The Binary Octahedral Group

29 August, 2019


The complex numbers together with infinity form a sphere called
the Riemann sphere. The 6 simplest numbers on this sphere lie at points we could call the north pole, the south pole, the east pole, the west pole, the front pole and the back pole. They’re the corners of an octahedron!

On the Earth, I’d say the “front pole” is where the prime meridian meets the equator at 0°N 0°E. It’s called Null Island, but there’s no island there—just a buoy. Here it is:

Where’s the back pole, the east pole and the west pole? I’ll leave two of these as puzzles, but I discovered that in Singapore I’m fairly close to the east pole:

If you think of the octahedron’s corners as the quaternions \pm i, \pm j, \pm k, you can look for unit quaternions q such that whenever x is one of these corners, so is qxq^{-1}. There are 48 of these! They form a group called the binary octahedral group.

By how we set it up, the binary octahedral group acts as rotational symmetries of the octahedron: any transformation sending x to qxq^{-1} is a rotation. But this group is a double cover of the octahedron’s rotational symmetry group! That is, pairs of elements of the binary octahedral group describe the same rotation of the octahedron.

If we go back and think of the Earth’s 6 poles as points 0, \pm 1,\pm i, \infty on the Riemann sphere instead of \pm i, \pm j, \pm k, we can think of the binary octahedral group as a subgroup of \mathrm{SL}(2,\mathbb{C}), since this acts as conformal transformations of the Riemann sphere!

If we do this, the binary octahedral group is actually a subgroup of \mathrm{SU}(2), the double cover of the rotation group—which is isomorphic to the group of unit quaternions. So it all hangs together.

It’s fun to actualy see the unit quaternions in the binary octahedral group. First we have 8 that form the corners of a cross-polytope (the 4d analogue of an octahedron):

\pm 1, \pm i , \pm j , \pm k

These form a group on their own, called the quaternion group. Then we have 16 that form the corners of a hypercube (the 4d analogue of a cube, also called a tesseract or 4-cube):

\displaystyle{ \frac{\pm 1 \pm i \pm j \pm k}{2} }

These don’t form a group, but if we take them together with the 8 previous ones we get a 24-element subgroup of the unit quaternions called the binary tetrahedral group. They’re also the vertices of a 24-cell, which is yet another highly symmetrical shape in 4 dimensions (a 4-dimensional regular polytope that doesn’t have a 3d analogue).

That accounts for half the quaternions in the binary octahedral group! Here are the other 24:

\displaystyle{  \frac{\pm 1 \pm i}{\sqrt{2}}, \frac{\pm 1 \pm j}{\sqrt{2}}, \frac{\pm 1 \pm k}{\sqrt{2}},  }

\displaystyle{  \frac{\pm i \pm j}{\sqrt{2}}, \frac{\pm j \pm k}{\sqrt{2}}, \frac{\pm k \pm i}{\sqrt{2}} }

These form the vertices of another 24-cell!

The first 24 quaternions, those in the binary tetrahedral group, give rotations that preserve each one of the two tetrahedra that you can fit around an octahedron like this:

while the second 24 switch these tetrahedra.

The 6 elements

\pm i , \pm j , \pm k

describe 180° rotations around the octahedron’s 3 axes, the 16 elements

\displaystyle{   \frac{\pm 1 \pm i \pm j \pm k}{2} }

describe 120° clockwise rotations of the octahedron’s 8 triangles, the 12 elements

\displaystyle{  \frac{\pm 1 \pm i}{\sqrt{2}}, \frac{\pm 1 \pm j}{\sqrt{2}}, \frac{\pm 1 \pm k}{\sqrt{2}} }

describe 90° clockwise rotations holding fixed one of the octahedron’s 6 vertices, and the 12 elements

\displaystyle{  \frac{\pm i \pm j}{\sqrt{2}}, \frac{\pm j \pm k}{\sqrt{2}}, \frac{\pm k \pm i}{\sqrt{2}} }

describe 180° clockwise rotations of the octahedron’s 6 opposite pairs of edges.

Finally, the two elements

\pm 1

do nothing!

So, we can have a lot of fun with the idea that a sphere has 6 poles.


Interview

1 January, 2019

Happy New Year! People like to ponder grand themes each time the Earth completes another orbit around the Sun, so let’s give that a try.

Maria Mannone is a musician who studies the relation between mathematics, music and the visual arts. We met at a conference on The Philosophy and Physics of Noether’s Theorems. Later she decided to interview me for the blog Math is in the Air. There’s a version in English and one in Italian.

