Symmetry and the Fourth Dimension (Part 11)

This is my favorite 2d view of a 4-dimensional cube, because it’s the most symmetrical. The 4 dimensions are depicted as going north, east, northeast and northwest.

Puzzle: count the cubes in this picture.

The cubes are a bit distorted, of course: they have to be because we’re squashing 4 dimensions down to 2! But they’re not very distorted.

Here’s the same picture rotated a bit, drawn by Tilman Piesk:

If we like, we can draw a 2d perspective picture of a 4-cube that’s so vivid it really looks 3-dimensional. There are lots of ways to do this. One clever way is to make the fourth dimension point inwards, towards the center:

This was drawn using Robert Webb’s Stella software, perhaps by Webb himself. I like the steel rods and brass balls.

On Wikicommons, someone who goes by the name “Maninthemasterplan” took another approach:

Cross your eyes so the two images overlap. If you get them to lie perfectly on top of each other, they’ll lock in place and you’ll see a 4-cube in 3d. This is called a stereogram.

Personally I don’t find this very helpful for understanding the 4-cube. Crossing my eyes and looking at this 3d image puts me into an altered state of mind which is fun but not good for doing mathematics!

We can also take a 3d perspective picture and set it rotating to make a movie, as Jason Hise did here:

But this is probably worth a discussion all its own! So, next time I’ll talk more about rotations in 4d.

Hinton and the tesseract

The 4-dimensional cube has many names. I’ll call it the 4-cube since this is short and generalizes to higher dimensions. But it’s also called a 4-dimensional hypercube or simply a hypercube—as well as an 8-cell or regular octachoron, for some reason.

When I was growing up, science fiction writers of a certain ilk called the 4-cube a tesseract. That name still persists in some quarters, though not among mathematicians.

The term ‘tesseract’ has an interesting origin. It’s from a rarely used prefix tessara- meaning ‘four’ and the Ancient Greek word ἀκτίς meaning ‘ray’ (as in the element ‘actinium’). So, the term means ‘four rays’, and it comes from the fact that there are 4 edges coming out of each vertex of the tesseract.

But that’s not the interesting part. What’s interesting is that is was coined in 1888 by Charles Howard Hinton, in a book called A New Era of Thought, which espoused some curious ideas about the spiritual benefits of visualizing 4-dimensional geometry. In his later book The Fourth Dimension, he developed a system of colored cubes to help people do this.

Rumors subsequently arose that these cubes had driven more than one person insane. The only substance to this that I can find is this letter appearing in Martin Gardner’s column after he wrote about Hinton:

Dear Mr. Gardner:

A shudder ran down my spine when I read your reference to Hinton’s cubes. I nearly got hooked on them myself in the nineteen-twenties. Please believe me when I say that they are completely mind-destroying. The only person I ever met who had worked with them seriously was Francis Sedlak, a Czech neo-Hegelian Philosopher (he wrote a book called The Creation of Heaven and Earth) who lived in an Oneida-like community near Stroud, in Gloucestershire.

As you must know, the technique consists essentially in the sequential visualizing of the adjoint internal faces of the poly-colored unit cubes making up the larger cube. It is not difficult to acquire considerable facility in this, but the process is one of autohypnosis and, after a while, the sequences begin to parade themselves through one’s mind of their own accord. This is pleasurable, in a way, and it was not until I went to see Sedlak in 1929 that I realized the dangers of setting up an autonomous process in one’s own brain. For the record, the way out is to establish consciously a countersystem differing from the first in that the core cube shows different colored faces, but withdrawal is slow and I wouldn’t recommend anyone to play around with the cubes at all.

Hiram Barton

Hinton also wrote science fiction stories, which he called Scientific Romances. I haven’t read them, though they’re available online.

