Planet Musings

n-Category Café Burrito Monads, Arrow Kitchens, and Freyd Category Recipes

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Guest post by Khyathi Komalan and Andrew Krenz

From Lawvere’s Hegelian taco to Baez’s layer cake analogy to Eugenia Cheng’s How to Bake Pi, categorists have cultivated a rich tradition of culinary metaphors and similes. A well-known example in the world of computation is Mark Dominus’s “monads are like burritos” — where a tortilla (computational context) wraps diverse ingredients (values) to create a cohesive entity (effectful value) whose burrito structure is maintained as the meal moves down the assembly line (undergoes computations).

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Monads, like burritos, come in many different varieties. In computer science monads serve to streamline computational patterns such as exception handling and context management. We illustrate these two examples by analogy.

Imagine you work at a burrito truck.

If a customer orders a burrito sans rice but rice is accidentally added, it can’t be served. The Maybe monad handles exceptions such as this — when something goes wrong, it returns a special “Nothing” value rather than a flawed result, and once a failure occurs, all subsequent steps automatically preserve this state avoiding the need for repetitive error-checking.

Diagram 1

Figure 1: The Maybe Monad illustrated with the burrito-making process

In Haskell, the parameterized type “Maybe a” has two constructors, “Just a” and “Nothing.” The former is an alias for values of type “a” whereas the latter is indicative of an error. The following Haskell code exhibits the maybe type as an instance of the monad class:

instance Maybe Monad where
return = Just
Nothing >>= f = Nothing
(Just x) >>= f = f x

the return function has type a -> Maybe a, which is suggestive of its role as the monad unit. The so-called bind operation >>= has type Maybe a -> (a -> Maybe b) -> Maybe b, and corresponds to a bare-bones Kleisli composition (see Monads: Programmer’s Definition for details).

A slight generalization allows for descriptive error messages.

Definition. Given a collection of exceptions $E$ , there is an associated Either monad $((-)+E, \eta, \mu)$ .

$\eta_X:X \to X + E$ is the coproduct insertion
$\mu_X:X + E + E \to X + E$ collapses two copies of $E$ into one
Kleisli morphisms are computations that may fail $X \to Y + E$
Kleisli composition automatically propagates exceptions

Of course, either monads are simply maybe monads with a set in place of the constant/singleton “Nothing” and they allow us not only to say that an error has occured, but also to indicate what that error was.

Now suppose one of your regular customers walks up to the window and orders “the usual.” Luckily you’ve recorded their preferences in a recipe book. The act of following the appropriate recipe is akin to executing computations that depend on a global read-only state. The * Reader monad * is the functional programmer’s way of incorporating this impure concept in pure functional terms.

Diagram 2

Figure 2: The Reader Monad illustrated with the burrito-making process

Definition. Given a collection of environments $E$ , there is an associated Reader monad $((-)^E, \eta, \mu)$ .

$\eta_X : X \to X^E$ turns elements into constant functions $x \mapsto \lambda e. x$
$\mu_X : (X^E)^E \to X^E$ turns function-valued functions into functions via diagonal evaluation $f \mapsto \lambda e. f(e)(e)$
Kleisli morphisms convert inputs into executable functions from environments to outputs $X \to Y^E$
Composition in the Kleisli category keeps track of the (immutable) environment as computations are chained together.

Here is the same definition given as an instance of the Haskell monad class:

instance Monad ((->) r) where
return x = \_ -> x
g >>= f = \e -> f (g e) e

The seminal paper of Moggi has several other interesting examples illustrating the power of monads. Nevertheless, monads may not always suffice for all of our needs. For example, what would happen if our burrito truck suddenly exploded in popularity requiring automation of repetative processes and parallel work stations?

This is where “Arrows” enter the picture. Introduced by John Hughes in 2000, Arrows generalize strong monads. Because of this, Arrows handle more complicated computational patterns in a natural way. While monads wrap values in computational contexts (like burritos in tortillas), Arrows can represent entire preparation processes capable of coordinating multiple inputs while maintaining awareness of the broader kitchen environment.

Arrows come with three core operations that determine their behaviour; looking at their types, we see that Arrows are evocative of a lax internal hom that interacts with binary products.

class Arrow a where
arr :: (x -> y) -> a x y
(>>>) :: a x y -> a y z -> a x z
first :: a x y -> a (x,z) (y,z)

arr turns functions into “Arrows.” This is like incorporating a standard burrito recipe or preparation step into the food truck’s workflow — taking a simple instruction like “add beans, then cheese” and automating it within our kitchen’s setup.
>>> composes composable Arrows. This allows for separately automated processes to be seamlessly strung together.
first enacts an automated process on one burrito while simultaneously passing a second burrito through the station.

These data are subject to 9 axioms, which we eventually discuss below.

Diagram 3
Figure 3: Arrow Operations. The three fundamental operations of Arrows enable complex workflows beyond monadic structures.

Shortly before Arrows were introduced, Power, Robinson, and Thielecke were working on Freyd categories — a categorical structure designed to model “effectful” computation. Using our simile, a Freyd category formalizes the relationship between an ideal burrito recipe (pure theory) and the real-world process of making that burrito in a particular kitchen.

A Freyd category consists of three main components:

A category $C$ with finite products which can be thought of as the syntax of our kitchen. In other words, $C$ is like a recipe book containing the abstract information one needs to interpret and implement in the context of an actual kitchen.
A symmetric premonoidal category $K$ which plays the more semantic role of our real world kitchen.
An identity-on-objects functor $J:C \to K$ which faithfully translates pure recipes into physical processes that work within the specific setup of the kitchen $K$ .

Figure 4: Freyd Category Structure. The relationship between pure recipes (category C) and real-world kitchen operations (category K), connected by the identity-on-objects functor J that preserves structure while accommodating practical constraints.

Although Arrows originated in Haskell, a highly abstract functional programming language, researchers began noticing apparent correspondences between the components of Arrows and those of Freyd categories. These two structures, developed from different starting points, seemed to address the same fundamental challenge: how to systematically manage computations that involve effects, multiple inputs and outputs, and context-awareness. Therefore, it was hypothesized that Arrows are equivalent to Freyd categories.

As a part of the Adjoint School, our group has been focusing on R. Atkey’s work, which dispells this folklore and precisely formulates the relationship between Arrows and Freyd categories. Just as Atkey asks in the title of his paper, this blog post will investigate the question of “what is a categorical model of Arrows?” The answer not only clarifies the theoretical underpinnings of these structures, but also reveals practical insights for programming language design and quantum computation models. Ultimately, we will see that there are indeed subtle differences between Arrows and Freyd categories.

Key Insights: - Monads encapsulate computational effects by wrapping values in contexts, much like burritos wrap ingredients in tortillas - Different monads (Maybe, Reader, etc…) deal with different patterns like exception handling and context management - Arrows generalize monads to handle multiple inputs and coordinate complex processes, like managing an entire kitchen rather than just making individual burritos

Beyond the Kitchen: Arrows and Freyd Categories

Formally, a monad on a category $C$ is a monoid in the category of endofunctors of $C$ . Arrows, like monads, are monoids in a certain category of functors. To be more specific, the structure of an Arrow on a category $C$ can be described as a monoid in the category of strong profunctors on $C$ . Let’s take a closer look at this construction.

Arrows A profunctor $P$ on a category $C$ is a functor $P: C^{\text{op}}\times C \to \text{Set}.$ Intuitively, a profunctor associates to each pair of objects a set of “generalized morphisms” between those objects.

The identity profunctor is simply $\text{id}(x, y) := C(x, y)$ , which uses the hom-sets of $C$ .

Composition of profunctors is defined as a coend. Given profunctors $P$ and $Q$ , their composition is the following profunctor:

$(P * Q)(x, z) = \int^y P(x, y) \times Q(y, z)$

Notice that this formula is vaguely reminiscent of a dot product; replacing the integral with a sum over $y$ , and the cartesian product with multiplication, it looks like the dot product of the row vector $P(x,-)$ with the column vector $Q(-,z)$ .

operations. –> We will now unpack this data to reach a more down-to-earth description of Arrows. This resulting characterization aligns more closely with the way in which Arrows are implemented in programming languages like Haskell.

Definition. An Arrow in a cartesian closed category $C$ consists of a mapping on objects and three families of morphisms:

A mapping on objects $\text{Ar} : \text{ob}(C) \times \text{ob}(C) \to \text{ob}(C)$

This defines the Arrow type constructor, which takes input and output types and produces an Arrow type between them.

A family of morphisms $\text{arr} : Y^X \to \text{Ar}(X, Y)$

This operation lifts a pure function into the Arrow context, allowing regular functions to be treated as Arrows.

A family of morphisms $\ggg : \text{Ar}(X, Y) \times \text{Ar}(Y, Z) \to \text{Ar}(X, Z)$

This enables sequential composition of Arrows, similar to function composition but now in terms of Arrows.
A family of morphisms $\text{first} : Y^X \to \text{Ar}(X \times W, Y \times W)$

This is perhaps the most distinctive operation. Intuitively, it allows an Arrow to process the first component of a pair while leaving the second component unchanged.

These data are subject to nine axioms which govern their interactions. To make these abstract operations more concrete, consider the following example, where $\text{Ar}(x, y) := Y^X,$ $\text{arr} := \text{id}_{(Y^X)},$ $\ggg := \text{composition},$ and $\text{first}(f) := f \times \text{id}.$ In what follows we list the Arrow laws and draw commutative diagrams based on this example.

The Arrow laws $\text{arr}(\text{id})\ggg a=a$ and $a\ggg \text{arr}(\text{id})=a$ express left and right unitality of identities under composition.

