Equidistribution of ornaments across your Christmas tree's lateral surface area

Introduction

I have been learning a lot of math lately. I now know more math than I ever fathomed I would, more than I knew there was out there to learn. This is great, and has given me a (much improved) altered view of the world. I now see and approach everything through mathematical lenses. Even the most mundane or routine day-to-day tasks can benefit from this outlook. For example: Decorating the Christmas tree.

The problem

It seems like just about everybody is concerned with ensuring the ornaments on the Christmas tree are placed evenly, and there is no bunching or grouping of ornaments on one side of the tree versus the other. This is usually achieved by eyeballing it; a totally subjective experience. Well, this concern came up again this year as it often does, except this time, I had my mathematical lenses...

Lets say you want to achieve near-perfect spacing of the ornaments on your Christmas tree, and all you have is a ruler, or maybe a carpenters square (apparently Sheldon does this on The Big Bang Theory, as my friend explained to me while I was writing this).

You can figure out the number of ornaments you have, since you can count them, so how do you know how far apart to space each ornament?

The strategy

Thinking about it, it seems the simplest way would be to divide the number of ornaments you have over the area of the ornament-hanging real estate, and we would get a result in terms of 1 ornament per X feet^2. I think we can work with that.

Surface area of the ornament-space of a standard tree

The formula for calculating the surface areas is well know for a wide variety of shapes. Given a shape, we can just google for the formula. Take a gander at your Christmas tree, what shape does it remind you of? Well, its conical. We can generalize a Christmas tree to the shape of a cone:

Now when I first did this, the result I got just didn't seem right, I was getting a result of ~91.5 square feet. There was no way my tree was exposing 91 square feet of ornament space! Well, the surface area of a cone includes the base, of course, and not too many people hang Christmas ornaments underneath of the tree. Whoops!

So after some googling, I found out that the surface area of the slanted part of the cone is called the Lateral Surface Area of a cone.

The formula

So all that is required is for you measure the radius of your Christmas tree, the height from the base, plug those numbers into the above formula, and that returns the surface area in square units. The units are whatever units you put into the formula, typically inches.

Note: The radius (r) is measured from the trunk out to the end of the branches at the base. The height (h) is measured from the base to the top. Do not measure from the floor to the tip, as this will include the length of the trunk, and since the surface area per unit of height is greatest at the bottom, this will throw your surface area calculation off by a lot.

Then, dividing the number of ornaments you have by the surface area gives you the number or ornaments per square unit. You can then at this point, cut out a square piece of paper to match the area that one ornament should occupy. Do this four times to make four squares. If you have thicker paper or card stock, use that. Tape the four squares together, two on top, two on bottom, offset by half a square, like seen here:

This will be your template. This gives you a frame of reference, so you can align ornaments in reference to ones already hung. You can even draw and cut out circles in the center to obviate any need for guesswork or eye-balling their placement.

Javascript inside svg?

Readers might or might not be aware that an SVG file is really just an XML document. And yes, as you suspected, it allows javascript and viewers are expected to support it, as if that was a sane thing to do. I'm sure this kind of thing is just about as smart as it sounds, in fact I believe I first discovered this foolhardy feature in a forum post displaying some obfuscated javascript inside an svg tag and it was in reference to a XSS attack that was leveraging Facebook.

I had to run a quick test, to see if it was true. I typed the following text to a file with the extension .svg, and the file opened up in internet explorer:

<svg version="1.1" xmlns="http://www.w3.org/2000/svg">
  <script type="text/javascript">alert('Hax!');</script>
</svg>

Now, I use chrome and keep my internet explorer locked down. I don't know what comes up on other people's machines, but at least mine prompted me, asking if I wanted to allow blocked ActiveX controls. However, chrome runs it no problem and without bothering to prompt me. Joy.

I don't know about you, but I don't want my images to be able to prompt me. It is not hard to imagine a scenario where an svg being served up by a server could contain javascript attempting to access the user's cookies for that server.

While the black hats of the world are busy thinking of new (ab)uses of this technology, it is interesting to consider the creative aspects. If an SVG contains only a JavaScript loop that draws the image, can the image be said to draw itself? Clearly, not in the most literal sense, as rather the image is interpreted, and it is the interpreter that does the drawing. Yet in some sense it does draw itself. At any rate, it can certainly create some intense images in only 1 kB or so.

