Wednesday, June 7, 2023

Bad Math and Poor Eyesight: Reconfiguring Dante’s Hellscape

March 31, 2014

by Arnaud Zimmern

Gustave Doré - Dante's Inferno Canto XXVIII

Gustave Doré (1832-1883)

Io vidi certo, e ancor par ch’io ‘l veggia,
un busto sanza capo andar si come
andavan li altri de la trista greggia;
e’l capo tronco tenea per le chiome,
pesol con mano a guisa di lanterna


“I truly saw, and seem to see it still,
a headless body make its way
like all the others in that dismal flock.
And by its hair he held his severed head
swinging in his hand as if it were a lantern.”

Inferno, XXVIII.117-122

Dante Alighieri’s description of the decapitated Bertran de Born in Inferno XXVIII is nothing short of scarring, as perhaps it was meant to be. It is after all at the end of this passage that Dante first discusses his famous contrapasso, the imaginative logic that matches infernal punishment to earthly vice and thereby undergirds Dante’s depiction of divine judgment. With so great a moral at stake, better drive the nail in hard. But the gruesomeness of the portrait – Bertran lolloping about, swinging his severed head along – scars not only the reader’s memory, but also the poet-pilgrim’s. “I truly saw, and seem to see it still,” the poet says of the headless body. It is this juxtaposition of the truth of the vision with its semblance through memory, united by and in the narrator’s “I,” that leads Suzanne Conklin Akbari to say: “Dante [the Pilgrim] saw it with the eye of the body; he [Dante the Poet] continues to see it with the eye of the mind”.[1] Her insight rests on one of the most confusing and most powerful aspects of the Commedia, namely that the narrator and the character of Dante are one and yet not quite the same. Each one is bound to his respective milieu and therefore to his respective vision, the poet existing in the ongoing present within which any narrator writes/is read, the pilgrim inhabiting the literary (to avoid saying fictional) past in which he putatively underwent “the struggle – of the way and of the pity of it – which memory, unerring, shall retrace.”[2]

This duality of vision and memory, pilgrim and poet, is one of the most contested facets of Dante’s text, and the debate has polarized over time along predictable battle lines. On the one hand are the traditional-conservative critics who, since Galileo Galilei and Jacopo Mazzoni’s time, have found in Dante “a ‘divine’ poet who… unveils to his readers ‘la mirabil fabbrica del cielo’” in all its allegorical exactitude.[3] On the other are the typically but not always postmodern critics who would humanize Dante so as to bump him off his pedestal by deconstructing his poem, poking at its ambiguities and flaws until it reveals the poet’s discontents with the divinity of his Commedia. John Kleiner, with his work on the cartography of Inferno and the “consistent inconsistencies of [its] terrain,” stands among the cleverest (and, to his credit, the least rhetorically polemical) of the second group. His study of the various ratios and proportions, circumferences and lengths of Hell – details found mostly in the descriptions and dialogues in the infernal zone of Fraud (Caïna) towards the end of Inferno – reads like a troubleshooting report, indexing problem after problem in Dante’s presumed-flawless architecture. Satan, for instance, at the center of the infernal cone, is at once too big for the space that surrounds him and too tall to permit the vivid, gory description Dante gives of Judas, Brutus, and Cassius caught in his three mouths.[4] Conversely, the bridges of the Malebolgia, those lower pockets of hell, are too small to span the gaps they are said to connect.[5]

Kleiner also takes a look at Dante’s sinners and the proportionality, the lex talionis of their contrapassi. Here disfigurement and disproportion are meticulously and intricately manipulated to great judicial and poetic effect. The head of the giant Nimrod, the Biblical architect of the tower of Babel, is structurally (and thus, for an architect, ironically) disproportionate to his torso, while the gouty disfigurement of the counterfeiter Master Adamo, changed from human to lute-shaped, is not merely juxtaposed with the harmonious body of the original man, Adam, but also contrasted by the harmony between Master Adamo’s sin and the space that surrounds him. As Christopher Hammond has noted, the ratio between the circumferences of the two rings of Hell above Adamo’s head is identical to the ratio of silver and non-precious metals he once used to make his counterfeit florins.[6]