She let me reprint the interview here… so with no further ado, here it is!

MM: You are one of the pioneers in using the internet and blogs for scientific education, with ‘This Week’s Finds.’ Which words would you use to feed the enthusiasm of young minds towards abstract mathematics?

JB: It seems only certain people are drawn to mathematics, and that’s fine: there are many wonderful things in life and there’s no need for everyone explore all of them. Mathematics seems to attract people who enjoy patterns, who enjoy precision, and who don’t want to remember lists of arbitrary facts, like the names of all 206 bones in the human body. In math everything has a reason and you can understand it, so you don’t really need to remember much. At first it may seem like there’s a lot to remember – for examples, lists of trig identities. But as you go deeper into math, and understand more, everything becomes simpler. These days I don’t bother to remember more than a couple of trigonometric identities; if I ever need them I can figure them out.

But the really surprising thing is that as you go deeper and deeper into mathematics, it keeps revealing more beauty, and more mysteries. You enter new worlds full of profound questions that are quite hard to explain to nonmathematicians. As the Fields medalist Maryam Mirzakhani said, “The beauty of mathematics only shows itself to more patient followers.”

MM: I love the reference to patterns, and the beauty to find. Thus, we can say that mathematical beauty is not ‘all out there’ as the beauty of a flower can be. Or, that some beautiful geometry present in nature can give a hint or can embody some mathematical beauty, but people have to work hard to find more of it—at least they have to learn how to look at things, and thus, how to mathematically think of them.

In the common opinion, a rose, or a water lily is beautiful (and it is!), but a bone is not ‘beautiful’ per se. Personally, each time I find patterns, regularities, hierarchical structures, I get excited and things seem to be at least mathematically interesting. I would like to ask you how would you relate the beauty in the natural world, both visible and ‘to discover,’ and the beauty of math. I’m wondering if they should be considered as two separate sets with occasional, random intersections, or as two displays of a generalized ‘beauty,’ as two different perspectives. Or, maybe, if the first can guide our search into math, or if math can teach us ‘how to look at things and finding beauty.’

JB: I think all forms of beauty are closely connected, and I think almost anything can be beautiful if it’s not the result of someone being heedless to their environment or deliberately hurtful.

It’s not surprising that flowers are very easy to find beautiful, since they evolved precisely to be attractive. Not to humans, at first, but to pollinators like birds and bees. It’s imaginable that what attracts those animals would not be attractive to us. But in fact there’s enough commonality that we enjoy flowers too! And then we bred them to please us even more; many of them are now symbiotic with us.

Something like a bone only becomes beautiful if you examine it carefully and think about how complex it is and how admirably it carries out its function.


From: http://acrmed.com/press-releases/

Bones are initially scary or ‘disgusting’ because when they’re doing their job they are hidden: we usually see them only when an animal is seriously injured or dead. So, you have to go past that instinctive reaction—which by the way serves a useful purpose—to see the beauty in a bone.

Mathematics is somewhere between a rose and a bone. Underlying all of nature there are mathematical patterns – but normally they are hidden from view, like bones in a body. Perhaps to some people they seem harsh or even disgusting when first revealed, but in fact they are extremely elegant. Even those who love mathematics find its patterns austere at first—but as we explore it more deeply, we see they connect in complicated delicate patterns that put the petals of a rose to shame.

MM: Thus, there seems to be an intimate dialogue between nature, both visible and hidden, and mathematical thinking. About nature and environment: in your Twitter image, there is a sketch of you as a superhero saving the planet, with the mathematical symbol ‘There is one and only one’ applied to our planet Earth. Can you tell the readers something about the way you combine your research in mathematics with your engagement for the environment?

Also, it is often said that beauty will save the world. Do you think that mathematical beauty can save the world?

JB: I mainly think of beauty—in all its forms—-as a reason why the world is worth saving. But we are very primitive when it comes to the economics of beauty. Paintings can sell for hundreds of millions of dollars, and we have a market for them. But nobody attaches any value to this critically endangered frog, Atelopus varius:


Atelopus varius,
from https://www.iucnredlist.org/species/54560/11167883

To my mind it’s more beautiful and precious than any painting. Not the individual, of course, but the species, which has taken millions of years to evolve. We are busy destroying species like this as if they were worthless trash. Our descendants, if we have any, will probably think we were barbaric idiots.

