Pentahedral Volume, Chaos, and Quantum Gravity

Pentahedral volume, chaos, and quantum gravity

Hal M. Haggard1 1Centre de Physique Th´eoriquede Luminy, Case 907, F-13288 Marseille, EU∗ We show that chaotic classical dynamics associated to the volume of discrete grains of space leads to quantal spectra that are gapped between zero and nonzero volume. This strengthens the connection between spectral discreteness in the quantum geometry of gravity and tame ultraviolet behavior. We complete a detailed analysis of the geometry of a pentahedron, providing new insights into the volume operator and evidence of classical chaos in the dynamics it generates. These results reveal an unexplored realm of application for chaos in quantum gravity.

A remarkable outgrowth of quantum gravity has been The area spectrum has long been known to be gapped. the discovery that convex polyhedra can be endowed with However, in the limit of large area eigenvalues, doubts a dynamical phase space structure [1]. In [2] this struc- have been raised as to whether there is a volume gap [6]. ture was utilized to perform a Bohr-Sommerfeld quanti- The detailed study of pentahedral geometry here pro- zation of the volume of a tetrahedron, yielding a novel vides an explanation for and alternative to these results. route to spatial discreteness and new insights into the We focus on H = V as a tool to gain insight into spectral properties of discrete grains of space. the spectrum of volume eigenvalues. The spatial volume Many approaches to quantum gravity rely on dis- also appears as a term in the Hamiltonian constraint of cretization of space or spacetime. This allows one to general relativity, the multiplier of the cosmological con- control, and limit, the number of degrees of freedom of stant; whether chaotic volume dynamics also plays an the gravitational field being studied [3]. In the simplest important role there is worthy of future investigation. approaches, such as Regge calculus, attention is often re- This work provides two lines of argument: The first stricted to simplices. However, it is not clear that such line is a detailed study of the classical volume associ- a restriction is appropriate for the general study of the ated to a single pentahedral grain of space. We provide gravitational field [4]. The present work takes up the preliminary evidence that the pentahedral volume dy- study of grains of space more complex than simplices. namics is chaotic and all of the analytic tools necessary The Bohr-Sommerfeld quantization of [2] relied on the for a thorough numerical investigation. In particular, we integrability of the underlying classical volume dynam- find a new formula for the volume of a pentahedron in ics, that is, the dynamics generated by taking as Hamil- terms of its face areas and normals, show that the vol- tonian the volume, H = Vtet. In general, integrability ume dynamics is adjacency changing, and for the first is a special property of a dynamical system exhibiting time analytically solve the Minkowski reconstruction of a high degree of symmetry. Instead, Hamiltonians with a non-trivial polyhedron, namely the pentahedron. In two or more degrees of freedom are generically chaotic the second line general results from random matrix the- [5]. Polyhedra with more than four faces are associated ory are used to argue that a chaotic volume dynamics to systems with two or more degrees of freedom and so it implies the generic presence of a volume gap. is natural to ask: “Are their volume dynamics chaotic?” We initiate the investigation of whether there is chaos Here we show that the answer to this question has in the volume dynamics of gravity by considering a sin- important physical consequences for quantum gravity. gle pentahedral grain of space. As in [2], examination Prominent among these is that chaotic volume dynam- of the classical volume dynamics of pentahedra relies on ics implies that there is generically a gap in the volume turning the space of convex polyhedra living in Euclidean spectrum separating the zero volume eigenvalue from its three-space into a phase space. This is accomplished with