Naturally, my mind jumped to the idea of making a prime number sieve, as a way to make a complex drawing with only a single loop. After a little bit of playing, I had something that looks a lot like Sieves of Chaos. After a little more tweaking, I had something that looks almost identical to Sieves of Chaos.

I encourage you to try it out for yourself; just copy and paste the below code to Notepad++, and save with a .SVG extension. You will notice the image is 8096 units wide. If your machine can handle it, there is no reason the width couldn't be extended. For some machines, this may already be a crippling number. The effect is very nice, and I even created a spanning desktop background out of it.

<svg version="1.1" xmlns="http://www.w3.org/2000/svg">
 <script type="text/javascript"><![CDATA[
  var width=8096;
  var height=900;
  var ns="http://www.w3.org/2000/svg";
  var svg = document.getElementsByTagName('svg')[0];
  svg.setAttribute("width", width);
  svg.setAttribute("height", height);
  
  var rect = document.createElementNS(ns, 'rect');
  rect.setAttribute("height", height);
  rect.setAttribute("width", width);
  rect.style.fill = "black";
  svg.appendChild(rect);

  for (var b = 2; b < width; b += 1)
  {
   for (var a = 1; a < width; a += b*2)
   {
    var cir = document.createElementNS(ns, 'circle');
    cir.setAttribute("shape-rendering", "geometricPrecision");
    cir.setAttribute("stroke-opacity", "0.07");
    cir.setAttribute("r",  b);
    cir.setAttribute("cx", a+b);
    cir.setAttribute("cy", height/2);
    cir.style.stroke = "red";
    cir.style.strokeWidth = "1";
    cir.style.fill = "none";
    svg.appendChild(cir);
   }
  }
 ]]></script> 
</svg>

Blogger is wise, and does not accept .svg files. There is hope. Here is the .png image that I am using for my background (below). Feel free to use that or render your own with the above code. Be warned, the below png file is 3 MB.

EntropyGlance

Entropy at a glance

In a hurry? Skip straight to the C# source code - EntropyGlance; Entropy at a glance - A C# WinForms project - https://github.com/AdamWhiteHat/EntropyGlance



So I wrote an file entropy analysis tool for my friend, who works in infosec. Here it is, hands-down the coolest feature this tool offers is a System.Windows.Forms.DataVisualization.Charting visualization that graphs how the entropy changes across a whole file:

This application provides both Shannon (data) entropy and entropy as a compression ratio.
Get a more intuitive feel for the overall entropy at a glance with by visualizing both measures of entropy as a percentage of a progress bar, instead of just numbers.

   However, for those who love numbers, standard measures of entropy are also given as well. Information entropy is expressed both as the quantity of bits/byte (on a range from 0 to 8), and as the 'normalized' value (range 0 to 1). High entropy means it the data is random-looking, like encrypted or compressed information.
   The Shannon 'specific' entropy calculation makes no assumptions about the type of message it is measuring. What this means is that while a message consisting of only 2 symbols will get a very low entropy score of 0.9/8, a message of 52 symbols (the alphabet, as lower case first, then upper) repeated in the same sequence one hundred times would be yield a higher-than-average score of 6/8.
   This is precisely why I included a compression ratio as a ranking of entropy that is much closer to notion of entropy that takes into account repeated patterns or predictable sequences, in the sense of Shannon's source coding theorem.



Dive deep into the symbol distribution and analysis. This screen gives you the per-symbol entropy value and the ability to sort by rank, symbol, ASCII value, count, entropy, and hex value:

As always, the C# source code is being provided, hosted on my GitHub:
EntropyGlance; Entropy at a glance - A C# WinForms project - https://github.com/AdamWhiteHat/EntropyGlance

RC4 stream cipher variants and visualization of table permutation state

Here, I present some work I have been doing on two RC4 stream cipher variants. The first variant, as seen below, I wrote to help me visualize and understand what the RC4 tables was doing, and help me understand its properties.

Identity Permutation

The class that contains it is called SimpleTable and is exactly that; The simplest R4C implementation possible. It is notable in the fact that it does not use key scheduling at all, and its starting state is that of the Identity Permutation. The identity permutation is where the value at index zero equals zero, the value at index one is one, and so on. An easy way to remember what the Identity Permutation is, just recall the notion of a Multiplicative Identity (which is 1), where by multiplying a number N by the Multiplicative Identity gives you back the value of N, also known as the identity. Similarly, the Identity Permutation of an array A just gives you A. This is the trivial permutation. That is, there is no permuting of the array at all!