Kleiner’s argument is subtler than what can be described in so short a compass, but suffice it to say that to the “list of arts that depend on number, measure, and ratio” including architecture (Nimrod), minting (Adamo), and metrical poetry, he adds the art of due justice. The implication is that Dante’s mis-measurements of Hell and his disfigurement of those characters whose sins relate particularly to disproportion are intimately related. The mis-measurements may affect carelessness and unintentionality, but they are actually deceptive and artful – fraudulent, Kleiner says, like the zone of Hell which they describe. They are there first to make us think Dante has made mistakes in that art most intimately related to the divine, ideal mathematics; and in a second measure, upon noticing the poetic-ironic precision of the contrapassi, to make us realize he may have planted those “mistakes” on purpose. Why? According to Kleiner, by laying bare his mensural faults for all to see, Dante wittingly undermines his own abilities, yes, as a geometer, and thus as a poet, but above all else as a judge, especially a judge over his fellow man. Consistently inconsistent cartography and intentionally unintentional disproportions constitute, for Kleiner, a self-parody, Dante’s quiet show of humility and deference, a sort of relativistic “know-nothing” fine print that exonerates the author from the charge of judicial hubris, and frees the characters in Inferno from the damnation he has written them into.

If, however, we recall Akbari’s distinction between the sight of the pilgrim and the memory of the poet, and if we take up her larger work in medieval theories of optics – according to which sight is associated with the intellect, and clear vision with moral purity – then the misshapenness of hell, the geometric errata of Inferno, ultimately force us to reconsider Dante’s hellscape, not his motives.

It is a commonplace of the critical discussion on the Commedia that the pilgrim’s journey is purifying, and the poet’s dangerous: as with St. Augustine in his Confessions, the narrator, in reviewing the course of his conversion and retelling past sins, risks lapsing right back into them, jeopardizing his present state of grace. The pilgrim then begins in sin and ends in purity; the newly-converted poet begins in purity and prays that he will continue in it. That the moral purity in question is interdependent on the clarity of one’s vision is nowhere more explicit than in the series of four purification rituals that the pilgrim undergoes, from the first books of Purgatorio to the end of Paradiso.[7] Indeed, with Saint Lucia, patron of eyesight, ever by his side, Dante is ceaselessly reminded (and as poet, he ceaselessly reminds us) that holiness, the intelligence of Truth, and thus the memory of Truth, are all contingent on lucidity.

Dante's Inferno, depicted by Bartolomeo di Fruosino (c. 1366-1441), Bib nat de France

Dante’s Inferno as depicted by Bartolomeo di Fruosino (c. 1366-1441).

Consider also Akbari’s increasingly popular distinction between “extramission”, Plato’s theory that light emanates from the eyes, and “intromission”, Aristotle’s theory that the eye receives and records light passively. Mapping Dante the poet’s use of one over the other, she convincingly argues that in the darkness of hell, Virgil and the pilgrim count on their own beams of light (extramission) to illuminate the road ahead, while in the overwhelming brightness of heaven, light finds its way into the eye (intromission). There “man is fully the object of God’s sight, human vision is perfected, and the intervening medium no longer has the capacity to deceive.”[8] Thus the architectural mis-measurements found in Inferno may be more symptomatic of the pilgrim’s poor eyesight and poor memory than of the self-parody that Kleiner sees. The poet, whose soul and eyesight admittedly have been cleared, is nevertheless dependent on the light of the pilgrim’s intellect; the written record of the text is manacled to the optical record. The memory (la mente) which Dante has sworn to preserve and transcribe ‘unerringly,’ is itself erroneous.

