Symmetry for Dummies: Noether’s Theorem

Einstein referred to her as the most important woman in the history of mathematics. Her theorem has been recognized as “one of the most important mathematical theorems ever proved in guiding the development of modern physics.” Yet many people haven’t the slightest clue of who this woman was, or what she did that was so significant to our understanding of how our world works. If you count yourself as one of those who have never heard of Emmy Noether and wish to enlighten yourself, please read on. I can only hope I do her memory justice. Not just by telling you who she was, but by also giving you an understanding of how her insight led to the coming together of symmetry and quantum theory, pointing academia’s arrow toward quantum electrodynamics.

Emmy Noether

Being a female in Germany in the late 1800s was not easy. She wasn’t allowed to register for math classes. Fortunately, her father happened to be a math professor, which allowed her to sit in on many of his classes. She took one of his final exams in 1904 and did so well that she was granted a bachelors degree. This allowed her to “officially” register in a math graduate program. Three years later, she earned one of the first PhD’s given to a woman in Germany. She was just 25 years old.

1907 was a very exciting time in theoretical physics, as scientists were hot on the heels of figuring out how light and atoms interact with each other. Emmy wanted in on the fun, but being a woman made this difficult. She wasn’t allowed to hold a teaching position, so she worked as an unpaid assistant, surviving on a small inheritance and under-the-table money that she earned sitting in for male professors when they were unable to teach. She was still able to do what professors are supposed to do, however – write papers. In 1916, she would pen the theorem that would have her rubbing shoulders with the other physics and mathematical giants of the era.

Noether’s Theorem – The Basics

Emmy Noether’s Theorem seems simple on the onset, but holds a fundamental truth that explains the fabric of our reality. It goes something like this:

For every symmetry, there is a corresponding conservation law.

We all have heard of laws such as Newton’s first law of motion, which is about the conservation of momentum. And the first law of thermodynamics, which is about the conservation of energy. Noether’s theorem tells us that there must be some type of symmetry that is related to these conservation laws. Before we get into the meaning, we must first understand a little known subject called The Principle of Least Action.

The Universe is Lazy

N_02I would wager a few Raspberry Pi Zeros that many of you already have an intuitive grasp of this principle, even if you’ve never heard of it before now. The principle of least action basically says that the universe has figured out the easiest way possible to get something done. Mathematically, it’s the sum over time of kinetic energy minus potential energy as the action occurs. Let us imagine that you’re trying to program an STM32 Discovery eval board in GCC. After about the 6,000th try, you toss the POS across the room and grab your trusty Uno. The graph depicts the STM32 moving through time and space.


The green points represent particular points of how how high the STM32 is at a given point in time. Note that there are no values for height and time – this example is meant to explain a principle. We can say that at these points (and all points along the curve), the SMT32 has both kinetic and potential energies. Let us call the kinetic energy (kt) and the potential energy (pt). The ‘t‘ subscript is for time, as both the energies are functions of time. The action for each point will be called s, and can be calculated as:

s = k_t-p_t

However, action is the total sum of the difference of energies at each point between t1 and t2. If you’ve read my integral post, you will know that we need to integrate in order to calculate the total action.

S = \int_{t_1}^{t_2} (k - p) dt

Now before you get your jumper wires in a bunch, all that is saying is that we’re taking the difference in potential (p) and kinetic (k) energies at each point along the curve between t1 and t2, and we’re adding them together. The elongated S symbol means a sum, and the (dt) means as it changes over time. The path that the STM32 will take will be the path where the action S is at its minimum value. Check out the video in the source section below if you’re confused. It’s only 10 minutes and goes into this concept in easy to follow details.

Noether’s Theorem – The Details

N_03Noether’s theorem is based upon a mathematical proof. It’s not a theory. Her proof can be applied to physics to develop theories, however. Now that we know what the principle of least action is, we can do just this.

Any law of nature can be traced back to a symmetry and the least action principle. Let’s consider two very simple examples – Newton’s first law of motion and the first law of thermodynamics.

Conservation of Momentum

Space has what is known as translational symmetry. That’s just fancy-pants talk for saying that what you do in one point in space is the same as what you do in another point in space. It doesn’t matter what hacker space you throw your STM32, it will act the same at all hacker spaces on earth. Space itself provides the symmetry. And because the principle of least action applies, you have a natural law – the first law of motion.