arXiv:1211.7311v2 [gr-qc] 17 Jan 2013 nearest neighbors. In loop gravity, it is convenient to the aid of two results: (1) Minkowski’s theorem [9] states work with a polyhedral discretization of space because it that the shape of a polyhedron is completely character- allows concrete study of a few degrees of freedom of the ized by the face areas A` and face normals ~n`. More gravitational ﬁeld; however, what is key is the spectral precisely, a convex polyhedron is uniquely determined, discreteness of the geometrical operators of the theory. up to rotations, by its area vectors A~` ≡ A`~n`, which This is because the partition functions and transition am- P ~ satisfy ` A` = 0, and we call the space of shapes of plitudes that deﬁne the theory are expressed as sums over polyhedra with N faces of given areas A`, these area and volume eigenvalues. The generic presence ~ P ~ ~ of gaps (above zero) in the spectra of these operators PN ≡ A`, ` = 1 ..N | ` A` = 0 , kA`k = A` /SO(3). ensures that these sums will not diverge as smaller and smaller quanta are considered; such a theory should be (2) The space PN naturally carries the structure of a well behaved in the ultraviolet regime. phase space [11], with Poisson brackets, P f, g = A~` · ∇ f × ∇ g , (1) ` A~` A~`

∗ [email protected] where f(A~`) and g(A~`) are functions on PN . This is the 2

FIG. 2. Illustrative Pachner move for a pentahedron (the FIG. 1. Closing a pentahedron into a tetrahedron. (a) A back face, 3, suppressed). (a) The Schlegel diagram of a ﬁducial face labeling of the pentahedron. (b) The closure of 54-pentahedron. (b) Schlegel diagram of a sub-dominant the pentahedron upon continuing its sides. (c) The resulting quadrilateral pyramid. (c) Resulting Schlegel diagram of a tetrahedron with face labels and associated scalings. 12-pentahedron upon completion of the move.

usual Lie-Poisson bracket if the A~` are interpreted physi- For the dominant class of pentahedra, like that pic- cally as angular momenta, i.e. as generators of rotations. tured in Fig. 1(a), there are ten distinct adjacencies and To study the pentahedral volume dynamics on P5, with two “orientations” per adjacency. A convenient method H = Vpent, it is first necessary to find this volume as a for referring to a pentahedron with a particular adja- function of the area vectors. Three auxilliary variables, cency is to state the labeling of the two triangular faces; α, β, and γ, defined presently, aid in the construction. this determines the adjacencies of all of the faces. By Assume that a pentahedron with the labels and face adja- orientation we mean a specification of which triangle is cencies depicted in Fig. 1(a) is given. This pentahedron consumed by the continuation process described above. can be closed into a tetrahedron by appropriately con- This triangle is referred to as the “upper” one. (It is tinuing the faces 1, 2, and 3; define the positive numbers tempting to think of it as the “smaller” triangle but this that scale the old face areas into the new ones to be α, is not generally true. Rather, the upper triangle can have β and γ respectively. Figures 1(b) and 1(c) depict the the larger area, which can be seen by imagining cutting continuation process. Note also that α, β, γ > 1. off the tip of Fig. 1 (b) at greater and greater angles with These scalings can be determined using the closure respect to the base giving upper triangles with larger and condition of the resulting tetrahedron, αA~1+βA~2+γA~3+ larger areas.) If we adopt the convention that the first la- A~4 = 0. By dotting in cross products of any two of A~1, bel of a pair denotes the upper triangle we completely set A~2, and A~3 and letting Wijk = A~i · (A~j × A~k) we find, the adjacency and orientation. Thus, Figure 1(a) depicts a 54-pentahedron. W W W While introducing α, β, and γ our choice of a 54- α = − 234 , β = 134 , γ = − 124 . (2) W123 W123 W123 pentahedron was fiducial. It is straightforward to list the analogous parameters for each type of pentahedron. For Furthermore, Fig. 1 suggests a formula for the volume of 0 0 example, a 53-pentahedron with closure α A~1 + β A~2 + the pentahedron: this volume is the difference of the large 0 0 0 A~3 + γ A~4 = 0 has α = W234/W124, β = −W134/W124 tetrahedron’s volume (Fig. 1(c)) and the small dashed 0 and γ = −W123/W124. Remarkably, these can be alge- tetrahedron’s volume (top of Fig. 1(b)). Thus we have, braically expressed in terms of the 54-parameters: √ 2 q p 0 0 0 V = pαβγ − α¯β¯γ¯ W , (3) α = α/γ, β = β/γ, γ = 1/γ. (4) pent 3 123 Now, note that if γ > 1 then necessarily γ0 < 1 and so whereα ¯ ≡ α − 1 and similarly for β¯ andγ ¯. Before inves- the constructability of 54- and 53-pentahedra are mutu- tigating the Hamiltonian flow of Vpent, it is necessary to ally exclusive. Thus we see that requiring the closure examine the role of the fiducial choice made in Fig. 1(a). scalings be greater than one is a strong condition on con- The Minkowski theorem mentioned above is only an structibility. By examining each case similarly, one finds existence and uniqueness theorem. That is, if one is given that this condition implies that constructability of 54- P5 ~ five vectors satisfying `=1 A` = 0, then Minkowski pentahedra is only consistent with the constructability guarantees that a unique pentahedron corresponding to of pentahedra of types {12, 21, 23, 32, 13, 31} and exclu- those vectors exists but tells one nothing about how to sive with all other types (see Appendix A). construct it. As demonstrated, e.g. numerically in [1], These results are more natural when interpreted in the reconstruction problem, i.e. building the polyhedron, light of Schlegel diagrams. A Schlegel diagram is a planar is difficult. Lasserre [10] was the first to appreciate that graph that represents a convex polyhedron P by project- reconstruction hinges on determining the adjacency of ing the whole polyhedron into one of its faces. A Pach- the faces of the end polyhedron from the given vectors. ner move generates a new Schlegel diagram from a given Remarkably, introducing α, β, and γ furnishes an ana- one by contracting an edge until its two vertices meet lytic solution to the adjacency problem, and subsequently and then re-expanding a new edge from this juncture in the Minkowski reconstruction for the pentahedron. a complementary direction, see Fig. 2. Note that the 3