The Galileos and Mazzonis in the crowd might at this point be tempted to do as the gates of Hell command and “abandon all hope”. ‘La mirabil fabbrica del cielo‘ that they sought in Dante’s work has not only been revealed to be crooked; it was never rightly recorded to begin with. But that is only true, presumably, of Inferno, and while that part of the text depicts Hell falsely, through the pilgrim’s impure and distorting lens, the cosmography of the Commedia as a whole still holds the door open for a new conceptualization of Hell’s geometry. We need only ask ourselves what the pit might have looked like if, from Paradiso – his eyes thoroughly washed clean – the pilgrim had thought to do like Orpheus and look back. Renaissance-era maps customarily depict the universe as geocentric, and heaven and hell as diametrically opposite. God repels Satan the way light repels darkness, always to the farthest imaginable point, thus to the very center of the Earth. A ‘God’s-eye view’ then of the iconic and obconic  – funnel  –  shape of Hell in Inferno transforms and reduces the space to a flat disk, with the area inside its perimeter included. The seeming flatness of the full moon, as seen from the Earth, is the result of the same perspectival shift. Thus in the omnivision of God and in the light of truth, Hell loses its appearance of three-dimensionality and is projected instead into a disk or, isomorphically, to a dot.[9]

Both shapes, disk and dot, are revealing in different ways. Perhaps because the obconicity of Inferno has steeped so long in the teapot of the Western imagination, we are prone to think of divine judgment as graduated or somehow situated on a spectrum, certain sins weighing more than others, usurers burning more than gluttons. Tellingly, we treat forgiveness in much the same way, only in reverse: certain sins demand wider mercy and more time to be pardoned, absolved, and pulled out of the depths. It is probably the single most important result of this alternative geometry of Hell (though to call it alternative is to forget that it has been hiding in plain sight): for the Christian God, judging and forgiving are not incremental or relative actions – as they often are for his creatures. They are absolute and indiscriminate actions. It is an anthropocentric fallacy – one which the traditional, conic reading of Dante’s stratified geography has thus far tacitly supported – to attribute to God what are ultimately human notions of piecewise clemency. A flattened hell reminds us that a mortal sin is a mortal sin is a mortal sin; that the willful rejection of God, regardless of form or expression (usury, sodomy, lukewarmness), so long as it is consciously chosen over the divine will, is itself the self-reducing decision that engenders hell. The choice of vice over virtue is a flattening one that projects to a single plane a two-fold creature called to play a part in a threefold love.

Hell figured as a point, meanwhile, accentuates its solipsistic and profoundly asocial nature.  A point, by definition, is a one-dimensional geometric fiction, occupying no space, existing merely as a placeholder. Accordingly, scholars have frequently noted the trend of gradually anti-social and self-absorbed sinners in Hell. Francesca da Rimini shares her story liberally and seems to preserve a sort of quiet communion with Paolo. A little lower in Hell, Guido da Montefeltro tells Dante his tale only with great reluctance, and ultimately retreats upon himself, preferring isolation to communication. The very lowest sinners say almost nothing at all, concentrating themselves instead on the perceived injustice they have been paid, nursing eternally their pain since that alone confirms the existence of their ego. Hell, in other words, results from a self-centeredness that borders on solipsism, indeed on the very edge of non-existence, what St. Bernard called a perpetual death. To think of the space and the souls within it as collapsing to a point calls to mind C. S. Lewis’ more modern portrait in The Great Divorce. The Dante-like protagonist and his Virgil-like guide bend down to inspect a small crack in the soil of Purgatory:

“Do you mean then that Hell – and all that infinite empty town – is down in some little crack like this?”
“Yes. All Hell is smaller than one pebble of your earthly world: but it is smaller than one atom of this world, the Real World. Look at yon butterfly. If it swallowed all Hell, Hell would not be big enough to do it any harm or to have any taste.”
“It seems big enough when you’re in it, Sir.”