Conservation of Energy

Time has the same translational symmetry as space does. If I toss the STM32 now, and toss it tomorrow, it will act the same. It doesn’t matter what point in time I toss it, the results will always be the same. Thus energy is conserved between different points in time. Time is our symmetry, and the 1st law of thermodynamics is the result.

Now, I realize these examples might seem a bit useless. But when you dig a bit deeper, things get interesting. Electrical charge is also conserved. Noether says there must then be some type of symmetry involved. What do you suppose that symmetry might be? Keep following that rabbit hole, and you’ll end up face to face with QED. We’ll get there in a future article, so for now just keep Noether’s Theorem in mind.


Physics Helps, The principle of least action, video link.

Ransom Stephens, Ph.D., Emmy Noether and The Fabric of Reality, video link

The Dark Arts: Hacking Humans

One of the biggest challenges for a company that holds invaluable data is protecting it. At first, this task would seem fairly straightforward. Keep the data on an encrypted server that’s only accessible via the internal network. The physical security of the server can be done with locks and other various degrees of physical security. One has to be thoughtful in how the security is structured, however. You need to allow authorized humans access to the data in order for the company to function, and there’s the rub. The skilled hacker is keenly aware of these people, and will use techniques under the envelope of Social Engineering along with her technical skills to gain access to your data.

Want to know how secure your house is? Lock yourself out. One of the best ways to test security is to try and break in. Large companies routinely hire hackers, known as penetration testers, to do just this. In this article, we’re going to dissect how a hired penetration tester was able to access data so valuable that it could have destroyed the company it belonged to.

Information Gathering


The start of any hack involves information gathering. This is usually pretty easy for larger companies. Their website along with a few phone calls can reveal quite a bit of useful information. However, you can be assured that any company who has hired a pen tester has taken the necessary precautions to limit such information.

And such was the case for our hacker trying to gain access to the ACME Corp. servers. Her first target was the dumpsters – dumpster dives have been proven to unearth a trove of valuable information in the past. But the dumpsters were inside the complex, which was guarded by a contracted security firm. Through a bit of website snooping and a few phone calls, she was able to find out the department that was in charge of trash removal for the company. She then placed a phone call to this department. Using a social engineering (SE) technique known as pretexting, she pretended to be with a trash removal company and wanted to submit a quote to service their business. Using another SE technique called elicitation, she was able to find out:

  • that trash collection took place on Wednesdays and Thursdays
  • the total number of dumpsters
  • that there was a special dumpster for paper and technology trash
  • the name of the current waste removal company – Waster’s Management
  • the name of the employee in charge of the waste removal – [Christie Smith]

Dumpster Dive

Armed with this information, she went to the Waster’s Management website and grabbed their JPEG logo. se_01Within a few days, she had a shirt and hat with the logo in her hands. She called the security department and said she was with Waster’s Management, and that [Christie Smith] had told her one of the dumpsters was damaged, and she needed to take a look at it before the next trash removal.

The next day, wearing the shirt and hat she had ordered online, she was given a badge from security and allowed access to the dumpsters. Now, any hacker worth her weight in PIC16F84’s already knows what dumpster she dove into. It didn’t take her long to walk away with several hard drives, a few USB drives and some useful documents. She was able to gain knowledge of an upcoming IT contract work, the name of the CFO, and the name of a server with some level of importance – prod23.

Hacking the Server

With some more SE, she was able to find out when the IT work was scheduled. It was after hours. She showed up a bit late and was able to walk right through the front door by claiming she worked for the IT contract company. She then shifted roles and pretended to be an employee. She approached one the real IT contract guys, and said she worked for the CFO, [Mr. Shiraz], and asked if he knew to be careful with the prod23 server. With more SE, she was able to find out the prod23 server was off-limits, encrypted, and only accessible by specific admins.


She was able to access an admin office, and it was there she would don her black hat. She booted the computer with BackTrack via USB and installed a key logger. She made an SSH tunnel to her personal server where she could dump the contents of the key logger, along with some other shells. Now, this is where things get interesting. She opened Virtual Box and used the computer’s hard drive as the boot medium. The VM booted the OS, and she hid all of the screen decorations to make it look like the target OS was running. The admin would log in without a clue, and our hacker would get their username and password through the key logger.

