphere intersect their neighbors.From our vantage point on Earth, at the center ofour horizon sphere we can see the same circle ofintersection sitting on opposite sides of the sky. If the observed CMB temperature fluctuations depended only on plasma density fluctuations,then the two images, one in front of us and one behind us, would display identical temperaturepatterns. Locating such pairs of matching circles

618 NOTICES OF THE AMS VOLUME 51, NUMBER 6

would conclusively prove the universe is finite andreveal its topology.

When the news appeared last October that the dodecahedral model accounts for the Mysteryof the Missing Fluctuations, another group of researchers was simultaneously searching formatching circles. Their massive computer searchwas expected to take months on a cluster of PCs,but by October their program, while not yet finishedwith the whole search, had finished checking forthe diametrically opposite circle pairs predicted bythe dodecahedral model and had found none [7].

The failure to find matching circles disappointedeveryone, but might not have dealt a fatal blow tothe idea of a finite universe. The observed CMBtemperature fluctuations depend not only onplasma density fluctuations but also on other factors such as the Doppler effect of the plasma’smotion and the gravitational influences the CMBphotons experience over the course of their 13.7 billion-year journey from the plasma to us. For themost part the circle searchers carefully accountedfor these various sources of contamination.However, they neglected residual contaminationfrom foreground sources within our own MilkyWay Galaxy. Total foreground contamination is intense, and the WMAP team devoted considerableeffort to subtracting as much of it as possible before releasing their cleaned CMB sky maps.Nevertheless, the remaining contamination is easily visible to the naked eye along the galacticequator and is likely strong enough to disrupt circle matching along a wider swath. Future workshould reveal the extent to which the residual foreground contamination may or may not obscurematching circles after resolving questions about how to model it correctly.

If the technical details get resolved and thematching circles really are not there, would weconclude that the universe is infinite? Not at all!First there is the possibility that the universe is finite but much larger than our horizon, in whichcase we could not detect its topology. More prac-tically, there is the possibility that the universeis comparable to, or slightly larger than, thehorizon. Such a universe would not generate de-tectable circles but might still account for theMystery of the Missing Fluctuations. All partic-ipants in recent discussions—circle searchers aswell as dodecahedral space modelers—agree thisis the obvious Plan B. But how can one hope todetect topology lying beyond our horizon? Itmay not be as hard as it seems. Just as one maydeduce the full length of a guitar string merelyby observing vibrations on its middle 80 per-cent (with the endpoints hidden from view!), onemay in principle deduce the topology of the uni-verse by observing density fluctuations within alimited volume.

ConclusionWhere will the conflicting pieces of evidence lead?To a “small” dodecahedral universe lying whollywithin our horizon? To a somewhat larger universelying just beyond our view but within our experi-mental grasp? Or to a presumed but unconfirmedinfinite flat space? Over the course of the decade im-proved measurements of curvature should providethe decisive clue. If Ω is sufficiently close to 1, thenthe dodecahedral model is dead. However, if Ω isfound to be near 1.02—and bounded away from1—then the dodecahedral model or some variationof it will almost surely prove correct, whether or notmatching circles are found. For now, the MissingFluctuations remain a Mystery.

Added in ProofStudies of the first-year WMAP data find that cer-tain features of the low-order CMB harmonics alignwith the ecliptic plane at roughly the 99.9% confi-dence level. Such alignments call into question thepresumed cosmic origins of the low-order har-monics, suggesting instead either some hitherto un-known solar system contribution to the CMB, or per-haps some error in the collection and processingof the data. The second-year WMAP data, origi-nally expected by February 2004 but delayed dueto unexpected surprises in the results, may soonshed additional light on these anomalies. If thetrue cosmological low-order harmonics, after sub-tracting off any solar system effects and/or pro-cessing errors, turn out to be even weaker than pre-viously believed, then the Mystery of the MissingFluctuations will deepen. However, the case forthe dodecahedral topology would vanish, forcingresearchers to reconsider other topologies—andperhaps other explanations—in light of the reviseddata.

AcknowledgmentI thank the MacArthur Foundation for its support.