In 1880 Hinton married Mary Ellen Boole, daughter of the famous logician. Her sister Alicia Boole Stott became an expert on 4-dimensional geometry, and we will meet her later in this series! But Hinton also married another woman, perhaps because his father was advocated of polygamy. He was convicted of bigamy and spent 3 days in prison for it. Not very long, but he also lost his job.

He later sailed to America, joined the mathematics faculty at Princeton, and developed a gunpowder-powered baseball pitching machine for the team there! If you don’t believe me, read this old article:

• The new baseball-pitching device: Prof. Hinton of Princeon explains his invention to the alumni, New York Times, 12 March 1897.

It begins:

Prof. Charles Howard Hinton, who holds the chair of mathematics at Princeton University, explained last night to members of the Princeton Club of this city the baseball gun which he has invented. About 300 Princeton men were present and manifested great interest in the operation of the machine.

It was actually able to throw curve-balls!

I’ve read claims that Hinton later resigned from Princeton due to injuries caused by this device. However, he re-introduced the machine at the University of Minnesota, where he worked as an assistant professor until 1900. He then resigned to move to the U.S. Naval Observatory in Washington, D.C.. (Why? More injured baseball players?) Near the end of his life, he worked as an examiner of chemical patents for the United States Patent Office. He died of a cerebral hemorrhage in 1907, at the age of 54.

For more on Hinton and his cubes see:

• Mark Blacklock, Cubic thought, The Fairyland of Geometry, 10 December 2009.

• Hinton’s cubes redux, Banubula, 9 November 2006.

For a guide to making the cubes, see:

• Hinton’s cubes.

Apparently these are based on color plates from Hinton’s book The Fourth Dimension. This book is reproduced in black and white here:

• Charles Howard Hinton, The Fourth Dimension, 1912 edition, Internet Archive.

but a modern edition including the color plates is freely available here:

• Charles Howard Hinton, The Fourth Dimension, to which is added The Language of Space, Celephaïs Press, 2004.

Symmetry and the Fourth Dimension (Part 10)

Some people say it’s impossible to visualize 4-dimensional things. But lots of people I know can do it.

How do they do it?

Well, how do you visualize 3-dimensional things? A computer screen is just 2d, but we can draw a 3d object on it by picking some diagonal direction—southeast in the picture below—to stand for ‘forwards, towards our eyes’. Similarly, we can draw a 4d object by picking another diagonal direction—northeast in this picture—to stand for ‘forwards in the 4th dimension’.

Here we are using this trick to draw 0d, 1d, 2d, 3d and 4d cubes. The first dimension, often called the x direction, points along the red arrow. The second, called the y direction, points along the green arrow. The third, the z direction, points diagonally along the blue arrow. And the fourth, sometimes called the w direction, points diagonally along the yellow arrow.

There’s nothing sacred about these names or these particular directions; we can implement this idea in lots of different ways. It’s ‘cheating’, but that’s okay. A vague, misty image can be a lot better than no image at all.

Of course, we need to think about math to keep straight which lines in our picture point in the w direction. But that’s okay too. A mixture of equations and visualization lets mathematicians and physicists make faster progress in understanding the 4th dimension than if they used only equations.

Physicists need to understand the 4th dimension because we live in 4d spacetime. Some mathematicians study much higher-dimensional spaces, even infinite-dimensional ones, and here visualization becomes more subtle, and perhaps more limited. But many mathematicians working on 4d topology take visualization very seriously. If you’ve ever seen the elaborate, detailed gestures they make when describing 4-dimensional constructions, you’ll know what I mean.

Visualizing shapes in 4 dimensions takes practice, but it’s lots of fun! As this series of posts continues, I want to give you some practice while talking about nice 4-dimensional shapes: the 4d relatives of the Platonic and Archimedean solids. In the series so far, we’ve spent a lot of time warming up by studying the 3d Platonic and Archimedean solids and developing a technique for classifying them, called Coxeter diagrams. All that will pay off soon!

Here’s a great series of videos that explains higher dimensions:

• Jos Leys, Étienne Ghys and Aurélien Alvarez, Dimensions.

