Monthly Archives: March 2007

The Bad Ballet of Regular Expressions: Pathological Programming in Thutu

For today’s installation of programming insanity, I decided to go with a relative of Thue, which is one of my favorite languages insane languages that I wrote about before. Thue is a language based on a rewriting system specified by a semi-Thue grammar. Todays language is called Thutu (pronounced tutu); it’s a string rewriting system like Thue, only it’s based on regular expressions instead of grammars, and it’s even got regular expression-based control flow mechanisms, making it a sort of hybrid language.

The scary thing about Thutu is that it’s not all that different from a language I’ve wanted to find some time to write myself – except that the one I want to write isn’t intended to be pathological. I’ve never stopped missing Teco for writing text processing programs; and since
my TECO programs tended to be roughly of the form: “Find something matching this pattern, and then take this action”, a regular-expression based language would make a lot of sense.

But anyway, today we’re looking at Thutu, which is a deliberately obscure version of this idea.

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Books for Young Mathgeeks: Rabbits, Rabbits, Everywhere

As promised, another review of a childrens math book. Tonight, my daughter and I read “Rabbits, Rabbits, Everywhere: a Fibonacci Tale” by Ann McCallum.

This time, I have absolutely no complaints. “Rabbits” is a beautifully told story, with delightful artwork, which makes the basic idea of the Fibonacci series understandable to a first grader. It’s a wonderful book, which I recommend absolutely without reservation. If you have a child around 1st grade age, buy this book.

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Basics: Axioms

Today’s basics topic was suggested to me by reading a crackpot rant sent to me by a reader. I’ll deal with said crackpot in a different post when I have time. But in the meantime, let’s take a look at axioms.

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Simplices and Simplicial Complexes

One thing that comes up a lot in homology is the idea of simplices and simplicial complexes. They’re interesting in their own right, and they’re one more thing that we can talk about
that will help make understanding the homology and the homological chain complexes easier when we get to them.

A simplex is a member of an interesting family of filled geometric figures. Basically, a simplex is an N-dimensional analogue of a triangle. So a 1-simplex is a line-segment; a 2-simplex is a triangle; a three simplex is a tetrahedron; a four-simplex is a pentachoron. (That cool image to the right is a projection of a rotating pentachoron from wikipedia.) If the lengths of the sides of the simplex are equal, it’s called a regular simplex.

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Sort-of Updating on George Shollenberger and His Book

So… Remember George Shollenberger? He’s the goofball who wrote a book allegedly containing the First Scientific Proof of God, which I dealt with here
and here.

Well, George has been continuing to babble away. He’s got his blog – and he continues to comment on a nearly daily basis on Amazon.com’s page for his book. In a particularly fascinating update, he speculates about why no one has posted any reviews of his book:

This book has now been on the market for six months. Its rank has oscillated monthly from a low rank to a popular rank. But, it has never been reviewed at Amazon.com or Barnes & Noble. Although I tried to simplify the contents, it seems apparent that the book’s unification of Science and Theology has made the book more difficult to understand. More recently, I have come to the conclusion that the absences of reviews are reflecting the awareness of the reader’s general acceptance of the book and their awareness of the potential major changes that could affect all humans, many business and industry, all governments, and the behavior of political and justice systems.

So, instead of sharing thoughts about the book through open reviews of the book on the Internet, I conclude that the thoughts of many readers about the book are being secured privately so that individuals and organizations can survive the potential changes. Clearly, the scientific proof of God will affect the whole world. This proof can expect to develop to a single worldwide religion, a virtual one-world government, and the worldwide sharing of natural resources. And, how will each individual handle the modern ideas of resurrection and reincarnation?

I also recognize that atheists are currently selling their best selling books For instance, the books by Sam Harris, Richard Dawkins, etc. are currently bestsellers for the book market for people who do not believe in God. For this reason, the publishers of these books do not want to open up a debate on my book with the best selling atheistic books now. So, any debate between atheists and theists is being delayed until this atheistic market is served. But, as you see in my Amazon.com blog of my book, I am preparing for this debate. With this blogging effort, I expect to reduce the market of atheistic books drastically.

Since the way individuals and organizations might handle this book was not predictable by me, I believe that this book will be compared with science books and scriptures for years.

Via some Pandas Thumb folks, I just heard that there’s a free copy of his book available. It’s a discard from the library of congress. You see, the way that the LoC works is that they receive a huge number of books every year. Most publishers send the LoC a copy of any book they publish; and virtually all sleezy self-publishing agencies support their false claim to be legit publishers by talking about how their publications are included in the collection of the LoC. So the LoC gets millions of books every year, most of which are garbage. So they periodically go through the slag heap, junking some of the worthless crap to make room for more of the worthless crap that they’re receiving every day. Part of the way that they recognize the junk is: if the book is never removed from its shelf during the first year at the library, it goes in the trash.

George’s book – the book that that is, according to George, one of the most important things ever published – is being thrown in the trash by the LoC because since they received it, no one has removed it from the shelf. Not once.

The book that’s going to single-handedly diminish the market for “pro-atheist” books, that’s going to trigger the creation of a single world-wide universal religion, that’s going to reinvigorate every major field of study from mathematics to nutrition – has never been looked at since it was received by the Library of Congress, and so they’re throwing it into the trash.

I suppose that George can still hope for it to reinvigorate the science of waste disposal.

