What Happens When Projective Geometry Meets Information Theory? (I) - An illustration of a mars drone on the top left. On the bottom left, there is an illustration of a Fano plane (explained later in the essay). On the right, there is a portrait of Claude Elwood Shannon.

To understand what happens when projective geometry meets information theory, we will start with a specific requirement. Imagine that you are in charge of a research mission on mars. You have managed to launch and land your space vehicle successfully on the red planet.

Following this, your drone sets about its mission flying toward Mount Olympus, the tallest mountain on Mars. However, you almost immediately notice something wrong with its trajectory. If it stays on course, it would collide with a rock and damage itself.

What Happens When Projective Geometry Meets Information Theory? (I) — An two-dimensional illustration of a mars drone. It seems to feature 2 rotos on either side. The centre looks like a circular disc with two legs for landing.
Mars Drone — Illustrative art created by the author

To avoid this situation, you need to act fast and send a corrective instruction signal to your Drone on Mars from your mission control on Earth. The challenge is that when you transmit signals over such distances through space, signal noise might corrupt the signal and cause errors.

So, your challenge is to make sure that your Mars drone gets the correct instruction signal at any cost. How would you go about solving this challenge?

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Cosmic Radiation Shields and Information Theory

A picture of Claude Shannon — Image credit: WikiCC

Most of the signal noise for your transmission would be caused by cosmic radiation. You could come up with equipment that essentially physically filters out such radiation. This was how this problem was being solved initially.

That was until the point when a young 21-year-old master’s degree student named Claude Elwood Shannon published his master thesis titled “A Symbolic Analysis of Relay and Switching Circuits.” Shannon then sealed the deal with his 1948 paper, “A Mathematical Theory of Communication.”

This man single-handedly started a new field of science that we called “Information Theory” today. We will go into more details about his work a bit later. For now, let us return to the problem at hand.

Error Detecting Codes

A signal is essentially just “information” that could be transmitted as binary zeroes and ones. Let us say that the following is the corrective signal that you wish to transmit:

10101

If cosmic radiation flips one of your bits (a term that Shannon published first after borrowing it from his colleague, John W. Tukey), it could end up like this:

11101

It might be just one “bit” for you and me, but it might mean “crash and burn” for your Mars drone. In order to avoid this, what if we repeated the original bits in blocks of two? Well, the following is the result:

10101 → 11 00 11 00 11

In this version of the signal, the drone would expect two identical bits in each block of code. As soon as a cosmic ray flips one bit, the code might look as follows:

11 01 11 00 11

The drone would not know what information was essentially transmitted in the second block. But it would know that something has gone wrong. All of a sudden, we have gone from transmitting just “information” to transmitting “error detecting” information.

This is a simple, yet profound transformation. Such is the genius of Claude Shannon. But he did not stop there; he went further into the rabbit hole.

The Magic of Error Correcting Code

To go one step further, Shannon asked the following question:

“What happens if we pack in three identical bits of information into each block?”

Then, the intended transmission signal would look like this:

111 000 111 000 111

If a cosmic ray flips one of the bits, the corrupted signal might be as follows:

111 010 111 000 111

Now, when the drone detects two zeroes and just one ‘one’ in the second block, the drone knows that the probability of it having been three identical zeroes is very high (because of the low cosmic ray frequency).

This is, of course, not fail-proof. But it is a good start in the direction of error-correcting code. But before we celebrate, did you notice that our information is taking up three times the space as it ideally should?

This was another one of Shannon’s genius discoveries: the more robust or error-resistant you want your signal to be, the slower your transmission gets (because of the redundancies introduced).

Shannon mathematically worked out the limit, which he termed the capacity of any given channel. In his research, he even made the profound discovery that one need not even try for a self-correcting signal.

It turns out that even unstructured “random” code exhibits self-correcting properties. So does that mean that we could turn to randomness for a solution? Well, not quite. Shannon’s colleague, Richard Hamming had other plans.

