## The Poincaré Homology Sphere, and Thinking What I’ll Do Next

Yenergy was good enough to write a comment about this, but people might have missed it. “Dodecahedral construction of the Poincaré homology sphere, part II” is up. The post is an illustration trying to describe several pages of the 1979 paper Eight Faces Of The Poincaré Homology 3-Sphere by R C Kirby and M G Sharlemann.

I admit I have to read it almost the same way a non-mathematician would. My education never took me into topology deep enough to be fluent in the notation or the working assumptions behind the paper. I may work my way farther than a non-mathematician, since I’ve been exposed to some of the symbols. The grammar of the argument is familiar. And many points of it are common to fields I did study. Nevertheless, even if you just skim the text, skipping over anything that seems too hard to follow, and look at the illustrations you’ll get something from it.

Past that, I wanted to thank everyone for seeing me into the start of May. I am figuring to give up the post-a-day schedule. It’s exciting to have three thousand-word and four posts of more variable lengths each week, but I need to relax that schedule some. I am considering, based on the conversation I got into with Elke Stangl about the Yukawa Potential, whether to do a string of essays about closed orbits. That would almost surely involve many more equations than is normal around here. But it could make for a nice change of pace.

## The Liquefaction of Gases – Part II

The CarnotCycle blog has a continuation of last month’s The Liquefaction of Gases, as you might expect, named The Liquefaction of Gases, Part II, and it’s another intriguing piece. The story here is about how the theory of cooling, and of phase changes — under what conditions gases will turn into liquids — was developed. There’s a fair bit of mathematics involved, although most of the important work is in in polynomials. If you remember in algebra (or in pre-algebra) drawing curves for functions that had x3 in them, and in finding how they sometimes had one and sometimes had three real roots, then you’re well on your way to understanding the work which earned Johannes van der Waals the 1910 Nobel Prize in Physics.

Future Nobel Prize winners both. Kamerlingh Onnes and Johannes van der Waals in 1908.

On Friday 10 July 1908, at Leiden in the Netherlands, Kamerlingh Onnes succeeded in liquefying the one remaining gas previously thought to be non-condensable – helium – using a sequential Joule-Thomson cooling technique to drive the temperature down to just 4 degrees above absolute zero. The event brought to a conclusion the race to liquefy the so-called permanent gases, following the revelation that all gases have a critical temperature below which they must be cooled before liquefaction is possible.

This crucial fact was established by Dr. Thomas Andrews, professor of chemistry at Queen’s College Belfast, in his groundbreaking study of the liquefaction of carbon dioxide, “On the Continuity of the Gaseous and Liquid States of Matter”, published in the Philosophical Transactions of the Royal Society of London in 1869.

As described in Part I of…

View original post 2,047 more words

## Reblog: Animated Sieve of Eratosthenes

The Math Less Traveled has a lovely video here, animating the Sieve of Eratosthenes, one of the classic methods of finding all of the prime numbers one wants. I suppose it won’t eliminate writing out and crossing off numbers for extra credit on a math test. I actually remember that being one test I had in, I believe, seventh grade, for reasons that I don’t think I ever got. Possibly the teacher wanted to have an easy time grading, or was giving everyone a break from too much computation by shifting to evaluation of our crossing-out abilities.

Here’s something I made yesterday! (Note, I strongly suggest watching it fullscreen, in HD if you have the bandwidth for it.)

Can you figure out what’s going on? The source code for the animation is here; I was inspired by Jason Davies’ visualization which was in turn inspired by this.

View original post

## Reblog: A quick guide to non-transitive Grime Dice

The bayesianbiologist blog here has an entry just about a special set of dice which allow for an intransitive game. Intransitivity is a neat little property, maybe most familiar from the rock-paper-scissors game, and it’s a property that sneaks into many practical applications, among the interesting ones voting preferences.

A very special package that I am rather excited about arrived in the mail recently. The package contained a set of 6-sided dice. These dice, however, don’t have the standard numbers one to six on their faces. Instead, they have assorted numbers between zero and nine. Here’s the exact configuration:

Aside from maybe making for a more interesting version of snakes and ladders, why the heck am I so excited about these wacky dice? To find out what makes them so interesting, lets start by just rolling one against another and seeing which one rolls the higher number. Simple enough. Lets roll Red against Blue. Until you get your own set, you can roll in silico.