The only problem is that it’s tough to navigate them. Click on your favorite language and you’ll see part 1 of the series. After you start playing it you’ll see an arrow at the lower right of the video that lets you jump to the next one. This is good if, like me, you’re impatient for the 4th dimension! That starts in part 3.

There’s a guide to all nine parts here:

• Jos Leys, Étienne Ghys and Aurélien Alvarez, Tour.

but you can’t get to the videos from there! They need a bit of help from a good website designer.

The picture above is a shot of the glorious 120-cell… one of the six Platonic solids in 4 dimensions. But more on that later! We’ll start with a simpler one: the 4-cube.

Symmetry and the Fourth Dimension (Part 9)

 

Last time in this series we completed the first phase of our quest: we got lots of polyhedra from Coxeter diagrams in a systematic way. But before we sail off into new seas, let’s review what we’ve done.

To spice things up, I’ll explain everything in a different way than before. If you don’t see how this new way relates to the old one, please ask! There’s a lot to say about this stuff that I’m not saying, so there are plenty of gaps left to fill.

And if you’re wondering what that thing is up there, read on.

Coxeter theory

Harold Scott MacDonald Coxeter, the ‘king of geometry’, developed a beautiful theory relating groups to highly symmetrical polyhedra and their higher-dimensional generalizations, called ‘polytopes’.

His book Regular Polytopes is required reading for anyone who wants to learn about these things. There’s a copy sitting by my bed, and there should be one by yours, too. It took him 24 years to write.

In his honor, finite groups generated by reflections in n-dimensional space are called Coxeter groups. They’re described by diagrams with n dots, called Coxeter diagrams. We’ve been looking at the 3d case, where there happen to be three such diagrams, with mysterious-sounding names:

A₃      o—3—o—3—o

B₃      o—3—o—4—o

H₃      o—3—o—5—o

I want to show you to use these diagrams. So let’s do an example, namely A₃:

o—3—o—3—o

This diagram is telling us to draw a bunch of triangles on a sphere, like this:

This shape is called a Coxeter complex. As you can see, it contains lots of great circles. For each one we get a symmetry: the reflection across that circle, as if a mirror cut our Coxeter complex in half through that circle. So, the symmetries of the Coxeter complex form a finite group, generated by reflections. This is a Coxeter group!

But how do we get the Coxeter complex from the diagram? The dots in the diagram are secretly called V, E, and F:

V—3—E—3—F

Among other things, these letters are names for the edges of our favorite triangle in the Coxeter complex:

It doesn’t matter which is our favorite triangle, so I just picked one. Each edge of this triangle gives a symmetry: the reflection that reflects the Coxeter complex across that edge! We call these symmetries V, E and F.

The 3 on the line going between V and E in this diagram:

V—3—E—3—F

is a quick way of saying that

(VE)³ = 1

In other words, doing the reflections V, E, V, E, V, E gets us back where we started.

To see this, let’s call our favorite triangle 1. Reflecting this triangle across the great circle containing the edge V, we get a new triangle, which we’ll call V. Reflecting that across the great circle containing the edge E, we get a new triangle, which we’ll call VE. And so on:

By the time we get to VEVEVE = (VE)³, we’re back where we started! That’s a total of 6 reflections, so the V and E edges of each triangle must meet at a 60° angle.

Similarly, the 3 on the line going between E and V says that

(EF)³ = 1

so doing the reflections E, F, E, F, E, F also gets us back where we started:

On the other hand, there’s no line connecting the dots V and F in our Coxeter diagram:

V—3—E—3—F

But this is an abbreviation for a line with a 2 on it! You see, lines with 2 on them occur so often that Coxeter decided to save time by not drawing such lines. So, we have

(VF)² = 1

and doing the reflections V, F, V, F also gets us back where started:

That’s a total of 4 reflections, so the V and F edges of each triangle must meet at a 90° angle.