References

[1] D. SPERGEL et al., First year Wilkinson MicrowaveAnisotropy Probe (WMAP 1) observations: Determi-nation of cosmological parameters, Astrophys. J. Suppl.Ser. 148 (2003), 175–194.

[2] C. BENNETT et al., First year Wilkinson MicrowaveAnisotropy Probe (WMAP 1) observations: Preliminarymaps and basic results, Astrophys. J. Suppl. Ser. 148(2003), 1–27.

[3] W. THRELFALL and H. SEIFERT, Topologische Untersuchungder Diskontinuitätsbereiche endlicher Bewegungs-gruppen des dreidimensionalen sphärischen Raumes,Math. Ann. 104 (1930), 1–70; and 107 (1932), 543–586.

[4] E. GAUSMANN, R. LEHOUCQ, J.-P. LUMINET, J.-P. UZAN, and J. WEEKS, Topological lensing in elliptical spaces, Classical Quantum Gravity 18 (2001), 5155–5186.

JUNE/JULY 2004 NOTICES OF THE AMS 619

[5] J. WEEKS, J.-P. LUMINET, A. RIAZUELO, and R. LEHOUCQ, Well-proportioned universes suppress CMB quadrupole,www.arxiv.org/astro-ph/0312312, to appear inMonthly Notices of the Royal Astronomical Society.

[6] J.-P. LUMINET, J. WEEKS, A. RIAZUELO, R. LEHOUCQ, and J.-P.UZAN, Dodecahedral space topology as an explanationfor weak wide-angle temperature correlations in thecosmic microwave background, Nature 425 (2003),593–595.

[7] N. CORNISH, D. SPERGEL, G. STARKMAN, and E. KOMATSU, Con-straining the topology of the universe, www.arxiv.org/astro-ph/0310233, to appear in Phys. Rev. Lett.

About the CoverMapping the Universe with HEALPixels

In Jeff Weeks’s article, the data that go into his analysis of the cosmic microwave background radiationhave been collected by the WMAP satellite, which scans different directions in the sky and records tem-peratures. Data are sampled and amalgamated according to a scheme called HEALPix (Hierarchical EqualArea iso-Latitude Pixelization), which has become the standard tool for such measurements—for theWMAP satellite as well as for its successor, Planck, scheduled to be launched in 2007. According to HEALPixthe sphere is first divided into twelve large zones (outlined in gray on the cover), and each of these inturn is parceled into N2 smaller “pixels” where N = 2n. For WMAP N = 512, which means 3,145,728pixels altogether, and Planck will have many more, so there is a lot of calculation to be done in dealingwith these data.

There is no canonical way to partition a sphere into uniform small regions, since there are only fiveregular solids. Any scheme used must therefore choose among various tradeoffs. The principal char-acteristic feature of HEALPix is that its pixels are all of equal area, more or less in the shape of rhombi,as the cover shows. Different formulas are used to generate the shapes in the polar and equatorial re-gions, but both depend on Archimedes’ Theorem that cylindrical projection is area preserving.

Several considerations in addition to efficiency go into devising such a scheme: regions of equal areaminimize the effect of noise, among other things. The distribution of regions along parallels of latitudeallows using the fast Fourier transform in decomposing the data into spherical harmonics. HEALPix isprobably close to being as fast as possible in facilitating spherical harmonic analysis of astrophysicaldata. Faster known mathematical techniques are not practical in this context, requiring as they do apartition of the sphere that does not deal well with noise.

The HEALPix suite of programs, written in FORTRAN,is publicly available from http://www.eso.org/science/healpix/. This package includes a number ofuseful peripheral tools as well as core routines. The doc-umentation explaining how to use it is good, although toa mathematician interested in understanding internals itwill likely seem that the algorithms it incorporates arenot well explained nor the programs themselves easy tofollow.

The data for the cover are obtained by degradationfrom those in the file of internal linear combination dataavailable at http://lambda.gsfc.nasa.gov/product/map/m_products.cfm.

I wish to thank William O’Mullane for making avail-able to me his Java versions of portions of the HEALPixcode, and Krzysztof Górski, the originator of HEALPix,for his spirited explanation of its role.

—Bill CasselmanGraphics Editor

(notices-covers@ams.org)