That was fun. We can do it over and over again and we’ll find that Red beats Blue more often than not. So it seems like Red is a pretty good…

View original post 485 more words

## Reblog: A visual proof of the Pythagoras theorem

“Notes On Mathematics” here presents a lovely, visual proof of the Pythagorean Theorem, that bit about the squares of the lengths of two sides of a right triangle adding up to the square of the hypotenuse’s length. I find particularly lovely about this that it can be done without words or explanatory text, which one can’t often get away with.

I think anyone staring at the two pictures would come away convinced of the theorem, but might still ask whether this is actually a logically rigorous proof. One can draw pictures showing all sorts of things which look like they’re so but actually aren’t. I’m thinking here of that puzzle where a grid of 49 squares is cut apart into polygons and rearranged and what do you know but there’s one missing square.

Generally, appeals to “just look at the picture” are a touch suspicious, since that carries along a lot of assumptions about what we see versus what we think, and the eye is pretty much in a continuous state of being fooled about everything we think it sees. I exaggerate but not that much.

But the argument presented here could be written out entirely in prose, without appealing to the physical intuition of what ought to happen if we move triangular blocks around. If you look at that one long enough, or work it out, you might get the same grin of cheerful accomplishment that this pair of pictures provides.

## Reblog: Kant & Leibniz on Space and Implications in Geometry

Mathematicians and philosophers are fairly content to share credit for Rene Descartes, possibly because he was able to provide catchy, easy-to-popularize cornerstones for both fields.

Immanuel Kant, these days at least, is almost exclusively known as a philosopher, and that he was also a mathematician and astronomer is buried in the footnotes. If you stick to math and science popularizations you’ll probably pick up (as I did) that Kant was one of the co-founders of the nebular hypothesis, the basic idea behind our present understanding of how solar systems form, and maybe, if the book has room, that Kant had the insight that knowing gravitation falls off by an inverse-square rule implies that we live in a three-dimensional space.

Frank DeVita here writes some about Kant (and Wilhelm Leibniz)’s model of how we understand space and geometry. It’s not technical in the mathematics sense, although I do appreciate the background in Kant’s philosophy which my Dearly Beloved has given me. In the event I’d like to offer it as a way for mathematically-minded people to understand more of an important thinker they may not have realized was in their field.

Kant’s account of space in the Prolegomena serves as a cornerstone for his thought and comes about in a discussion of the transcendental principles of mathematics that precedes remarks on the possibility of natural science and metaphysics. Kant begins his inquiry concerning the possibility of ‘pure’ mathematics with an appeal to the nature of mathematical knowledge, asserting that it rests upon no empirical basis, and thus is a purely synthetic product of pure reason (§6). He also argues that mathematical knowledge (pure mathematics) has the unique feature of first exhibiting its concepts in a priori intuition which in turn makes judgments in mathematics ‘intuitive’ (§7.281). For Kant, intuition is prior to our sensibility and the activity of reason since the former does not grasp ‘things in themselves,’ but rather only the things that can be perceived by the senses. Thus, what we can perceive is based…

View original post 700 more words

## Reblog: Infinity Day

I hadn’t thought of this as “infinity day” coming up, but, why not? The Sciencelens blog here offers some comfortable familiar comments introducing the modern mathematical construction of infinitely large sets and how to compare sizes of infinitely large sets.

Today is 8 August, the eighth of the eighth, 8-8.  Or, if you turn it on it’s side, a couple of infinity signs stacked on top of each other… Yep, it’s Infinity Day!

The concept of infinity refers to something that is without limits. It has application in various fields such as mathematics, physics, logic and computing. Infinite sets can be either countably infinite (for example the set of integers – you can count the individual numbers, even though they go on forever) or uncountably infinite (e.g. real numbers – there are also infinitely many of them, but you cannot count the individual numbers because they are not discreet entities).