So, I’ve sketched how the Coxeter group and the Coxeter complex arise from the Coxeter diagram. To be a bit more precise, the Coxeter group A₃ has generators V, E, F obeying relations

(VE)³ = (EF)³ = (VF)² = 1

and also

V² = E² = F² = 1

since V, E, and F are reflections. If we draw a sphere with one great circle serving as the ‘mirror’ for each reflection in the Coxeter group, we get the Coxeter complex.

What makes Coxeter complexes special, compared to other ways of tiling a sphere with triangles? One thing is that they have exactly as many symmetries as triangles. If you pick any triangle and call it your favorite, there’s exactly one symmetry—that is, rotation and/or reflection—sending this triangle to any other.

So, the Coxeter complex is actually a picture of its own symmetry group!

The particular Coxeter complex we’ve been looking at has 24 triangles. So, its symmetry group has 24 elements. This group is called the tetrahedral finite reflection group, or A₃ for short. There’s a lot to say about it, but not now! There’s another more urgent question.

Polyhedra

How do we get polyhedra from our Coxeter complex?

The easiest one works like this. We take the Coxeter complex and create a polyhedron with a corner in the middle of each triangle, connecting two corners with an edge whenever their triangles touch:

The polyhedron looks better if we view it from a slightly different angle, and let a skilled artist like Tom Ruen do the drawing:

This polyhedron is called the Poincaré dual of the Coxeter complex.

There’s a notation for this particular polyhedron:

•—3—•—3—•

All the dots are black, because this is the fanciest, most interesting polyhedron that comes from our Coxeter complex. We get other polyhedra by blackening just some of the dots.

Earlier in this series, I described how to build these other polyhedra using the concept of ‘flag’. But there’s another way, using the Coxeter complex, which I’ll sketch here.

Suppose we blacken just some of the dots in our Coxeter diagram. Then our favorite triangle belongs to a bunch of triangles, all related by reflections corresponding to the dots we left white. And indeed, all the triangles can be grouped into bunches that all look the same… and there’s a polyhedron that has a corner in the middle of each bunch of triangles.

For example, suppose we leave the E dot white:

•—3—o—3—•

Then our favorite triangle belongs to a bunch of triangles—in this case, just a pair!—that are related by reflections along the E edge of our favorite triangle. If we group our triangles into pairs like this, we get a polyhedron with a corner in the middle of each pair:

Again, it looks better if we let Tom Ruen draw it:

I should do more examples, but I think I’ll wrap up by describing the procedure in more highbrow language. Skip the next paragraph if you don’t know group theory, and move on to the complete list of examples!

If we call the Coxeter group G, blackening all the dots of the Coxeter diagram gives a polyhedron with one corner for each element of G. But if we don’t blacken all of them, the reflections corresponding to the white dots generate a subgroup B of G. Then we get a polyhedron with one corner for each element of G/B. Last time I said each way of blackening some dots describes some sort of ‘flag’. In these terms, B is the subgroup fixing your favorite flag of this sort, and G/B is the set of all flags of this sort. Each of those flags corresponds to what I’m calling a ‘bunch of triangles’ in my current story.

But now let’s see what we get from all this! We get three families of polyhedra, and these are almost all the ‘Archimedean solids’ in 3 dimensions.

The A₃ family:     o—3—o—3—o

This family of polyhedra can all be gotten from the tetrahedron by chopping off vertices, edges or faces. They’re associated to the Coxeter complex we’ve just been looking at:

It’s built from triangles whose interior angles are and . These numbers come from taking π and dividing it by the numbers on the edges of the Coxeter diagram: 3, 3, and the invisible edge labelled 2.

As mentioned, this Coxeter complex has 24 triangles, and its symmetry group, with 24 elements, is called the tetrahedral finite reflection group, or A₃.

