The first (and at the time of writing, only) course offered on PhysicsGraph is Introduction to Quantum Computing. I decided to take advantage of a promotional offer (see below) and recently finished the entire course. And as I did with Math Academy, I decided to write a review.

PhysicsGraph compared to Math Academy

As noted above (and freely confirmed by Chris Sutherland, one of its two founders), PhysicsGraph is very much modeled on Math Academy. In particular, it shares the following features:

A finely-scaffolded knowledge graph. Physics, like mathematics, is a hierarchical field: you must know elementary topics by heart in order to understand more advanced ones. Both PhysicsGraph and Math Academy embody these dependencies in a “knowledge graph” for each course, a directed acyclic graph of individual topics (e.g., “quantum operations and quantum circuits”) showing which topics are prerequisites for other topics (e.g., “multi-qubit operations and quantum circuits”). As a student progresses through the knowledge graph, fulfillment of prerequisites (i.e., completing the corresponding lessons) unlocks lessons for more advanced topics.

To reduce cognitive load on students, ideally the topics in the knowledge graph should be “bite-sized,” each containing the minimum of material needed to advance to the next topics in the graph. This help ensure that students can fully master a given topic before going on to the next. (Justin Skycak of Math Academy refers to the ideal knowledge graph as being “finely scaffolded.”) For the most part this seems to be the case with PhysicsGraph, although there were some longer lessons (like quantum teleportation and superdense coding) where some might have preferred their being split in two.

As does Math Academy, PhysicsGraph publishes the knowledge graphs for topics in its course(s), so you can see the scaffolding for yourself. For example, see the knowledge graph for the Introduction to Quantum Computing course.

Experience Points (XP). Like Math Academy, PhysicsGraph assigns each lesson an XP value that presumably reflects the amount and difficulty of the material in that lesson. Students then earn XP for themselves by successfully answering questions, with XP bonuses handed out for successfully answering all questions in a lesson or review.

I found accumulating XP on PhysicsGraph to be easier than on Math Academy: I was able to do over a thousand XP in a week and well over a hundred XP per day. (I did 354 XP on one particularly productive day.) In contrast, I keep myself busy trying to maintain a pace of 40 XP per day on Math Academy. Part of the difference may be that I was already familiar with many topics in the Introduction to Quantum Computing course (e.g., matrix multiplication and vector spaces) from learning and practicing them on Math Academy.

Adaptive diagnostics. Like Math Academy, PhysicsGraph has students take a diagnostic exam before starting a course. The exam is used to judge the level of the student’s knowledge and which prerequisite topics they can safely skip. That frees the student from getting bored covering material that they already know.

I found there were only a couple of times where I felt PhysicsGraph assumed knowledge I wasn’t fully solid on, one being expressing complex numbers in polar form. (I had some problems with this in Math Academy too.)

Spaced repetition review. Like Math Academy, PhysicsGraph prompts students to periodically review previously-covered (and hopefully mastered) material, scheduled in such a way that a student reviews the material for a given topic at the point at which they’re in danger of forgetting it.

Math Academy has an elaborate algorithm for doing this (see my discussion of it), one that incorporates “implicit review” of material, i.e., reducing the overall number of reviews based on the fact that successfully reviewing a topic also implicitly reviews prerequisites for that topic. The PhysicsGraph website doesn’t contain detailed information about its SRS technology, but I wouldn’t be surprised to learn it has a similar scheme.

Im Math Academy the reviews tend to predominate as you reach the end of a course, so that you spend as much or more time reviewing previous material as you do learning the last few topics in the course. That wasn’t my experience with PhysicsGraph, which may be a good thing or bad thing: It’s possible that PhysicsGraph is not scheduling enough reviews to promote long-term retention.

Gamification. Like Math Academy, PhysicsGraph allows students to compare their progress to others by looking at a leaderboard displaying XP totals for the highest-ranking students. However, unlike Math Academy, participating in the leaderboard is not optional, and there is only a single leaderboard instead of the multiple “leagues” of Math Academy. I think participation in gamification schemes should be optional, since some students don’t want to compare themselves to others and don’t need the motivation provided by such comparisons. As for having multiple leagues, I don’t think that makes sense until the number of active PhysicsGraph students is one or two orders of magnitude larger than it is now.

