Orthogonal by Greg Egan

Oliver Brown
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I was inspired to write this review after watching Christopher Nolan’s movie Tenet. The central idea of Tenet is explored in the Orthogonal series (amongst many other things) but with more rigor and detail. As such, this review contains spoilers about the premise of Tenet. I won’t cover the details of Tenet’s plot, but if you were already going to watch Tenet, do that first.

Despite ostensibly being a review, this is mostly a piece designed to persuade you to read these books. It contains a fairly detailed explanation of the initial premise for the book (while avoiding spoilers for the plot), because talking about the things that are good is difficult otherwise.


The movie Tenet follows in the wake of Interstellar by taking a confusing scientific concept and creating a movie that can be compelling for a mainstream audience. In Interstellar this was the idea of time dilation, in Tenet it is the idea of the “arrow of time”, and whether, perhaps, it is not as immutable as it seems.

The one line conceptual spoiler for Tenet is the following: there are machines that allow people and objects to travel backwards in time, while in the same space as people traveling forwards. Interactions between normal people and “inverted” people are strange and unintuitive.

In the movie, the details of “how” are basically ignored. And although the plot is clever and demonstrates much of the oddness that would be present, there is little detail.

Riemannian space

The premise for Orthogonal is a simple change to a single formula that defines the geometry of space-time. There is a detailed explanation by the author himself, but a summary is as follows:

  • In two dimensional space, the distance between two points can be found using the Pythagorean theorem: d = √(x² + y²), where x and y are the differences in the points’ positions in the x-axis and y-axis respectively.
  • A very similar formula also works for three (and higher) dimensions: d = √(x² + y² + z²).
  • General relativity combines time and space together to form a single thing called spacetime.
  • It is possible to calculate the “distance” between two points in spacetime. However, in this case the formula features a minus sign: d = √(x² + y² + z² - t²).

This kind of spacetime is described by the author as “Lorentzian”. That minus sign is a consequence (or a cause, depending on your philosophy) of the speed of light being constant, and appears to describe our universe.

In the universe of Orthogonal however, the spacetime is “Riemannian”. That is, the distance between two points in spacetime is just: d = √(x² + y² + z² + t²).

This is summarized by saying the universes have metrics of +, +, +, − and +, +, +, + respectively.

The Arrow of Time

This small change has a lot of consequences. The biggest is that there is no longer a universal speed limit. You can go as fast as you want, even reaching a sort of infinite velocity. Just like in our universe there is time dilation at high velocities. At “infinite velocity”, time stops passing completely for a stationary observer. In the universe of Orthogonal however, if you were to continue accelerating you would find that time has started to progress backwards for you, relative to a stationary observer.

The exact mechanics of this may sound confusing, but are explained well in the book, and it more detail on the author’s site.

An actual review

With all the preliminaries out of the way, it’s time to explain why I think Orthogonal is so good. It basically does three things, and it probably could have stopped at the first one and still have been good.

1. Hard sci-fi with a relevant plot

The story is “hard sci-fi”. It has a universe that has solid rules that are as scientifically rigorous as possible and tries to avoid deviating from them. In order to also be a good sci-fi story though (as opposed to just good science), it also needs a plot that allows the reader to explore that universe.

Orthogonal presents the main characters with a world ending disaster. Not a terribly unique idea in the abstract, but in this case the disaster is one that could only happen in the Riemannian spacetime it presents.

Very quickly, the characters devise a plan to avert disaster. In the best traditions of disaster movies, it initially sounds crazy, but it just might work. And like the disaster itself, the proposed solution would only be possible in Riemannian spacetime.

2. Explore some further detail of the universe

Orthogonal is a trilogy of novels, spanning a significant time period. A lot of less apocalyptic consequences of Riemannian spacetime are explored. One of the more surprising is a running story of social development.

In the world of Orthogonal, women are not treated well. The way this manifests is very similar to our own world (so similar that it reminds me of classic allegorical Star Trek episodes). However the reason for the treatment in the story is, again, something that could only happen the world of Orthogonal (and is in fact a direct result of of the physics possible in Riemannian spacetime).

3. First principles

The final thing Orthogonal does, and the thing that takes the story from being “good” to being absolutely staggering, is how the details of the world are presented.

It starts in a world with a similar level of technology to us at the beginning of the 20th century. The author makes no overt attempt to explain the world narratively. For the reader it appears initially that the characters inhabit a world that behaves very differently to our own for no reason. But, the main characters in the story are striving for knowledge (after all they soon have a world ending disaster to avert), and so they start to learn how the world works.