Anyways, this is done to see the perfectly ordered state, and how each round effects that state. In this way, we can visually check for the avalanche effect. In order to visualize the table, i just assign each value 0 to 255 a different shade of grey (I also have a rainbow-colored option that might be easier to tell apart similar values). At each step I create a Bitmap by looping through the table. Below, you can find an animated GIF of the first 100 steps of this cipher being applied to the identity permutation:


Notice how it takes a while to get going, and the first several values don't move much at all. After 256 steps, or one round, the cursor arrives back at index zero. Because the location of the first several values have not moved much or at all, we can clearly see that a mere 256 steps is insufficient at permuting the state enough to avoid leaking the first part of your key. Therefore it it is vital to permute the table for several rounds (256 steps per round) before you start using the stream.

Wired Equivalent Protocol

As some of you may know, WEP used RC4 with a weak key schedule. The key is spread out over 256 bytes using the following approach:

j = (j + table[i] + key[i mod keylength]) % 256;
SwapValues(table[i], table[j]);

and then it began streaming bytes from the table. Typically a nonce is concatenated to the key. Every time the table is set up and/or the nonce changes, some information about the key is leaked. Obviously a more secure procedure would use a hash of the key and the nonce, instead of the plain-text key, and to toss away the first 1024 bytes or so.

Cycle Length

Because each step in an RC4 cipher is a permutation, there is a limit to the number of unique bytes that can be produced before it begins repeating. This is called the permutation cycle. The length of the permutation cycle depends on the exact starting state, but we can get an upper bound.

Since there are 256 elements in the array, and two indices into the array (i and j), there is a maximum of
256! * 256^2 = 5.62 * 10^512 = 2^1700
possible states. That's 4.6 * 10^488 yottabytes!

This is the maximum possible states, however, and other starting states could have less. If the RC4 algorithm performed as a random permutation (which it does not, it performs worse), the cycle length would be half of the theoretical maximum above. Luckily the number above is so vast, that even some faction of it is still so many bytes that all of humanity has never and likely will never have that much total storage.

One thing to watch out for, however is something called Finney States. If an RC4 is started in one of these Finney States, the length of the cycle is much, much reduced. The chance of randomly generating one of these starting states, however, is VERY, very low.

Strengthening RC4

As stated, and visualized, above, it is vital to permute the table for several rounds (at 256 steps per round) after the key schedule, discarding the bytes, before you start using the stream. Also, it would be foolish to use the actual bytes of the key for permuting the starting state. It would be instead better to use a hash of the key + nonce or a key derivation function from the key instead the actual value of the key itself.

Another idea is, after shuffling the table enough rounds to hide the key, scramble the table an additional number of rounds, that value being some function of the key. This increases the possible starting states by whatever your range is.

In the classic RC4, each step would return one byte. The number of steps taken before returning each byte is configurable in my implementation.

Memory hardening

Check out the experimental branch for a memory hardened version. It stores the key class in memory, with the key XORed with a one-time pad, and then is protected in memory from access with the System.Security.Cryptography.ProtectedMemory class.

Other uses

The pseudo-random byte stream from the RC4 table is deterministic. Therefore if two remote computers with a shared secret, both computers can independently set up an RC4 table with exactly the same starting state and will get the same sequence of bytes which would be difficult to guess, given just the stream of bytes. If the plain text is XORed by the pseudorandom byte stream, then it can be decrypted by XORing it by the same byte steam.

The project includes 2 variants: 1) A simple table with a method to visualize the permutation state of the table and the avalanche effect as a bitmap 2) A more serious attempt at a secure implementation.


NOTE: THIS HAS NOT BEEN CRYPTO-ANALYZED AND PROBABLY NOT ACTUALLY SECURE, SO DO NOT TRUST IT!

Screenshots

Source code

Here is the GitHub page to the project (master branch).
Or just directly download the Zip file (experimental branch).