Community forum. Like Math Academy, PhysicsGraph has a Discord server in which students can (at least in theory) share experiences, help each other with problems, provide feedback to the service’s founders, and so on. At present the PhysicsGraph server is relatively inactive, presumably because the number of students is still relatively small. (Even with Math Academy, which presumably has hundreds if not thousands of active students, only a relatively few people post to the Discord server; this is typical of social media in general, and I wouldn’t expect PhysicsGraph to be any different.)

Target market. PhysicsGraph will likely appeal to the same types of students that Math Academy does, namely homeschoolers, advanced K-12 students, university students, and adults of any age learning new fields for personal enjoyment (like me) or professional enrichment. In particular, based on discussions on the Math Academy discord server and elsewhere, taking a Math Academy course appears to be a popular way to prepare for a formal instructor-led course (e.g., at a college or university) or to supplement such a course while taking it. I expect PhysicsGraph to have a similar appeal, and in fact their proposed next courses (“Physics I” and “Physics II”) appear to directly address that use case.

No “freemium” option. Like Math Academy, PhysicsGraph is a paid service with no free offering (although it does offer a free trial with a money-back guarantee). In my opinion this is exactly the way it should be: These services offer real value to students, and students should be willing to pay for that value.

PhysicsGraph’s subscription pricing is currently $29 US per month, compared to $49 US per month for Math Academy. I think this is a fair price given the paucity of courses at present compared to what Math Academy offers; however I think there’s room for raising the price in future if PhysicsGraph can built out a complete set of undergraduate physics courses and even extend into adjacent areas like chemistry. (I should also add that PhysicsGraph’s current annual price of $199 US is an absolute bargain, representing a 43% discount off paying per month; with Math Academy, paying by the year gets you only a 15% discount.)

The lack of a free offering also means that PhysicsGraph, like Math Academy, has an easier path to becoming a sustainable business by bringing in revenue from day one. The long-term financial health of a service like PhysicsGraph is presumably important for its founders and employees. But it’s also important for those of its customers who are lifelong learners, who want to take more courses over time and continue to get reviews for those previously taken. And unlike a typical VC-funded “freemium” educational offering, a service that can sustain itself solely on revenue from students will also be resistant to adding intrusive ads, employing deceptive marketing techniques, and generally implementing other forms of “enshittification.”

There are also areas where PhysicsGraph differs from Math Academy, for better or worse:

No timed quizzes. Math Academy periodically tests students’ mastery of previously-covered material using timed quizzes, requiring the student to answer a series of questions within a 15-minute period. The results of the quizzes are then used to schedule additional reviews and to modify the difficulty of subsequent quizzes. Despite finding these quizzes to be stressful, I think they’re useful in exposing gaps in learning and determining how best to correct them. I hope PhysicsGraph sees fit to offer a similar feature in future.

No “free response” questions. Math Academy lessons, reviews, and quizzes typically use multiple-choice questions with five possible answers. However some questions require responses where the student types in the answer. (Math Academy provides some UI widgets to make this easier when entering formulas.) As with quizzes, I find these types of questions more stressful to do, because I can’t just test my tentative answer by comparing it with the presented list of choices; instead I have to double- or even triple-check my work. But, also as with quizzes, I think free-response questions help in learning the material. This is another feature that PhysicsGraph might consider for future implementation in cases where it makes sense.

No question answering when revisiting lessons. This is a subtle point, but one that I think is important: In both Math Academy and PhysicsGraph you can go back to previous lessons and go through the material again. In Math Academy you can also answer the questions again in the same manner as when taking the lesson originally, except that you do not receive any further XP. However, this is not possible in PhysicsGraph; when revisiting a lesson you see all the questions presented along with your previous answers (correct or incorrect) and the accompanying explanations. I think this is a mistake: a major reason to revisit a lesson is to solidify your understanding of that lesson’s topics, and working through the exercises again helps contribute to that. I think PhysicsGraph should consider implementing Math Academy’s approach.

No “AI-powered” marketing. Math Academy advertises itself as “AI-Powered,” and its website has a “How our AI works” web page. As I discussed in my review, I am not a fan of this strategy, which presumably seeks to exploit the current hype around ChatGPT and other LLM-based services to help market Math Academy. PhysicsGraph is blessedly free of this conflation of LLMs with the machine learning-based analytics underlying Math Academy and (presumably) PhysicsGraph. It’s not trying to advertise itself as A Young Lady’s Illustrated Feynman Lectures, and thank goodness for that.