As the plot develops over several generations, we follow scientists and engineers as they untangle the physics of the universe. We learn with them as the details of how the Riemmannian equivalents of electromagnetism, thermodynamics, relativity and quantum mechanics work.

That is the what makes Orthogonal really special. Not only working out how this strange universe works, but coming with a plausible way that characters in the universe might be able to work it out for themselves.


Stephen Hawking said that he was warned every equation added to a popular science book would halve the number of sales. Despite being a work of fiction, much of Orthogonal feels like a popular science book but the author has not let such an idea concern him. It is full of equations and diagrams. I wouldn’t say it is impossible to enjoy the book if you chose to ignore them (or at least not completely engage with them), but you would be missing out on a lot.

If, on the other hand, that kind of detailed explanation appeals to you, there is a site (all created personally by Greg Egan himself) dedicated to providing way more detail on all the physics and maths explored in the book.

Orthogonal is hard sci-fi. Conceptually it may be the least accessible of Egan’s works. It is, however, spread over three novels to avoid overwhelming the reader and plenty of time is dedicated to exploring how the characters deal with this world, and that can be enjoyed independently of understand why and how it works.


Actually, the least conceptually accessible work from Greg Egan is probably Dichronauts, a story set in a universe with the metric +, +, −, −. That universe is so weird that you can’t turn past ninety degrees, and as you get closer to it, you stretch towards infinity.

Buy it on Amazon


Buy it on Amazon


Buy it on Amazon


Buy it on Amazon


Mobile apps, ads and consent

Oliver Brown
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I’ve recently spent some time adding the Google User Messaging Platform (UMP) to Tic-tac-toe Collection. The motivating factor was needing to support App Tracking Transparency in iOS 14 when using AdMob, but more generally, due to the GDPR, apps have significant requirements around consent and data management (especially with ads) and my haphazard custom implementation was a pain and probably not entirely compliant.

My effort to do this led me to learning quite a bit about the current state of this in apps and on the web.

Aside - Technical implementation

UMP is provided as a native library for both Android and iOS. There are official Xamarin bindings available for each (they are very new, and for a while I had to bind the iOS implementation myself).

I created a cross-platform wrapper to make using it easier in a Xamarin Forms app.

Google UMP is an implementation of a Consent Management Platform (CMP) as defined by the IAB (I tried to find a link to a neutral definition, but if you search Google for “Consent Management Platform”, you get a whole load of ads for companies wanting to help you).

This will look very familiar to most European users (and probably others) since most websites display a popup with a similar setup. What I did not realize until investigating this is that the information and settings it contains (down to the exact wording in some cases) is very proscribed and there it not much leeway about how the information is presented (possibly explaining why they all feel so universally unpleasant to use).

More surprising, at least initially, is that the way this information is stored is actually really well defined as part of shared API - the Transparency & Consent Framework by the International Advertising Bureau Europe. The popup which you provide consent through makes the information available to the hosting site (as well as advertising components on the page) in a predictable way.

A lot of the information is contained in this GitHub repo, with the API defined here.

The web API is really well specified and provides a JavaScript object available for anything on the page to query to get current consent status. The mobile API is a bit less so. Basically the data is stored in the shared preferences for the app (SharedPreferences on Android and NSUserDefaults on iOS) with well known keys you have to query yourself.

(Mainly just for fun, I created a package for querying this data with Xamarin.)

There a few things about the spec that I found interesting:

  • The “purposes” for which data can be used are very specifically defined, and numbered. Many things refer to them by number.
  • The list of vendors is also centrally determined and numbered.
  • There is what appears to be a typo in the spec: property names include VendorConsents, PurposeConsents and then PublisherConsent. This last one is structured the same as the others, but named in the singular. And since, like HTTP referer, it wasn’t caught before publication, it is now set in stone.

Some problems

There are some problems with this system. Some are general and some are specific to Google’s implementation.

The biggest is the language presented to the user is still very verbose. The whole point of the GDPR was to give users more insight and control about how their data is used. I seriously doubt many people will actually make an effort to understand all the purposes and the consequences of them.

One weird limitation of Google’s implementation is that it is impossible for the user to see their previous consent status and to tweak it slightly. When you first display the form, everything is turned off. The user can select to accept everything or dig through and accept individual permissions. If the form is presented again, everything is turned off again. If the user wanted to revoke or grant a specific permission, they’d have to remember themselves what they chose last time.