Lorenz Chaos Attractor

This project was inspired by one of Daniel Shiffman's 10 minute coding challenge YouTube videos, The Lorenz Attractor in Processing.

        dX = ((A * y)  -  (A * x)) * time;
        dY = ((B * x) -y -(x * z)) * time;
        dZ = ((x * y)  -  (c * z)) * time;

So it turns out this it not too terribly exciting. While its true that adjusting the starting values by a small amount change the behavior, if you go much outside the values its currently set for, you will end up with a pattern that quickly degenerates to a single, boring point. Personally, I was hoping for a more chaotic system. You might notice I am not using the 3rd point. I have yet to find a 3D drawing library that I like, though I need one for visualizing other projects. Anyways, since this was an experiment, I did the pragmatic thing and just made it 2D since I already knew how to do that.

Here is the result:

GitHub project

It wanted to draw the pattern very small, so I had to scale up the image by multiplying each number by some scale number.

One possibly useful idea is to use the cosine of the tangent of each number. This has the effect of canceling out the spiral and spreading the numbers out over a field. If you use just the tangent, you get a gradient from the top left corner. Perhaps you could use this as a pseudo-random noise source.

public static void TanCos(Lorenz system)
{
        system.x = 16 * (decimal)Math.Tan(Math.Cos((double)system.x));
        system.y = 16 * (decimal)Math.Tan(Math.Cos((double)system.y));
}

public static void Tan(Lorenz system)
{
        system.x = 6 * (decimal)Math.Tan((double)system.x);
        system.y = 6 * (decimal)Math.Tan((double)system.y);
}

True hardware random number generator with the Raspberry PI

So, I have been getting into cryptology lately, (For my most recent projects that I may or may not have blogged about at this point, see Bloom Filter and RC4Ever on GitHub).

The other day, I had a need for a TRUE random number generator, so I was searching the web for a hardware random number generator, when I found some very pleasant information: I already own one!

As it turns out, the Raspberry Pi (A/A+/B/B+/and 2) includes a hardware based random number generator, and according to many sources, its a very good source of truly random bytes. Yay!


To get this working on your own Pi, its a breeze:
1) Install the RasPi's random number generator tools: sudo apt-get install rng-tools.
2) Add to the boot process file (/etc/modules.conf) the command to run the hwrng module: bcm2708-rng.
3) Reboot the Pi.

Now, /dev/hwrng is available for reading. Its treated like a device, and you can use the dd command to copy bytes from the device stream to a file (like examples you might have seen doing the same from /dev/random).

NOTE: /dev/hwrng is accessible by the user root only.

But we can change that! The following command gives the user level read access: sudo chmod a+r /dev/hwrng

NOTE: This setting gets reset upon every reboot.

Again, we can change that: Add the following line to /etc/rc.local file, just above the exit 0 line:
chmod a+r /dev/hwrng

And its just that easy!

Now, say if you want to generate 1 megabyte worth of random bytes to a file in /tmp, simply enter the following command into a terminal:
dd if=/dev/hwrng of=hwrng-test-data.bin bs=1024 count=1024

The bs argument specifies the size to buffer before writing to disk. You probably want to leave that at or around 1024. Its the count argument that specifies the size, or amount, of data you want to copy from /dev/hwrng, in Kilobytes. So 1024 == 1 MB, where as 1 == 1KB.

Now, its time for Step 4) Create a C# helper library to simplify the retrieval of random bytes from /dev/hwrng.


So there is two ways to approach this. One is to make a C++ library that makes native calls and then write a .NET interop library to wrap that Or, if you are like me, a little lazy, and find that ever since transitioning to C# you find it difficult to write anything in C or C++ that compiles, you may opt to just issue the above commands to the shell and just read in the resulting file from the tmp directory.

As hackey as this second option might seem, it works remarkably well and I have written a GitHub project doing just that. It consists of a library to return random bytes, and a console executable exercising said library to get random bytes. Links below!


- The PiRngWrapper GitHub Project
- The PiRngWrapperLibrary.cs wrapper code file

Bloom Filter - A novel, space efficient data structure like a hash-table for billions of values.

Introduction

A bloom filter is a truly novel data structure. Similar to a hash table, it can tell you if you've hashed a particular value previously. You can add many, many more values to a bloom filter than you can to a hash table, does not degrade performance as the number of values in the set grows large, and requires only a fraction of the space of a hash table to store it!

This is not just an academic exercise, or something that only works in theory or in special cases. Indeed, companies like google use bloom filters to quickly determine if it has never seen that value before, thus avoiding a more costly lookup against a database every time the bloomfilter returns false.