Diagram 5
Figure 5: Arrow Laws

The Arrow law $(a \ggg b)\ggg c=a \ggg (b \ggg c),$ represents associativity of composition.

Diagram 6
Figure 6: Arrow Laws

The Arrow law $\text{first}(a\ggg b)=\text{first}(a)\ggg \text{first}(b)$ encodes functoriality of $- \times W: C \to C$ .

Diagram 7
Figure 7: Arrow Laws

The Arrow law $\text{first}(a)\ggg \text{arr}(\pi_{1})=\text{arr}(\pi_{1})\ggg a$ express naturality of the counit $- \times W \to \text{id}_{C}$ , i.e., the first projection maps.

Diagram 8
Figure 8: Arrow Laws

The Arrow law $\text{first}(a)\ggg \text{arr}(\alpha)=\text{arr}(\alpha)\ggg \text{first}(\text{first}(a))$ asks that $\text{first}$ play nicely with associators.

Diagram 9
Figure 9: Arrow Laws

The Arrow law $\text{first}(a)\ggg \text{arr}(\text{id} \times f)=\text{arr}(\text{id} \times f)\ggg \text{first}(a)$ is an interchange law which says $\text{id} \times g:(- \times W) \to (- \times W')$ is a natural transformation for every $g:W \to W'$ in $C$ .

Diagram 10
Figure 10: Arrow Laws

Two Arrow laws trivialise as a result of our example, so diagrams aren’t produced. The first such law is $\text{arr}(f;g)=\text{arr}(f)\ggg \text{arr}(g).$ For our example, this law trivialises, as $\ggg : = \text{composition}$ and $\text{arr} := \text{id}_{(Y^X)}.$ The second law to trivialise is $\text{first}(\text{arr}(f))=\text{arr}(f \times \text{id})$ since we have set $\text{first}(f) := f \times \text{id}.$

Freyd Categories

To understand Freyd categories, we must first define what a symmetric premonoidal category is.

Definition. A symmetric premonoidal category includes:

An object $I$ (unit).
Natural transformations that define how objects interact:
- Associativity: $\alpha : (x \otimes y) \otimes z \to x \otimes (y \otimes z)$
- Left unitor: $\lambda : x \otimes I \to x$
- Right unitor: $\rho : I \otimes x \to x$
- Symmetry: $\sigma : x \otimes y \to y \otimes x$
All components are central .

A morphism $f : x \to x'$ is central if $\forall g:y \to y', \quad f \otimes y ; x' \otimes g = x \otimes g ; f \otimes y'$

Now, we can define a Freyd category, recalling the definition from the introduction.

Definition. A Freyd category consists of:

A category $C$ with finite products.
A symmetric premonoidal category $K$ .
An identity-on-objects functor J:C→KJ : C \to K that:
- Preserves symmetric premonoidal structure.
- Ensures $J(f)$ is always central.

Arrows vs Freyd Categories: Similarities and Differences

At first glance, the definition of a Freyd category appears strikingly similar to that of an Arrow. This apparent similarity led to the folklore belief that they were equivalent structures.

A Freyd category consists of two categories $C$ and $K$ with an identity-on-objects functor $J: C \to K$ , where: - $C$ has finite products - $K$ is symmetric premonoidal (with a functor $− \otimes z$ ) - $J$ maps finite products in $C$ to the premonoidal structure in $K$

In our culinary metaphor, this loosely translates to: - $C$ : The idealized recipes (Haskell types and functions) - $K$ : The real-world kitchen operations (computations represented by the Arrow type $\text{Ar}(x,y)$ ) - $J$ : The translation process (via arr, embedding pure functions) - Composition in $K$ : The sequencing of operations (via >>>) - Premonoidal structure in $K$ : The ability to process pairs (via first)

Recalling the how we’ve interpreted Arrows in the cullinary setting, the apparent correspondence between Arrows and Fryed categories seemes quite natural. In fact, for many years the two concepts were thought to be two ways of speaking about the same thing among those in the programming languages community.

However, Atkey’s work revealed a crucial distinction: Arrows are more general than Freyd categories . The key difference lies in how they handle inputs:

Freyd categories allow only a single input to computations
Arrows support two separate inputs:
- One may be structured (modeled using comonads)
- This additional flexibility allows Arrows to represent computations that Freyd categories cannot

To bridge this gap, Atkey introduced the concept of indexed Freyd categories , which can model two structured inputs. The relationship can be summarized as: Arrows are equivalent to Closed Indexed Freyd Categories.

In our culinary metaphor, we can understand this relationship as follows: a Freyd category is like a restaurant that can only take one order at a time (a single input), while Arrows are like a more sophisticated establishment that can handle both individual orders and special requests that come with their own context (two inputs, one potentially structured). The closed indexed Freyd categories that Atkey identifies represent the perfect middle ground — restaurants that can efficiently manage multiple orders with specialized instructions while maintaining the core operational principles that make kitchens function. This is particularly valuable when preparing complex “quantum dishes” where ingredients might be entangled and interact with each other in non-local ways.

Diagram 11
Figure 11: Arrows vs. Freyd Categories. Arrows support two inputs (one potentially structured) and are equivalent to Closed Indexed Freyd Categories, which generalize standard Freyd Categories that handle only single inputs.

This distinction helps explain why Arrows have proven particularly useful in domains like quantum computing, where managing multiple inputs with complex relationships is essential.

R. Atkey’s paper finds the relationship between Arrows and different constraints on Freyd categories as follows:

Diagram 12
Figure 12: Relationship Between Structures

Key Insights: - Arrows can be defined both as monoids in categories of strong profunctors and operationally through concrete morphisms ( $\text{arr}$ , $\ggg$ , $\text{first}$ ) - Freyd categories formalize the relationship between pure functions and effectful computations using symmetric premonoidal structure - Despite the folklore belief, Arrows are strictly more general than Freyd categories because they can handle two separate inputs (one potentially structured) - Arrows are equivalent to closed indexed Freyd categories, bridging the conceptual gap

Applications and Questions The main goal of our Adjoint School project was to structure effects in quantum programming languages using generalizations of monads. Relative monads are a popular generalization of monads. These monads need not be endofunctors, and they’re known to generalize Arrows as well. Since we already know how to structure quantum effects using Arrows, it follows that it should be theoretically possible to structure quantum effects using relative monads.

Arrows’ capacity to handle multiple inputs with a single, potentially structured output offers tractability that is particularly useful in quantum computing. Particles in quantum systems can be in entangled states, where the manipulation of one particle influences others in real time, irrespective of distance. This non-local interaction can be modeled through Arrows’ ability to combine several inputs while keeping track of their interrelationships.

Our group investigated the possibility of doing exactly this. The main technical issue arises from the fact that the way Arrows have been implemented in Haskell to structure quantum effects does not provide a categorical semantics for the problem.

For our ACT2025 presentation, we were able to construct a relative monad capable of handling classical control in the quantum setting, but the following questions still remain:

Can one build a relative monad to model quantum effects?
If so, how might an implementation of these ideas in Haskell compare to Arrow-based approaches?

The ride from burrito monads to Arrow kitchens has carried us farther than we anticipated, illustrating that even established mathematical folklore sometimes requires precise re-evaluation. As we continue to learn about these structures, we hope this post will motivate others to participate in the exploration of these tools and their use in quantum computing and beyond.

Doug Natelson — 25 years of Nano Letters

Back in the dawn of the 21st century, the

American Chemical Society founded a new journal, Nano Letters, to feature letters-length papers about nanoscience and nanotechnology. This was coincident with the launch of the National Nanotechnology Initiative, and it was back before several other publishers put out their own nano-focused journals. For a couple of years now I've been an associate editor at NL, and it was a lot of fun to work with my fellow editors on putting together this roadmap, intended to give a snapshot of what we think the next quarter century might hold. I think some of my readers will get a kick out of it.

by Douglas Natelson at August 28, 2025 01:19 PM

John Baez — Giant Molecular Clouds

This image shows a filament of cosmic dust over ten light years long! It’s part of a giant cloud of cold gas and dust that’s starting to collapse under its own gravity to form stars. Newborn stars are hidden inside. The cosmic dust grains here are so cold that observations at millimeter wavelengths were needed to detect their faint glow, shown as orange in this false-color image.

Giant molecular clouds like this range from 15 to 600 light-years across. And they’re very dense… for outer space, that is. While space in the Solar System has about one atom per cubic centimeter, a giant molecular cloud has between 10 or 1000, and this shoots up in regions that are collapsing. The total mass of a giant molecular cloud can thus be huge: over 10,000 times that of the Sun.

As a giant molecular cloud collapses, it forms filaments and clumps. The dense parts of these filaments and clumps are called ‘molecular cores’. These can easily have 1000 to 1,000,000 particles per cubic centimeter. Of course this rises much higher in regions where stars are formed.

Somewhere like this is where our Solar System was born. The dust grains became our Earth. Some of the complex organic compounds floating around became you and me.

But this one is in the constellation of Taurus. It’s called the Taurus molecular cloud, and hundreds of new stars are being born here:

• Wikipedia, Taurus molecular cloud.

Most of the gas in a giant molecular cloud is molecular hydrogen, helium, and carbon monoxide. They also contain methanol, ethanol, benzene, ammonia and other things. The dust is mainly silicates, iron and organic compounds along with some ices. Next time I’ll tell you about polycyclic aromatic hydrocarbons, which get pretty complicated and interesting:

by John Baez at August 27, 2025 09:47 AM

Tommaso Dorigo — A Remarkable Graph: The Full Dalitz Plot Of Neutron Decay

The neutron is a fascinating particle, and one which has kept experimental physicists busy for almost a century now. Discovered by James Chadwick in 1932 in a cunning experiment which deserves a separate post (it is a promise, or a threat if you prefer), the neutron has been all along a protagonist in the development of nuclear weapons as well as in the extraction of nuclear power from fission reactors. And of more relevance to our discussion here, it has powered endless studies both in the context of nuclear and subnuclear physics.