The Hamming Code

Both Claude Shannon and Richard Hamming worked at Bell Labs back in the day (this was THE tech heavyweight of the time). Hamming was a brilliant mathematician. For his research work, Hamming had acquired “weekend-only” low-priority access to Bell’s Model V mechanical relay computer.

The challenge he faced was that during weekends, there was no computer operator doing shifts. Consequently, he had to wait until Monday if any mechanical errors halted the computation. And this bothered Hamming.

He came up with a brilliant solution to scratch his own itch. He developed what we know today as the Hamming code. He essentially encoded each possible three-bit combination as a seven-bit string. This was his code combination:

000 → 0000000

001 → 0010111

010 → 0101011

011 → 0111100

101 → 1011010

110 → 1100110

100 → 1001101

111 → 1110001

So, you would transform your corrective transmission signal using the Hamming code as follows:

10101 → 010 101 → 0101011 1011010

The Hamming code might look arbitrary, but it has the same error correcting property as our “repeat each bit thrice” strategy. Furthermore, the Hamming code’s redundancy-to-information ratio is 2.33 as compared to the 3 from our previous strategy.

What Happens When Projective Geometry Meets Information Theory

In order to understand how the Hamming code achieves this, we shall discuss a very specific projective geometry known as the Fano Plane.

The Fano Plane

Gino Fano was a nineteenth/twentieth century mathematician who was one of the pioneers of the notion of finite geometries. He came up with the notion of a small geometry known as the Fano plane. It looks like so:

What Happens When Projective Geometry Meets Information Theory? — An illustration of the Fano plane. The plane consits of an equilateral triangle with an inscribed circle. The vertices of the triangle and the contact points of the circle and the trinagle generate points. Each vertex is joined by a line segment with the centre of the opposite vertex, such that these are also the contact points between the circle and the triangle.
The Fano Plane — Illustration created by the author

What is so special about this, you ask? Well, this geometry features seven points and seven lines. If you are a typical person, this would be your response to this:

“Hold on a second! Where is the seventh line? I see only six.”

Well, the Fano plane features six lines that look like the typical Euclidean line. The seventh line just happens to be curved. In my essay on How To Actually Make Parallel Lines Intersect, I listed the governing axioms for projective geometry as follows:

1. Every pair of points creates a unique line.

2. Every pair of lines intersects exactly at one point.

The Fano plane does not violate any of these axioms. Hence, as per Fano’s logic, the curved line is indistinguishable from the straight line. I love how Jordon Ellenberg describes this logic:

“If it walks like geometry, and it quacks like geometry, we call it geometry.”

— Jordon Ellenberg.

Projective Geometry Meets Information Theory

All of this is great. But what does the Fano plane have to do with the Hamming code or your Mars drone? Well, it turns out that they are conceptually the same thing.

In order to understand how this is, I am afraid that you will have to wait until I publish the second part of this essay series. The concept that bridges the Fano plane with the Hamming code is so nuanced that it deserves more space than we have in this essay.

Projective Geometry Meets Information Theory – Closing Comments

Before we close, I thought I would go over some interesting applications of error correcting codes. The Hamming code started a revolution of increasingly advanced error correcting technology.

If you have ever used a Compact Disc (CD), the reason that (most) CDs would still play fine after enduring a physical scratch or two was due to an error correcting algorithm known as the Reed-Solomon code.

For those among you who belong to newer generations, your typical flash drive uses the so-called Bose-Chaudhuri-Hockenghem code to battle data corruption. The applications for such technologies range from Mars rovers to banking.

Speaking of mars rovers, whatever is going to happen to your mars drone? As it is heading at rapid pace towards the rock, can you come up with an error-proof transmission on time and save the Martian day?

I apologise to end this essay on such a cliff-hanger, but we will find out in the second instalment of this essay series. Until then, I hope you enjoyed our journey so far!

Update: I have now published the follow-up essay!

Reference and credit: Jordon Ellenberg.

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