question of chaos. 5 The shape space P5 has dimension dim P5 = 4, i.e. two degrees of freedom, and again generic Hamiltonians with two degrees of freedom are chaotic [5]. Further, 4 note that the boundaries between the adjacency regions above correspond to quadrilateral pyramids, for example the one of Fig. 2(b) in the case of a 54/12 boundary. 3 Because the closure scaling works for both pentahedra at a boundary pyramid the volume formulae must agree on the boundary and Vpent is clearly continuous. In fact, see 2 Appendix C, it can be shown that Vpent is not smooth but C2, that is, its first and second derivatives are continuous. This surprising result strengthens the expectation that 1 1 2 3 4 5 the volume dynamics is chaotic: Dynamical billiards with boundaries that have limited smoothness, an analogous property, frequently exhibit chaos [12]. FIG. 3. A partial phase diagram for the adjacency classes of a pentahedron with fixed γ = 1.7. The central, shaded re- The analytic formulae presented in this work make it gion is the parameter space in which the 54-pentahedron is possible to numerically establish chaos but this will be constructible. Neighboring regions are labelled by the penta- a lengthy procedure. Here we present early numerical hedral classes that are constructible within them. evidence that strengthens the general arguments given above. We have numerically implemented the volume dynamics for H = Vpent on the oscillator phase space de- central diagram of Figure 2(b) is also the Schlegel dia- veloped in [13], where the effective tool of symplectic in- gram of a pentahedron, in this case a pyramid built on a tegrators can be applied [14]. We use an implicit Runge- quadrilateral base. These are referred to as sub-dominant Kutta symplectic integrator with Gauss coefficients. The pentahedra because they are of co-dimension one in the harmonic oscillator description can be connected, via space of all pentahedra; this is due to the fact that four symplectic reduction, to the shape space P5, studied in planes meet at the apex of such a pentahedron, a non- detail in [15]. generic intersection in three dimensions. We will have We can report several interesting pieces of evidence more to say about these pyramids briefly. for chaos: The majority of initial conditions investigated The pentahedral types compatible with a 54- lead to trajectories that cross adjacency boundaries and pentahedron (listed below (4)) are precisely those that quickly destabilize the integrator. Adjacency crossing is are reachable by a Pachner move not degenerating a face. an entirely new feature, not present (nor possible) in the The α, β, and γ parameters can be used to extend the tetrahedral case. The instability of these trajectories is classification further: ordering the set {α, β, γ} by mag- in accordance with the limited smoothness of Vpent at nitude the compatible types can be narrowed to just one, these boundaries. By contrast, only a small set of initial e.g. if α > β > γ then only 12 type pentahedra are com- conditions give rise to trajectories that do not cross ad- patible with type 54. Finally, the last two cases can be jacency boundaries. These trajectories can be integrated distinguished; if γ ≥ αβ/(α + β − 1) then the type 54 is for long times and Poincarésections created; they corre- constructible, else the type 12 is and for equality we have spond to stable tori. Thus the dynamics is certainly of a pyramid, which is a limiting case consistent with both mixed type but instability (chaos) appears to dominate 54- and 12-scalings (see Appendix B). This provides a the phase space. These results are being strengthened by complete solution to the adjacency problem and leads to another group [16]. In the special case of equi-area pen- analytic formulae for the full Minkowski reconstruction tahedra, they find evidence of chaos, characterized by (see Appendix D). positive intermediate Lyapunov exponents, throughout a These findings are summarized in a pentahedral phase large fraction of the phase space. The analytic results diagram, Fig. 3. A similar diagram applies with any presented here will allow a complete resolution (for all pentahedral type as the central region, and so, by patch- areas) of the question soon. Therefore it is important to ing these diagrams together one obtains a phase portrait address what chaotic volume dynamics implies about the spanning all pentahedral adjacencies and orientations. volume spectrum, which we turn to now. The importance of all of this is that the formula (3) One route to carry the integrable-chaotic distinction only applies to 54-pentahedra. Define Vpent as the func- into quantum theory has been provided by an elegant tion whose level contours consist of all continuously con- synthesis of semiclassical and matrix theoretic arguments nected equivolume pentahedra in P5. We have shown [5, 8]. Generic quantum systems with chaotic classical that this function is piecewise defined with a different limits have different spectral properties than their classi- formula in each adjacency region; each formula obtain- cally integrable counterparts. Among these differences is able in the same manner as (3). This provides a complete a different expected behavior for the spacing s between description of Vpent and allows us to begin to address the neighboring levels, see Fig. 4. Classically chaotic systems 4