It is a similar insight, that Hell is vivid and expansive only when we inhabit it, that must have driven Dante, centuries earlier, to paint in Inferno a space so vast, colorful, and distorted, so seemingly concrete and yet patently deceptive, only to reveal its true nature sotto voce.

There is no doubt more interpretation to be done with these two different geometries of Hell, but it might be more suitable to close with a crucial caveat: neither of these alternative geometries should supersede the traditional mapping of Dante’s vision. They should not (and do not) in any way contradict the moral poetics of the conic shape – Hell’s architecture is, whichever way it is seen, ultimately the result of unbounded and unforgiving pride. It is carved or drilled out of an enormously good universe to make room for those free beings whose self-absorption blinds them to the sad, tiny solitude of their rebellion. In that respect, these two new geometries depend entirely upon the older: to agree with the logic that undergirds the new visualizations is to acknowledge the importance of, as well as the symbiosis with, the traditional.

Chart of Hell by Sandro Botticielli (1445-1510)

Chart of Hell by Sandro Botticielli (1445-1510).

Scholars have made much of a set of errors that Virgil makes in Inferno XI in Dante’s “preliminary charting of Hell, arguing that the poet Dante is making a point in this canto about the limited capacity of both pagan and Christian rationalist epistemologies for an understanding of evil.”[10] And indeed Hell remains a sufficiently ungraspable notion that poetic visions, pagan or otherwise, remain some of our best efforts to fathom. If, however, Dante’s text offers readings that are seemingly in tension, proposing all at once a stratified, a flattened, and a punctal geography of punishment and sin, it is not a sign of textual or epistemological failure, nor, as Kleiner put it, a self-parody on Dante’s part, but yet another example of the text’s abundance. The poet has, like guides are prone to say when they find they have led their group astray, only offered us the scenic view – a more holistic scope of the landscape of damnation. For those still seeking ‘la mirabil fabbrica del cielo’ in Dante, there is work to be done, it seems, in perspectival geometry.


Arnaud Zimmern is a student of mathematics and literature at Southern Methodist University in Texas, USA.



1. Akbari, Suzanne Conklin. Seeing through the Veil: Optical Theory and Medieval Allegory. Toronto: University of Toronto, 2004, p. 152.

2. Alighieri, Dante. Robert Hollander and Jean Hollander. Inferno. New York: Anchor, 2002, II.5-6.

3. Kleiner, John. Mismapping the Underworld: Daring and Error in Dante’s Comedy. Stanford, CA: Stanford UP, 1994, 4.

4. Assuming a 6 foot tall Dante, the ratio of proportions between Dante and the giants, and the giants and Satan, results in a 2500 ft. tall Satan. Even with half his body lodged in ice, Satan’s mouths would still be too far away for Dante to catch sight of the souls of Judas, Brutus, and Cassius, let alone describe them in detail. Kleiner adds, to illustrate, that Dante’s claim is analogous to that of a tourist at the foot of the Empire State Building claiming he can see the smiling faces of friends on the top floor. Hence, doubtful.

5. Kleiner’s explanation: “Since we are told in Canto XXX that the ninth ditch of the Malebolge is a half-mile across, it follows that the bridges spanning the ditch are also a half-mile long. Similarly, since the giant Antaeus picks up Virgil and Dante and lowers them into the central well, it follows that the reach of the giants arms should determine the depth of the pit; the pit should have a depth equal to about half the height of the seventy-foot giant. What we learn from these calculations is clearly inconsistent with the image of the terrain originally suggested by Dante in Canto XVIII.  According to the measurements, the little bridges are almost as long as the Brooklyn Bridge and the very deep central well is barely twice the  depth of a good-sized suburban swimming pool.”