Once the login information came in, she was able to access the admin’s computer, and from there the prod23 server. You can imagine the look on the faces of the top executives for ACME Corp when our hacker handed them a copy of the keys to their kingdom.

Social engineering is human hacking, and a dark art in itself. Our hacker in this story would have never been able to even get close to the server if she did not have SE skills. No matter how secure you make something, so long as you allow humans access to it, it’s vulnerable to attack. And then it’s down to how well-trained your people are in repelling these kinds of intrusions.Just ask Target.

You can find the full story in the source below.


Social Engineering, The Art of Human Hacking, Chapter 8, by Christopher Hadnagy, ISBN-13: 860-1300286532

Adventures of ArduinoMan – The Rudis

Rudis – A small wooden sword given to a Gladiator as proof of his achieved freedom. It signifies his ascent from being a slave to becoming a free man.

One thing is certain – anything that runs on electricity can be connected to the internet. The only obstacle is cost. And as costs come down, the reality of The Internet of Things will be upon us. Everything from cars to curling irons will be connected to the Internet. With this newly connected world will come a new breed of hacker. The Black Hats will move out from behind their keyboards and spill into the streets, only to be met by the White Hats as they battle for control over our endlessly connected world.

And such was the case on the morning of October 16, 2029. The air was cool and breezy when Randall C. Tubbs, a senior police officer at the Bronx 49th precinct, received a call over his radio to check out a tripped alarm at a nearby cell tower. Barely a minute had passed by when he pulled in to the Tower Road cul-de-sac on the day our story begins. The cell tower dominated the horizon, and was silhouetted against a cloudless blue sky. The trees of the forest surrounding the area were just starting to show their colors, with the yellow oak leaves being most vibrant. A narrow gravel driveway led to a small, brown, nondescript building at the base of the tower. At first glance, Officer Tubbs could see no sign of anything unusual. There was no service truck in sight, and the gate to the ten-foot-tall chain link fence surrounding the tower was latched shut and securely locked.

It wasn’t until he unlocked the gate that he first noticed something odd. A security camera on the right corner of the building was pointing toward the forest. He glanced around and quickly spotted two other cameras, each of them pointing away from the tower building. Clearly, they should have been pointing toward the tower and the door to the building… a door that Officer Tubbs now realized was slightly open. He could barely make out shadows moving around from the small sliver of light that was peeking ominously through the opening, suggesting someone was inside. Suddenly, the sound of his footsteps on the gravel seemed to become amplified, and his breathing so loud that for a split second he held his breath. He reached down and turned the volume of his radio to silent, and slowly began making his way to the open door.

Continue reading “Adventures of ArduinoMan – The Rudis”

Nanowire Batteries Never Need Replacing

In this day and age we’re consistently surrounded with portable electronic devices. In order for them to be called “portable”, they must run on batteries. Most, if not all, use rechargeable batteries. These batteries have a finite lifespan, and will eventually need to be replaced. UCI chemist [Reginald Penner] and doctoral candidate [Mya Le Thai] have been hard at work on making rechargeable batteries that last forever.

Nanowires are great candidates for rechargeable battery technology because the wires, thousands of times thinner than a human hair, are great conductors of electricity. The problem is repeated charging and discharging makes them brittle, which causes them to eventually fail. Typically, the researchers at UCI could get 5000 to 7000 cycles in before they failed. After some trial and error, they found that if they coat a gold nanowire with an acrylic-like gel, they can get up to 200,000 charge/discharge cycles through it before failure.

We’ve seen rechargeable battery hacks before, but making a battery that never needs replacing is sure to get everyone excited.

Symmetry for Beginners – The Rubik’s Cube

Symmetry is everywhere in our natural world. Just take a look at your hands, a butterfly, or a sunflower.  It’s easy to pass off the idea of symmetry and symmetric structures as a simple quirk of existence, and to pay it little mind. If this is your view, I can assure you it will no longer be by the end of this series. If we force ourselves to look beyond the grade school applications of symmetry, we find a world rich in connections via many different types of symmetric identities. One of the most interesting is Gauge Symmetry, which lies at the heart of Quantum Electrodynamics, or QED (we’ll get into this a bit later in the series). Several branches of higher level mathematics study symmetry in detail, allowing a host of sciences, from physics to chemistry,  to view difficult problems and theories from a different perspective.