Even worse, there is no way to exit the form without making changes. If you launch them form, and select “Manage options”, your only way out saves the “everything off” state.

Purpose one

The first numbered purpose is “Store and/or access information on a device”. This is essentially a technology agnostic way of referring to a cookie. It doesn’t actually allow an app or website to do anything itself, it has to be combined with another purpose. And interesting things happen if the user does not grant it.

Most significantly, Google Mobile Ads (whether using AdMob or Google Ad Manager) will refuse to display any ads at all. In theory any app using Google Ads should be using a system that checks if the user is subject to the GDPR, request consent for purpose one and then respect it in this way.

The new update to Tic-tac-toe Collection does that. Which means European users who refuse consent will disables ads completely, for free. Since my ad revenue is low, and I’m already not a big fan of how mobile advertising tends to work… I’m kind of okay with it.

Interestingly, there is some debate about whether the GDPR actually requires this. If an app requires ads to be financially viable, and storing cookies is required to technically deliver ads (say for basic stats tracking or fraud prevention - not personalisation) that sounds very much like a legitimate interest, which means it can be done without consent. Google does address this here with the following quote

Google uses cookies or mobile ad identifiers to support ads measurement. Existing ePrivacy laws require consent for such uses, for users in countries where local law requires such consent. Accordingly, our policy requires consent for ads personalisation and ads measurement where applicable, even if ads measurement can, for GDPR purposes, be supported under a controller’s legitimate interests.

New site for my music YouTube channel

Oliver Brown
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I created a new website to go with the music visualization YouTube channel I launched.

There is not much content there yet, besides details of everything that is on the channel. But those details are probably the most interesting part.

I created a tool using the YouTube data API that generates Hugo content based on the channel.

The process started simple and got more sophisticated over a few days. First I generate a page for each video. Then I get a list of all the channel playlists, and add a tag to each video page for each playlist. I also create a page for the tag that links to the actual YouTube playlist. Finally, for each video I parse the description adding YouTube chapter links where appropriate.

It highlights another interesting advantage of static site generators. It is really easy to create content with external tools. You don’t need to access any kind of API, you just generate text files.

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Blog moved to Hugo

Oliver Brown
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After spending most of its time on Wordpress (initially a self-hosted version and then Wordpress.com), the blog is now a static site generated with Hugo and running as an Azure Static Web App.

Ever since I created the Tic-tac-toe Collection site on Hugo I’ve wanted to move the blog, but I never had the time. The recent events related to COVID-19 have, however, given me random bits of free time, so here we are.

Tic-tac-toe Collection is being run as a static website on Azure Blob Storage, which is a way of connecting various bits of Azure together to host a static site. The new Azure Static Web App Service joins them all for you in a much more straightforward fashion.

Under ideal circumstances, the minimal steps to set up a static web app are:

  • Upload your site to Github.
  • Create a new Static Web App pointing at your Github.

And, if you’re lucky, that’s it. It generates a GitHub action for you that does the actual building, and it’s based on Microsoft Oryx. Oryx is a slightly magical, slightly scary tool designed to auto-detect and then build static web apps using a range of different technologies.

Unfortunately, for me, the auto generated Hugo script did not actually work. I ultimately used the GitHub action from here.

Theme for 2020 - Land of Shame and Despair

Oliver Brown
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One of the reasons I started retuning pieces to different modes was the hope that something good might come out of it by chance. One example has appeared. I don’t think “good” is the right word, but it at least sounds intentional (most pieces just sound accidentally out of tune).

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Land of Hope and Glory retuned into Ultra Locrian.

It is dark and disharmonious. Hitting a note on nearly every beat combined with a lack of any real resolution gives it an unrelenting monotony. And somehow, halfway through when the higher octave repeat starts you still end up with the feeling of: “It’s still going? And somehow it’s worse?”.

I would like to end on a cheerful note. Nothing with “Locrian” in the name really does that. So instead, have some Sonic the Hedgehog music:

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Visualization of the Duke Nukem 3D theme

Oliver Brown
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One big advantage of doing this to the Duke Nukem 3D theme is that the original was already a MIDI file. That means people familiar with it are already used to hearing slightly different versions, and the version I generate is no more or less authentic than any of the others.

Although I have to admit I do not remember hearing good quality tubular bells on the computer I first played it on (but they have since grown on me while making this).