presented here provides a very general mechanism that would ensure a volume gap for all discrete grains of space. One might worry that this result could be spoiled by the unfolding process mentioned above. This concern is justified because more and more states crowd into the interval of allowed volume in the limit of large areas. How- ever, we are interested in the density of states at small volumes (analogously low energies), in fact zero volume, and in this limit the density of states must smoothly go to zero. This is because zero volume polyhedra correspond to collinear configurations of area vectors [2], which are individual points of the phase space. These points are not as large as a Planck cell and so cannot support many FIG. 4. The unfolded spectrum for ∼ 1000 consecutive eigen- quantum states. Thus, taking into account the average values of Sinai’s billiard (a prototypical chaotic billiard). The behavior of the spectrum near zero energy, “refolding” solid (dashed) curve is the prediction for a classically chaotic if you like, may cause some squeezing of states but will (integrable) system: Wigner’s surmise or Poisson’s distribu- not destroy a volume gap. The numerical results of [16] tion respectively. Adapted from [7]. illustrate this general argument in the equi-area case. Previous numerical investigations of the quantum vol- exhibit level repulsion; at s = 0, which means degenerate ume for 5-valent spin networks (corresponding to the eigenvalues, we see that generically the probability for pentahedra here) have, in stark contrast, found evidence Wigner’s surmise vanishes. In sharp contrast, classically of an accumulation at zero volume of the volume eigen- integrable systems exhibit level bunching, the probability values [6]. There are multiple reasons for this disagree- is finite (even maximal) at s = 0. ment. Two important reasons are clear in the present To arrive at these results two main steps are taken: framework: the well studied Ashtekar-Lewandowski and First, in order to compare spectra across different sys- Rovelli-Smolin volume operators [18] both assume that tems, the spectra are “unfolded.” Unfolding is a normal- the volume of a region can (i) be broken into a sum of ization procedure whereby the average level spacing in a parallelepiped contributions that are of the same form given energy interval is brought to one. Once the spectra and (ii) these contributions are minimally coupled. The have been unfolded they can be characterized by their pentahedral volume formula, Eq. (3), suggests that (i) is statistical fluctuations, e.g. by the probability of a given not a good approximation and that there is a strong cou- level spacing P (s). The second step then is to find P (s) pling between the area vectors (fluxes) for more generally for different types of systems. Random matrix theory has shaped regions. Furthermore, the parallepiped volume is established, for the chaotic case, Wigner’s surmise proportional to the tetrahedral one studied in [2] and there found to be integrable. So the minimal coupling π P (s) = s exp(−πs2/4), (5) assumption of these proposals may keep the classical dy- 2 namics near integrability and lead to the accumulations as an excellent approximation for Hermitian Hamiltonian found in [6]. The polyhedral volume operator studied systems having time reversal symmetry. By contrast here is semiclassically consistent and lucidly exposes ge- semiclassical results give P (s) = e−s for the integrable ometrical structures, like the strong coupling of Eq. (3). case. The generic validity of the connection between ran- In this note we have completely solved the geometry of dom matrices and chaos, originally an empirical observa- a pentahedron specified by its area vectors and defined its tion, has been demonstrated semiclassically [17]. volume as a function of these variables. By performing It is the level repulsion of the Wigner surmise that a numerical integration of the corresponding volume dy- guarantees the presence of a volume gap when the clas- namics we have given early indicators that it generates a sical limit of the volume dynamics is chaotic: if there is chaotic flow in phase space. These results uncover a new a zero volume eigenvalue, repulsion forbids the accumu- mechanism for the presence of a volume gap in the spec- lation of further eigenvalues on top of it. Also, repulsion trum of quantum gravity: the level repulsion of quantum continues to hold for mixed phase spaces, such as the systems corresponding to classically chaotic dynamics. pentahedral one, with a sufficiently large proportion of The generic presence of a volume gap further strength- chaotic orbits [8]. ens the expected ultraviolet finiteness of quantum gravity The presence of a volume gap in the (integrable) case theories built on spectral discreteness. of a tetrahedron has already been established in [6] and [2]. Furthermore, a chaotic pentahedral volume dynamics Thank you to E. Bianchi for insights and generous en- would strongly suggest that there will be chaos for poly- couragement. Also to R. Littlejohn, A. Essin and D. Beke hedra with more faces, which have an even richer struc- for early interest and ideas. This work was supported by ture in their phase space [1]. Consequently, the argument the NSF IRFP grant OISE-1159218. 5