(To be clear, I think LLMs may have a place in a service like PhysicsGraph or Math Academy. However I think they’re an unneeded distraction from the core functioning of the service, and figuring out when and where their use does make sense will require some thought and experimentation.)

Per-class pricing. For both Math Academy and PhysicsGraph the main offering is via a monthly or yearly subscription; I think this is the right choice for both services, since it provides ongoing revenue with which to improve the service and add more courses. However, as I previously noted with Math Academy, I think it’s useful to also offer students the option to pay a one-time fee for an individual course, including the ability to review material for that course after completing it.

PhysicsGraph has done this with the Introduction to Quantum Computing course, offering lifetime access for $99 US, discounted from $199 US as an early-bird promotion. The promotional price thus equates to over three months of the subscription pricing, with the regular pricing (when it goes into effect) amounting to over six months of a subscription.

I could have opted for a $29 per month subscription, knocked out the Introduction to Quantum Computing course in less than a month, and then cancelled the service. However, I went for the per-class price because I wanted to be able to review topics in the course on an ongoing basis without having to maintain a subscription.

PhysicsGraph compared to Quantum Country

PhysicsGraph is not the first attempt to teach the basics of quantum computing and leverage spaced repetition to help the reader remember them. A couple of years ago Andy Matuschak and Michael Nielsen created the Quantum Country website, billed as “a free introduction to quantum computing and quantum mechanics . . . presented in a new mnemonic medium which makes it almost effortless to remember what you read.” It covers much of the same material as the PhysicsGraph Introduction to Quantum Computing course, including a discussion of quantum teleportation.

How does Quantum Country compare to the PhysicsGraph quantum computing course? My snap judgment is that Quantum Country is superior as a scientific essay for a knowledgeable audience, and makes an honest attempt to sweat the details (as opposed to hand-waving over them). However, I think it’s inferior as a way to actually learn quantum computing basics.

First, and most trivially, Quantum Country assumes a basic level of knowledge that some readers won’t have, to be “comfortable with complex numbers and with linear algebra—vectors, matrices, and so on . . . [and] with the logic gates used in conventional computers—gates such as AND, OR, NOT, and so on.” That’s not a problem with a scientific essay, and as the authors note there are other resources to learn those details. But it is a problem for readers coming into the topic relatively cold.

Quantum Country also spends some time on the history of the ideas behind quantum computing and the context for its development, and includes a fair amount of material on the principles and benefits of spaced repetition. If you subscribe to Justin Skycak’s theory of skill development then a lot of that is arguably superfluous—like teaching someone how to play basketball by starting out with a discussion of the high-level principles that make it a great game.² PhysicsGraph’s (and Math Academy’s) approach is more like “let’s have you first learn how to dribble.”

Next, Quantum Country material is not divided up and scaffolded as it would be in PhysicsGraph or Math Academy; it’s more like an essay where the reader is continually led along. This may be a small point, but I think it’s important for a student to have a small sense of accomplishment after finishing one lesson and before beginning the rest.

I think it’s also important for the system to gate access to further lessons on successfully completing the current one; with Quantum Country you can completely skip (or wrongly answer) all the spaced repetition questions and just go on to read further without necessarily understanding what you’ve read already.

Finally, in various places Quantum Country mimics the “this exercise is left to the reader” practice typical of traditional textbooks. This is contrary to the instructional philosophy behind both Math Academy and PhysicsGraph, in which all topics should be fully explained (even over-explained), with the role of the student then being to practice on their own applying the explained techniques to problems that are very similar (sometimes only trivially different) to those covered in the instructional material.

However, Quantum Country does have one feature that I think might be profitably considered for PhysicsGraph (and Math Academy): it sends email messages to remind readers when reviews are due. Math Academy assumes a model in which students are accessing the system every day, will see offered reviews in the normal course of events, and do not need additional reminders.

PhysicsGraph also assumes that students are motivated to check the system on a regular basis, in part because they’re paying for a continuing subscription. But what about students (like me) who pay for lifetime access to a single PhysicsGraph course? There may come a time when reviews for that course are due only every few days, or even every few weeks or months. In that case it would be useful to get email reminders when reviews are due.