Here are all the polyhedra in this family. The list has some repeats, because this Coxeter diagram is its own mirror image!

tetrahedron		•—3—o—3—o
truncated tetrahedron		•—3—•—3—o
octahedron		o—3—•—3—o
truncated tetrahedron		o—3—•—3—•
tetrahedron		o—3—o—3—•
cuboctahedron		•—3—o—3—•
truncated octahedron		•—3—•—3—•

The B₃ family:     o—3—o—4—o

This family of polyhedra can all be gotten by taking the cube or the octahedron and chopping off vertices, edges or faces. They come from the Coxeter complex whose triangles have interior angles and :

This Coxeter complex has 48 triangles, and its symmetry group is a Coxeter group with 48 elements, called the octahedral finite reflection group, or B₃.

Here are all the polyhedra in this family. We’ve seen some of these already in the A₃ family, and there’s a reason for that: the group B₃ happens to contain A₃ as a subgroup! It’s twice as big.

cube		•—4—o—3—o
truncated cube		•—4—•—3—o
cuboctahedron		o—4—•—3—o
truncated octahedron		o—4—•—3—•
octahedron		o—4—o—3—•
rhombicuboctahedron		•—4—o—3—•
truncated cuboctahedron		•—4—•—3—•

The H₃ family:     o—3—o—5—o

This family of polyhedra can all be gotten by taking the dodecahedron or icosahedron and chopping off vertices, edges and faces. They come from the Coxeter complex built from triangles whose interior angles are and :

This Coxeter complex has 120 triangles, and its symmetry group is a Coxeter group with 120 elements, called the icosahedral finite reflection group, or H₃.

Here are all the polyhedra coming from this Coxeter complex. These are my favorites, because they look the most fancy, and I have a fondness for the quirky charm of 5-fold symmetry:

dodecahedron		•—5—o—3—o
truncated dodecahedron		•—5—•—3—o
icosidodecahedron		o—5—•—3—o
truncated icosahedron		o—5—•—3—•
icosahedron		o—5—o—3—•
rhombicosidodecahedron		•—5—o—3—•
truncated icosidodecahedron		•—5—•—3—•

Afterword

The picture at the start of this post was taken by my friend Allen Knutson:

It’s the H₃ Coxeter complex, and for some reason it’s in the music library at Cornell University, where Allen teaches math.

The picture of H. S. M. Coxeter is from the cover of a book about him:

• Siobhan Roberts, King of Infinite Space, Walker & Company, 2006.

As usual, the pretty pictures of solids with brass balls at the vertices were made by Tom Ruen using Robert Webb’s Stella software. Tom Ruen also drew the Coxeter complexes; I’m to blame for the crude extra stuff added on.

You can see the previous episodes of this series here:

• Part 1: Platonic solids and Coxeter complexes.

• Part 2: Coxeter groups.

• Part 3: Coxeter diagrams.

• Part 4: duals of Platonic solids.

• Part 5: Interpolating between a Platonic solid and its dual, and how to describe this using Coxeter diagrams. Example: the cube/octahedron family.

• Part 6: Interpolating between a Platonic solid and its dual. Example: the dodecahedron/icosahedron family.

• Part 7: Interpolating between a Platonic solid and its dual. Example: the tetrahedron family.

• Part 8: The missing solids, coming from Coxeter diagrams with both ends blackened.

42

In The Hitchhiker’s Guide to the Galaxy by Douglas Adams, the number 42 is the “Answer to the Ultimate Question of Life, the Universe, and Everything”. But he didn’t say what the question was!

Since today is Towel Day, let me reveal that now.

If you try to get several regular polygons to meet snugly at a point in the plane, what’s the most sides any of the polygons can have? The answer is 42.

The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point. If you do the math, you’ll see the reason this works is that

There are actually 10 solutions of

with and each of them gives a way for three regular polygons to snugly meet at a point. But this particular solution features the biggest number possible!