Appendix A: Table of closure scalings main text about the mutually exclusive constructibility of different adjacency classes. For example, if α, β, and γ are all greater than 1, then necessarily γ2 is less then This appendix contains an exhaustive list of the closure 1 and the 53-pentahedron is not constructible, see Eq. relations for pentahedra. Each closure relation is also (A2). In this manner one can check that a pentahedron algebraically related to the scaling parameters α ≡ α1, whose constructibility is mutually consistent with that of β ≡ β1, and γ ≡ γ1 of the first case and once again the a 54-pentahedron is from the set {12, 21, 23, 32, 13, 31}. shorthand Wijk ≡ A~i·(A~j ×A~k) is used. The pentahedron Furthermore, if the parameters α, β, and γ are ordered corresponding to each case can be read off by first noting the mutually consistent set can be further narrowed. For which vector doesn’t appear in the closure relation, this example, assume that α > β > γ > 1 then only 54- and vector corresponds to the upper triangle and then noting 12-pentahedra are mutually consistent; as an illustrative which vector has no multiplier, this vector corresponds check note that under this assumption α15 < 0 and so the to the lower triangle. Thus case 1. gives the parameters 23-pentahedron is no longer constructible. The equations for a 54-pentahedron. necessary to resolve this final ambiguity (e.g. between 54- and 12-pentahedra) are described in the next section. This list can be used to confirm the claims in the

