iiinova — where your digital self keeps evolving

Steve Trettel @stevejtrettel.bsky.social · 20/05/2025

And then, after tracing several billion more rays, the resulting film looks like this 🤯🤩

Steve Trettel @stevejtrettel.bsky.social · 20/05/2025

Running the simulation for several million photons, you can start to see the Dino resolving (upside down!) on the simulated film

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Steve Trettel @stevejtrettel.bsky.social · 20/05/2025

Here’s the view of the Dino from each of several thousand pixels: can you start to see its shape if you hold the phone far away? 🤓

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Steve Trettel @stevejtrettel.bsky.social · 20/05/2025

Heuristically , we expect a "blurry" image to form: a bright outline of the dino on the area exposed to einstein rings, and a decaying green 'halo' as the lensed dino moves outside the black holes line of sight. Here’s the light contributing to several hundred pixels

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Steve Trettel @stevejtrettel.bsky.social · 20/05/2025

When a piece of the film is not lined up directly with a portion of the dinosaur, it does not form a ring, and so contributes less overall light

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Steve Trettel @stevejtrettel.bsky.social · 20/05/2025

The intuition: when a piece of the film is cloned with the dinosaur and black hole, from that point on the films perspective the dinosaur will be distorted into an einstein ring, taking up a large portion of the field of view, imparting that pixel with extra green light

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Steve Trettel @stevejtrettel.bsky.social · 20/05/2025

I had to know if it worked, so I built a little simulator! Here's the setup: a little toy dinosaur and a conical spotlight, then a black hole, then a simulated piece of film (that will record when a simulated photon hits it, and accumulate them)

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Steve Trettel @stevejtrettel.bsky.social · 20/05/2025

Summers here - time to catch up sharing some of what I’ve been up to! First up - *gravitational photography* - using (simulated) black holes instead of lenses to focus an image onto a screen. Can you see the ghostly dinosaur? 1/n

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Steve Trettel @stevejtrettel.bsky.social · 16/05/2025

Also @motivickyle.bsky.social - my collaborator Nadir Hajouji is a Reed alum! (Jerry Shurman was his ugrad advisor 😁)

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Steve Trettel @stevejtrettel.bsky.social · 16/05/2025

We’re going to make the website better (ie, actually informative 🤓) soon! For now just threw up a bunch of beautiful renders that came out of writing the paper!

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Steve Trettel @stevejtrettel.bsky.social · 16/05/2025

Not that I can think of - they both involve Tori but are *very different* incarnations of Tori. Ours here is the 2d surface with its conformal geometry, SLView is a solid Torus with the geometry of SL2R (but this viz is closely related to the “starscapes” project of me Edmund and Kate)

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Steve Trettel @stevejtrettel.bsky.social · 09/03/2025

I actually recently drew some of these slices! Here’s the real points, and what they look like on the complex elliptic curve (well, a surface in R3 whose conformal structure gives a Riemann surface isomorphic to the complex elliptic curve)

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Steve Trettel @stevejtrettel.bsky.social · 20/01/2025

The regular hexagon can be rolled up isometrically into a flat torus in 4 dimensional space: here’s a stereographic projection of the result. (Can you see the hexagonal pattern in the spheres covering its surface? 😁)

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Steve Trettel @stevejtrettel.bsky.social · 19/01/2025

Fun pic from a new project! Drawing lots of donuts today 😁

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Steve Trettel @stevejtrettel.bsky.social · 07/12/2024

Best of luck to all taking the Putnam today, from our team here at USF! 🤓🤘

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Steve Trettel @stevejtrettel.bsky.social · 07/12/2024

If I do this I'll send you the pic :)

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Steve Trettel @stevejtrettel.bsky.social · 07/12/2024

I like this idea!! I wonder if there's any nice math to compute the area (complex dynamics isn't my field of research so I only know the very basics). Otherwise could just estimate it computationally (draw the Julia set, count the black pixels...ha)

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

The parameter space of "c" for which the Julia set is "big" (connected) is the famous Mandelbrot Fractal - visible as an emergent image here from the collective behavior of tens of thousands of Julia sets.

