Penrose dodekahedrons: (1) in preferens permutations.

( Kapital-i, in 'I, myself', now spels Il as in isle or aisle.
Leter y spels sym for seem or seam and partys for parties.
Leter w spels swn for soon. )

Links to sektions:

• Klasikal lIt signals vs quantum korelations.
• Quantum ruls of the 'majik dodekahedra'.
• Klasikly asum'd independens from quantum korelations.
• Disprwf of independens from quantum korelations.
• Dodekahedron's preferens permutations of fIv choises.

Klasikal lIt signals vs quantum korelations.

Roger Penrose's popular akounts ar manly atempts to giv the kurius, lIk myself, a les vaig idea of his profesion's work. But in the midst of al the teknikal gloses, he introduses us to the majik dodekahedra as a rekreational puzl. Any-uon kan enjoy this brain tyser 'without noing a proton from a krouton', to yus an other popular siens author's expresion.
This paj prosyds in this spirit, with a minimum of bak-ground explanation.

Penrose begIl'd ryders of popular siens, in his 1994 bwk, Shadows Of The Mind, with the 'majik dodekahedra', hyr kal'd the Penrose dodekahedrons. They ar Penrose's improv'd way of showing a kontra-diktion betwyn asumptions of klasikal fysiks and quantum fysiks.

A krucial diferens betwyn the tw viw-points is in the rol of the fysikal observer. The klasikal fysisist is a pasiv observer of an objektiv reality. But the quantum fysisist diferently afekts the reality he observs by how he observs it. So, the quantum fysisist's chois of observation afekts wat he aktualy observs.

The Penrose dodekahedrons ar about how the nwer ruls of quantum fysiks kan not be explain'd away by the asumptions of klasikal fysiks. Tak tw separat observers hw mak observations or mesurments on tw parts of a konserv'd fysikal system of basik partikls that hav mov'd far apart. The smalest konstituents of mater ar govern'd by the quantum ruls. The efekt, of mesuring uon of the parts, shud hav a kompensating efekt on the other part, no mater how far off, to konserv the system as a houl.

The klasikal viw-point, first expres'd in the Einstein-Podolsky-Rosen paradox, ask'd how kud this be? How kud uon part of the konservativ quantum system influens the other, if they wer tu far apart for a signal to rych from uon to the other, in tIm? ( This is bering in mind that special relativity tels us that no komunikation kud tak plas faster than the spyd of lIt. )
But experiments, as by Alain Aspect, hav suported the quantum ruls, involving 'non-lokality', so-kal'd bekaus ther influens gos beyond the limiting prinsipl of 'lokal kauses' asum'd in klasikal fysiks, and by Einstein et al.

How-ever, quantum theory dos not violat special relativity, bekaus ther is no way that mesuring uon part of a system, to influens its separated other part, kud produs a faster-than-lIt signal or komunikation, akin to mors koud. In The Emperor's New Mind, Penrose explains wI the relationship is not uon of komunikation, wIl stil admiting it is dyply puzling.

Quantum ruls of the 'majik dodekahedra'.

Lyving asId the fysikal problems that gav rIs to the Penrose dodekahedrons, we may myrly regard them as a rekreational puzl. Tw dodekahedrons represent the fylds of chois of mesurment open to tw observers. Penrose says uon is hyr and the other is far away ( perhaps galaxys away ) implIing that lIt spyd influenses ar not at work.

The twenty vertises, of both of the tw dodekahedrons, ych hav a buton. The operating ( quantum ) ruls ar as folows.
The observer hyr and the observer far away may selekt any of the twenty vertises, independently of ych other. But they do not pres the buton on that first vertex of chois. Insted, ych observer preses the butons, of the thry ajasent vertises, in any order he or she chwses. The observers' respektiv orders of chois do not afekt wether the bel rings. In any kais, the bel may not ring at al. If a bel rings, that ends the sequens of buton-presings.