Learning to teleport quantum states

I think there are multiple areas where PhysicsGraph could be improved. In particular, after I completed the course and started to get reviews for various topics, I had real difficulty completing the reviews for the last topics in the course, superdense coding and quantum teleportation. Above and beyond any intellectual deficiencies on my part, I think this may be due to at least two factors:

First, and most important, I think not having periodic timed quizzes is a real lack. There have been multiple occasions in Math Academy where I didn’t do well on quizzes and had to go back and do additional reviews and even revisit lessons. I think this really helped me better understand and retain the material. This is missing in PhysicsGraph, and as a result I wasn’t doing as much review as I should have been.

I also think some of the topics in the Introduction to Quantum Computing course could be even more finely scaffolded than they are. As one example, I think the lesson on separable and entangled states and Bell states might be better split into two. I’ve had major problems remembering the symbols for the Bell states and their associated quantum states; that might have been at least partially remedied by having a separate lesson (along with associated questions and reviews) dedicated to learning about those states.

But even in its current somewhat embryonic state, PhysicsGraph is a fun and effective way to learn physics. The choice of quantum computing for the first course is an excellent one: The material is understandable with only a minimum of mathematics knowledge, and the course itself teaches any additional mathematics needed.

The topic is also intrinsically interesting. Quantum mechanics, on which quantum computing depends, is a shining example of our ability to understand the world and manipulate it, the theory enabling everything from your cat’s laser pointer to the laptop on which I’m writing this article. Quantum entanglement is a phenomenon that Einstein found baffling (he called it “spukhafte Fernwirkung,” “spooky action at a distance”) and the ability to teleport quantum states is one of its most surprising consequences.

Yet at the end of the Introduction to Quantum Computing course, after all the preliminary lessons about Kronecker products and Bell states, PhysicsGraph will take you to a point where you can not only understand how quantum teleportation works, but why it must work exactly that way. Doing it in practice is far from trivial—real magic is hard—but you will have unlocked the spell that underlies the trick.

I myself am not planning to subscribe to PhysicsGraph on an ongoing basis. My personal interest is in learning more about data science, and to that end I’m working through the relevant Math Academy courses to learn the needed background material. My interest in physics these days is a relatively casual one that can easily be satisfied through other means.

However, if you are interested in really learning physics, and not just reading popular articles or watching videos about it, I strongly recommend that you check out PhysicsGraph. It’s a worthy attempt to apply to learning physics the proven educational techniques—direct instruction, spaced repetition, and so on— implemented in the Math Academy service. PhysicsGraph is still under construction, but I have confidence that the people behind it will end up building something great.

[This post was originally published on Cohost.]

I‘ve been around long enough to notice that stories involving magic seem to be more popular now than they used to be, and I’ve often wondered why that might be. I was reminded of that once again while reading one of the most delightful and charming stories about magic, Kamome Shirahama’s manga series Witch Hat Atelier. In the manga the young girl Coco is initiated into the world of witches—a world she thought had been closed to her from birth, but which she discovers is open to anyone who can learn to draw the intricate designs by which spells are cast.

The world of Witch Hat Atelier is of course fictional, and its magic likewise. But here in the real world we can also perform magic by making drawings, like so-called Feynman diagrams (see above), and manipulating symbols in other ways. These manipulations have a deep connection to physical reality, and enable us to divine the secrets of the universe and shape matter and energy to our will. They are “magic that actually works.”

So why don’t people pay more attention to real magic than fake magic? Of course, fake magic is embedded in stories that are more entertaining than real life. But beyond that, I think there are two key reasons:

First, real magic is hard. Not as hard as people think—it’s possible to grasp the basics of something like quantum computing or even quantum mechanics with no more mathematics than what is taught in high school (algebra, complex numbers, and matrix multiplication). But it is indeed true that applying that knowledge to real physical systems calls for much more knowledge and expertise.

More important, I think, is that though our diagrams and calculations may capture the universe precisely, in and of themselves they are powerless to change it. To do that requires advanced engineering and precision manufacturing, using techniques refined over multiple generations by thousands of people. No one person can totally comprehend everything that goes in making, say, a modern smartphone; even mundane products like LED light bulbs and your cat’s laser pointer are incredibly sophisticated at their core. Unable to understand how these devices work and what goes into making them, we simply take them for granted.