Probabilistic

First off, its important to understand that a bloom filter is NOT a hash table, it operates in an entirely different way. A bloom-filter is what is known as a probabilistic data structure. What this means is, that it can tell you to within a certain probability, if an element exists in a set. In other words, false positive matches ARE possible, but false negative matches ARE NOT possible. For example, if you check a bloom filter for the existence of a value, and it returns false, you can know with 100% certainty that the bloom filter does not contain that value value before. However, if you test a value against the filter and it returns true, there is a small probability that it has in fact not seen that value before, but is returning a false positive. How big of a probability? Here's the beauty: It can be as small as you want it to be. It depends on a few factors, including the size of the filter, how full it is, and how many bits you use to store each value in the filter.

In a HashTable class, each item is stored as a key value pair, so the size of your object plus a 32 bit integer. Contrast that to a bloom filter, which stores only about 3-7 bits per value hashed. Also, my implementation applies compression when saving the filter to disk, providing even more space savings. A bloom filter with 160,000 values hashed and a 1% collision probability results in a filter that is 235KB uncompressed, and a whopping 54KB when compressed! Remember the filter is an array of bits. The entropy of the array is going to be at its greatest, and thus the compression ratio lowest, when exactly 1/2 of the bits are flipped, or the filter is half-way 'full'. This has the unusual property of getting smaller as you add more hashes to the filter. Actually, this is misleading--the actual filter itself never changes size, its only the compressed version that varies in size.

To handle the compression I just used the System.IO.Compression.DeflateStream class. An important note about working with this class: build an array of bytes and send your entire file in one go. In this way it will compress the whole file as one chunk. If you sent data to this stream piecemeal, it will compress each piece separately and you will get a poor compression ratio.

How it works

So how does this all work? The filter part of a bloom filter is just a large array of bits. You also require several different hash functions that each return a unique result for the same input value. When you add a value to the filter, the value is sent to about 3-7 different hash functions. Each hash function will return a value that is between 0 and the number of bits in the filter. Each value is used as an index to access and element on the array of bits that is your filter. When hashing a value, you just set the bit at each index location in the array to 1. Then testing for the presence of a value in the filter, you pass the value to the hash functions the same way as above, then visit each index, checking to see if any of them are 0. If even one bit at one of those index positions are zero, it means the filter has never seen that value before, because it would have set all those bits to 1. If all the bits at the index locations are 1, then it is likely that the filter has seen that value before. However,there is a chance that it is a false positive, because it could be that that value's different hashes all mapped to bits from other values. As the filter becomes more full, more bits are set to 1, and so the odds of a false positive go up. To build your filter by supplying the estimated number of values you think you are likely to store in the filter, and don't go above a certain ratio of 1 bits to 0 bits. If you were to let your filter hash so many values that every bit got set to 1, then the probability of receiving a false positive for a random value becomes 100%.

Solving the many hash problem

As I mentioned before, this requires several different hash functions that each return a unique result for the same input value. Although I said 3-7 hash functions, you might require 14 or more, when working with filters that can handle large number of hashes or a low false positive likelihood or both.

Instead of writing a bunch of separate hash algorithms, I implemented a stream cipher where in I just scramble the cipher table by a number of rounds that is unique to that input. Then, I can return as many indices as the filter is configured for. This sets up the table once per value. It needs to reset the table or else the indices that we mark will depend on every value that came before it, and in that particular order. Currently the bottle-neck is how many times it has the scramble the table for each value. If you need to hash really long values, you'll want to lower the number of rounds it scrambles the table.

Variations

In this implementation, the bloom-filter size is set once you create it, meaning that it cannot grow bigger if it gets too full, nor can you resize this bloom filter to become smaller if you sized it too big. Because multiple values could rely on the same bit, this implementation does not support removal of items, because to do so would cause several values to begin reporting false negatives.

In order to make a bloom filter that supports deletion, use a number like a byte instead of bits in your filter, and each time you visit an index in the filter while adding values, increment the number you find there. Then, to delete a value, visit each index as you did before, but decrement the number there. This way, if two values map to the same index, that information is tracked by incrementing the value. This is what is known as a Counting Bloom Filter.

There are other variants of bloom filters out there, including bloom filters that can grow in size if it gets too full, but such a thing is beyond the scope of my needs. In essence, when the filter gets too full, you create another separate filter, and add new values by first checking the first filter to see if it exists, and if not, adding the value to the second filter. Checking for the presence of a value requires checking both (and other) filters. For information on scalable bloom filters, please see this whitepaper.