$The mandelbrot fractal: a black collection of circles with infinite complexity leading to jagged edges, on a white backgroound$

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

Its clear theres some sort of region containing all the "big" Julia sets, and outside that in all directions they burst into dust (mathematically - into totally disconnected cantor sets). But to get a better view we need to zoom out

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

Some Julia sets are "large" and some are "small": can we tell which values of "c" lead to which? One way to try and get a sense of this experimentally is by just drawing a lot of Julia sets! Let's draw the Julia set for "c" right where "c" is in the complex plane

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

Something strange is going on at a couple points along the animation: near the center the fractal's roughly disk like (recall its a perfect disk at c=0), but if c strays too far in certain directions it bursts into a constellation of tiny dots and almost disappears

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

Indeed, we can associate every point c in the complex plane to a fractal in this way, by drawing the Julia set corresponding to z^2+c. Here's a quick animation moving through these (the red dot shows the associated point c)

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

And here's the Julia set for f(z)=z^2-1

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

But things quickly get more interesting: here's the Julia set for f(z)=z^2-1/2

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

The set of all points that stay bounded is the (filled) Julia set for the function f. Let's start with a boring example - if f(z)=z^2, then points inside the unit disk stay bounded (in fact, they converge to zero) and points outside go to infinity: the Julia set is a disk

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

Complex dynamics is all about taking a complex function f(z) and looking at what happens when you iterate, computing f(f(z)), f(f(f(z))) and so on. For any given starting number z, this produces a sequence, and this sequence can either blow up to infinity, or stay bounded.

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

Each pixel of the ghostly image is itself an infinitely intricate fractal! There are many fractal - producing processes in mathematics, but these particular ones (called Julia sets) arise from a really simple procedure in the complex plane.

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

This was inspired by a beautiful animation by @matthen.com I remember seeing some years ago, and wanted to try something similar. Some of you may recognize a familiar ghostly image if you squint - more on this later. But the story starts with the details, so here's a deep zoom

$A grid of black and white fractals: the left half of the image contains only lighter colored fractals and the right half bigger, darker fractals.$

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Steve Trettel @stevejtrettel.bsky.social · 06/12/2024

I have a second piece display at O'Hanlon Center for the Arts in Mill Valley CA this month, entitled "Julia". Here it is, and a short math-explainer of what we're looking at

$A hexagonal grid in the plane with each cell filled with a small image of a black and white fractal. Some of the fractals are larger (and darker) than the others, so when looking from a distance another image emerges (as the fractals themselves act like dark/light pixels for the larger image)$

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

Math is full of these beautiful pictures that don't always make it out of our heads and into the world. Thanks for reading along; I'm glad I could share a glimpse of this one with you :)

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

And of course, showing a wormhole like object means we have got to fly through it! Here's the view of an intrepid astronaut who enters the connect sum tube to emerge over an island: can you make sense of what they see when looking back through the wormhole?

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

Here's a view of this world from the torus component, with an infinite number of paths to the beach visible as a lattice of blue orbs. (Experts may realize this lattice looks a whole lot like the fundamental group of the torus, and that this is not a coincidence 😀)

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

But light can also wrap around the torus many times before encountering the connect sum tube, and spiraling up to the beach. This means in these *other* directions you should also see a little bubble containing the beach. But light traveled longer to get there, so perhaps it should appear smaller.

The same image as the previous two posts, now with an orange geodesic starting at the stick figure and spiraling many times before going up the tube and emerging on the horizontal plane.

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

Light in this space can take many paths to the beach: looking directly up the tube you expect to see something like the Paris wormhole: a distorted bubble containing a beach.

