In other words, as promised, and without venturing beyond the borders of the Annulus of Infamy, residents are equipped to estimate the local value of π empirically.
Out of curiosity, the author has done this, using facts provided in the cantos. It can be reported here for the first2 time that the magnitude of Satanic π within the Inferno and its secular value diverge after four decimal places. (Which is the greater?) It should be concerning that the architecture of the underworld rests, in fact, on circles that are fundamentally defective. (The risk of catastrophic geometric failure under dynamic stress may be greater than we think if Wallis Pioniv’s suggestion that the rings are rapidly rotating3 proves correct.) It also serves as a warning to those considering a Faustian bargain4 for superior knowledge, that their gain may be false or illusory.
[EXERCISE Direct Line to Satan] Immediately upon entering the Annulus of Infamy, measure (‘Take Theta’s strength’) the internal angle θ1 between the two Secret Chords. Show that the distance from the portal inscription to the centre of the Inferno is
R1 being the radius of C1. Thus the length of the dreaded journey to the Satanic centre is computable, again without leaving the Annulus of Infamy.
To advance our exploration of the underworld, and evade Satan, we require further notation. It will prove convenient to consider the Satanic centre as a point circle, C10, having radius R10 = 0. We can now view the innermost region as Annulus 9 rather than a disc. Let Tk be any point on Ck, for k = 1, 2, …, 10. (It follows that T10 is the centre.) Then Tk, k = 2, 3, …, 9, is a tangent point defining (as midpoint) a Secret Chord in Ak − 1, which in turn defines (by endpoints) two points Tk − 1 on Ck − 1. All points T1 so defined on C1 permit exit from the underworld. Also, Tk, k = 2, 3, …, 9, defines as origin two Secret Chords in Ak. (In A9, these are collinear, and equal to a diameter.)
[EXERCISE Escape from the Underworld] The above equation for R1 defines the shortest route to Satan but offers no possibility of escape. This author holds the view that the reader who has persevered to this point deserves a chance of immortality, and here it is. Enter A1 (Infamy), proceed a distance l1 along one of the two Secret Chords (say, the right-hand) and enter A2, if compelled, at the chord’s midpoint (a tangent point, T2). But if the Counting Owl invigilating at this gate is distracted at that moment, you can instead sneak along the second half of the chord to the exit (T1, in this case the Garden of Renunciation). If unsuccessful on the first occasion, repeat the strategy for each of the next seven Circles in the hope of working your way back to the first.
It should be apparent that if (say) the right-hand Secret Chord is chosen at consecutive points of entry, and retreat has nowhere been possible, the route to Satan will be a piecewise linear function spiralling anticlockwise into the centre. It is almost certain that this shape is the explanation for the Winding the Curse allusions in traditional eschatology, and that the tangent points represent the mysterious Eight Kisses of Redemption5 in the Gospel of St Ignorius.
The reader now instructed in this strategy would be well advised, for planning and provisioning purposes, to inform herself of the least distance she must travel should immortality not be her lot. Verify that the soul’s journey of the damned in this case has length6
noting that θ9 = 0, l9 = R9. It is an intriguing fact, evident from this expression, that even in that sorry wilderness of imperfect circles, her winding path is independent of the Satanic value of π. (There is a subtlety to observe here, discussed in the secret content of Appendix D.) This strongly suggests that at least some route planning in the underworld can be successfully undertaken using geometry familiar to us.
[REMARK Behold the Power of Two] Consider such a soul, who everywhere progresses though the Circles without reversal. Beginning at the (inscription) portal of A1, she can choose left or right Secret Chords, giving rise to two possible entry points T2 into A2. Each of these is the origin of two Secret Chords (left and right) in A2, each of which, in turn, defines an entry point T3 on C3. It will be apparent that there are 2k ways of exiting Ak (to enter Ak + 1). In particular, she enjoys a choice of 28 = 256 pathways into A9 (all have the same length L, given above). That such binomial decision trees might similarly prescribe our destiny in the living world has long been a human preoccupation. The idea is discussed by Milton Noyes in Quincunx and Can’t, a fascinating study of its special significance in the Syllabine civilization. A recent metaphysical critiquing of the egregious deaths of nineteenth century American desperados Rigo Mortiss and Kevin Fister (described in Chapter 1 of the present work) recapitulates both underworld design and divine retributive torment, suggesting their ordeal was a spirit-gift of prescience ministered through the preacher (and sheriff) of Dante. (Some view this episode in history more melodramatically, as an occasion on which Satan visited America.)