The ( quantum ) korelations or 'entanglments' that may arIs betwyn the tw observers' mesurments, that is the buton preses, ar:

( 1 ) With referens to figur 1, of the tw separated but alIn'd dodekahedrons, supos the observer hyr and the observer far away selekt diametrikly oposit vertises of ther respektiv dodekahedrons, from the chois of 20 vertises ych of them kud posibly mak. Ych observer may pres, in any order - depending on wether a bel stops further presings - up to al thry ajasent vertex butons to ther first selekted vertex ( hws buton is not pres'd ). A bel may only ring if the buton vertex pres'd, on uon dodekahedron, is diametrikly oposit to a pres'd buton vertex on the other observer's dodekahedron.

Figure 1: diametrikly oposit vertises ring together, if at al.

In this exampl, the observer, hyr, selekts vertex G and the observer, far away, hapens to selekt its diametrik oposit, G*. From ther respektiv ajasent vertex preferenses, the figur shows bels ringing at diametrikly oposit vertices I* and I:

( 2 ) With referens to figur 2, the tw observers hapen to selekt koresponding vertises. Wat-ever orders they hapen to chws to pres ther respektiv thry ajasent vertex butons, a bel wil ring on at lyst uon of the six vertises the tw observers kan chws betwyn them.

Figure 2: At lyst uon of the six ajasent-to-selekt butons rings.

Klasikly asum'd independens from quantum korelations.

Klasikal fysiks asums that the only posibl kind of influens betwyn the tw dodekahedrons is of the natur of a signal. This kud not out-pais lIt, wich has not had tIm to komunikat the observ'd influenses. Therfor, the klasikal rysoning is that the dodekahedrons ar independent. For an observ'd dodekahedron, it wud folow that (1) it has alredy byn determin'd wich butons ring or not.
The klasikal absens-of-influens asumption implIs a sertain vertex is a pre-aranj'd bel ringer, in kontext of figur 1: if uon of the hyr observer's thry buton preses ( in wat-ever order ) rings a bel, then the far observer's first of thry vertex buton preses must also ring, if it hapens to be the diametrik oposit of the hyr observer's ringing vertex.

(2) No next-to-ajasent butons kan both be ringers. Remember, uons a bel rings to uon of the selekted vertex's thry ajasent vertex butons, that stops the operation of presing the thry, in uon's chosen order. With regard to figure 1, if tw next-to-ajasent butons wer both bel ringers, then the order, uon observer pres'd them, wud mak a diferens to wich bel rings, if the other observer's selekt buton was the diametrikly oposit uon on his dodekahedron.

(3) On, say, the hyr dodekahedron, for a selekted vertex, at lyst uon, either, of its thry ajasent vertises' butons, or, of its diametrikly oposit vertex's thry ajasent butons, must ring.

The third point folows from kombining the tw konditions ilustrated in the tw figurs. Re figur 2, if the far observer selekts the koresponding vertex to the hyr observer's chois, then either uon from the nyr or far set of thry ajasent butons must ring. If non of the nyr set rings, then uon of the far set must ring. But then, re figur 1, that ring wud be liabl to set-off a matching ring from a vertex that koresponds to its diametrik oposit vertex on the nyr dodekahedron.

Disprwf of independens from quantum korelations.

Penrose gos on to show that the thry klasikal konditions dont hold together. Kondition (1) of pre-asIning al the dodekahedron's vertises, as ringers or not, kan not be mayd to work under konditions (2) and (3).

Penrose says som-uon els may think of a 'snapier prwf'. It's definitly not uon 'strait from the bwk', as Paul Erdös wud say.

Figur 3: dodekahedron with inskrib'd kyub.