The subject matter of the ideas explored in symmetry is complicated, and not well known outside of academia and the theoretical sciences. It is the goal of this series of articles to simplify some of the concepts that underpin the study of symmetry, so that the average hacker can gain a basic (and I mean basic) understanding of this fascinating body of knowledge, and put it to use in future projects.  We’ll start things off by taking a look at a machine that has crossed the Hackaday server many times – those nifty Rubik’s Cube solvers. Just how do those things work anyway?

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The Dark Arts: Anonymity

Love him or hate him, Edward Snowden knew a thing or two about anonymity. In June of 2013, he blew the whistle on the NSA’s out-of-control programs that can target virtually anyone plugged into the digital age. The former CIA employee was working as a contractor for the NSA, where he had access to highly classified documents for many of these general populace surveillance programs. He eventually took off to Hong Kong and released the documents to a handful of reporters. One of these documents was a power point presentation of the NSA complaining about how the TAILS operating system was a major thorn in their side. Naturally, Snowden insisted that the reporters and himself only communicate via the TAILS O/S. He used PGP, which is an encryption method with the highly sophisticated title of “Pretty Good Privacy”, and asked not to be quoted at length for fear of identification via stylometry.

In this article, we’re going to go over the basics of anonymity, and introduce you to methods of staying anonymous while online.

Continue reading “The Dark Arts: Anonymity”

Printing Magnetic Fields

We told you about these “printable” magnets a while back. When you have the ability to squeeze many smaller magnets into a tiny spot and adjust their north/south orientations at will, you can not only control the strength of the overall magnetic field, you can construct new and seemingly physics-defying widgets. This article will not focus on the magnets themselves, but instead we’re going to peel away the closed source shroud that hides the inner workings of that nifty little printer of theirs. There has been a lot of talk about these printable magnets, but very little about how they’re made. This changes today. We’ll show you how this magnetic field printer works so you can get busy making your own.


Several years ago, a company called Correlated Magnetic Research introduced to the world the idea of a magnetic field printer with the Mini MagPrinter. It sold for a whopping $45,000, which limited it to businesses and well-funded universities. They eventually changed their company to Polymagnet and now focus on making the magnets themselves. It appears, however, that they’ve refined their printer for a higher resolution. Skip to 2:45 in this video to see the Mini MagPrinter in action. Now skip to 7.25 in this video to see their next generation printer. Now lets figure out how they work.

What We Know

Original Mini MagPrinter

Firstly, you can toss your Kickstarter idea in the recycle bin because they hold several patents for their printer. But that doesn’t mean you can’t make one in your garage or for your hackerspace. Their machine might have cost $45k, but we’d be willing to bet a dozen Raspberry Pi Zeros that you could make one for two orders of magnitude less. But first we need to know how it works. Let’s look at the science first.

The Curie Point

The Curie Point is a temperature where a magnet loses its magnetic field. It is theorized that magnetism arises from the spin and angular momentum of electrons. If you get them lined up correctly, you get a magnet. When you heat the metal past the Curie Point, this alignment gets all messed up and you lose the magnetic properties. And, of course, you can align the atoms back up by introducing the metal to a strong magnetic field.

Halbach Array

Halbach array is created when smaller magnets are arranged so their magnetic fields are focused in a particular direction and cancel out in another direction. The magnets made by the magnetic field printer can be considered Halbach arrays.

How It Works

Everything begins with a blank Neodymium magnet. We’re all familiar with CNC technology, so we’ll focus on the magnetic field printing head itself. Reading through the comments of the original article, many believe that it uses a combination of heating to exceed the Curie Point and a high strength electromagnet to “write” the magnetic field into the blank. However, after looking closely at this patent, it appears this is not the case. There is no heating involved. The printer head consists of “an inductor coil having multiple layers and a hole extending through the multiple layers” and works by “emitting from the inductor coil a magnetic field that magnetizes an area on a surface of the magnetizable material…”. In short, it’s just a strong, local magnetic field.

Left – Magnetic field print head. Right – Drawing of internal structure of the print head.

Make Your Own

Now that you have a basic idea of how to print magnetic fields, you can start working on one of your own design. You already know how to make 3d printers and laser cutters. Just take one of these designs and replace the head with your custom-built magnetic printer head, whip up some software and bring this technology into the open source community. Blank Neodymium magnets and magnetic field viewing film are fairly cheap. First one to print the skull and wrenches logo gets a free t-shirt!