Since infinity is really, really big – incomprehensibly so – it can lead to some amusing paradoxical scenarios; things that don’t make sense, by making complete sense.

An example of this is the Galileo Paradox, which states that

View original post 369 more words

## Reblog: Multivariable Calculus

I believe that I used this textbook, or at least one similar to it, in learning Multivariable Calculus. (Admittedly there are certain similarities among introductory textbooks on this subject which can’t be avoided.) Still, it should be useful for people not sure how you get from coordinate pairs to proving swiftly that the force of gravity on the inside of a solid-shell style Dyson sphere is zero.

## ASCII-Art Math

I had forgotten the challenges of doing more than the most basic mathematics expressions in ASCII art, and had completely forgotten there were tools that tried to make it a bit easier. Enteropia here’s put forth a script which ought to make it a bit easier to go from LaTeX into an ASCII representation, and I have the feeling I’m going to want to find this again later on, so I’d best make some kind of link I can locate when I do.

Many of computer algebra systems date back to the times when GUI machines were rare and expensive, if were present at all. Thus command line was a standard interface. Unfortunately text terminal doesn’t fit very well for displaying mathematical expressions which demand for rich typesetting. To represent math formulas CAS’s resorted to some kind of ASCII art:

` inf 1 ==== / n n \ [ x log (x) > I ---------- dx / ] gamma(n x) ==== / n = 0…`

That’s the output of Maxima. Some of the systems went further and don’t restrict themselves to plain ASCII. Axiom can produce such a nice output:

```                         2
x   - %A  ┌──┐
┌┐  %e     \│%A
│   ──────────── d%A
└┘    tan(%A) + 2
```

Even now most CAS’s retain command line interface, for example Mathematica 8’s terminal session:

`In[14]:= Pi*(a+b^2/(Exp[12]+3/2 ">2)) 2 b Out[14]= (a + -------) Pi 3 12 - + E…`

View original post 427 more words

## Philosophical Origins of Computers

Scott Pellegrino here talks a bit about Boole’s laws, logic, set theory, and is building up into computation and information theory if his writing continues along the line promised here.

As indicated by my last post, I’d really like to tie in philosophical contributions to mathematics to the rise of the computer.  I’d like to jump from Leibniz to Boole, since Boole got the ball rolling to finally bring to fruition what Leibniz first speculated on the possibility.

In graduate school, I came across a series of lectures by a former head of the entire research and development division of IBM, which covered, in surprising level of detail, the philosophical origins of the computer industry. To be honest, it’s the sort of subject that really should be book in length.  But I think it really is a great contemporary example of exactly what philosophy is supposed to be, discovering new methods of analysis that as they develop are spun out of philosophy and are given birth as a new independent (or semi-independent) field their philosophical origins.  Theoretical linguistics is a…

View original post 771 more words

## Reblog: Why are Hot Dogs So Inexpensive?

Here’s a cute little observation about presentation and the power of those volume formulas that kind of get looked at when we’re in the chapter about the volumes of basic solids (circular cylinders, in this case) and not afterwards. It’s also for hot dog fans.

Memorial Day is the unofficial start of grilling season. According to the National Hot Dog and Sausage Council (yes, it exists), Americans will consume about 7 billion hot dogs between now and Labor Day–that’s about 818 per second! The estimated cost of all this is about \$1.7 billion, or less than 25 cents per serving.

A large part of this low price is probably due to the quality of the ingredients, but I want to focus on hot dogs purchased from vendors rather than at supermarkets. Street corner hot dog stands have been cropping up around Durham for the last several weeks, and while I haven’t purchased from any, I get the impression that they are quite inexpensive.

A nice stylized example for us to consider comes from a new book entitled X and the City: Modeling Aspects of Urban Life by John Adam. In chapter 4, “Eating in…

View original post 183 more words

## Where Rap Music and Discrete Mathematics meet.

It’s the weekend; why not spread a bit of mathematics humor, using the basic element of mathematics humor, the Venn diagram?

Interestingly, Venn diagrams are also an overlap between Mathematics Humor and Philosophy Humor.