The code

My C# Bloom Filter project on GitHub
Or download zip here.

The biggest problem with Mathematics today

I start this time with a little disclaimer: This is, after all, a blog, and most blogs still fit the original definition, which is a public forum used for a person to state or explain their beliefs and/or feelings. So it is without further ado, that I proceed, unabashed:

The biggest problem with Mathematics today, particularly around people approaching, or new to the field, is the conventions of naming things.

Historically, and seemingly by convention, mathematical concepts are named after the first person to define or make serious contributions to that field. This is what is known as an eponym.

This is terrible practice, even tear-able! I am bitter about the amount of time I waste trying to find the 'name' of the mathmatical concept I wish to express or research, or when I have to derail my research to go define a term or concept I don't recognize, only to find out that I am already know what it was.

If people would just name stuff after what it actually fucking does, instead of some person's last name, we would all be a lot better off (except for maybe the aforementioned person).

In software development, we have this concept known as refactoring. This term includes fundamental re-structuring of the code, but also can be as simple as a bunch of renaming of everything to fit a more consistent, or holistic, view or concept. A similar thing needs to happen to the field of mathematics, and sooner rather than later!

No more eponyms in Mathematics!

Trick: How to mentally convert and calculate rate of pay

I have been real busy lately, so I will share only a short tip this week. However, I have some cool new projects/concepts I have been working on, such as a term rewriting system, so keep checking back.

I have been interviewing for a new job, its time to move up. Often I am quoted an annual salary, and I want to see how that compares with hourly wage. I do this calculation in my head, on the spot, and so can you. This technique leverages Estimation/Approximation.

Since 40hrs/week times 52wks/year = 40 * 52 = 2080 full-time work hours in a year.
We can use approximation by multiplying or dividing by 2000. Since we know that 2 * 1000 = 2000, multiplying/diving by 2000 is trivial. Remember, to multiply or divide by any power of ten, now matter how great, just count the zeros and just shift the decimal place once to the right, or towards a smaller quantity, that number of times. e.g. 43.50 * 1000 = 43,500





'So how large is the error from estimating?', One might ponder... Well ponder no more! Hark:

Estimation Error - From hourly to yearly--

$12/HR @ 40HR/WK
 
  $24,000/YR - Estimated
  $24,960/YR - Actual
  -------
     -$960  - Difference

$25/HR @ 40HR/WK
 
  $50,000/YR - Estimated
  $52,000/YR - Actual
  -------
   $2,000/YR - Difference

$50/HR @ 40HR/WK
 
 $100,000/YR - Estimated
 $104,000/YR - Actual
  --------
   $4,000  - Difference
 
 
Estimation Error - From yearly to hourly--
 
 $25K/YR @ 40HR/WK
  
   $12.50/HR - Estimated
   $12.02/HR - Actual
   ------
  -$0.48/HR  - Difference
 
 $50K/YR @ 40HR/WK
  
   $25.00/HR - Estimated
   $24.04/HR - Actual
   ------
   -$0.96/HR
     
 $100K/YR @ 40HR/WK
  
   $50.00/HR - Estimated
   $48.07/HR - Actual
   ------
   -$1.92/HR

Infix Notation Parser via Shunting-Yard Algorithm

Infix notation is the typical notation for writing equations in algebra.
An example would be: 7 - (2 * 5)

Parsing such an equation is not a trivial task, but I wanted one for my EquationFinder project, as I wanted to respect order of operations.

Strategies include substitution/replacement algorithms, recursion to parse into a tree and then tree traversal, or converting the infix notation to reverse polish notation (RPN), also known as post-fix notation, then using a stack based postfix notation evaluator. I choose the latter, as such algorithms are well defined in many places on the web.

My code consists of 3 classes, all static:
(Links go to the .cs file on GitHub)

1. InfixNotation - this simply holds a few public variables and calls the public methods on the below two classes.
2. ShuntingYardAlgorithm - this converts an equation in infix notation into postfix notation (aka RPN).
3. PostfixNotation - this evaluates the equation in postfix notation and returns a numerical result value.

In order to implement the shunting-yard algorithm and the postfix evaluator, I simply wrote the steps to the algorithms as written on Wikipedia:
(Links go to the Wikipedia article)
Link to the Shunting-Yard Algorithm to convert Infix notation to Postfix notation.
Link to the Postfix Notation Evaluation Algorithm.