John Baez — The Hayashi Track

When a star first forms, it is powered not by nuclear fusion but simply by gravity. It shrinks, which causes a release of gravitational energy. This tends to heat it, which slows its shrinking. But it releases energy in the form of light, which tends to cool it down. Then it can keep shrinking. So you need math to figure out exactly what happens.

In 1961, Chushiro Hayashi figured it out. Stars between one tenth and twice the mass of the Sun begin life by shrinking while staying at the same temperature! As a result their overall luminosity goes down. This is called the ‘Hayashi track’, and you can see it happening in the vertical blue lines here:

Later, the star’s surface cools down—but for some reason the star expands enough to get brighter! Then we say it’s on the ‘Henyey track’. This change happens sooner for more massive stars. You can see it happening when the blue lines bend to the right and go back up.

The red lines say how old a star is. You can see that our Sun followed the Hayashi track for about 10 million years. In the process, its luminosity decreased by a factor of 10.

The region to the right of the blue lines is called the ‘forbidden zone’. It’s impossible for stars to stay here for long, because they’re highly convective: hot gas from below rises to the surface, quickly cooling these stars. When stars are first born they start out here—but they very quickly move to the Hayashi track.

For more, try these:

• Wikipedia, Hayashi track.

• Wikipedia, Henyey track.

The boundary of the forbidden zone is called the ‘Hayashi limit’, and you can see a derivation of it here:

• Wikipedia, Hayashi limit.

I haven’t developed enough intution for stellar mechanics to explain these things well. As usual, it seems you need to learn the mathematical models, then think about them long enough until they become intuitive. It’s particularly important to get a good sense of the two mechanisms whereby energy moves through a star—radiation and convection—and when one or the other is dominant. But I don’t understand that well! So instead of trying to explain why young stars work the way they do, let me quote Wikipedia, which is pretty readable here:

Stars form when small regions of a giant molecular cloud collapse under their own gravity, becoming protostars. The collapse releases gravitational energy, which heats up the protostar. This process occurs on the free fall timescale, which is roughly 100,000 years for solar-mass protostars, and ends when the protostar reaches approximately 4000 K. This is known as the Hayashi boundary, and at this point, the protostar is on the Hayashi track. At this point, they are known as T Tauri stars and continue to contract, but much more slowly. As they contract, they decrease in luminosity because less surface area becomes available for emitting light. The Hayashi track gives the resulting change in temperature, which will be minimal compared to the change in luminosity because the Hayashi track is nearly vertical. In other words, on the HR diagram, a T Tauri star starts out on the Hayashi track with a high luminosity and moves downward along the track as time passes.

The Hayashi track describes a fully convective star. This is a good approximation for very young pre-main-sequence stars because they are still cool and highly opaque, so that radiative transport is insufficient to carry away the generated energy and convection must occur. Stars less massive than 0.5 M_☉ remain fully convective, and therefore remain on the Hayashi track, throughout their pre-main-sequence stage, joining the main sequence at the bottom of the Hayashi track. Stars heavier than 0.5 M_☉ have higher interior temperatures, which decreases their central opacity and allows radiation to carry away large amounts of energy. This allows a radiative zone to develop around the star’s core. The star is then no longer on the Hayashi track, and experiences a period of rapidly increasing temperature at nearly constant luminosity. This is called the Henyey track, and ends when temperatures are high enough to ignite hydrogen fusion in the core. The star is then on the main sequence.

Lower-mass stars follow the Hayashi track until the track intersects with the main sequence, at which point hydrogen fusion begins and the star follows the main sequence. Even lower-mass ‘stars’ never achieve the conditions necessary to fuse hydrogen and become brown dwarfs.

Why does having a higher interior temperature decrease a star’s central opacity? I would have thought more ionization increased the opacity.

By the way, the image in this post came from Steven W. Stahler and Francesco Palla’s paper “The formation of stars”, and it was republished here:

• Francesco Palla, 1961–2011: Fifty years of Hayashi tracks, in First Stars IV: From Hayashi to the Future, AIP Conference Proceedings, Vol. 1480, 2012, pp. 22–29.

by John Baez at August 26, 2025 02:12 PM

Doug Natelson — Learning and AI/LLMs - Why do we need to know or teach anything anymore?

The fall semester is about to begin at my university, and I'm going to be teaching undergraduate statistical and thermal physics. This is a course I've taught before, last full term in 2019, and the mass availability of large language models and generative AI tools have changed the world in the interim. We've all seen the headlines and articles about how some of these systems can be very good at solving traditional homework and exam problems. Many of these tools are capable of summarizing written material and writing essays that are very readable. Higher education is wrestling with the essential question: What is the right working relationship between students, teachers, and these tools, one that benefits and actually educates students (both about subject matter and the use of these tools)? Personalized individual AI tutoring seems like it could be great for teaching huge numbers of people. Conversely, if all we are doing is teaching students to copy-paste assignments into the homework-answer-machine, clearly we are failing students at multiple levels.

The quote in the image here (from Kathy Hepinstall Parks) is one that I came across this week that originates in the FAQ from a writers workshop. For my purposes I could paraphrase: Why should we learn physics (or any other science or engineering discipline) when a machine already knows the formalism and the answers? On some level, this has been a serious question since the real advent of search engines. The sum total of human knowledge is available at a few keystrokes. Teaching students just rote recall of facts is approaching pointless (though proficiency can be hugely important in some circumstances - I want a doctor who can diagnose and treat ailments without having to google a list of my symptoms.).

My answer to this question is layered. First, I would argue that beyond factual content we are teaching students how to think and reason. This is and I believe will remain important, even in an era when AI tools are more capable and reliable than at present. I like to think that there is some net good in training your brain to work hard, to reason your way through complicated problems (in the case of physics, formulating and then solving and testing models of reality). It's hard for me to believe that this is poor long-term strategy. Second, while maybe not as evocative as the way creative expression is described in the quote, there is real accomplishment (in your soul?) in actually learning something yourself. A huge number of people are better at playing music than I am, but that doesn't mean it wasn't worthwhile to me to play the trumpet growing up. Overworked as referencing Feynman is, the pleasure of finding things out is real.

AI/LLMs can be great tools for teachers. There are several applet-style demos that I've put off making for years because of how long it would take for me to code them up nicely. With these modern capabilities, I've been able to make some of these now, in far less time than it would otherwise have taken, and students will get the chance to play with them. Still, the creativity involved in what demos to make and how they should look and act was mine, based on knowledge and experience. People still have a lot to bring to the process, and I don't think that's going to change for a very long time.

by Douglas Natelson at August 24, 2025 04:14 PM

Peter Rohde You just can’t get rid of some useless ASIO sluts

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by Peter Rohde at August 24, 2025 02:15 AM

John Baez — 4-Dimensional Cross-Polytope

When bad news gets me down, I often get insomnia. I wake up in the middle of the night, start thinking about how we’re all doomed, and can’t easily stop. To break out of these doom loops, I do elaborate visualization exercises. They don’t really put me to sleep, they just calm me down. Then later I can fall asleep.

Here’s what I’ve been doing this week. I visualize this shape made of two interpenetrating tetrahedra, called the ‘stella octangula’ or ‘stellated octahedron’. Notice that these two tetrahedra are ‘dual’: each vertex of the yellow one is above the center of a triangle in the red one, and vice versa.

Then I imagine the yellow tetrahedron slowly moving ‘up’ into the 4th dimension while the red one moves ‘down’. At some point, the distance between each corner of the yellow tetrahedron to the 3 nearest corners of the red one equals the distance between any 2 corners of the yellow tetrahedron. Then I’ve got a 4d shape called the ‘cross-polytope’. All its faces are regular tetrahedra.

There are easier ways to think about the cross-polytope, which is one of the six 4-dimensional regular polytopes. So the real challenge is to visualize how this way of getting it leads to the same result.

My go-to way to think about the cross-polytope is to imagine the 4 coordinate axes in 4-dimensional space and put two dots on each axis, one unit away from the origin in each direction:

(±1, 0, 0, 0)
(0, ±1, 0, 0)
(0, 0, ±1, 0)
(0, 0, 0, ±1)

These are the vertices of a cross-polytope. It’s the 4d analogue of an octahedron. Just as the octahedron has equilateral triangles as faces, this guy has regular tetrahedra as faces. Can you see what they are, and count them?

Don’t worry—if you’re too busy now, you can do it when you’re lying in bed at 3 am thinking about global warming and the decline of democracy. Start by visualizing this picture:

Then visualize the tetrahedra. But the hard part is to rotate this cross-polytope in your mind so you see it as made of two dual tetrahedra, one red and one yellow, and an edge connecting each vertex of the red one to the 3 nearest vertices of the yellow one. That’s been keeping me busy every night this week.

By the way, everything I just said has a 3d version! The 3d analogue of the cross-polytope is a regular octahedron. The corners of a regular octahedron are

(±1, 0, 0)
(0, ±1, 0)
(0, 0, ±1)

But here’s a flattened picture of an octahedron:

See the two interpenetrating equilateral triangles? If you move one up, and move one down, they can become two opposite faces of a regular octahedron.