But I digress! I switched from pure mathematics and highly theoretical physics to more practical concerns around 2010, when I spent two years at the Centre for Quantum Technologies, in Singapore. I was very lucky that the director encouraged me to think about whatever I wanted. I was wanting a change in direction, and I soon realized that mathematicians, like everyone else, need to think about global warming and what we can do about it: it’s the crisis of our time. I spent some time learning the basics of climate science and working on some projects connected to that. It became clear that to do anything about global warming we need new ideas in politics and economics. Unfortunately, I’m not especially good at those things. So I decided to do something I can actually do, namely to get mathematicians to turn their attention from math inspired by the physics of the microworld—for example string theory—toward math inspired by the visible world around us: biology, ecology, engineering, economics and the like. I’m hoping that mathematicians can solve some problems by thinking more abstractly than anyone else can.

So to finally answer your last question: I’m not sure the beauty of mathematics can save the world, but its beauty is closely connected to clear thinking, and we really need clear thinking.

MM: Yes, in a certain sense, despite culture, technology, and thousands of years of human history, people are quite primitive when it comes to evaluating beauty as detached from the economy.

You brought up an important point: the research focus of mathematicians. This is a tricky point because young researchers are kind of split between following new ideas and projects, and the search for funds, that often leads them to join existing projects or just well-funded areas and to put aside their more ‘visionary’ ideas. What would be your suggestion to find a balance?

JB: I don’t know if I can give advice here: I’ve never needed to search for funds, I get paid to teach calculus and other courses, so I always just do the best research I can. That’s already quite hard—I could talk all day about that!

I suppose if you’re struggling for funds you have to fight to remember your dreams, and try to work your way into a situation where you can pursue these dreams. I imagine this is also true for any entrepreneur with a visionary idea. Academics struggling to get grants really aren’t all that different from executives in a large corporation trying to get funding for their projects.

MM: My last question is about the theme of peace, very important to the Baez family. Many innovations are related to the military. Do you think that the needed clear thinking you mentioned, can first of all come from times, themes, and ideas of peace?

JB: We are currently in a struggle that’s much bigger, and more inspiring, than any war between human tribes. We’re struggling to come to terms with the Anthropocene: the epoch where the Earth’s ecosystems and even geology are being transformed by humans. We are used to treating our impact on nature as negligible. This is no longer true! The Arctic is rapidly melting:

And since 1970, the abundance of many vertebrate species worldwide has dropped 60%. You can see it in this chart prepared by the Worldwide Wildlife Fund:

If this were a war, and these were humans dying, this would be the worst war the world has ever seen! But these changes will not merely affect other species; they are starting to hit us too. We need to wake up. We will either deliberately change our civilization, quite quickly, or we will watch as our cities burn and drown. Isn’t it better to use that intelligence we humans love to boast about, and take action?

MM: Thank you Professor, I hope these words will enlighten many people.


Compositionality – Now Open For Submissions

24 August, 2018

Our new journal Compositionality is now open for submissions!

It’s an open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Topics may concern foundational structures, an organizing principle, or a powerful tool. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

Compositionality is free of cost for both readers and authors.



CALL FOR PAPERS

We invite you to submit a manuscript for publication in the first issue of Compositionality (ISSN: 2631-4444), a new open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline.

To submit a manuscript, please visit http://www.compositionality-journal.org/for-authors/.

SCOPE

Compositionality refers to complex things that can be built by sticking together simpler parts. We welcome papers using compositional ideas, most notably of a category-theoretic origin, in any discipline. This may concern foundational structures, an organising principle, a powerful tool, or an important application. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

Related conferences and workshops that fall within the scope of Compositionality include the Symposium on Compositional Structures (SYCO), Categories, Logic and Physics (CLP), String Diagrams in Computation, Logic and Physics (STRING), Applied Category Theory (ACT), Algebra and Coalgebra in Computer Science (CALCO), and the Simons Workshop on Compositionality.

SUBMISSION AND PUBLICATION

Submissions should be original contributions of previously unpublished work, and may be of any length. Work previously published in conferences and workshops must be significantly expanded or contain significant new results to be accepted. There is no deadline for submission. There is no processing charge for accepted publications; Compositionality is free to read and free to publish in. More details can be found in our editorial policies at http://www.compositionality-journal.org/editorial-policies/.