6. Christopher Hammond is a professor of Mathematics at Connecticut College with a keen interest in Dantistics. He points out: “Master Adam makes a curious reference to his circle of hell being eleven miles in circumference and half a mile across. Dante takes numbers seriously, and does not tend to make numerical references without a specific intent.  Nevertheless, eleven has no discernible numerological significance.  I argue that, if one takes the approximation of π as 22/7, one can actually draw a picture of two circles, whose diameters are in the ratio 7:8, the same ratio as the amount of gold Master Adam included in his counterfeit florins. In other words, the very geometry of his place in hell corresponds to the sin he committed on earth.”

7. Akbari 138-150.

8. Peter Brown’s 2006 review of Akbari’s work, from which this quote is drawn, is more concise in its formulation of the argument than the original. See Brown, Peter. “Review of Seeing through the Veil: Optical Theory and Medieval Allegory by Suzanne Akbari.” Speculum 81.2 (Apr., 2006): 463-464.

9. The term isomorphic makes complicated that which is quite simple. Imagine a disk and zoom out as far away as you can. The pixel, the dot you see now is fundamentally the same shape, iso-morphic.

10. Cogan, Marc. The Design in the Wax: The Structure of the Divine Comedy and Its Meaning. Notre Dame: University of Notre Dame, 1999, p. 7.


3 Responses to “Bad Math and Poor Eyesight: Reconfiguring Dante’s Hellscape”
  1. David L. Rosa II says:

    This is a wonderful article. The idea that from God’s perspective hell is a disk or a even a point is great food for thought. I never figured to view the Comedy from a mathematical point of view. My only question, however, is: how is it that Dante (poet) could misrepresent hell when he (pilgrim) went on the journey himself? Wouldn’t he be lying by misrepresenting hell, thus committing a sin?

  2. Arnaud Zimmern says:

    Hi David, thanks for the comment, and it’s a good question that the article could have done a better job of clarifying. Dante the poet isn’t willfully misrepresenting Hell in this argument — he is rather manacled to the optical record, the vision of Hell which the at-the-time unpurified pilgrim recorded. That lack of purification imposes itself on the pilgrim’s eyesight, says Suzanne Akbari, and may therefore be responsible for the distorted space that John Kleiner has called our attention to. So Dante the poet isn’t lying, because this isn’t willful and because it surpasses the bounds of his vincible ignorance, but he also isn’t giving us what we might call a (w)holy true depiction of Hell’s configuration. Does that help make some sense?

  3. Father Gregory Schweers says:

    Having read this delightful article of one of my former students, I would be less than gracious if I did not show both enthusiasm for, and encouragement to, Mr. Zimmern at the beginning of a (hopefully) long academic and scholarly career!

    It may prove helpful to those who read his excellent article to recall a citation from the Inferno not cited above which states:

    O Voi che avete gl’intelletti sani,
    Mirate la dottrina che s’asconde
    Sotto il velame delli versi strani

    ‘O You that have wise/strong intellects
    Look upon the teaching/doctrine which hides itself
    Under the veil of these strange verses.’

    Here, already in Canto IX, v. 61-63, Dante the Poet reminds all of his readers that they will have to approach the ENTIRE Commedia and all its multiform descriptions with an eye that is sharp enough to see what is there before him, and what is hidden under the veiling of poetry. Is this not, inter alia, Dante’s friendly warning that verbal, literal lines of the poem will demand much by way of interpretive skill and reconsideration? Mr. Zimmern rightly underlines this dualism in his article; indeed, Dante himself acknowledges and even stresses this notion by telling us early on to ‘pay attention’.

    So, when Prof. Kleiner writes about the ‘mismapping’ in the Inferno of Dante, I can only cite another text wherein ‘me thinks he doth protest too much!’ What Kleiner so frequently does is take the ‘versi strani’ and turn them into some kind of text that is ripe for Deconstruction. Sadly, Dante is no post-Derrida poet – he wrote in 1300 AD — but for all ages, even our own sadly deconstructed one.

    Once we see the light that moves the Sun and all the other stars at the end of Paradiso, the Inferno will not be so difficult to understand either, mapped or mismapped!

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