1. α1A~1 + β1A~2 + γ1A~3 + A~4 = 0,

W234 W134 W124 α ≡ α1 = − β ≡ β1 = γ ≡ γ1 = − . (A1) W123 W123 W123

2. α2A~1 + β2A~2 + A~3 + γ2A~4 = 0,

W234 α W134 β W123 1 α2 = = β2 = − = γ2 = − = . (A2) W124 γ W124 γ W124 γ

3. α3A~1 + A~2 + β3A~3 + γ3A~4 = 0,

W234 α W124 γ W123 1 α3 = − = β3 = − = γ3 = = . (A3) W134 β W134 β W134 β

4. A~1 + α4A~2 + β4A~3 + γ4A~4 = 0,

W134 β W124 γ W123 1 α4 = − = β4 = = γ4 = − = . (A4) W234 α W234 α W234 α

5. α5A~1 + β5A~2 + γ5A~3 + A~5 = 0,

W235 W135 W125 α5 = − = 1 − α β5 = = 1 − β γ5 = − = 1 − γ. (A5) W123 W123 W123

6. α6A~1 + β6A~2 + A~3 + γ6A~5 = 0,

W235 1 − α W135 1 − β W123 1 α6 = = β6 = − = γ6 = − = . (A6) W125 1 − γ W125 1 − γ W125 1 − γ

7. α7A~1 + A~2 + β7A~3 + γ7A~5 = 0,

W235 1 − α W125 1 − γ W123 1 α7 = − = β7 = − = γ7 = = . (A7) W135 1 − β W135 1 − β W135 1 − β

8. A~1 + α8A~2 + β8A~3 + γ8A~5 = 0,

W135 1 − β W125 1 − γ W123 1 α8 = − = β8 = = γ8 = − = . (A8) W235 1 − α W235 1 − α W235 1 − α

9. α9A~1 + β9A~2 + γ9A~4 + A~5 = 0,

W245 α W145 β W125 1 α9 = − = 1 − β9 = = 1 − γ9 = − = 1 − . (A9) W124 γ W124 γ W124 γ 6

10. α10A~1 + β10A~2 + A~4 + γ10A~5 = 0,

W245 γ − α W145 γ − β W124 γ α10 = = β10 = − = γ10 = − = . (A10) W125 γ − 1 W125 γ − 1 W125 γ − 1

11. α11A~1 + A~2 + β11A~4 + γ11A~5 = 0,

W245 γ − α W125 γ − 1 W124 γ α11 = − = β11 = − = γ11 = = . (A11) W145 γ − β W145 γ − β W145 γ − β

12. A~1 + α12A~2 + β12A~4 + γ12A~5 = 0,

W145 γ − β W125 γ − 1 W124 γ α12 = − = β12 = = γ12 = − = . (A12) W245 γ − α W245 γ − α W245 γ − α

13. α13A~1 + β13A~3 + γ13A~4 + A~5 = 0,

W345 α W145 γ W135 1 α13 = − = 1 − β13 = = 1 − γ13 = − = 1 − . (A13) W134 β W134 β W134 β

