Disguised Geometry in Fra Angelico’s Annunciation

Detail of Fra Angelico’s Cortona Annunciation.

Fra Angelico (1395–1455) was a Dominican monk-artist known for his visionary dreams and advanced understanding of mathematics. He is featured in The Geometry of Angels (1998), where mathematician Ralph Abraham teams up with William Irwin Thompson to analyze one of the friar’s visionary paintings from 1435. It’s the Annunciation. The angel Gabriel informs Mary she’s pregnant with a god, and his golden wings create a series of ellipses. If we “pull” the 2D ellipses into 3D spheres, Abraham wonders if the wings could represent Dante’s 3-sphere universe — two intersecting onion-like wholes— an image that was on everyone’s mind at the time.

In the context of Renaissance math and linear perspective, Abraham adds that the flapping wings generate a family of toroids:

Each wing represents a circle. By stretching out the wings and flapping them, the angel carves out the tori of the cosmic three-sphere. As two spatial dimensions plus movement makes three dimensions, the flight of angels may be seen as a construction of the world, and a means of navigating between the Ball of Heaven and the Ball of Earth.

What a vision! This is really exciting because that family of toroids creates a Hopf map, a topological shape not uncovered in mathematics until the 1930s! You can create your own Hopf map here. Essentially, it’s a series of rings that show how a 4D sphere (aka 3-sphere, hypersphere, or “real” sphere) can be mapped by a collection of 2D ellipses. One interesting property of this shape is that each ring or “fiber” is linked with the others exactly once.

Physicist Roger Penrose marked the Hopf fibration “an element of the architecture of our world,” and mathematician Eric Weinstein said to Joe Rogan “it’s the most important object in the entire universe.” It’s also called a “principal bundle,” what the universe “is based around,” a shape not really understood until the 1970s, yet Fra Angelico painted one in 1435 disguised as some angel wings!?

This isn’t the first time an artistic vision predicted a scientific discovery. See Leonard Shlain’s book Art and Physics or Jonah Lehrer’s Proust was a Neuroscientist. Dante pre-cognized the 3-sphere in the 14th century with his poetic comedy. Check out Dante and the 3-sphere by Mark Peterson, and read more about fibrations here.

Stephen Lipscomb’s illustration of Dante’s “3-sphere Universe.”
The Hopf map or 3-sphere (S3), composed of fibers, “where each fiber is a circle — one for each point of the 2-sphere.” Mathematician Nico Belmonte, describing the image above: “You are probably now thinking ‘but what does a 4D sphere actually look like?’ You can’t really see it, but here we use a stereographical projection to show some points of a 4D sphere into 3D space.”

Visualizing the Invisible

As a 4D sphere, the Hopf fibration models the “foundation” for the 4D world (space + time). Humans are at least 4D beings (with a “long self”), even if we feel locked in 3D. We can’t see 4D with our eyes, but we can fantasize.

In his book Art Meets Mathematics in the Fourth Dimension, Steven Lipscomb tells the story of the “quest to imagine the elusive 3-sphere.” Gustave Doré tried in the 1830s, Edwin Abbott Abbott tried with Flatland (1884), and in 1917, Albert Einstein visualized the universe as a 3-sphere (without the aid of Planet Hopf) describing the image as “the place where the reader’s imagination boggles. Nobody can imagine this thing.”

Fra Angelico can. In this version of the Annunciation, he added sand to the paint so that in candlelight the fiber-wings sparkle like star fields. In this version, a black-and-white figure living above the world offers a golden orb that floats down to Mary and Gabriel—reminiscent of the annunciations in Part 8 of Twin Peaks. I’m also reminded of the fictional sophons from The Three-Body Problem — a family of proton-sized supercomputers capable of unfolding and infolding dimensionally — sent to earth by aliens.

In this version, a tiny angel dressed in red expels people from Eden, while the Father, disguised as black-and-white art, watches over his mystical insemination. Mary looks like a giant.

What’s noteworthy about Fra Angelico’s Annunciations — besides the embedded Hopf bundles — is that angel and human mirror each other, arms crossed like they’re dead, or, like they’re both holding invisible babies. The liturgical gesture can also signify mourning and baptizing.

Details from a Fra Angelico Annunciations.

Tori, with their ascending/descending currents, appear in Alex’s Grey’s paintings of our subtle and causal bodies. Grey calls one a “guardian angel,” “toroidal ball of spiritual light,” and “Mobius Sphere.” Is it the With?

Astrophysicists point out that our sun also has a large toroidal field surrounding it that is itself embedded inside a larger torus encompassing our galaxy. Some cosmologists go as far as to claim that the spinning 4D torus, symbolized by a 2D figure 8 or ∞, represents “the fundamental form of balanced energy flow found in sustainable systems at all scales, from micro-atomic to macro-galactic.” Designer Jois Pon envisions more practical applications with his Torii Pods, always motivated by the natural flow of things. Philosopher Peter Sloterdjik also has a lot to say about tori and “biunities” and “sphereologies” in his book, Bubbles (2011), the first volume in his Spheres trilogy. Sloterdjik:

All births are twin births; no one comes into the world unaccompanied or unattached.

Alex Grey, Yogi and Mobius Sphere, 1987.
Hopf fibration overlaid on Fra Angelico’s Gabriel.

Further, to describe the feeling of “being-in-the-world,” Sloterjik uses Thomas Aquinas’s idea of angels — that they are “not like physical creatures in space but that they created out of themselves the space that they illuminate and animate with their essence”(2017:190). It’s like what Maurice Merleau-Ponty says in The Primacy of Perception (1964:5): “The body is not in space, it inhabits or haunts space.

It applies itself to space like a hand to an instrument, and when we wish to move about we do not move the body as we move an object. We transport it without instruments as if by magic.

What this all means is that Fra Angelico paints invisible shapes to be ‘seen’ or rather enacted by the viewer’s own imagination, aided by perception’s inclination towards closure and boundary completion (see Finding out about filling-in). It also means that, according to Fra Angelico, angels create three-dimensional space by beating their wings, and in a sense so do humans.

“Planet Hopf”




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David Titterington

David Titterington


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