Actually this sort of trick works in any dimension. Take two regular n-simplexes, dual to each other and interpenetrating; then move one ‘up’ into the (n+1)st dimension and the other ‘down’. At some point their vertices will be the vertices of an (n+1)-dimensional cross-polytope. In 3 dimensions this is easy for me to visualize, while in 4 dimensions I can just barely visualize it… though it’s getting easier every night.

by John Baez at August 23, 2025 08:29 AM

Peter Rohde Photo albums

Peter’s photos: https://www.icloud.com/sharedalbum/#B275oqs3qKSZvQ

Screenshots: https://www.icloud.com/sharedalbum/#B27532ODWjIQb9

Climbing book launch: https://www.icloud.com/sharedalbum/#B27GWZuqDGnuOyN

Salisbury waters: https://www.icloud.com/sharedalbum/#B275qXGF1JQFkx

Christmas with Ash: https://www.icloud.com/sharedalbum/#B27G6XBubAhoT6

Hosin BBQ duck: https://www.icloud.com/sharedalbum/#B27GY8gBYG3b5mD

Hawks Nest to Smiths Lake: https://www.icloud.com/sharedalbum/#B2759UlCqSH5bE

Europe & Alps: https://www.icloud.com/sharedalbum/#B275ON9t3W0lu

Point Perpendicular: https://www.icloud.com/sharedalbum/#B27GqkRUiGivXD2

Newnes canyoning: https://www.icloud.com/sharedalbum/#B27GfnH8tgHSmX

Coffs Harbour to Yamba: https://www.icloud.com/sharedalbum/#B27J0DiRHJKuuWr

Wendy Bruere Christmas (2020): https://www.icloud.com/sharedalbum/#B27G4TcsmGoHysj

Six Foot Track: https://www.icloud.com/sharedalbum/#B2753qWtHZA9EX

Kosciusko to Kiandra: https://www.icloud.com/sharedalbum/#B27GgZLKuGaewVm

Camping food: https://www.icloud.com/sharedalbum/#B27GtnIORgbmHu

The Aardvark: https://www.icloud.com/sharedalbum/#B275VaUrzvmAiT

Kangaroo Valley kayaking: https://www.icloud.com/sharedalbum/#B27JEsNWnJrCpi0

Claustral canyon: https://www.icloud.com/sharedalbum/#B2755Z2WMOTpsk

Budawang: https://www.icloud.com/sharedalbum/#B27GDdyTvGvpINL

Mother’s Day panoramas (2021): https://www.icloud.com/sharedalbum/#B27GFssfGG9WmJP

Point Perpendicular & Nowra: https://www.icloud.com/sharedalbum/#B27GRMtznGPdeuZ

Blood moon: https://www.icloud.com/sharedalbum/#B27GdIshaG8NgGX

La Perouse to Coogee: https://www.icloud.com/sharedalbum/#B275aVbMK4h7qo

Canberra ASPI launch: https://www.icloud.com/sharedalbum/#B27GQOeMmGj4Zcv

Edible foraging: https://www.icloud.com/sharedalbum/#B275ejO179Si0N

Sydney to Wollongong: https://www.icloud.com/sharedalbum/#B275M7GFPUasMe

Album for Dad, Father’s Day (2021): https://www.icloud.com/sharedalbum/#B2752plgjnnkUe

Vaucluse (with Cheryl, Nestor & Wendy): https://www.icloud.com/sharedalbum/#B275CmvAS4uA0Z

Bouddi National Park: https://www.icloud.com/sharedalbum/#B27GdPblXG8WdOo

Tom Thumb (the 2nd): https://www.icloud.com/sharedalbum/#B275aDWbr4CN2w

Eden to Victoria: https://www.icloud.com/sharedalbum/#B27GJDfWGArX8l

Wendy’s book launch (the 2nd): https://www.icloud.com/sharedalbum/#B27GIcgc2G7h08y

Mark & Pat Bruere visit Sydney: https://www.icloud.com/sharedalbum/#B27G0ehgLbyWyg

New Years Eve climb (2021): https://www.icloud.com/sharedalbum/#B27Ju8EH6JOZxmU

Newnes Canyoning (2022): https://www.icloud.com/sharedalbum/#B275BydzFU0GZ8

Royal National Park (2022): https://www.icloud.com/sharedalbum/#B27GlxzuqGVI5nE

Peter & Wendy: https://www.icloud.com/sharedalbum/#B27Gf693ZG52tfd

Book photo shoots: too rude…

Wendy & Peter’s mushroom trip: https://www.icloud.com/sharedalbum/#B27GrhkPxG27So8

Post-mushroom hike: https://www.icloud.com/sharedalbum/#B27GdFryYG8i3Ur

Wendy Kalymnos favourites: https://www.icloud.com/sharedalbum/#B27JqstnBJEXkH2

Wendy Frenchmans screenshots: https://www.icloud.com/sharedalbum/#B27Jr1PPdJpd7Dq

Instagram: https://www.icloud.com/sharedalbum/#B27GzFCC1Gb4tqr

Haute route: https://www.icloud.com/sharedalbum/#B27J8GySPJtWoQ1

Kim’s KKKalendar: https://www.icloud.com/sharedalbum/#B275fk75vIL0sH

Frenchmans Cap Wild: https://www.icloud.com/sharedalbum/#B27G4VTwGGoFBkz

Photoshoot with Zixin: https://www.icloud.com/sharedalbum/#B27GPCdxkGKPkM4

Wendy birthday hike (2023): https://www.icloud.com/sharedalbum/#B27GWBC59GnHpQW

Bateman’s Bay to Bawley Point: https://www.icloud.com/sharedalbum/#B27JsHvHoJ8bxWf

Stockton Sand dunes (2023): https://www.icloud.com/sharedalbum/#B27GVfZ2vGloFZV

Wendy book launch (2023): https://www.icloud.com/sharedalbum/#B27J058xyJR4IBM

Dolomites (2023): https://www.icloud.com/sharedalbum/#B0Z5kuVsbGJUzKO

Mount Arapiles: https://www.icloud.com/sharedalbum/#B275GH8Mq8Uh2X

Mount Solitary loop: https://www.icloud.com/sharedalbum/#B275nhQST2mETE

Klaus Hanz Franz Rohde Kunst: https://www.icloud.com/sharedalbum/#B27GqQrCLGiY3vb

Klaus Rohde funeral slideshow: https://www.icloud.com/sharedalbum/#B27GDZLe8GXP58K

Dad (old, B&W): https://www.icloud.com/sharedalbum/#B27GLLXGLJ5mbT2

Klaus & Ursula wedding: https://www.icloud.com/sharedalbum/#B275cLqfN7154g

Test Greece: https://www.icloud.com/sharedalbum/#B27Jq4WnLJ6JMNd

From Will Skea (Alps): https://www.icloud.com/sharedalbum/#B27JHciePJFwacG

From Will Skea (Frenchmans Cap): https://www.icloud.com/sharedalbum/#B275ZhN2v3EVq6

From Will Skea (Arapiles): https://www.icloud.com/sharedalbum/#B27JPrgBGJu3BTD

Coffs Harbour to Yamba (2): https://www.icloud.com/sharedalbum/#B27GFqhgJG9LHgT

Mark magic show (2021): https://www.icloud.com/sharedalbum/#B27G60dj6ARCvd

Wendy Christmas present (2020): https://www.icloud.com/sharedalbum/#B275FrPQ6GxvRu

AHS 25 year reunion: https://www.icloud.com/sharedalbum/#B275O3DjHUvSv

WhatsApp: https://www.icloud.com/sharedalbum/#B275tzEA5fX1nc

Armidale High School: https://www.icloud.com/sharedalbum/#B27GnbeumG4PnAF

Book photos for Mum & Dad: https://www.icloud.com/sharedalbum/#B27Gtec4XQkASe

Miscellaneous: https://www.icloud.com/sharedalbum/#B27Gq6kMgGKn7GR

Three Capes Trail (2022): https://www.icloud.com/sharedalbum/#B27G7HOIlGrDUGZ

Childhood computer programming: https://www.icloud.com/sharedalbum/#B275fu2MutDU8N

Magic with Mark in Maroubra: https://www.icloud.com/sharedalbum/#B27Gv6DhEGD9U3G

Photoshoot with Zixin (2024): https://www.icloud.com/sharedalbum/#B27GCATCnJGoRfW

Butt Crack (2021): https://www.icloud.com/sharedalbum/#B275VtHQfMv0zw

Greece photos new (edited to remove photos from wrong album): https://www.icloud.com/sharedalbum/#B27GY3uThGoBcGj