[EXERCISE Random Walk] Marigold enters the underworld through the portal of the Annulus of Infamy. She interprets the inscription correctly and adopts the strategy described above. From each tangent point Tk, k = 2, 3, …, 9, she moves to a point Tk + 1 on Ck + 1 with probability p or, with probability 1 − p proceeds unchallenged by the Counting Owl along the Secret Chord in Ak − 1 to a point Tk − 1 on Ck − 1. Then Marigold inhabits a Markov chain having state space C1 (escape from the underworld), C2, C3, …, C10 (Satan), where states C1 and C10 are absorbing barriers. On average, how many visits to states C2, C3, …, C9 does she make before entering state C1 or state C10? What is the probability that Marigold returns to the living?
[EXERCISE The Prophets Rejoice] Marigold enters the underworld through the portal of the Annulus of Infamy. This time, she is mistaken for Theta. What is the probability that Marigold returns to the living?
Apart from spirit-world mathematics, there are many things we can learn from Moreish’s reporting. Perhaps, though, the least surprising should be her description of the state of permanent unrest in which an earthly prophet7 ultimately finds himself. Our expectations around this have been radically reset by the brilliant studies on corrupted exaltation conducted by Timothy Bystander. In The Prophet of the One False God he combined scriptural historiography, forensic accounting, interviews with clerics, and victim statements, together with appeal to the obvious, to determine that the universal motivation of prophets has been aggrandizement. (This is a somewhat sharper conclusion than comes from earlier research, where too often a model of delusional illness is implicitly forgiving.) An exception, he concedes, may prove to be the swint. For the rest of us, he concurs with Moreish in her beautifully evoked personal enlightenment that the route to salvation is geometry. To enter Paradise, however, requires more: the disciple must submit to evidence-based atheism8, and dedicate her worship to the contemplation of probability.
For both entertainment and utility (to render sensible her later passage through the annuli), this might include a geometric-probabilistic method of estimating Satanic π. An adaptation of Buffon’s Needle problem is suggested, using concentric (underworld, defective) circles drawn on the plane in place of parallel lines.
It does need to be mentioned that some influential geometers propose that the Circles of Hell are so damaged, so distorted, as to more resemble, in fact, squares. Interestingly, as noted in TWF Compendium, the square providing accommodation equivalent to that of a unit circle will have sides of length √π, likely to be very unstable indeed9 (particularly under Pioniv rotation). To inform ourselves more reliably on structural integrity matters will require a third10 emissary having Moreish’s courage and empiricist curiosity, and that would seem a circumstance too fortunate to hope for.
1 In researching scriptural and historical primary sources for this work, the author has been surprised at the frequency of mention, across millennia, of Standing Giant. The epithet is a colloquial convergence of Indo-European and Semitic ancestors, in both cases meaning ‘Unsubmitting’ (hence the earliest identification with free will) more than ‘standing’ in a literal sense—though the iconography usually has it so. Beneficent,
just, naturally powerful, and apparently the sole survivor of every superseded pantheon in history, it would seem logical to admit this figure (perhaps with some rider gifting omnipotence and a role in Creation) as the residue Deity of our monotheistic corporations. That God, as such, would take colossus form and maintain a lodging (well distanced from the squabbling prophets, we see) in the First Annulus should not surprise: the entry to Infamy is the busiest spirit passage in the underworld, requiring considerable seniority of oversight.
Note that the large Neolithic shadow photograph discovered by Nicholas in the Medallion Caves is a rare example of Standing Giant depicted female. The pudendal placement of the tunnel entrance in her composition, initially thought fertility signifying (or pornographic—a view much favoured by Professor Lecémot and colleagues), has been reinterpreted non-sexually: as in the underworld, she is the guardian of the threshold (limen), one chamber to the next.
