Steve Trettel
@stevejtrettel.bsky.social
Math Prof: Geometry, Topology and Illustration at University of San Francisco. Minnesotan, from the occupied lands of the Dakota people.
And then, after tracing several billion more rays, the resulting film looks like this 🤯🤩
May 20, 2025 at 9:34 PM
And then, after tracing several billion more rays, the resulting film looks like this 🤯🤩
Running the simulation for several million photons, you can start to see the Dino resolving (upside down!) on the simulated film
May 20, 2025 at 9:33 PM
Running the simulation for several million photons, you can start to see the Dino resolving (upside down!) on the simulated film
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? 🤓
May 20, 2025 at 9:32 PM
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? 🤓
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
Here’s the light contributing to several hundred pixels
May 20, 2025 at 9:32 PM
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
Here’s the light contributing to several hundred pixels
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
May 20, 2025 at 9:32 PM
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
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
May 20, 2025 at 9:31 PM
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
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)
May 20, 2025 at 9:30 PM
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)
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
May 20, 2025 at 9:29 PM
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
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)
March 9, 2025 at 9:45 PM
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)
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? 😁)
January 20, 2025 at 10:12 PM
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? 😁)
Here’s a cute lil donut!
January 19, 2025 at 11:58 PM
Here’s a cute lil donut!
Fun pic from a new project! Drawing lots of donuts today 😁
January 19, 2025 at 11:26 PM
Fun pic from a new project! Drawing lots of donuts today 😁
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.
December 6, 2024 at 9:27 PM
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.
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
December 6, 2024 at 9:23 PM
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
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
December 6, 2024 at 9:14 PM
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
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
December 6, 2024 at 9:09 PM
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
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)
December 6, 2024 at 8:46 PM
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)
And here's the Julia set for f(z)=z^2-1
December 6, 2024 at 8:43 PM
And here's the Julia set for f(z)=z^2-1
But things quickly get more interesting: here's the Julia set for f(z)=z^2-1/2
December 6, 2024 at 8:43 PM
But things quickly get more interesting: here's the Julia set for f(z)=z^2-1/2
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
December 6, 2024 at 8:40 PM
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
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
@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
December 6, 2024 at 8:20 PM
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
@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
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
December 6, 2024 at 8:13 PM
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
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?
December 4, 2024 at 9:01 PM
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?