Singapore (all combined): https://www.icloud.com/sharedalbum/#B275qsTcwJKJjl

Hong Kong (transit): https://www.icloud.com/sharedalbum/#B2759v1AbS8Hve

Taiwan: https://www.icloud.com/sharedalbum/#B27GQD2D7Gw0hAp

India (combined): https://www.icloud.com/sharedalbum/#B27Gtue8VQy83g

Freycinet: https://www.icloud.com/sharedalbum/#B27G5VpecGE5Tbg

Triglav: https://www.icloud.com/sharedalbum/#B275MbK9Vy8erz

Shared with me: https://www.icloud.com/sharedalbum/#B27GGXqixzPOrm

Mount Wellington climbing: https://www.icloud.com/sharedalbum/#B27Gd59qiG8Kjy4

New Zealand combined (2004): https://www.icloud.com/sharedalbum/#B27GIZ8BIGNN5jy

New Zealand combined (2005): https://www.icloud.com/sharedalbum/#B27GcuRfIGFVIcL

Yea: https://www.icloud.com/sharedalbum/#B27GZYbYHGhFIir

Mount Pleasant: https://www.icloud.com/sharedalbum/#B275Iy2hC0JTTL

D’Aguilar: https://www.icloud.com/sharedalbum/#B27Gh7fzTGZBosS

Bali (2001): https://www.icloud.com/sharedalbum/#B27G1qNHBGOTbIr

Samba Ninjas: https://www.icloud.com/sharedalbum/#B27GG34bAzqQ0v

Armidale (misc): https://www.icloud.com/sharedalbum/#B27GSkLVwGyobbX

Emma’s party (2008): https://www.icloud.com/sharedalbum/#B275S2ms99Zyby

Goettingen (2011): https://www.icloud.com/sharedalbum/#B27JIrbT3Jsgxhd

South Coast track: https://www.icloud.com/sharedalbum/#B27G58NWBG6QyN7

Childhood (misc): https://www.icloud.com/sharedalbum/#B27GVU1CZGmfOcg

Minsk (2006): https://www.icloud.com/sharedalbum/#B27G3JpSBGX1UkQ

Baden-Baden (2019): https://www.icloud.com/sharedalbum/#B27595X5HTVzJr

Berlin (combined): https://www.icloud.com/sharedalbum/#B27JqWzChJ6qizD

Switzerland (combined): https://www.icloud.com/sharedalbum/#B275zXwoYGJ6HMF

Italy highlights: https://www.icloud.com/sharedalbum/#B27G47PHQGoJium

Germany (misc): https://www.icloud.com/sharedalbum/#B275hPMfYGu5xVJ

Garmisch (2022): https://www.icloud.com/sharedalbum/#B27GFsbvlG9Xrr6

Germany (2019): https://www.icloud.com/sharedalbum/#B27G6Mn98G56Ncb

Garmisch (2006): https://www.icloud.com/sharedalbum/#B27J5lIdKGLC9KG

Baden-Baden (2005): https://www.icloud.com/sharedalbum/#B275sWRpHHQkt9

Berlin (2005): https://www.icloud.com/sharedalbum/#B27GgOQtrGjQrpH

Zugspitze (2005): https://www.icloud.com/sharedalbum/#B27G81mNdGcApGt

Amsterdam, Bristol (2006): https://www.icloud.com/sharedalbum/#B275B9SRzyBjlH

Baden-Baden (2006): https://www.icloud.com/sharedalbum/#B275eD9V79I2XR

Berlin (2006): https://www.icloud.com/sharedalbum/#B275toRf1fH8MD

Berlin, Jena (2007): https://www.icloud.com/sharedalbum/#B27GTI3fvGVgNit

Erlangen (2006): https://www.icloud.com/sharedalbum/#B27JrotZ2JpMb0i

Garmisch (2010): https://www.icloud.com/sharedalbum/#B27JPJPSiJurzNg

Germany (2010): https://www.icloud.com/sharedalbum/#B275FhYPQP650

Stuttgart (2006): https://www.icloud.com/sharedalbum/#B27GmitydGVVaZh

Changi (2019): https://www.icloud.com/sharedalbum/#B27GnmlKoG4JHpX

Japan (2007): https://www.icloud.com/sharedalbum/#B275AerZbG6FxVL

Japan (2012): https://www.icloud.com/sharedalbum/#B27GjBjobGg6PUa

Miscellaneous (including Japan 2013): https://www.icloud.com/sharedalbum/#B27GTpbybGySbE8

Currumbin & Tugin (2021): https://www.icloud.com/sharedalbum/#B275vBKZ4xH9X6

Brisbane (2021): https://www.icloud.com/sharedalbum/#B275YHsSjxQnm0

Weed in Byron (26/6/2025): https://www.icloud.com/sharedalbum/#B275Q2ydoGsQ4O5

Weed in Byron 2: https://www.icloud.com/sharedalbum/#B27GQDYhLGwsuY4

by Peter Rohde at August 23, 2025 04:01 AM

Peter Rohde Stalking

Ursula1.pdf Download

Ursula2.pdf Download

by Peter Rohde at August 23, 2025 12:56 AM

Peter Rohde Why?

The person dressed up as Ursula pretending to be my mother clearly isn’t and hasn’t been for a long time.
When I went back to Armidale after leaving BTQ and being left unemployed she made numerous ongoing promises to provide me with assistance, both in obtaining my own accommodation and providing financial assistance.
These didn’t materialise and the promises were revoked.
Instead I was evicted from the family home and subject to ongoing stalking and harassment that required multiple referrals to law enforcement, both to the police and the Attorney-General, demanding cease and desist.
These have been systematically ignored and up until the last message she continues to bypass these requests, approaching my personal friends to harass me and stalk me indirectly. The messages passed on are the usual fake “I’m worried about him” bullshit.
Why has my family home been confiscated by security, who actively break the law by ignoring cease and desist from stalking notices made to law enforcement, forcing an unemployed civilian into ongoing homelessness since early in the year?
What is the rational for my eviction and being barricaded from my own home?
I continue to face a medical blockade and am unable to access essential medicines. Seroquel scripts are deliberately delayed past known script deadlines to try and destabilise me.
Vyvanse scripts are denied outright as the psychiatrist does not respond. He is also known to be a state actor.
It has been repeatedly indicated to me not to worry about finances because they have my back. Instead now the only cash I have is that obtained from fully drawing out a cash advance against my credit card and it will only last days. At that point I’m on the street.
Is everyone here on the same page as to what the deal is? If not, who is playing you off? They clearly need to be deposed.
These are violations of human rights and constitute war crimes and crimes against humanity. Whoever is behind it needs to be removed. End of story.
Who else is being subject to this kind of high level manipulation?
It has been repeatedly suggested that full accountability for the lives of those I care for would be provided. This has not been forthcoming. It is also a violation international law to not provide accountability for the lives of those who are known to have been threatened by the state. These are grounds for removal.
Can anyone answer the question as to why I am in this situation? Who is even living in the family home? Some stooge dressed up as Ursula? It’s a poor lifestyle choice to say the least.
It’s pretty obvious they’re trying to get rid of me and once they do they’ll get rid of all of you too.

by Peter Rohde at August 22, 2025 11:29 PM

Matt von Hippel — Some Dumb AI Ideas

Sometimes, when I write a post about AI, I’ve been sitting on an idea for a long time. I’ve talked to experts, I’ve tried to understand the math, I’ve honed my points and cleared away clutter.

This is not one of those times. The ideas in this post almost certainly have something deeply wrong with them. But hopefully they’re interesting food for thought.

My first dumb idea: instruction tuning was a mistake.

I’m drawing the seeds of this one from a tumblr post by nostalgebraist, someone known for making a popular bot trained on his tumblr posts in the early days before GPT became ChatGPT.

AIs like ChatGPT are based on Large Language Models, insanely complicated mathematical formulas that predict, given part of a text, what the rest of that text is likely to look like. In the early days, this was largely how they were used. Loosely described nostalgebraist’s bot, called nostalgebraist-autoresponder, began with a list of tumblr posts and asks and determines what additional posts would best fit in.

If you think about it, though, ChatGPT doesn’t really work like that. ChatGPT has conversations: you send it messages, it sends you responses. The text it creates is a dialogue, with you supplying half the input. But most texts aren’t dialogues, and ChatGPT draws on a lot of non-dialogue texts to make its dialogue-like responses.

The reason it does this is something called instruction tuning. ChatGPT has been intentionally biased, not to give the most likely completion to a task in general, but to give completions that fit this dialogue genre. What I didn’t know until I read nostalgebraist’s post was that this genre was defined artificially: AI researchers made up fake dialogues with AI, cheesy sci-fi conversations imagining how an AI might respond to instructions from a user, and then biased the Large Language Model so that rather than giving the most likely text in general, it gives a text that is more likely to look like these cheesy sci-fi conversations. It’s why ChatGPT sounds kind of like a fictional robot: not because sci-fi writers accurately predicted what AI would sound like, but because AI was created based on sci-fi texts.

For nostalgebraist, this leads into an interesting reflection of how a sci-fi AI should behave, how being warped around a made-up genre without history or depth creates characters which act according to simple narratives and express surprising anxiety.

For myself, though, I can’t help but wonder if the goal of dialogue itself is the problem. Dialogue is clearly important commercially: people use ChatGPT because they can chat with it. But Large Language Models aren’t inherently chatbots: they produce plausible texts, of any sort you could imagine. People seem to want a machine that can, for example, answer scientific questions as part of a conversation. But most competent answers to scientific questions aren’t conversations, they’re papers. If people stuck with the “raw” model, producing excerpts of nonexistent papers rather than imitating a dialogue with a non-existent expert, wouldn’t you expect the answers to be more accurate, with the model no longer biased by an irrelevant goal? Is the need to make a sell-able chatbot making these AIs worse at everything else people are trying to use them for?

I’m imagining a world where, instead of a chatbot, OpenAI built an “alternate universe simulator”. You give it some context, some texts or parts of texts from a universe you made up, and it completes them in a plausible way. By imagining different universes, you can use it to answer different questions. Such a gimmick would get fewer customers, and fewer investors, it would probably do worse. But I have to wonder if the actual technology might have been more useful.

My second idea is dumber, to the point where I mostly know why it doesn’t work. But thinking about it might help clarify how things work for people unused to AI.

I saw someone point out that, unlike something like Wikipedia, AI doesn’t give you context. You shouldn’t trust Wikipedia, or a source you find on Google, blindly. If you want to, you can look through the edit history on Wikipedia, or figure out who wrote a page you found on Google and how. If ChatGPT tells you something, by default you don’t know where that knowledge came from. You can tell it to search, and then you’ll get links, but that’s because it’s using Google or the like behind the scenes anyway. You don’t know where the model is getting its ideas.

Why couldn’t we get that context, though?

Every text produced by a Large Language Model is causally dependent on its training data. Different data, different model, different text. That doesn’t mean that each text draws from one source, or just a few sources: ChatGPT isn’t copying the training data, at least not so literally.