14. α14A~1 + β14A~3 + A~4 + γ14A~5 = 0,

W345 β − α W145 β − γ W134 β α14 = = β14 = − = γ14 = − = . (A14) W135 β − 1 W135 β − 1 W135 β − 1

15. α15A~1 + A~3 + β15A~4 + γ15A~5 = 0,

W345 β − α W135 β − 1 W134 β α15 = − = β15 = − = γ15 = = . (A15) W145 β − γ W145 β − γ W145 β − γ

16. A~1 + α16A~3 + β16A~4 + γ16A~5 = 0,

W145 β − γ W135 β − 1 W134 β α16 = − = β16 = = γ16 = − = . (A16) W345 β − α W345 β − α W345 β − α

17. α17A~2 + β17A~3 + γ17A~4 + A~5 = 0,

W345 β W245 γ W235 1 α17 = − = 1 − β17 = = 1 − γ17 = − = 1 − . (A17) W234 α W234 α W234 α

18. α18A~2 + β18A~3 + A~4 + γ18A~5 = 0,

W345 α − β W245 α − γ W234 α α18 = = β18 = − = γ18 = − = . (A18) W235 α − 1 W235 α − 1 W235 α − 1

19. α19A~2 + A~3 + β19A~4 + γ19A~5 = 0,

W345 α − β W235 α − 1 W234 α α19 = − = β19 = − = γ19 = = . (A19) W245 α − γ W245 α − γ W245 α − γ

20. A~2 + α20A~3 + β20A~4 + γ20A~5 = 0,

W245 α − γ W235 α − 1 W234 α α20 = − = β20 = = γ20 = − = . (A20) W345 α − β W345 α − β W345 α − β

Appendix B: Pyramidal pentahedra tahedra. For these pentahedra both of the closing scalings of the two neighboring adjacency regions are valid As briefly observed in the main text, the boundary be- and thus the volume of the pentahedral pyramids can tween two adjacency regions consists of pyramidal pen- 7 be calculated in two distinct manners. Setting these two the quadrilateral of the diagram, volumes equal one finds a constraint satisfied amongst the three scaling parameters. For example, at the adja- A1 = A − Ad = A[1 − (1 − λ)(1 − µ)]. (B3) cency between a 54-pentahedron and a 12-pentahedron the following constraint is satisfied, Setting the two expressions for A1 equal yields a relation between the face scalings and the edge scalings: αβ γ = . (B1) 1 α + β − 1 α = . (B4) λ + µ − λµ While the argument outlined above is geometrically ob- A structurally identical argument applied to the faces 2 vious the algebraic manipulations are unnecessarily com- and 3 yields the two further relations plex. To avoid this complexity we provide another even simpler geometric argument here. 1 1 β = , γ = , (B5) To fix notations consider the transition from a 54- µ + ν − µν ν + λ − νλ pentahedron to 12-pentahedron. As discussed in the main text this transition occurs through a Pachner move, where ν is a final edge scaling used to bring the third see Fig. 3 of the main text. Notice that during this move edge that meets at the apex of the scaled tetrahedron the edge bordering faces 1 and 2 goes from having non- down to its length in the pentahedron (this is the edge zero length when the 54-pentahedron is constructible, to where faces 2 and 3 meet). having zero length when the pentahedron is pyramidal, During the Pachner move depicted in Fig. 3, the edge and then vanishes altogether when the 12-pentahedron joining faces 2 and 3 degenerates into a point. This can is constructible. This is the observation we will use to only happen if µ = 0 and hence at the pyramidal config- derive Eq. (B1). uration we have α = 1/λ and β = 1/ν. Putting these re- Figure 5 shows the face corresponding to A~1 of the lations into the expression for γ yields a relation between initial 54-pentahedron, along with the larger, triangular the area vectors at the 12/54 pyramidal configuration: face that appears in Fig. 2(b). Let the edge lengths of αβ the triangle be `1, `2, and `3. Introduce two parameters γ = . (B6) λ and µ that scale the edges `1 and `2 to give the corre- α + β − 1 sponding edge lengths of the unscaled pentahedron face Furthermore, for the 54-pentahedron to be constructible λ` and µ` ; these paremeters are necessarily less than 1 2 we must have µ > 0 and a straightforward algebraic in- one, λ, µ < 1. version of the equations above shows that this occurs when γ is greater than the right hand side of Eq. (B6). These are the results quoted in the main text.