Books for Young Mathgeeks: "A Place for Zero"

I recently had the opportunity to get hold of a collection of children’s picture books with math stories. A fellow scienceblogger had been contacted by a publisher, who offered to send review copies of their books to interested SBers.

The publisher turned out to be the folks who publish the “Sir Cumference” books. My wife bought me a copy of the first of that series as a joke, and my daughter immediately appropriated it, and absolutely loved it. So I requested copies of a large bunch of their math adventures, and I’ll be posting reviews as my daughter and I finish them.

The first one that we read together is “A Place for Zero”:, by Angeline Sparagna Lopresti. My daughter picked this one because of the artwork: it’s done in a really attractive style – simple enough to be engaging, and yet complex enough to really be a part of the story.

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Mathematical proof that God Spoke Creation (if you buy his book)

One of my fellow SBers, Kevin over at Dr. Joan Bushwell’s Chimpanzee Refuge wrote a scathing article reviewing an incredibly bad anti-evolution blog. There’s no way that I can compete with Kevin’s writing on the topic – you should really check it out for a great example of just how to take a moronic creationist, and reduce him to a whimpering puddle of protoplasm.

But while looking at the site that Kevin shredded, I can across a link to another really, really bad site, and this one is clearly in my territory:
Science Proves Creation, a site set up by an individual named “Samuel J. Hunt”. Mr. Hunt claims to have developed mathematical proof that the universe was created by Gods words.

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Homotopy

I’ve been working on a couple of articles talking about homology, which is an interesting (but difficult) topic in algebraic topology. While I was writing, I used a metaphor with a technique that’s used in homotopy, and realized that while I’ve referred to it obliquely, I’ve never actually talked about homotopy.

When we talked about homeomorphisms, we talked about how two spaces are homeomorphic (aka topologically equivalent) if and only if one can be continuously deformed into the other – that is, roughly speaking, transformed by bending, twisting, stretching, or squashing, so long as nothing gets torn.

Homotopy is a formal equivalent of homeomorphism for functions between topological spaces, rather than between the spaces themselves. Two continuous functions f and g are homotopic if and only if f can be continuously transformed into g.

The neat thing about the formal definition of homotopy is that it finally gives us a strong formal handle on what this continuous deformation stuff means in strictly formal terms.

So, let’s dive in and hit the formalism.

Suppose we’ve got two topological spaces, S and T, and two continuous functions f,g:ST. A homotopy is a function h which associates every value in the unit interval [0,1] with a function from S to T. So we can treat h as a function from S×[0,1]→T, where ∀x:h(x,0)=f(x) and h(x,1)=g(x). For any given value x, then, h(x,·) is a curve from f(x) to g(x).

Thus – expressed simply, the homotopy is a function that precisely describes the transformation between the two homotopical functions. Homotopy defines an equivalence relation between continuous functions: continuous functions between topological spaces are topologically equivalent if there is a homotopy between them. (This paragraph originally included an extremely confusing typo – in the first sentence, I repeatedly wrote “homology” where I meant “homotopy”. Thanks to commenter elspi for the catch!)

We can also define a type of homotopy equivalence between topological spaces. Suppose again that we have two topological spaces S and T. S and T are homotopically equivalent if there are continuous functions f:ST and g:TS where gºf is homotopic to the identity function for T, 1T, and fºg is homotopic to the identity function for S, 1S. The functions f and g are called homotopy equivalences.

This gives us a nice way of really formalizing the idea of continuous deformation of spaces in homeomorphism – every homeomorphism is also a homotopy equivalence. But it’s not both ways – there are homotopy equivalences that are not homeomorphisms.

The reason why is interesting: if you look at our homotopy definition, the equivalence is based on a continuous deformations – including contraction. So, for example, a ball is not homeomorphic to a point – but it is homotopically equivalent. The contraction all the way from the ball to the point doesn’t violate anything about the homotopical equivalence. In fact, there’s a special name for the set of topological spaces that are homotopically equivalent to a single point: they’re called contractible spaces. (Originally, I erroneously wrote “sphere” instead of “ball” in this paragraph. I can’t even blame it on a typo – I just screwed up. Thanks to commenter John Armstrong for the catch.

Addendum: Commenter elspi mentioned another wonderful example of a homotopy that isn’t a homeomorphism, and I thought it was a good enough example that I wish I’d included it in the original post, so I’m promoting it here. The mobius band is homotopically equivalent to a circle – compact the band down to a line, and the twist “disappears” and you’ve got a circle. But it’s pretty obvious that the mobius is not homeomorphic to a circle!. Thanks again, elspi – great example!

Clear Object-Oriented Programming? Not in Glass

Todays bit of programming insanity is a bit of a novelty: it’s an object-oriented programming language called Glass, with an interpreter available here. So far in all of my Friday Pathological Programming columns, I haven’t written about a single object-oriented language. But Glass is something
special. It’s actually sort of a warped cross between Smalltalk and Forth – two things that should never have gotten together; in the words of the language designer, Gregor Richards, “No other language is implemented like this, because it would be idiotic to do so.”

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Basics: Discrete vs Continuous

One thing that I frequently touch on casually as I’m writing this blog is the distinction between continuous mathematics, and discrete mathematics. As people who’ve been watching some of my mistakes in the topology posts can attest, I’m much more comfortable with discrete math than continuous.

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