2 The author accepts that the claim to priority is problematic. It is said that Martin Allegorio’s Hadean Symphony has concealed within its phrasing the expansion of Satanic π, though the composer is unforthcoming on the question. Musicians believe that a faultless performance will invoke the Apocalypse: to be entrusted with the score, conductors must swear a special oath to introduce an error or grace note somewhere before the chilling Zēkrit chord in the sonata movement. Eugénie Vardov’s tone poem Misconduct is composed entirely from such error notes; increasingly, it is finding an audience as an overture played before the symphony for palliation, or even exorcism.
3 Here it is opportune to consider one of the two11 (for physicists) most puzzling ontological facts of the underworld: How do we explain the ambient temperature for which the Inferno is named? Classical imagery of burning brimstone, exotic timbers or even hydrocarbon oils may dramatize verse and etching, but these are hardly sustainable for eternity. Moreish (who for protection wore a Carpasian cloak, and by disposition rejects the supernatural) sought but found no trace of radioactivity in atmosphere or soil, making nuclear reactions unlikely as a mechanism. One interesting (and oddly parsimonious) post-mediaeval theory has been that spirits themselves are combustible, providing an inexhaustible supply of fuel for whatever duration the underworld has reason to exist, and alleviating overcrowding at the same time. However, the thermodynamic properties (enthalpy, and so on) of a human soul in oxidation have never been measured, and the feasibility of spirit fires remains uncertain. (Sufficient, nevertheless, to explain sectarian aversions to cremation.) A recent and appealingly realistic hypothesis relates to the concept of Pioniv rotation, positing that small (sub-seismic) chaotic imbalances in ring angular velocities will generate frictional heat at annular boundaries (namely, C2 through C9). Preliminary calculations based on dimensions obtainable from Inferno and using geophysical parameter estimates from earth science indicate that extreme temperatures are possible with negligible dissipation of rotational energy. (For a moderately non-technical discussion, see ‘Melt-zones threaten Counting Owl’, in J. Numerical Ornith.) To test the friction theory, a systematic thermal and acoustic mapping programme is suggested as part of future explorations, ideally with laser theodolite stations set up across the rings to detect and record the actual fluctuations.
4 Such pacts with the devil are more frequent than is commonly supposed. Most of us have encountered others whose glance or stare is indefinably disturbing, who conceal their eyes with incongruous sunglasses, or avert their gaze when photographed. These are, we now know, the Luciferans—Fausts, Mephistopheleans and Satans incarnate—living in our midst. What they seek to hide is the one publically examinable circle in primate anatomy: the pupil. Fortunately, advances in digital infrared pupillometry promise a simple point-of-wonder diagnostic test to identify imposter humans and the otherwise damned. In the coming months, readers can anticipate the author’s smartphone camera graphing application (patent pending, trademarked Eye’sPi®) that calculates pupillary π to the required fifth decimal place and triggers an alarm if a Luciferan is detected.
5 Octo Oscula Redemptionis. The Secret Chord is a special case of an osculating curve ‘kissing’ Ck at Tk, k = 2, 3, …, 9. From any such tangent point, at the Counting Owl’s discretion, the redeemed may advance along the chord to a point Tk − 1 more distant from Satan. This is the ‘deliverance’ of Leonardo (see epigraph). The unredeemed (Leonardo’s infidel—a deictic term, in the relativism of faiths) are turned inward ‘from bad to worse’, closer to Satan. The argument for this interpretation of the Gospel is strong on several grounds.
6 To better understand this sum, note that (for fixed R1) as θ1 → 0, the radii Rk, k = 2, 3, …, 9 are forced (by successive containment) to approach zero (the cosines approach unity); therefore L → R1, the length of the direct line to C10, the Satanic centre. It hardly needs pointing out that in the limit the kissing points vanish, and with them the possibility of redemptive escape. This can be seen as a mathematical statement of what early Plutonic cosmographers called Theta Collapse, an event mentioned by St Ignorius and others as a pagan Judgement prophecy: In a single calamitous night, all of humanity12 is consigned to the First Annulus, and
Eight Rings Eight Kisses Gone
His Spiralled Curse Undone
Into Th’Adversary’s Throne
The Counting Owl Is Flown.