But why is this so important? Well, it turns out that if you look for natural numbers that make

as close to 1 as possible, while still less than 1, the very best you can do is It comes within of equalling 1, since

And why is this important? Well, suppose you’re trying to make a doughnut with at least two holes that has the maximum number of symmetries. More precisely, suppose you’re trying to make a Riemann surface with genus that has the maximum number of symmetries. Then you need to find a highly symmetrical tiling of the hyperbolic plane by triangles whose interior angles are and and you need

for these triangles to fit on the hyperbolic plane.

For example, if you take you get this tiling:

A clever trick then lets you curl up the hyperbolic plane and get a Riemann surface with at most

symmetries.

So, to get as many symmetries as possible, you want to make as small as possible! And thanks to what I said, the best you can do is

So, your surface can have at most

symmetries. This is called Hurwitz’s automorphism theorem. The number 84 looks mysterious when you first see it — but it’s there because it’s twice 42.

In particular, the famous mathematician Felix Klein studied the most symmetrical doughnut with 3 holes. It’s a really amazing thing, called Klein’s quartic curve:

It has

symmetries. That number also looks mysterious when you first see it. Of course it’s the number of hours in a week, but the real reason it’s there is because it’s four times 42.

If you carefully count the triangles in the picture above, you’ll get 56. However, these triangles are equilateral, or at least they would be if we could embed Klein’s quartic curve in 3d space without distorting it. If we drew all the smaller triangles whose interior angles are and each equilateral triangle would get subdivided into 6 smaller triangles, and there would be a total of triangles. But of course

Half of these smaller triangles would be ‘left-handed’ and half would be ‘right-handed’, and there’d be a symmetry sending a chosen triangle to any other triangle of the same handedness, for a total of

symmetries (that is, conformal transformations, not counting reflections).

But why is this stuff the answer to the ultimate question of life, the universe, and everything? I’m not sure, but I have a crazy theory. Maybe all matter and forces are made of tiny little strings! As they move around, they trace out Riemann surfaces in spacetime. And when these surfaces are as symmetrical as possible, reaching the limit set by Hurwitz’s automorphism theorem, the size of their symmetry group is a multiple of 42, thanks to the math I just described.

Puzzles

Puzzle 1. Consider solutions of

with positive integers and show that the largest possible value of is

Puzzle 2. Consider solutions of

with positive integers and show that the largest possible value of is

Acknowledgments and references

For more details, see my page on Klein’s quartic curve, and especially the section on incredibly symmetrical surfaces. Sums of reciprocals of natural numbers are called ‘Egyptian fractions’, and they have deep connections to geometry; for more on this see my article on Archimedean tilings and Egyptian fractions.

The picture of a triangle, heptagon and 42-gon (also known as a tetracontakaidigon) was made by Tyler, and you can see all 17 ways to get 3 regular polygons to meet snugly at a vertex on Wikipedia. Of these, only 11 can occur in a uniform tiling of the plane. The triangle, heptagon and 42-gon do not tile the plane, but you can see some charming attempts to do something with them on Kevin Jardine’s website Imperfect Congruence:

The picture of Klein’s quartic curve was made by Greg Egan, and you should also check out his page on Klein’s quartic curve.

---

“Good Morning,” said Deep Thought at last.
“Er…good morning, O Deep Thought” said Loonquawl nervously, “do you have…er, that is…”
“An Answer for you?” interrupted Deep Thought majestically. “Yes, I have.”
The two men shivered with expectancy. Their waiting had not been in vain.
“There really is one?” breathed Phouchg.
“There really is one,” confirmed Deep Thought.
“To Everything? To the great Question of Life, the Universe and everything?”
“Yes.”
Both of the men had been trained for this moment, their lives had been a preparation for it, they had been selected at birth as those who would witness the answer, but even so they found themselves gasping and squirming like excited children.
“And you’re ready to give it to us?” urged Loonsuawl.
“I am.”
“Now?”
“Now,” said Deep Thought.
They both licked their dry lips.
“Though I don’t think,” added Deep Thought. “that you’re going to like it.”
“Doesn’t matter!” said Phouchg. “We must know it! Now!”
“Now?” inquired Deep Thought.
“Yes! Now…”
“All right,” said the computer, and settled into silence again. The two men fidgeted. The tension was unbearable.
“You’re really not going to like it,” observed Deep Thought.
“Tell us!”
“All right,” said Deep Thought. “The Answer to the Great Question…”
“Yes…!”
“Of Life, the Universe and Everything…” said Deep Thought.
“Yes…!”
“Is…” said Deep Thought, and paused.
“Yes…!”
“Is…”
“Yes…!!!…?”
“Forty-two,” said Deep Thought, with infinite majesty and calm. – Douglas Adams