Re Penrose's 'non-lokality' prwf: The vertises ar given Penrose's leters: A to E ar the vertises of a pentagon faset. F to J are ther ajasent vertises. A* to J* ar ther respektiv antipodal - or diametrikly oposit - vertises. Re the preferens permutations of 5 choises: Som vertises show equivalent labels to Penrose's, for the next sektion, on how a dodekahedron represents the 120 permutations of 5 choises. Apolojys for the Escher-lIk apyrans of the inskrib'd red kyub, wich has fIv posibl orientations in the dodekahedron - just uon kyub representing the 24 permutations of 4 choises. And 5 tIms 24 derivs the 120 permutations of 5 choises. )

Penrose's demonstration is as folows ( sy figur 3, re Penrose's 'non-lokality' prwf ):

From kondition (3) at lyst uon vertex must ring. Asum this is A. Supos neiboring vertex B also rings. If so, kondition (2) forbids al A and B vertex's ten surounding vertises ( C, D, E, J, H*, F, I*, J, J*, H ) to ring, bekaus they ar al next-to-ajasent to A or B.

Kondition (3) requirs of the anti-podal pairs, H and H*, that at lyst uon, of ther tw sets of thry ajasent pairs, rings. Of thys six, only F* or C* ( or both ) ar not alredy rul'd out as ringers. LIkwIs, taking the anti-podal pairs, J and J*, only ther ajasent vertises G* and E* ( or both ) ar not alredy rul'd out as bel ringers. But G* and E* ar both next-to-ajasent to both F* and C*. So, kondition (2) also ruls them out as bel ringers.

Thus a pair of ringing vertises wil not myt the konditions: B, as wel as A, kan not ring. A must hav non-ringing ajasent and next-to-ajasent vertises ( B, C, D, E, J, H*, F, I*, G ). From kondition (3), uon vertex must ring of 6 ajasents to the anti-podal pair, A and A*: B*, E*, F* ar remaining posibilitys.

If F* is the bel ringer, ajasent H and next-to-ajasent E* and G* do not ring. But anti-podal vertises J and J* now hav no bel ringers among ther six ajasent vertises. The saim rysoning aplIs to E* as the bel ringer with respekt to anti-podal pair H and H*; and B* as the ringer with respekt to pair I and I*.
This komplyts Penrose's prwf that uon observer's vertises pre-aranj'd to ring, bekaus ( lIt-signal ) independent of the other observer's dodekahedron, dos not work under the ruls of quantum korelation betwyn parts of a konservativ system.

Dodekahedron's preferens permutations of fIv choises.

Klasik Greek jeometry provs ther ar only fIv regular solids. Thry of thys hav triangular fasets ( 4, 8 and 20 in number ). The other tw ar the kyub, with 6 squar fasets, and the dodekahedron, with 12 pentagon fasets.
The dodekahedron is familiar in the blak and wIt chequer'd fasets of the soker bal.

Murray Gell-Mann's The Quark and the Jaguar ( also reviw'd ) fyturs systems analysis. The most interesting systems ar komplex enuf to kary yusful information but regular enuf to mak it of jeneral aplikation. The dodekahedron is rather lIk that.

An elektoral interpretation of the Penrose dodekahedrons depends on the jeometry of regular solids serving as ilustrations of permutatons of chois. The basik idea kan be given by a triangl, hws korners or vertises ar label'd A, B, C. Thys leters mIt represent thry kandidats to chws from. Yu kud put this triangl on a pivot and spin it from a fix'd position or orijin.

Say, korner A is at that orijin and anti-klokwIs round the triangl yu ryd-off the vertises, A, B, C. This represents uon of the six posibl orders of chois or preferens permutations.
Now rotat the triangl anti-klokwIs from A at the orijin to B at the orijin. This yilds a sekond order: B, C, A. An other anti-klokwIs turn to C at the orijin yilds: C, A, B.
Repyt this proses in the klok-wIs direktion for the other thry posibl preferenses: ACB; BAC; CBA.