The code for this is pretty extensive, but I will prettify it and present it below. Alternatively, you can view and download the code from the MathNotationConverter project on my GitHub.

InfixNotationParser:

public static class InfixNotation
{
   public static string Numbers = "0123456789";
   public static string Operators = "+-*/^";

   public static bool IsNumeric(string text)
   {
      return string.IsNullOrWhiteSpace(text) ? false : text.All(c => Numbers.Contains(c));
   }

  public static int Evaluate(string infixNotationString)
  {
    string postFixNotationString = ShuntingYardConverter.Convert(infixNotationString);
    int result = PostfixNotation.Evaluate(postFixNotationString);
    return result;
  }
}

ShuntingYardConverter 

(converts an equation from infix notation into postfix notation):

public static class ShuntingYardAlgorithm
{
   private static string AllowedCharacters = InfixNotation.Numbers + InfixNotation.Operators + "()";

   private enum Associativity
   {
      Left, Right
   }
   private static Dictionary<char, int> PrecedenceDictionary = new Dictionary<char, int>()
   {
      {'(', 0}, {')', 0},
      {'+', 1}, {'-', 1},
      {'*', 2}, {'/', 2},
      {'^', 3}
   };
   private static Dictionary<char, Associativity> AssociativityDictionary = new Dictionary<char, Associativity>()
   {
      {'+', Associativity.Left},
      {'-', Associativity.Left},
      {'*', Associativity.Left},
      {'/', Associativity.Left},
      {'^', Associativity.Right}
   };

   private static void AddToOutput(List<char> output, params char[] chars)
   {
      if (chars != null && chars.Length > 0)
      {
         foreach (char c in chars)
         {
            output.Add(c);
         }
         output.Add(' ');
      }
   }
   
   public static string Convert(string infixNotationString)
   {
      if (string.IsNullOrWhiteSpace(infixNotationString))
      {
         throw new ArgumentException("Argument infixNotationString must not be null, empty or whitespace.", "infixNotationString");
      }

      List<char> output = new List<char>();
      Stack<char> operatorStack = new Stack<char>();
      string sanitizedString = new string(infixNotationString.Where(c => AllowedCharacters.Contains(c)).ToArray());

      string number = string.Empty;
      List<string> enumerableInfixTokens = new List<string>();
      foreach (char c in sanitizedString)
      {
         if (InfixNotation.Operators.Contains(c) || "()".Contains(c))
         {
            if (number.Length > 0)
            {
               enumerableInfixTokens.Add(number);
               number = string.Empty;
            }
            enumerableInfixTokens.Add(c.ToString());
         }
         else if (InfixNotation.Numbers.Contains(c))
         {
            number += c.ToString();
         }
         else
         {
            throw new Exception(string.Format("Unexpected character '{0}'.", c));
         }
      }

      if (number.Length > 0)
      {
         enumerableInfixTokens.Add(number);
         number = string.Empty;
      }

      foreach (string token in enumerableInfixTokens)
      {
         if (InfixNotation.IsNumeric(token))
         {
            AddToOutput(output, token.ToArray());
         }
         else if (token.Length == 1)
         {
            char c = token[0];

            if (InfixNotation.Numbers.Contains(c)) // Numbers (operands)
            {
               AddToOutput(output, c);
            }
            else if (InfixNotation.Operators.Contains(c)) // Operators
               if (operatorStack.Count > 0)
               {
                  char o = operatorStack.Peek();
                  if ((AssociativityDictionary[c] == Associativity.Left &&
                     PrecedenceDictionary[c] <= PrecedenceDictionary[o])
                        ||
                     (AssociativityDictionary[c] == Associativity.Right &&
                     PrecedenceDictionary[c] < PrecedenceDictionary[o]))
                  {
                     AddToOutput(output, operatorStack.Pop());
                  }
               }
               operatorStack.Push(c);
            }
            else if (c == '(') // open brace
            {
               operatorStack.Push(c);
            }
            else if (c == ')') // close brace
            {
               bool leftParenthesisFound = false;
               while (operatorStack.Count > 0 )
               {
                  char o = operatorStack.Peek();
                  if (o != '(')
                  {
                     AddToOutput(output, operatorStack.Pop());
                  }
                  else
                  {
                     operatorStack.Pop();
                     leftParenthesisFound = true;
                     break;
                  }
               }

