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Dante’s universe as 3-sphere April 22, 2008

Posted by David Pierce in Uncategorized.
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The quotations about the universe as an infinite sphere remind me of another kind of sphere that is beyond our usual powers of imagination.
In the one-hundred cantos of the Divine Comedy, Dante travels from the central point of the earth out into the heavens. He passes through the spheres of

  1. the moon,
  2. Mercury,
  3. Venus,
  4. the sun,
  5. Mars,
  6. Jupiter,
  7. Saturn,
  8. the fixed stars,
  9. the Primum Mobile.

In this ninth sphere, he sees another point, surrounded by circles or spheres. These spheres are angels, named from inside to outside:

  1. Seraphim,
  2. Cherubim,
  3. Thrones,
  4. Dominions,
  5. Virtues,
  6. Powers,
  7. Principalities,
  8. Archangels,
  9. Angels.

These are named in Canto XXVIII of the Paradiso, where Dante reports (text from the Princeton Dante Project):

16 I saw a point that flashed a beam of light
17 so sharp the eye on which it burns
18 must close against its piercing brightness.
19 The star that, seen from here below, seems smallest
20 would seem a moon if put beside it,
21 as when one star is set beside another.
22 As near, perhaps, as a halo seems to be
23 when it encircles the light that colors it,
24 where the vapor that forms it is most dense,
25 there whirled about that point a ring of fire
26 so quick it would have easily outsped
27 the swiftest sphere circling the universe.
28 This point was encircled by another ring,
29 and that by the third, the third by the fourth,
30 the fourth by the fifth, and the fifth by the sixth.
31 Higher there followed the seventh, now spread so wide
32 that the messenger of Juno, in full circle,
33 would be unable to contain its size.
34 And so, too, the eighth and ninth,
35 each one revolving with diminished speed
36 the farther it was wheeling from the first.
37 And that one least removed from the blazing point of light
38 possessed the clearest flame, because, I think,
39 it was the one that is the most intruthed by it.
40 My lady, who saw me in grave doubt
41 yet eager to know and comprehend, said:
42 ‘From that point depend the heavens and all nature.
43 ‘Observe that circle nearest it,
44 and understand its motion is so swift
45 because it is spurred on by flaming love.’
46 And I to her: ‘If the universe were arranged
47 in the order I see here among these wheels
48 I would be content with what you’ve set before me.
49 ‘However, in the world of sense we see
50 the farther from the center they revolve
51 the more divinity is in their orbits.

Beatrice then explains that the innermost rings of angels correspond to the outermost heavenly spheres: so n corresponds to 10-n in the lists above.

In a lecture given in Ankara a while back, Piergiorgio Odifreddi suggested (as I recall) that the two spheres or rather balls—of the heavens, and of the angels—should be considered as identified along the 2-spheres that are their boundaries, so that a 3-sphere is obtained, with Lucifer and God as antipodal points.
I note Dante’s description of the latter Point in lines 11–12 of Canto XXX:

1 About six thousand miles away from here
2 the sixth hour burns and even now this world
3 inclines its shadow almost to a level bed,
4 when, deep in intervening air, above us,
5 begins such change that here and there,
6 at our depth, a star is lost to sight.
7 And, as that brightest handmaid of the sun advances,
8 the sky extinguishes its lights,
9 even the most beautiful, one by one.
10 Not otherwise the victory that revels
11 in eternal joy around the point that overcame me
12 and seems enclosed by that which it encloses
13 little by little faded from my sight,
14 so that, compelled by seeing nothing and by love,
15 I turned my eyes to gaze on Beatrice.

As a symbol for a world that cannot be fully comprehended, the three-dimensional surface of a four-dimensional body may serve as well as something infinite in extent.

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Comments»

1. You can call me Steph - May 8, 2008

OK. Say you have a sphere (uninturupted surface, constant curvature, convex or concave your choice). You’re either in or out sort of thing. What’s in between? A boundary. To cross it, and with no reference point to orient yourself (that is the nature of the sphere), how do you know the direction in which you left the sphere? The only gauge you have is a trajectory, towards or away, duality sort of thing. Two spheres? That would mean two trajectories at once. Blasphamy is how I sees it, like putting a sphere in an other sphere, and not saying: Look, I created an imaginary boundary. In a nut shell, to me, as far as spheres are concerned, there is only one other.

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