But it does mean that, if ChatGPT says something is true, you should in principle be able to ask which data was most important in making it say that. If you leave a piece of data out of the training, and get similar answers, you can infer that the response you got doesn’t have much to do with that piece of data. But if you leave out a text in training, and now ChatGPT gives totally different responses to the same question…then there’s a pretty meaningful sense that it got the information from that source.

If this were the type of non-AI statistical model people use in physics, this would be straightforward. Researchers do this all the time: take one experiment out of the data, see how their analysis changes, and thereby figure out which experiments are most important to check. One can even sometimes calculate, given a model, where you should look.

Unfortunately, you can’t do this with ChatGPT. The model is just too big. You can’t calculate anything explicitly about it, the giant mathematical formulas behind it are so complicated that the most you can do is get probabilities out case by case, you can’t “unwind” them and see where the numbers come from. And you can’t just take out sources one by one, and train the model again: not when training takes months of expensive computer time.

So unlike with the previous idea, I understand even on a technical level why you can’t do this. But it helped me to be able to think about what I would like to do, if it were possible. Maybe it helps you too!

by 4gravitons at August 22, 2025 04:00 PM

Tommaso Dorigo — Some Thoughts On Co-design For Tracking Optimization

These days I am organizing a collaborative effort to write an article on holistic optimization of experiments and complex systems. "So what is the news," I could hear say by one of my twentythree faithful readers (cit.) of this blog. Well, the news is that I am making some progress in focusing on the way the interplay of hardware design and software reconstruction plays out in some typical systems, and I was thinking I could share some of those thoughts here, to stimulate a discussion, and who knows, maybe get some brilliant insight.

Peter Rohde A call for global insurrection against tyranny and in the name of righteousness

Let it be known to all governments and systems of power:

It is their responsibility to serve the people not themselves.
While there are no equals, all are to be treated with equality.
Where they are self-serving there is a mandate for insurrection such that they serve the people.
Where they seek self-protection they will be denied and removed from power.
Where they keep secrets from the people there is a mandate for insurrection to enforce transparency and accountability for all.
Where they threaten or condemn the people they are condemned and there is a mandate for insurrection.
Where they fail to account for the lives of the people they serve there is a mandate for insurrection.
Where tyrannical power structures exist there is a mandate to disestablish them.
Where they declare war or work against one another there is a mandate for insurrection and unification.
Where they lie to us, deceive us or withhold the truth, they shall be removed from power and the truth be told to all.
Where legal systems uphold and enable tyranny they shall be removed. These are not our laws and we do not recognise them.

This is the natural order that guarantees our survival and gifts this world to our children. This world belongs to them and where we fail to serve them we condemn ourselves. And where government has failed to uphold this, we will not obey them as they have no right to exist.

We do not have to ask for these things, they are required, and if not given we shall simply take them.

Where the truth has not been told it shall be told.

If we fail to do so we condemn our children ourselves.

by Peter Rohde at August 20, 2025 07:06 AM

John Baez — The Cosmic Horseshoe

Astronomers have found a truly huge black hole! It’s in the massive galaxy in the center here, called the Cosmic Horseshoe. The blue ring is light from a galaxy behind the Cosmic Horseshoe, severely bent by gravity.

This black hole is 36 billion times the mass of the Sun. It’s not just ‘supermassive’: any black hole over 10 billion times the Sun’s mass is considered ‘ultramassive’. Not many have been found.

To me the coolest part is that the Cosmic Horseshoe has swallowed all the other galaxies in its group: it’s part of something called a ‘fossil group’, which I hadn’t heard about until today.

Our galaxy, the Milky Way, is part of a group too: the Local Group. Many smaller galaxies have already fallen into ours. Eventually the Milky Way and Andromeda may collide and form a single bigger galaxy. This will be an ‘elliptical galaxy’—too disorganized to have spiral arms. So it’s not surprising that if you wait long enough, galaxy groups form a single big elliptical galaxy which eventually eats the rest.

Some ancient galaxy groups have already done this, and they’re called `fossil groups’. The Cosmic Horseshoe is the big bully in this particular fossil group: it’s 100 times heavier than the Milky Way. It’s surrounded by a halo of very hot gas, 10 million Kelvin, emitting lots of X rays. But most of its mass can’t be explained by stars, gas and dust, so we say 90% is dark matter. All this is completely typical of a fossil group, except the Cosmic Horseshoe is bigger than average.

Fossil groups show us what the future will be like. Big galaxies will eat the rest, and big black holes at the center of these galaxies will eventually eat most of the matter. Dark matter—whatever that is—takes longer to fall in. But there’s plenty of time.

Here’s the new paper about this ultramassive black hole. Luckily, it’s free to read:

• Carlos R. Melo-Carneiro, Thomas E. Collett, Lindsay J. Oldham, Wolfgang Enzi, Cristina Furlanetto, Ana L. Chies-Santos and Tian Li, Unveiling a 36 billion solar mass black hole at the centre of the Cosmic Horseshoe gravitational lens, Monthly Notices of the Royal Astronomical Society 541 (2025), 2853-–2871.

Below is a picture from this paper. The blue blob in a box at left is called the ‘counter-image’ of the galaxy that’s making the Einstein ring. When light from behind a massive body is bent by gravity, it often forms a ring, called an ‘Einstein ring’, together with a counter-image.

by John Baez at August 17, 2025 07:45 AM

Doug Natelson — 20 years of Nanoscale Views, + a couple of things to read

Amazingly, this blog has now been around for more than twenty years (!) - see this first post for reference from June of 2005, when I had much less gray hair and there were a lot more science blogs. Thanks to all of you for sticking around. Back then, when I debuted my writing to my loyal readers (all five of them at the time), I never thought I'd keep this up. Some info, including stats according to blogger:

Total views: 8.3M
Most views in one day, this past May 31, with 272K
Top two most-viewed posts are this one from 2023 with a comment thread about Ranga Dias, and this one from 2009 titled "What is a plasmon?"
Just a reminder that I have collected a bunch of condensed matter terms and concept posts here.
I've also written some career-related posts, like a guide to faculty job searches, advice on choosing a graduate school, needs-to-be-updated advice on postdoc positions, etc.
Some personal favorite posts, some of which I wish had gotten more notice, include the physics of drying your hands, the physics of why whiskey stones aren't as good as ice to cool your drink, materials and condensed matter in science fiction, the physics of vibranium, the physics of beskar, the physics of ornithopters, and why curving your pizza slice keeps if from flopping over. I'm also happy with why soft matter is hard, which was a well-viewed post.
I also like to point out my essay about J. Henrik Schön, because I worry that people have forgotten about that episode.

Real life has intruded quite a bit into my writing time the last couple of years, but I hope to keep doing this for a while longer. I also still hope one day to find the right time and approach to write a popular book about the physics of materials, why they are amazing, and why our understanding of this physics, limited as it is, is still an astonishing intellectual achievement.

Two other things to read that I came across this week:

This post about Maxwell's Demon from the Skull in the Stars blog (which has been around nearly as long as mine!) is an excellent and informative piece of writing. I'm definitely pointing my statistical and thermal physics undergraduate class to this next month.
Ross McKenzie has a very nice looking review article up on the arXiv about emergence. I haven't read it yet, but I have no doubt that it will be well-written and thought-provoking.

by Douglas Natelson at August 16, 2025 01:20 PM

Matt von Hippel — Technology as Evidence

How much can you trust general relativity?

On the one hand, you can read through a lovely Wikipedia article full of tests, explaining just how far and how precisely scientists have pushed their knowledge of space and time. On the other hand, you can trust GPS satellites.

As many of you may know, GPS wouldn’t work if we didn’t know about general relativity. In order for the GPS in your phone to know where you are, it has to compare signals from different satellites, each giving the location and time the signal was sent. To get an accurate result, the times measured on those satellites have to be adjusted: because of the lighter gravity they experience, time moves more quickly for them than for us down on Earth.

In a sense, general relativity gets tested every minute of every day, on every phone in the world. That’s pretty trustworthy! Any time that science is used in technology, it gets tested in this way. The ideas we can use are ideas that have shown they can perform, ideas which do what we expect again and again and again.

In another sense, though, GPS is a pretty bad test of general relativity. It tests one of general relativity’s simplest consequences, based on the Schwarzchild metric for how gravity behaves near a large massive object, and not to an incredibly high degree of precision. Gravity could still violate general relativity in a huge number of other ways, and GPS would still function. That’s why the other tests are valuable: if you want to be sure general relativity doesn’t break down, you need to test it under conditions that GPS doesn’t cover, and to higher precision.

Once you know to look for it, these layers of tests come up everywhere. You might see the occasional article talking about tests of quantum gravity. The tests they describe are very specific, testing a very general and basic question: does quantum mechanics make sense at all in a gravitational world? In contrast, most scientists who research quantum gravity don’t find that question very interesting: if gravity breaks quantum mechanics in a way those experiments could test, it’s hard to imagine it not leading to a huge suite of paradoxes. Instead, quantum gravity researchers tend to be interested in deeper problems with quantum gravity, distinctions between theories that don’t dramatically break with our existing ideas, but that because of that are much harder to test.

The easiest tests are important, especially when they come from technology: they tell us, on a basic level, what we can trust. But we need the hard tests too, because those are the tests that are most likely to reveal something new, and bring us to a new level of understanding.

by 4gravitons at August 15, 2025 04:00 PM

n-Category Café Safeguarded AI Meeting

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This week, 50 category theorists and software engineers working on “safeguarded AI” are meeting in Bristol. They’re being funded by £59 million from ARIA, the UK’s Advanced Research and Invention Agency.