Appendix C: Smoothness of the pentahedral volume

As explained in the main text Vpent is certainly continuous. However, in order for Vpent to define a Hamiltonian flow it is important to check that at least its first derivatives are continuous as well. It turns out that Vpent is C2, that is, its first and second derivatives are both continuous. Because Vpent is defined in a piecewise manner over the different adjacency regions and the formulae in each region are different it is plausible that Vpent is not smooth. 2 FIG. 5. The first face of a 54-pentahedron and its scaling into To illustrate the demonstration that Vpent is C we fo- a triangle through the closure process described in the main cus on just two formulae expressing it in the 54 and 12 text. regions respectively as:  √ √ √ From the definition of α the area of the large triangle 2 αβγ − p(α − 1)(β − 1)(γ − 1) W ,  3 123 is A = αA1. On the other hand, this area can also be  √ q α−γ q β−γ β−1 β √ Vpent = 2 α−1 α expressed in terms of the angle θ3, A = 1/2`1`2 sin θ3. 3 α−β α−β α−β − α−β α−β α−β W345,  Similarly, the area of the small dashed triangle is .... 1 √ (C1) Ad = (1 − λ)`1(1 − µ)`2 sin θ3 = (1 − λ)(1 − µ)A, (B2) It will be easier to work with the function V / W , 2 pent 123 which will have the same smoothness as Vpent as long and this yields a second formula for the area of face 1, as W123 = 0 is avoided. The advantage of the latter 8 function is that it can be viewed as a function of α, β, and as can be confirmed by dotting in ~nr, ~ns, and ~nt. This γ alone because W345/W123 = (α − β). Again we gloss formula can be used to find the edge lengths of face 4 over consideration of points where these coordinates fail, and they are all proportional to h1. Let these three i.e. where the denominators in their definitions are zero. edge lengths be h1e1, h1e2, and h1e3 where the dimen- Now, we√ just check that the first and second derivatives sionless lengths (e1, e2, e3) are completely determined by of Vpent/ W123 with respect to α, β, and γ and evaluated the given area vectors and the formula (D1). Then using at γ = αβ/(α + β − 1) are the same when calculated in Heron’s formula for the area ∆ of a triangle given its edge 2 region 54 and in region 12. At third derivatives the two lengths, A4 = h1∆(e1, e2, e3), and this can be solved for calculations begin to disagree. Finally, this calculation h1, is carried out for each of the boundaries connecting two adjacency regions.

s Appendix D: Minkowski reconstruction A4 h1 = (D2) ∆(e1, e2, e3) Having determined the adjacency class of a pentahedron it is straightforward to explicitly reconstruct it. For deﬁniteness, once again, assume that we have established that the adjacency is that of a 54-pentahedron. Choose Finally, h can be extracted from the relation of the vol- the origin of coordinates at the vertex where the faces 2, 5 ume to the heights, 3, and 4 meet. In these coordinates, only two of the ﬁve perpendicular heights to the faces are left to be found as h2 = h3 = h4 = 0. Now, the vector pointing to the intersection of three planes can be expressed in terms of the normals to the 1 Vpent = (A1h1 + A5h5), (D3) planes ~nr, ~ns, and ~nt by, 3 1 xrst = [hr(~ns × ~nt) |~nr · (~ns × ~nt)|

+ ht(~nr × ~ns) + hs(~nt × ~nr)], and the value of the volume determined by Eq. (5) of the (D1) main text. This completes the Minkowski reconstruction.

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