STEERING BOARD

John Baez, University of California, Riverside, USA
Bob Coecke, University of Oxford, UK
Kathryn Hess, EPFL, Switzerland
Steve Lack, Macquarie University, Australia
Valeria de Paiva, Nuance Communications, USA

EDITORIAL BOARD

Corina Cirstea, University of Southampton, UK
Ross Duncan, University of Strathclyde, UK
Andree Ehresmann, University of Picardie Jules Verne, France
Tobias Fritz, Max Planck Institute, Germany
Neil Ghani, University of Strathclyde, UK
Dan Ghica, University of Birmingham, UK
Jeremy Gibbons, University of Oxford, UK
Nick Gurski, Case Western Reserve University, USA
Helle Hvid Hansen, Delft University of Technology, Netherlands
Chris Heunen, University of Edinburgh, UK
Aleks Kissinger, Radboud University, Netherlands
Joachim Kock, Universitat Autonoma de Barcelona, Spain
Martha Lewis, University of Amsterdam, Netherlands
Samuel Mimram, Ecole Polytechnique, France
Simona Paoli, University of Leicester, UK
Dusko Pavlovic, University of Hawaii, USA
Christian Retore, Universite de Montpellier, France
Mehrnoosh Sadrzadeh, Queen Mary University, UK
Peter Selinger, Dalhousie University, Canada
Pawel Sobocinski, University of Southampton, UK
David Spivak, MIT, USA
Jamie Vicary, University of Birmingham and University of Oxford, UK
Simon Willerton, University of Sheffield, UK

Sincerely,

The Editorial Board of Compositionality


Applied Category Theory 2018/2019

15 June, 2018

A lot happened at Applied Category Theory 2018. Even as it’s still winding down, we’re already starting to plan a followup in 2019, to be held in Oxford. Here are some notes Joshua Tan sent out:

  1. Discussions: Minutes from the discussions can be found here.
  2. Photos: Ross Duncan took some very glamorous photos of the conference, which you can find here.

  3. Videos: Videos of talks are online here: courtesy of Jelle Herold and Fabrizio Genovese.

  4. Next year’s workshop: Bob Coecke will be organizing ACT 2019, to be hosted in Oxford sometime spring/summer. There will be a call for papers.

  5. Next year’s school: Daniel Cicala is helping organize next year’s ACT school. Please contact him at if you would like to get involved.

  6. Look forward to the official call for submissions, coming soon, for the first issue of Compositionality!

The minutes mentioned above contain interesting thoughts on these topics:

• Day 1: Causality
• Day 2: AI & Cognition
• Day 3: Dynamical Systems
• Day 4: Systems Biology
• Day 5: Closing


Applied Category Theory Course: Databases

6 June, 2018

 

In my online course on applied category theory we’re now into the third chapter of Fong and Spivak’s book Seven Sketches. Now we’re talking about databases!

To some extent this is just an excuse to (finally) introduce categories, functors, natural transformations, adjoint functors and Kan extensions. Great stuff, and databases are a great source of easy examples.

But it’s also true that Spivak helps run a company called Categorical Informatics that actually helps design databases using category theory! And his partner, Ryan Wisnesky, would be happy to talk to people about it. If you’re interested, click the link: he’s attending my course.

To read and join discussions on Chapter 3 go here:

Chapter 3

You can also do exercises and puzzles, and see other people’s answers to these.

Here are the lectures I’ve given so far:

Lecture 34 – Chapter 3: Categories
Lecture 35 – Chapter 3: Categories versus Preorders
Lecture 36 – Chapter 3: Categories from Graphs
Lecture 37 – Chapter 3: Presentations of Categories
Lecture 38 – Chapter 3: Functors
Lecture 39 – Chapter 3: Databases
Lecture 40 – Chapter 3: Relations
Lecture 41 – Chapter 3: Composing Functors
Lecture 42 – Chapter 3: Transforming Databases
Lecture 43 – Chapter 3: Natural Transformations
Lecture 44 – Chapter 3: Categories, Functors and Natural Transformations
Lecture 45 – Chapter 3: Composing Natural Transformations
Lecture 46 – Chapter 3: Isomorphisms
Lecture 47 – Chapter 3: Adjoint Functors
Lecture 48 – Chapter 3: Adjoint Functors
Lecture 49 – Chapter 3: Kan Extensions
Lecture 50 – Chapter 3: Kan Extensions
Lecture 51 – Chapter 3: Right Kan Extensions
Lecture 52 – Chapter 3: The Hom-Functor
Lecture 53 – Chapter 3: Free and Forgetful Functors
Lecture 54 – Chapter 3: Tying Up Loose Ends


Applied Category Theory Course: Resource Theories

12 May, 2018

 

My course on applied category theory is continuing! After a two-week break where the students did exercises, I’m back to lecturing about Fong and Spivak’s book Seven Sketches. Now we’re talking about “resource theories”. Resource theories help us answer questions like this:

  1. Given what I have, is it possible to get what I want?
  2. Given what I have, how much will it cost to get what I want?
  3. Given what I have, how long will it take to get what I want?
  4. Given what I have, what is the set of ways to get what I want?