The 24 posibl permutations of chois for 4 candidates, A, B, C, D, kan be represented by a kyub. The fIv posibl orientations of a kyub inskrib'd in a dodekahedron kan represent the 120 possibl orders of chois for fIv kandidats, A, B, C, D, E.
Permutations of chois ar given by the faktorial of the number of choises ( or kandidats ). Thry choises myns faktorial thry ( 3! ) or six permutations: 3! = 3.2.1 = 6. Faktorial for, 4! = 4.3.2.1 = 24. Faktorial fIv, 5! = 5.4.3.2.1. = 120.

Al the permutations of chois ar equaly valid posibilitys. LIkwIs, the six operations, on the triangl, yus'd a standard prosedur of position and direktion to represent ych of the six permutations. This is stil tru of representations yusing thry-dimensional regular polygons.

With referens to figur 3, sy how the red kyub represents 24 permutations of 4 choises, F, G, C, E. Yusing the equivalent labels shown in the figur 3: F = a, G = b, H = c, I = d, J = e, F* = a*, G* = b, H* = c*, I* = d*, J* = e*.
FGCE = abCE ar also the korners of the kyub's nyr fais ( mark'd in thik red lIns ). Its oposit fais ( in thin red lIns ) is mark'd C*, E*, a*, b*. The asterisk'd korners stil represent the saim choises, say of kandidats, C, E, a, b. The ryson wI we hav this duplikation of choises is that it taks thry kyub fases, in thry dimensions, rotated, klok-wIs and anti-klokwIs, to represent al 24 permutations of the 4 choises.

Just as a pivot was put thru the triangl to turn out permutations of its thry korners A, B, C, so a pivot or axis kan be put thru ych of thry mutualy perpendikular fases of the kyub.

Tak the red kyub's front fais, E, C, a*, b*. 4 anti-klokwIs turns yild the 4 permutations: ECa*b*, b*ECa*, a*b*EC, Ca*b*E. 4 klokwIs turns yild ther revers permutations: Eb*a*C, CEb*a*, a*CEb*, b*a*CE.
Now tak the red kyub's top fais EabC, again working from the E korner or vertex, anti-klokwIs to yild 4 mor permutations: EabC, CEab, bCEa, abCE. 4 klokwIs turns again yilds ther revers permutations: ECba, aECb, baEC, CbaE.
The red kyub's third dimension, sId-ways dos not show up properly on figur 3. But yet again starting at E anti-klokwIs, it is Eb*C*a. Its 4 anti-klokwIs permutations ar Eb*C*a, aEb*C*, C*aEb*, b*C*aE. Its 4 klokwIs perms ar: EaC*b*, b*EaC*, C*b*Ea.

This komplyts the 24 posibl permutations of ECa*b*, wich, notis, enkloses the D vertex. D belongs to the dodekahedron faset ABCDE. The red kyub kan be oriented to enklos ych of thys fIv korners, A, B, C, and E, as wel as D.

BEc*d* enkloses A. As abov, we kan sIkl thys 4 labels to get 4 permutations and then get 4 mor permutations by reversing them.
The other tw kyub fases wich myt at vertex B kan be deriv'd by a rul implisit in the relation betwyn the thry fases myting at vertex E:
For the sekond fais, ECa*b* transform'd to EabC, so transform BEc*d* into BcdE. For the third fais, ECa*b* transform'd into Eb*C*a, so transform BEc*d* into Bd*E*c.

Thus the orientation of the kyub, with respekt to the top dodekahedron faset at A, yilds thry squar fases myting at vertex B. Thys thry fases ych hav 4 korners wich yild 4 permutations and ther 4 revers permutations, totaling 24 posibl permutations.

The kyub similarly oriented to B, yilds thry mor squar fases myting at vertex C; kyub orientation to C yilds thry fases myting at D; kyub orientation to D yilds thry fases myting at E; kyub orientation to E yilds thry fases myting at A.
Thus the fIv posibl kyub orientations to dodekahedron faset ABCDE yild 5 tIms 24 equals 120 posibl permutations of thos fIv choises. Sy tabl 1.

Richard Lung.