The same image as the post above, with a single yellow line leading from the stick figure on the donut, directly up the tube, and onto the flat plane above.

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

Alright: so this is what a connect sum tube looks like. Now let's leverage this to help us understand the space in the art piece. Here you (the observer) are in a rolled up toroidal space, connected by a tube to a flat space (which we'll imagine to be a beach)

A horizontal blue plane connected to a torus below by a vertical tube, against a black background. On the torus there is a cartoon stick figure, and on the plane a cartoon palm tree representing a beach.

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

I've been doing math on the internet long enough to know that you can't post a still pic of a wormhole like object without people (rightly) begging to fly through it. So here you go!

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

The cartoon suggests geodesics in most directions are approximately straight lines in the plane: you mostly see SF. But within a small sphere in the center of your vision light rays fall down the wormhole tube, spiraling around until they exit in Paris. Can you spot the Eiffel tower to the right?

The golden gate bridge on a clear bluesky day surrounds a distorted gray sphere in the center of the image. Within this sphere one can see a gray plaza, a stone building, and to the right a very distorted and compressed image of the Eiffel tower.

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

To build up an understanding of the space in the artwork, let's start simple and consider two otherwise unremarkable flat spaces joined by a single wormhole. Here in the 2D cartoon I've imagined one of the spaces to be SF and the other Paris. What do you see when standing in SF?

Two parallel horizontal planes connected by a vertical tube, against a black background. The top plane has a cartoon of the golden gate bridge on it, and the bottom a cartoon of the Eiffel tower. A small stick figure stands on the top plane, and three geodesics emanate from them: two travel along relatively straight lines in the plane (remaining in SF) whereas the other falls down the tube, spirals around, and emerges in paris.

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

To render images in such a world, we trace light rays from objects to our eyes along geodesics. The geometry of space controls the behavior of such geodesics, affecting what we see. Here's another two dimensional example, showing a geodesic spiraling down the throat of a wormhole.

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

This is the "inside view" of a 3 manifold: what you'd see inside of one of the spaces mathematicians like me study. This manifold is built as a connect sum (non-mathematicians, think "wormhole") from a flat 3-torus to Euclidean 3-space. Here's a 2-D analog:

A blue donut connected by a tube to a flat plane against a black background

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

My artwork "Many Paths to the Beach" is currently on display at O'Hanlon Center for the Arts in Mill Valley CA. here's a math explainer thread: what are we looking at here?

A repeating 3D lattice of blue orbs against a black background. Looking closely into any one of the orbs reveals a distorted view of sea, sky and sand dunes. Each orb gives a slightly different perspective on the scene.

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Steve Trettel @stevejtrettel.bsky.social · 04/12/2024

So honored to have some of my pieces on display at O’Hanlon Center for the Arts in their exhibition “Chaos” this month. This one is titled “Many Paths to the Beach” (explanatory thread to follow once I’m to my office)

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Steve Trettel @stevejtrettel.bsky.social · 03/12/2024

Bay Area people: I’m really excited to have two pieces of mathematical art on display at O’Hanlon Center for the Arts in Mill Valley, for their exhibition “Chaos”. The opening reception is today 530-7pm! www.ohanloncenter.org/event/chaos-...

ohanloncenter.org

CHAOS Gallery Show Art Talk and Reception | O'Hanlon Center for the Arts

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Steve Trettel @stevejtrettel.bsky.social · 03/12/2024

Oh! No but this is a good idea - when I got time to think thru how to write it I’ll make some 😃

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Steve Trettel @stevejtrettel.bsky.social · 02/12/2024

The hopf fibers corresponding to the vertices of a regular dodecahedron

Pastel colored circles in space against a white background , linked together in a tangled blob with smaller circles in the center and larger reaching outwards. Accompanying this to the left is a smaller glass sphere with a dodecahedron inscribed on its surface: pastel vertices (one corresponding to each of the circles) connected by blue edges.

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