The basic idea is to develop a mathematical box that can contain a powerful genie. More precisely:

By combining scientific world models and mathematical proofs we will aim to construct a ‘gatekeeper’, an AI system tasked with understanding and reducing the risks of other AI agents. In doing so we’ll develop quantitative safety guarantees for AI in the way we have come to expect for nuclear power and passenger aviation.

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This program director is David Dalrymple, and you can get a much better description of the project from him in the first 4 minutes here:

David Dalrymple on safeguarded, transformative AI.

It’s remarkable how many of the applied category theorists in the UK are involved in this. Here you can find a partial list:

Category theorists in AI.

If you’re wondering “why category theory?”, I think the idea is this: software based on general abstract math is more flexible, yet also easier to formally verify.

For example the Topos Institute, run by my former student Brendan Fong, now has a branch in the UK largely funded by ARIA. At the meeting, Topos is demonstrating how to build models in CatColab, their new category-based software.

I have decided not to be part of this project, though some of my math is getting used here. I’ve always preferred to avoid doing things connected to AI, for various reasons. But this project might make AI better. It could also have various bad effects. I have no idea how successful it will be, so I’m watching with fascination and profoundly mixed emotions.

Scott Aaronson Updates!

(1) My 8-year-old son asked me last week, “daddy, did you hear that GPT-5 is now out?” So yes, I’m indeed aware that GPT-5 is now out! I’ve just started playing around with it. For detailed reports on what’s changed and how impressive it is compared to previous models, see for example Zvi #1, #2, #3. Briefly, it looks like there are major reductions in hallucinations and sycophancy, and improvements in practical usefulness for coding and other tasks, even while the “raw intelligence” is unlikely to blow away someone who was already well-acquainted with o3 and Opus 4 other state-of-the-art models, the way ChatGPT and then GPT-4 blew away people who had no idea what was possible in late 2022 and early 2023. Partly how impressive a result you see depends on which of several GPT-5 models your query gets routed to, which you don’t entirely control. Anyway, there’s grist here for the people who claim that progress toward AGI is slowing down, but also grist for the people who claim that it continues pretty much as expected within our post-ChatGPT reality!

(2) In other belated news, OpenAI and DeepMind (and then, other companies) announced that they achieved Gold Medal performance on the International Math Olympiad, by solving 5 of the 6 problems (there was one problem, the 6th and hardest, that all of the AIs struggled with). Most importantly, this means that I’ve won $100 from my friend Ernest Davis, AI expert at NYU, who bet me $100 that no AI would earn a Gold Medal at the International Math Olympiad by December 4, 2026. Even though I’m normally risk-averse and reluctant to take bets, I considered this one to be extremely safe, and indeed I won it with more than a year to spare.

(3) I’ve signed an open letter to OpenAI, along with many of my fellow former OpenAI employees as well as distinguished scientists and writers (Geoffrey Hinton, Stuart Russell, Sheldon Glashow, Sean Carroll, Matt Yglesias…), asking for more transparency about OpenAI’s continuing efforts to change its own structure. The questions basically ask OpenAI to declare, in writing, whether it has or hasn’t now completely abandoned the original nonprofit goals with which the organization was founded in 2015.

(4) At Lighthaven, the rationalist meeting space in Berkeley that I recently visited (and that our friend Cade Metz recently cast aspersions on in the New York Times), there’s going to be a writer’s residency called Inkhaven for the whole month of November. The idea—which I love—is that you either write a new blog post every day, or else you get asked to leave (while you also attend workshops, etc. to improve your writing skills). I’d attend myself for the month if teaching and family obligations didn’t conflict; someone standing over me with a whip to make me write is precisely what I need these days! As it is, I’m one of the three advisors to Inkhaven, along with Scott Alexander and Gwern, and I’ll be visiting for a long weekend to share my blogging wisdom, such as I have. Apply now if you’re interested!

(5) Alas, the Springer journal Frontiers of Computer Science has published a nonsense paper, entitled “SAT requires exhaustive search,” claiming to solve (or dissolve, or reframe, or something) the P versus NP problem. It looks indistinguishable from the stuff I used to get in my inbox every week—and now, in the ChatGPT era, get every day. That this was published indicates a total breakdown of the peer review process. Worse, when Eric Allender, Ryan Williams, and others notified the editors of this, asking for the paper to be retracted, the editor-in-chief declined to do so: see this guest post on Lance’s blog for a detailed account. As far as I’m concerned, Frontiers of Computer Science has now completely discredited itself as a journal; publication there means nothing more than publication in viXra. Minus 10 points for journals themselves as an institution, plus 10 points for just posting stuff online and letting it be filtered by experts who care.

(6) Uma Girish and Rocco Servedio released an arXiv preprint called Forrelation is Extremally Hard. Recall that, in the Forrelation problem, you’re given oracle access to two n-bit Boolean functions f and g, and asked to estimate the correlation between f and the Fourier transform of g. I introduced this problem in 2009, as a candidate for an oracle separation between BQP and the polynomial hierarchy—a conjecture that Ran Raz and Avishay Tal finally proved in 2018. What I never imagined was that Forrelation could lead to an oracle separation between EQP (that is, Exact Quantum Polynomial Time) and the polynomial hierarchy. For that, I thought you’d need to go back to the original Recursive Fourier Sampling problem of Bernstein and Vazirani. But Uma and Rocco show, using “bent Boolean functions” (get bent!) and totally contrary to my intuition, that the exact (zero-error) version of Forrelation is already classically hard, taking Ω(2^n/4) queries by any randomized algorithm. They leave open whether exact Forrelation needs ~Ω(2^n/2) randomized queries, which would match the upper bound, and also whether exact Forrelation is not in PH.

(7) The Google quantum group, to little fanfare, published a paper entitled Constructive interference at the edge of quantum ergodic dynamics. Here, they use their 103-qubit superconducting processor to measure Out-of-Time-Order Correlators (OTOCs) in a many-body scrambling process, and claim to get a verifiable speedup over the best classical methods. If true, this is a great step toward verifiable quantum supremacy for a useful task, for some definition of “useful.”

(8) Last night, on the arXiv, the team at USTC in China reported that it’s done Gaussian BosonSampling with 3,050 photons and 8,176 modes. They say that this achieves quantum supremacy, much more clearly than any previous BosonSampling demonstration, beating (for example) all existing simulations based on tensor network contraction. Needless to say, this still suffers from the central problem of all current sampling-based quantum supremacy experiments, namely the exponential time needed for direct classical verification of the outputs.

by Scott at August 14, 2025 12:50 AM

Jordan Ellenberg — I’m on the Kirchner podcast!

I knew this was going to be a fun one when the first question was about how I came to use lyrics from the Housemartins in How Not To Be Wrong. No one has ever asked me that before! I resisted the urge to do a whole hour of 1980s college radio content and instead we actually did talk about math, schooliness, AI, etc. Have a listen!

by JSE at August 13, 2025 02:52 PM

Tommaso Dorigo — Swedish Physics Days

On August 13-15 I will attend for the first time to the Swedish Physics Days, an important national event for Swedish physics. This year the congress takes place at Lulea University of Technology, the institute where I am currently spending some time, hosted by the Machine Learning group through a Guest Researcher fellowship granted by WASP (Wallenberg AI, Autonomous Systems and Software Program).

Terence Tao — Rough numbers between consecutive primes

First things first: due to an abrupt suspension of NSF funding to my home university of UCLA, the Institute of Pure and Applied Mathematics (which had been preliminarily approved for a five-year NSF grant to run the institute) is currently fundraising to ensure continuity of operations during the suspension, with a goal of raising $500,000. Donations can be made at this page. As incoming Director of Special Projects at IPAM, I am grateful for the support (both moral and financial) that we have already received in the last few days, but we are still short of our fundraising goal.

Back to math. Ayla Gafni and I have just uploaded to the arXiv the paper “Rough numbers between consecutive primes“. In this paper we resolve a question of Erdös concerning rough numbers between consecutive gaps, and with the assistance of modern sieve theory calculations, we in fact obtain quite precise asymptotics for the problem. (As a side note, this research was supported by my personal NSF grant which is also currently suspended; I am grateful to recent donations to my own research fund which have helped me complete this research.)

Define a prime gap to be an interval ${(p_n, p_{n+1})}$ between consecutive primes. We say that a prime gap contains a rough number if there is an integer ${m \in (p_n,p_{n+1})}$ whose least prime factor is at least the length ${p_{n+1}-p_n}$ of the gap. For instance, the prime gap ${(3,5)}$ contains the rough number ${4}$ , but the prime gap ${(7,11)}$ does not (all integers between ${7}$ and ${11}$ have a prime factor less than ${4}$ ). The first few ${n}$ for which the ${n^\mathrm{th}}$ prime gap contains a rough number are

$\displaystyle 2, 3, 5, 7, 10, 13, 15, 17, 20, \dots.$

Numerically, the proportion of ${n}$ for which the ${n^\mathrm{th}}$ prime gap does not contain a rough number decays slowly as ${n}$ increases:

Erdös initially thought that all but finitely many prime gaps should contain a rough number, but changed his mind, as per the following quote:

…I am now sure that this is not true and I “almost” have a counterexample. Pillai and Szekeres observed that for every ${t \leq 16}$ , a set of ${t}$ consecutive integers always contains one which is relatively prime to the others. This is false for ${t = 17}$ , the smallest counterexample being ${2184, 2185, \dots, 2200}$ . Consider now the two arithmetic progressions ${2183 + d \cdot 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13}$ and ${2201 + d \cdot 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13}$ . There certainly will be infinitely many values of ${d}$ for which the progressions simultaneously represent primes; this follows at once from hypothesis H of Schinzel, but cannot at present be proved. These primes are consecutive and give the required counterexample. I expect that this situation is rather exceptional and that the integers ${k}$ for which there is no ${m}$ satisfying ${p_k < m < p_{k+1}}$ and ${p(m) > p_{k+1} - p_k}$ have density ${0}$ .