               if (!leftParenthesisFound)
               {
                  throw new FormatException("The algebraic string contains mismatched parentheses (missing a left parenthesis).");
               }
            }
            else // wtf?
            {
               throw new Exception(string.Format("Unrecognized character '{0}'.", c));
            }
         }
         else
         {
            throw new Exception(string.Format("String '{0}' is not numeric and has a length greater than 1.", token));
         }
      } // end foreach

      while (operatorStack.Count > 0)
      {
         char o = operatorStack.Pop();
         if (o == '(')
         {
            throw new FormatException("The algebraic string contains mismatched parentheses (extra left parenthesis).");
         }
         else if (o == ')')
         {
            throw new FormatException("The algebraic string contains mismatched parentheses (extra right parenthesis).");
         }
         else
         {
            AddToOutput(output, o);
         }
      }

      return new string(output.ToArray());
   }
}

PostfixNotation

(evaluates the postfix notation and returns a numerical result):

public static class PostfixNotation
{
   private static string AllowedCharacters = InfixNotation.Numbers + InfixNotation.Operators + " ";

   public static int Evaluate(string postfixNotationString)
   {
      if (string.IsNullOrWhiteSpace(postfixNotationString))
      {
         throw new ArgumentException("Argument postfixNotationString must not be null, empty or whitespace.", "postfixNotationString");
      }

      Stack<string> stack = new Stack<string>();
      string sanitizedString = new string(postfixNotationString.Where(c => AllowedCharacters.Contains(c)).ToArray());
      List<string> enumerablePostfixTokens = sanitizedString.Split(new char[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).ToList();

      foreach (string token in enumerablePostfixTokens)
      {
         if (token.Length > 0)
         {
            if (token.Length > 1)
            {
               if (InfixNotation.IsNumeric(token))
               {
                  stack.Push(token);
               }
               else
               {
                  throw new Exception("Operators and operands must be separated by a space.");
               }
            }
            else
            {
               char tokenChar = token[0];

               if (InfixNotation.Numbers.Contains(tokenChar))
               {
                  stack.Push(tokenChar.ToString());
               }
               else if (InfixNotation.Operators.Contains(tokenChar))
               {
                  if (stack.Count < 2)
                  {
                     throw new FormatException("The algebraic string has not sufficient values in the expression for the number of operators.");
                  }

                  string r = stack.Pop();
                  string l = stack.Pop();

                  int rhs = int.MinValue;
                  int lhs = int.MinValue;

                  bool parseSuccess = int.TryParse(r, out rhs);
                  parseSuccess &= int.TryParse(l, out lhs);
                  parseSuccess &= (rhs != int.MinValue && lhs != int.MinValue);

                  if (!parseSuccess)
                  {
                     throw new Exception("Unable to parse valueStack characters to Int32.");
                  }

                  int value = int.MinValue;
                  if (tokenChar == '+')
                  {
                     value = lhs + rhs;
                  }
                  else if (tokenChar == '-')
                  {
                     value = lhs - rhs;
                  }
                  else if (tokenChar == '*')
                  {
                     value = lhs * rhs;
                  }
                  else if (tokenChar == '/')
                  {
                     value = lhs / rhs;
                  }
                  else if (tokenChar == '^')
                  {
                     value = (int)Math.Pow(lhs, rhs);
                  }

                  if (value != int.MinValue)
                  {
                     stack.Push(value.ToString());
                  }
                  else
                  {
                     throw new Exception("Value never got set.");
                  }
               }
               else
               {
                  throw new Exception(string.Format("Unrecognized character '{0}'.", tokenChar));
               }
            }
         }
         else
         {
            throw new Exception("Token length is less than one.");
         }
      }

      if (stack.Count == 1)
      {
         int result = 0;
         if (!int.TryParse(stack.Pop(), out result))
         {
            throw new Exception("Last value on stack could not be parsed into an integer.");
         }
         else
         {
            return result;
         }
      }
      else
      {
         throw new Exception("The input has too many values for the number of operators.");
      }

   } // method
} // class

Another alternative technique is to using the Shunting-Yard Algorithm to turn infix notation into an abstract syntax tree (Linq.Expressions anyone?). I will likely post this technique later.


Other blog posts by me that are related to this article are the Threaded Equation Finder, a Mixed Radix System Calulator and Drawing Text Along a Bezier Spline.