Once upon a time technology was simpler to understand. I think one of the attractions of the steampunk genre is that it harks back to the last time technology was truly legible, all puffing steam engines and rotating gears. Even electric motors and generators are not incomprehensible, although they mark the transition from the intuitive physics of Michael Faraday (famed for giving lectures and demonstrations to enthusiastic crowds of Londoners) to the mathematical physics of James Clerk Maxwell and his equations of electromagnetism.

Maxwell published his key papers in the 1860s. Perhaps not so coincidentally, 1870 is when the economist Brad DeLong sees everything changing: “In 1870 industrial research labs, modern corporations, globalization, and the market economy . . . proved keys to the lock that had kept humanity in its desperately poor iron cage . . . . And previously unimaginable economic growth revolutionized human life over and over, generation by generation.”

Those developments brought advanced technology to millions and ultimately billions, but they also killed its magic, as the heroic lone inventor in their workshop (another steampunk staple) gave way to the corporate R&D department and globalized supply chains. So we seek the magic we crave in novels, comics, and films and TV—the latter themselves benefiting from the use of computer graphics to create increasingly fantastical special effects.

Now we can be (fake) magicians ourselves, thanks to software and its ability to create virtual worlds in which our actions can be translated instantly into effects, whether that be digging a hole in Minecraft or casting an elaborate spell in a fantasy MMO. I could be snobbish and dismiss all this as inauthentic wish fulfillment (for example, comparing “survival mode” in Minecraft to a true survival experience in a wilderness), but I think that’s a fundamental mistake: we have always created new environments for us to live in, and (for example) an urban street with quaint shops and comfy apartments is just as much an artificial creation as the latest AAA title. There is no virtue in elevating the former over the latter.

And just as science drives technology, so technology drives science in a feedback loop, for example from the early microscopes that discovered bacteria to the scanning tunneling microscopes that can visualize and manipulate individual atoms. It’s possible that software worlds and the tools used to create them will in turn enable new ways to do science and engineering, so that in the future the diagrams drawn by a real-life Coco can create real-life magic.

Mightfo (@Mightfo) - 2022-11-19 20:31

Nice thoughts! Theres a lot of interesting things to talk about here, but one aspect that comes to mind is the way that this relates to the role of agency in fiction.

For example, why are medieval fantasy themes popular? I suspect that part of this is because they are useful for both the writer and the reader in terms of providing a setting that allows for more flexible agency.

If you compare a medieval fantasy setting to a realistic modern setting, I think you’ll find differences such as:

• The state is less powerful(except at the author’s whims) and has less ability to distribute its power effectively. This means several things—such as that the interactions between friends and foes is less cordoned off, and the onus of fixing a problem can be relocated more to individuals.

• Warfare can conceivably make individual skill/effort a lot more relevant, especially when you introduce magic and unrealistic strength etc, which gives characters more agency and room for expression of their efforts

• There is more nature to act against and with- even without the fantasy element, nature is wild and untamed, back to when it was more of a threat to humans.

• There’s more unknowns, because of less state/more nature/less information/less ability to completely overwhelm individuals with modern structures and technologies, even prior to magic.

And there’s probably more that I’m just not remembering from the last time I thought about this, haha.

Because of these, the writer has more avenues to create and unfold conflicts dynamically to create drama, back and forth, themes, gravitas, specific ways things play out, etc without having to fight constantly to create plausibility. I guess the “Wild West” setting has a lot of similarities to it, now that I think about it.

Fictional magic, similarly, is an individual-scale avenue for potential/unknowns/agency that the writers can get more freedom with and readers can be connected to particular things they enjoy seeing expressed(again, agency of individuals).

Frank Hecker (@hecker) - 2022-11-20 00:54

Thank you for the comment! I think you are correct about historical/fantasy settings providing more agency to the characters. I’m guessing that a part of this is also that rule was/is personalized, being centered in the persons of the monarch and their courtiers/vassals/etc., as opposed to being exercised via a more impersonal bureaucracy.

squircular reasoning (@spiders) - 2023-07-24 23:19

i think an aspect of “real magic”, as you put (sub)atomic sciences, that makes it disinteresting to a large number of people (including myself) is how divorced from everyday sensory reality it is.