Resource theories in their modern form were arguably born in these papers:

• Bob Coecke, Tobias Fritz and Robert W. Spekkens, A mathematical theory of resources.

• Tobias Fritz, Resource convertibility and ordered commutative monoids.

We are lucky to have Tobias in our course, helping the discussions along! He’s already posted some articles on resource theory here on this blog:

• Tobias Fritz, Resource convertibility (part 1), Azimuth, 7 April 2015.

• Tobias Fritz, Resource convertibility (part 2), Azimuth, 10 April 2015.

• Tobias Fritz, Resource convertibility (part 3), Azimuth, 13 April 2015.

We’re having fun bouncing between the relatively abstract world of monoidal preorders and their very concrete real-world applications to chemistry, scheduling, manufacturing and other topics. Here are the lectures so far:

Lecture 18 – Chapter 2: Resource Theories
Lecture 19 – Chapter 2: Chemistry and Scheduling
Lecture 20 – Chapter 2: Manufacturing
Lecture 21 – Chapter 2: Monoidal Preorders
Lecture 22 – Chapter 2: Symmetric Monoidal Preorders
Lecture 23 – Chapter 2: Commutative Monoidal Posets
Lecture 24 – Chapter 2: Pricing Resources
Lecture 25 – Chapter 2: Reaction Networks
Lecture 26 – Chapter 2: Monoidal Monotones
Lecture 27 – Chapter 2: Adjoints of Monoidal Monotones
Lecture 28 – Chapter 2: Ignoring Externalities
Lecture 29 – Chapter 2: Enriched Categories
Lecture 30 – Chapter 2: Preorders as Enriched Categories
Lecture 31 – Chapter 2: Lawvere Metric Spaces
Lecture 32 – Chapter 2: Enriched Functors
Lecture 33 – Chapter 2: Tying Up Loose Ends

 


Applied Category Theory Course: Ordered Sets

7 April, 2018

My applied category theory course based on Fong and Spivak’s book Seven Sketches is going well. Over 250 people have registered for the course, which allows them to ask question and discuss things. But even if you don’t register you can read my “lectures”.

We study the applications to logic—both classical logic based on subsets, and a nonstandard version of logic based on partitions. And we show how this math can be used to understand “generative effects”: situations where the whole is more than the sum of its parts. But the real payoff comes in Chapter 2, where we discuss “resource theories”.

Lecture 1 – Introduction
Lecture 2 – What is Applied Category Theory?
Lecture 3 – Chapter 1: Preorders
Lecture 4 – Chapter 1: Galois Connections
Lecture 5 – Chapter 1: Galois Connections
Lecture 6 – Chapter 1: Computing Adjoints
Lecture 7 – Chapter 1: Logic
Lecture 8 – Chapter 1: The Logic of Subsets
Lecture 9 – Chapter 1: Adjoints and the Logic of Subsets
Lecture 10 – Chapter 1: The Logic of Partitions
Lecture 11 – Chapter 1: The Poset of Partitions
Lecture 12 – Chapter 1: Generative Effects
Lecture 13 – Chapter 1: Pulling Back Partitions
Lecture 14 – Chapter 1: Adjoints, Joins and Meets
Lecture 15 – Chapter 1: Preserving Joins and Meets
Lecture 16 – Chapter 1: The Adjoint Functor Theorem for Posets
Lecture 17 – Chapter 1: The Grand Synthesis

If you want to discuss these things, please visit the Azimuth Forum and register! Use your full real name as your username, with no spaces, and use a real working email address. If you don’t, I won’t be able to register you. Your email address will be kept confidential.

I’m finding this course a great excuse to put my thoughts about category theory into a more organized form, and it’s displaced most of the time I used to spend on Google+. That’s what I wanted: the conversations in the course are more interesting!