Category Theory for Scientists

 

At last—a textbook on category theory for scientists! And it’s free!

• David Spivak, Category Theory for Scientists.

It’s based on a course the author taught:

This course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I actually mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth.

I believe that the language and toolset of category theory can be useful throughout science. We build scientific understanding by developing models, and category theory is the study of basic conceptual building blocks and how they cleanly fit together to make such models. Certain structures and conceptual frameworks show up again and again in our understanding of reality. No one would dispute that vector spaces are ubiquitous. But so are hierarchies, symmetries, actions of agents on objects, data models, global behavior emerging as the aggregate of local behavior, self-similarity, and the effect of methodological context.

Some ideas are so common that our use of them goes virtually undetected, such as set-theoretic intersections. For example, when we speak of a material that is both lightweight and ductile, we are intersecting two sets. But what is the use of even mentioning this set-theoretic fact? The answer is that when we formalize our ideas, our understanding is almost always clarified. Our ability to communicate with others is enhanced, and the possibility for developing new insights expands. And if we are ever to get to the point that we can input our ideas into computers, we will need to be able to formalize these ideas first.

It is my hope that this course will offer scientists a new vocabulary in which to think and communicate, and a new pipeline to the vast array of theorems that exist and are considered immensely powerful within mathematics. These theorems have not made their way out into the world of science, but they are directly applicable there. Hierarchies are partial orders, symmetries are group elements, data models are categories, agent actions are monoid actions, local-to-global principles are sheaves, self-similarity is modeled by operads, context can be modeled by monads.

He asks readers from different subjects for help in finding new ways to apply category theory to those subjects. And that’s the right attitude to take when reading this book. I’ve found categories immensely valuable in my work. But it took effort to learn category theory and see how it can apply to different subjects. People are just starting to figure out these things, so don’t expect instant solutions to the problems in your own favorite field.

But Spivak does the best job I’ve seen so far at explaining category theory as a general-purpose tool for thinking clearly. Since I’m busy using category theory to clarify the relationships between fields like chemistry, population biology, electrical engineering and control theory, this subject is very much on my mind.

Graph Laplacians

There’s been some new progress on graph Laplacians!

As a mathematical physicist, I’ve always been in love with the Laplacian:

   

It shows up in many of the most fundamental equations of physics: the wave equation, the heat equation, Schrödinger’s equation… and Poisson’s equation:

which says how the density of matter, affects the gravitational potential

As I’ve grown interested in network theory, I’ve gotten more and more interested in ‘graph Laplacians’. These are discretized versions of the Laplacian, where we replace 3-dimensional space by a ‘graph’, meaning something like this:

You can get a lot of interesting information about a graph from its Laplacian. You can also set up discretized versions of all the famous equations I mentioned.

The new progress is a simple algorithm for very efficiently solving Poisson’s equation for graph Laplacians:

• Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, Zeyuan Allen Zhu, A simple, combinatorial algorithm for solving SDD systems in nearly-linear time.

Here’s a very clear explanation of the general idea, conveying some sense of why it’s so important, without any nasty equations:

• Larry Hardesty, Short algorithm, long-range consequences, MIT News, 1 March 2013.