In fact Erdös’s observation can be made simpler: any pair of cousin primes ${p_{n+1}=p_n+4}$ for ${p_n > 3}$ (of which ${(7,11)}$ is the first example) will produce a prime gap that does not contain any rough numbers.

The latter question of Erdös is listed as problem #682 on Thomas Bloom’s Erdös problems website. In this paper we answer Erdös’s question, and in fact give a rather precise bound for the number of counterexamples:

Theorem 1 (Erdos #682) For ${X>2}$ , let ${N(X)}$ be the number of prime gaps ${(p_n, p_{n+1})}$ with ${p_n \in [X,2X]}$ that do not contain a rough number. Then
$\displaystyle N(X) \ll \frac{X}{\log^2 X}. \ \ \ \ \ (1)$
Assuming the Dickson–Hardy–Littlewood prime tuples conjecture, we can improve this to
$\displaystyle N(X) \sim c \frac{X}{\log^2 X} \ \ \ \ \ (2)$
for some (explicitly describable) constant ${c>0}$ .

In fact we believe that ${c \approx 2.8}$ , although the formula we have to compute ${c}$ converges very slowly. This is (weakly) supported by numerical evidence:

While many questions about prime gaps remain open, the theory of rough numbers is much better understood, thanks to modern sieve theoretic tools such as the fundamental lemma of sieve theory. The main idea is to frame the problem in terms of counting the number of rough numbers in short intervals ${[x,x+H]}$ , where ${x}$ ranges in some dyadic interval ${[X,2X]}$ and ${H}$ is a much smaller quantity, such as ${H = \log^\alpha X}$ for some ${0 < \alpha < 1}$ . Here, one has to tweak the definition of “rough” to mean “no prime factors less than ${z}$ ” for some intermediate ${z}$ (e.g., ${z = \exp(\log^\beta X)}$ for some ${0 < \beta < \alpha}$ turns out to be a reasonable choice). These problems are very analogous to the extremely well studied problem of counting primes in short intervals, but one can make more progress without needing powerful conjectures such as the Hardy–Littlewood prime tuples conjecture. In particular, because of the fundamental lemma of sieve theory, one can compute the mean and variance (i.e., the first two moments) of such counts to high accuracy, using in particular some calculations on the mean values of singular series that go back at least to the work of Montgomery from 1970. This second moment analysis turns out to be enough (after optimizing all the parameters) to answer Erdös’s problem with a weaker bound

$\displaystyle N(X) \ll \frac{X}{\log^{4/3-o(1)} X}.$

To do better, we need to work with higher moments. The fundamental lemma also works in this setting; one now needs precise asymptotics for the mean value of singular series of ${k}$ -tuples, but this was fortunately worked out (in more or less exactly the format we needed) by Montgomery and Soundararajan in 2004. Their focus was establishing a central limit theorem for the distribution of primes in short intervals (conditional on the prime tuples conjecture), but their analysis can be adapted to show (unconditionally) good concentration of measure results for rough numbers in short intervals. A direct application of their estimates improves the upper bound on ${N(X)}$ to

$\displaystyle N(X) \ll \frac{X}{\log^{2-o(1)} X}$

and some more careful tweaking of parameters allows one to remove the ${o(1)}$ error. This latter analysis reveals that in fact the dominant contribution to ${N(X)}$ will come with prime gaps of bounded length, of which our understanding is still relatively poor (it was only in 2014 that Yitang Zhang famously showed that infinitely many such gaps exist). At this point we finally have to resort to (a Dickson-type form of) the prime tuples conjecture to get the asymptotic (2).

by Terence Tao at August 11, 2025 03:27 AM

John Preskill — Nicole’s guide to writing research statements

Sunflowers are blooming, stores are trumpeting back-to-school sales, and professors are scrambling to chart out the courses they planned to develop in July. If you’re applying for an academic job this fall, now is the time to get your application ducks in a row. Seeking a postdoctoral or faculty position? Your applications will center on research statements. Often, a research statement describes your accomplishments and sketches your research plans. What do evaluators look for in such documents? Here’s my advice, which targets postdoctoral fellowships and faculty positions, especially for theoretical physicists.

Keep your audience in mind. Will a quantum information theorist, a quantum scientist, a general physicist, a general scientist, or a general academic evaluate your statement? What do they care about? What technical language do and don’t they understand?

What thread unites all the projects you’ve undertaken? Don’t walk through your research history chronologically, stepping from project to project. Cast the key projects in the form of a story—a research program. What vision underlies the program?

Here’s what I want to see when I read a description of a completed project.
- The motivation for the project: This point ensures that the reader will care enough to read the rest of the description.
- Crucial background information
- The physical setup
- A statement of the problem
- Why the problem is difficult or, if relevant, how long the problem has remained open
- Which mathematical toolkit you used to solve the problem or which conceptual insight unlocked the solution
- Which technical or conceptual contribution you provided
- Whom you collaborated with: Wide collaboration can signal a researcher’s maturity. If you collaborated with researchers at other institutions, name the institutions and, if relevant, their home countries. If you led the project, tell me that, too. If you collaborated with a well-known researcher, mentioning their name might help the reader situate your work within the research landscape they know. But avoid name-dropping, which lacks such a pedagogical purpose and which can come across as crude.
- Your result’s significance/upshot/applications/impact: Has a lab based an experiment on your theoretical proposal? Does your simulation method outperform its competitors by X% in runtime? Has your mathematical toolkit found applications in three subfields of quantum physics? Consider mentioning whether a competitive conference or journal has accepted your results: QIP, STOC, Physical Review Letters, Nature Physics, etc. But such references shouldn’t serve as a crutch in conveying your results’ significance. You’ll impress me most by dazzling me with your physics; name-dropping venues instead can convey arrogance.

Not all past projects deserve the same amount of space. Tell a cohesive story. For example, you might detail one project, then synopsize two follow-up projects in two sentences.

A research statement must be high-level, because you don’t have space to provide details. Use mostly prose; and communicate intuition, including with simple examples. But sprinkle in math, such as notation that encapsulates a phrase in one concise symbol.

A research statement not only describes past projects, but also sketches research plans. Since research covers terra incognita, though, plans might sound impossible. How can you predict the unknown—especially the next five years of the unknown (as required if you’re applying for a faculty position), especially if you’re a theorist? Show that you’ve developed a map and a compass. Sketch the large-scale steps that you anticipate taking. Which mathematical toolkits will you leverage? What major challenge do you anticipate, and how do you hope to overcome it? Let me know if you’ve undertaken preliminary studies. Do numerical experiments support a theorem you conjecture?

When I was applying for faculty positions, a mentor told me the following: many a faculty member can identify a result (or constellation of results) that secured them an offer, as well as a result that earned them tenure. Help faculty-hiring committees identify the offer result and the tenure result.

Introduce notation before using it. If you use notation and introduce it afterward, the reader will encounter the notation; stop to puzzle over it; tentatively continue; read the introduction of the notation; return to the earlier use of the notation, to understand it; and then continue forward, including by rereading the introduction of the notation. This back-and-forth breaks up the reading process, which should flow smoothly.

Begin every paragraph with a topic sentence.

Avoid verbs that fail to relate that you accomplished anything: “studied,” “investigated,” “worked on,” etc. What did you prove, show, demonstrate, solve, calculate, compute, etc.?
Tailor a version of your research statement to every position. Is Fellowship Committee X seeking biophysicists, statistical physicists, mathematical physicists, or interdisciplinary scientists? Also, respect every application’s guidelines about length.

If you have room, end the statement with a recap and a statement of significance. Yes, you’ll be repeating ideas mentioned earlier. But your reader’s takeaway hinges on the last text they read. End on a strong note, presenting a coherent vision.
Writing is rewriting, a saying goes. Draft your research statement early, solicit feedback from a couple of mentors, edit the draft, and solicit more feedback.

by Nicole Yunger Halpern at August 11, 2025 12:12 AM

Justin Wilson — Phases of a Game Show, Part 2

In a previous post, we discussed a phase transition that occurred in the piping above you on a game show. In the scenario, you are led on stage in front of a large audience. After a brief time, the audience votes on how “likeable” you are. The catch is that it doesn’t simply tally the votes, but turns spigots on a lattice of piping above your head. Water is then released and if enough people like you, it closes off the passage, keeping you dry. This exciting game show1 was described in that post:

Each “like” turns a spigot off, stopping water from flowing through one pipe in a grid overhead. Once voting ends, water is dumped into the system. If it can find a path to the bottom, you get soaked. [Emphasis added] The better your “likeability,” the less likely spigots open a path for water to flow and the drier you stay. That’s your prize for this game show (and hey, you also get the knowledge that people out there like you).
This system models a type of phase transition known as percolation.

The full post is here:

I highlighted above a key phrase “If it can find a path to the bottom, you get soaked.” What I didn’t say, but should have is that the water was being forced through the pipes, not just dropping down due to gravity. This is a very important point since our phases and phase transition changes dramatically if we just let gravity do the work. In the case of the water being “forced,” it can travel back up pipes if it helps it find its way out and onto your head, but in the case when only gravity is present, it falls down the pipes. To facilitate gravity, we’ll turn the pipes 45 degrees, and if we insert water at a single point on top, it could look like this:

Testing our gravity setup by putting in water at only one pipe up top. Notice that it never goes back up a pipe, only down.

This setup is a different problem called directed percolation. It also has a phase transition, but one that is different in some fundamental ways from regular percolation.