(Apposite as it is to the angle terminology of modern analysis, the classical identification of theta with this apocalyptic vision of abrupt inner Circle degeneracy has unknown origins. Its first usage in the riddle described above has not been dated.) The hermeneutics here is often confused by conflation with the Crucifixion earthquake, which was reputed to have penetrated to the underworld. In fact, the scales of destruction of the two are not comparable, leaving no doubt that the true Theta Collapse belongs in our future.
7 As a general rule, the bleeding preciousness of some in a particular matter is best congealed, in the author’s experience, by respectful adherence to historical accuracy and, if argument proves unavoidable, a sensitively restrained recitation of clotted truisms. (The alternative, ligature, method should be reserved for cases of extreme provocation.) In accord with the hierarchy given in the authoritative Register of Claimant Visionaries, only St Ignorius and Princess Periphereia (who enjoy Legitimus status as validated prophets; see above, note to Chapter 7) are excluded from the Piazza fate outlined above (we know this from Virgil himself). Apart from Bystander, the leading scholarship here originates with Kuklosian’s group at the Mount Sycamore School of Theology, which also coordinates the multicentre editorship of the Register. That document’s most recent revision (RCV-5) has a newly created Superlativus category, currently empty but presumably awaiting the expected ascension of swints.
8 At this point, some might wonder how the atheist can reconcile a Satanic understorey. The rationale appears to be that evil is evidenced in believers, whereas a grace of God is not. Furthermore, whilst Luciferans are demonstrably amongst us (footnote 4 above), the heavenly host is everywhere absent. The irony is exquisite: Paradise, should it exist, receives the unbeliever. (A corollary, not apparent to the criminally dull, is that the fantasist narcissism of religious ‘martyrdom’ condemns the perpetrator straight to Satan. See also footnote 9 to Chapter 7.)
9 Not that the transcendental in itself should disconcert the underworld. (But see ‘Is Satanic π transcendental?’ communicated by Wallis Pioniv in Proceedings of the Lindenblüten Society. Here is found the first treatment of polynomial equations with Diabolo group coefficients.)
10 That is, after Dante Alighieri and Moreish.
11 The other relates to the phases of water under Inferno conditions. It is easy to state that the triple point is shifted on the absolute scale, but why this should be so is not explained.
12 Geometers of today can be thankful that a qualifier, ‘save issue of Euclid’, was inserted by mediaeval catastrophists, probably in the interests of their own salvation.
[REMARK Alien Life] Professor Pioniv conveys for interest
the following argument, given for analysis in a recent Cambridge philosophy paper.
PREMISE: There is one God, one Creation, one Satan and one Inferno.
THEREFORE IF spirited mortal alien beings exist, their souls descend to this one Inferno.
BUT exhaustive exploration of this one Inferno (Dante, Moreish) revealed no alien souls.
THEREFORE EITHER spirited mortal alien beings do not exist
OR spirited mortal alien beings exist
AND there is more than one Inferno, more than one Satan, more than one Creation, and more than one God.
CONCLUSION: The search for extraterrestrial intelligence is driven by polytheists seeking confirmation of their faith13.
13 For the purist, this outcome would be deeply satisfying linguistically (see comment on plurality in footnote 8 to Chapter 7). However, as the reader may infer from footnote 8 above, atheism can readily entertain multiple underworlds, in which case the searchers could discover many Satans but no Gods. For purists who are also mathematicians, ‘more than one’ raises questions of precise cardinality (and finiteness), a possible ordering principle, and whether Satanic π might be uniquely valued in each instance. If the latter were so, Infernos could easily be hierarchized by magnitude of π. The journal Pious has editorialized on this subject, suggesting a more intriguing measure based on prime string properties in the decimal expansions. The question is important for believers, as the ordering will naturally (by transference) rank respective Gods—humanity may not have been given the best. (Nor, possibly, the worst of Satans.)
Wallis Pioniv is Rede Professor in Logic at Nazarene College.
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