i say this as somepony who is intensely interested in earth sciences; geology, botany, paleontology, zoology, even microbiology and some aspects of astronomy, to a certain degree, are all fairly accessible, and influence ones life in very visible ways. i may not be able to see tectonic plates moving but i can feel earthquakes, and i can see rock structures that are only possible because of uplift. i don’t have a microscope, but if i wanted one, i could get one, and suddenly the cellular structures of the organisms around me would be open to me

even molecular biology, to a certain degree, is somewhat accessible. like, PCR is revolutionary not just because it is a powerful tool for multiplying and sequencing dna but also because it is such a wildly simple technique that you can literally do it over a fucking campfire, and someone has. the barrier to entry for dna sequencing as an amateur biologist is mostly the cost of the chemical components—you don’t need a college degree to understand and carry out the process.

quantum physics, and to a strong degree also much of astrophysics, by contrast, is not something that can readily be understood if you didn’t specialize in it. people can sell you analogies in pop sci books but my trust of such books has been basically irrevocably damaged after i’ve been burned too many times finding out that i’ve been fed a false explanation of how something worked and that the true explanation is just way over my head completely incomprehensible. and if i did manage to understand it, it would feel about as real to me as fictional magic.

i feel like i couldn’t go applying the things i learned in any real way, not the way that knowing about the genetic history of plant families makes you understand what your senses are experiencing better.

the closest i ever came to feeling like i got that kind of everyday application of quantum mechanics to my life was reading QED by richard feynman and him adding up arrows to explain why light bent on a hot road. which is pretty interesting, if a bit abstract! (and there’s that thing again, the real magic doesn’t actually feel particularly real or connected to reality) but meanwhile, i have never actually been able to “understand” quantum computing. ppl make videos trying to explain it and i’ve watched many of them, i’ve tried to even write qasm, but the conclusion i came to is that you can’t actually understand it unless you go into heavy maths and write code, and you can’t write code that actually does anything discernable on a quantum computer unless you have a lot of money reply

Frank Hecker (@hecker) - 2023-07-25 08:00

Thanks for stopping by to comment! You’re right about quantum phenomena being hard to see and visualize, though people do try: here’s an article about seeing single photons with the naked eye, and another one about replicating the famous double-slit experiment with a cheap laser pointer. You’re also right about Feynmann’s QED: it’s an interesting and fun book.

rose (@rosieposie) - 2023-07-25 03:05

for me the closest thing to ’fictional magic’ is ’cooking’. you follow a recipe passed down by the sages of old, you improvise a little every once in a while to make it better, and voila, you turned things that people don’t like so much into a magical substance that enamors everyone at the party and has them begging for you to do that trick again next month. and you can just do it in real life any time you want to.

and really if you think about it, the only difference between chemistry and cooking is that cooking is that cooking is limited (mostly) to human-safe substances. and has significantly less extreme effects, like your product probably won’t kill most people. also something something potion brewing. reply

Frank Hecker (@hecker) - 2023-07-25 08:01

Thanks for commenting! You’re right, cooking can be pretty magic at times, especially in the hands of a master chef.

ArBe (@arbe) - 2023-07-25 05:46

I’ve thought about these comparisons(often from slightly different angles, but definitely in the same vein, and often also inspired by fiction like WHA) many times, especially the frustrating disparity between [ the ability to understand the fascinating systems underlying everything ] and [ the ridiculously disproportional amount of time and effort required to actually utilise that knowledge practically, especially as an individual ].

This post really nicely expands on those thoughts in ways I never would’ve thought of, and introduces some entirely new threads of thought that feel really enjoyable to explore, like the point about artificial environments. Great post!

brackets added for readability, maybe I wouldn’t need this crutch if my punctuation were better. Sorry

Frank Hecker (@hecker) - 2023-07-25 08:02

Thanks for your comment! I’m glad you found the post interesting.

Maynard (@Quelklef) - 2023-07-25 12:00

this feels like a very wise and true analysis

i’ll add that programming is one field where you can still be a rag-tag do-it-yourself-magician. the stuff you produce will not have the production value and polish of a megacorporation, but it still just might be functional

Frank Hecker (@hecker) - 2023-07-25 12:53

Thanks for stopping by to comment! I apologize for being so late to follow-up. Yes, you’re absolutely right, software is the one place a sole creator can make a difference, especially given all the (mostly) free tools at their disposal nowadays.