It begins:

In the last decade, theoretical computer science has seen remarkable progress on the problem of solving graph Laplacians — the esoteric name for a calculation with hordes of familiar applications in scheduling, image processing, online product recommendation, network analysis, and scientific computing, to name just a few. Only in 2004 did researchers first propose an algorithm that solved graph Laplacians in “nearly linear time,” meaning that the algorithm’s running time didn’t increase exponentially with the size of the problem.

At this year’s ACM Symposium on the Theory of Computing, MIT researchers will present a new algorithm for solving graph Laplacians that is not only faster than its predecessors, but also drastically simpler.

This animation shows two different “spanning trees” for a simple graph, a grid like those used in much scientific computing. The speedups promised by a new MIT algorithm require “low-stretch” spanning trees (green), in which the paths between neighboring nodes don’t become excessively long (red).

I can’t beat this article at its own game… except to clarify that ‘solving graph Laplacians’ means solving Poisson’s equation with a graph Laplacian replacing the usual Laplacian.

So, let me just supplement this article with the nasty equations saying what a graph Laplacian actually is. Start with a graph. More precisely, start with a simple graph. Such a graph has a set of vertices and a set of edges such that

which says the edges are undirected, and

which says there are no loops.

The graph Laplacian is an operator that takes a function on the vertices of our graph,

and gives a new such function as follows:

The version of Poisson’s equation for this graph Laplacian is thus

But I should warn you: this operator has eigenvalues that are less than equal to zero, like the usual Laplacian People often insert a minus sign to make the eigenvalues ≥ 0.

There is a huge amount to say about graph Laplacians! If you want, you can learn more here:

• Michael William Newman, The Laplacian Spectrum of Graphs, Masters Thesis, Department of Mathematics, University of Manitoba, 2000.

I’ve been learning about some of their applications here:

• Ernesto Estrada, The Structure of Complex Networks: Theory and Applications, Oxford University Press, Oxford, 2011.

I hope sometime to summarize a bit of this material and push the math forward a bit. So, it was nice to see new progress on efficient algorithms for doing computations with graph Laplacians!

Quantum Techniques for Studying Equilibrium in Reaction Networks

 

The summer before last, I invited Brendan Fong to Singapore to work with me on my new ‘network theory’ project. He quickly came up with a nice new proof of a result about mathematical chemistry. We blogged about it, and I added it to my book, but then he became a grad student at Oxford and got distracted by other kinds of networks—namely, Bayesian networks.

So, we’ve just now finally written up this result as a self-contained paper:

• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks.

Check it out and let us know if you spot mistakes or stuff that’s not clear!

The idea, in brief, is to use math from quantum field theory to give a somewhat new proof of the Anderson–Craciun–Kurtz theorem.

This remarkable result says that in many cases, we can start with an equilibrium solution of the ‘rate equation’ which describes the behavior of chemical reactions in a deterministic way in the limit of a large numbers of molecules, and get an equilibrium solution of the ‘master equation’ which describes chemical reactions probabilistically for any number of molecules.

The trick, in our approach, is to start with a chemical reaction network, which is something like this:

and use it to write down a Hamiltonian describing the time evolution of the probability that you have various numbers of each kind of molecule: A, B, C, D, E, … Using ideas from quantum mechanics, we can write this Hamiltonian in terms of annihilation and creation operators—even though our problem involves probability theory, not quantum mechanics! Then we can write down the equilibrium solution as a ‘coherent state’. In quantum mechanics, that’s a quantum state that approximates a classical one as well as possible.

All this is part of a larger plan to take tricks from quantum mechanics and apply them to ‘stochastic mechanics’, simply by working with real numbers representing probabilities instead of complex numbers representing amplitudes!

I should add that Brendan’s work on Bayesian networks is also very cool, and I plan to talk about it here and even work it into the grand network theory project I have in mind. But this may take quite a long time, so for now you should read his paper:

• Brendan Fong, Causal theories: a categorical perspective on Bayesian networks.