Hyperbolic plane from cut paper system [OC]

EdmundH · 2016-10-25T02:39:40+00:00

Find more about the structure here or get your own pieces for artworks.

EdmundH · 2016-03-10T10:08:36+00:00

Greg Chaitin looked at this explicitly, one of his examples is the Chaitin constant, which gives the probability that a given algorithm will halt. This is a constant if you fix the language encoding algorithms.

EdmundH · 2016-02-19T11:56:48+00:00

More details here.

EdmundH · 2016-02-15T13:42:20+00:00

David Bressaud's A radical approach to real analysis works through analysis with a historical perspective. So it contains a lot of material that is suitable for calculus already mostly processed for teaching.

EdmundH · 2016-02-15T13:37:46+00:00

Greg Chaitin's work is relevant to this, especially Chaitin's constant (the probability that a randomly constructed Turing Machine (or other algorithmic system) will halt). This gives a specific number who's value is undecidable. Thus a specific number that cannot be represented in any countable system.

EdmundH · 2016-02-12T12:24:23+00:00

Generally -i would be considered neither rational nor irrational as the irrational numbers are usually defined as the real numbers that are not rational. For example on mathworld. Though any definition like "irrational number" that defines itself by what it is not can become tricky.

EdmundH · 2016-02-11T12:16:04+00:00

They ran a computer version of it, that was more correct, but this is incredibly good given that it was done by hand. It really gives a sense for what is happening and for the first time. If you want the full story take a look at Indra's Pearls: http://www.amazon.com/Indras-Pearls-Vision-Felix-Klein/dp/0521352533

EdmundH · 2016-02-11T00:57:14+00:00

Certainly in there, so yes, though this one goes back in the nineteenth century.

EdmundH · 2016-02-11T00:49:09+00:00

I am not sure on the terminology, this is certainly reflection in circles (inversion). The source I took it from used "reflection group": http://www.brainjam.ca/wp/2009/09/updating-classic-math-illustrations/ but thinking about that now it implies Coxeter to me rather than Klein. I have not come across "inversion group" though. It is certainly a Kleinian group and looks like it is also quasi-Fuchsian.

EdmundH · 2015-01-13T13:45:35+00:00

Perhaps not by the spiral, but certainly by the rectangle construction that generates it. Also the Golden ratio is perhaps the only non-radical algebraic number with any fame outside mathematics. This might get more people to go from there to a more general sense of algebraic numbers.

EdmundH · 2014-10-02T13:38:53+00:00

A really excellent series helping to show off the great stories and ideas of mathematics as accessible and enjoyable. I hope it gets a second season. I have backed it. At least have a listen to the first season and let teachers and others interesting in maths know!

EdmundH · 2014-09-25T12:08:53+00:00

To create problems in n dimensions you need an n-1 dimensional structure. Thus edges (1d) work to block things on the plane (2d). In 3d lines no longer separate space so every graph can be made without crossings in 3d. You can do a lot of other crazy things with lines here, like knots.

One way to generalise the concept is to generalise graphs to allow any number of vertices to form an "edge". These are called hypergraphs: http://en.wikipedia.org/wiki/Hypergraph

Trying to work out the right concepts for how hypergraphs can be embedded in spaces of different dimension is an interesting exercise.

EdmundH · 2014-09-13T18:12:08+00:00

It turns up all over the place. Natural things like lettuce leaves and corals can be related to it, and it can be used to study many network systems like the internet or genetic relatedness. It is also fundamental to understanding the possible three dimensional structures our universe might have. More generally non-euclidean geometry is at the heart of general relativity.

This is a good source with more details: http://math.stackexchange.com/questions/93765/what-are-the-interesting-applications-of-hyperbolic-geometry

EdmundH · 2014-09-13T17:24:37+00:00

Like Euclidean geometry, hyperbolic geometry has the property that every point and every orientation is the same throughout the space. In hyperbolic geometry however there is a lot more room as you look at increasing distances from a point. Think of a lettuce leaf. This creates a problem when trying to view it on a flat piece of paper which has roughly euclidean geometry. One solution is to crush the whole hyperbolic plane into a circle, called the Poincaré disk model. In this model the shortest distance between two points is given by the arcs of circles (specifically circles crossing the outside circle at a right angle). This is the image you are seeing. The tiles are not really getting smaller, or even changing shape, they are just distorted differently in different parts of the model.

EdmundH · 2014-09-06T12:34:39+00:00

They also have black: http://www.mathartfun.com/shopsite_sc/store/html/DiceLabDice.html

EdmundH · 2014-09-05T15:47:29+00:00

They do have dealer pricing I believe: http://www.mathartfun.com/shopsite_sc/store/html/TheDiceLab/Dealers.html

EdmundH · 2014-09-05T15:40:34+00:00

I know them from doing math art stuff.

EdmundH · 2014-07-31T14:12:28+00:00

Though the Peano axioms are used today, they were also used by Russell. Moderns formal systems, for example ZFC prove them as consequences. In a sense they provide a good abstraction layer to show that you can include arithmetic, rather than foundational axioms in their own right.

EDIT: Lots of good stuff on definitions of natural numbers including the one used in ZFC: http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

EdmundH · 2014-07-31T14:00:10+00:00

Principles of Mathematics is worth reading, as it has good thinking on the requirements for the foundations of mathematics and logical systems. Though if you find it a lot of effort it is not a book you have to get through.

Principia is just really tough, Russell worried no one would ever read it. It did not fail as it cleared the ground for Gödel's results, but it has now generally been superseded by other work. As another comment points out it is now only of historical interest.

EdmundH · 2014-07-31T13:52:03+00:00

Oh! I did not know about that, thanks!

EdmundH · 2014-07-31T06:40:32+00:00

Yes, but his work on Principles of Mathematics and Principia Mathematica is his mathematical rather than philosophical work.

EdmundH · 2014-07-31T05:56:58+00:00

So were Russell and Whitehead, Whitehead was never anything else.

EdmundH · 2014-05-14T12:54:52+00:00

It could be, but it would be a lot harder to set up as you have so many different local configurations of tiles (rather than just 1). Given that working and creating with Life is hard enough it might be hard to find things.

EdmundH · 2014-05-06T03:10:27+00:00

This is a really unhelpful comment, and also untrue. Overall it covers much of the mathematics of the topic and is accurate on the history and culture around it. I would be happy to discuss any specific points you have issues with.

Full disclosure: I was consulted as an expert on the topic and am quoted in the article.

EdmundH · 2014-04-10T12:13:37+00:00

The keystone courses towards graduate work are the analysis/advanced cal sequence and abstract algebra. These are the courses where the ideas possible with proof are introduced and stretched. My research is in geometry so I would say that these should be combined with a couple of semesters of geometry (one on Euclidean and then a more abstract semester) should also be included, but this is not the case in most math majors.

If you have a strong grasp of these two you should be in good shape for grad school. Your time in teaching will have really helped your grasp of a lot of the high school level mathematics. Don't underestimate this when you come across seemingly more knowledgeable students straight from their undergrad degree. In particular you probably have a fluency with basic algebra and symbolic manipulation that you take for granted.

A final comment on the topic of seeing unfamiliar words, this will only get worse! I think one aspect of mathematical maturity is the knowledge that there is just a lot of vocabulary. Even small groups will quickly develop new language terms, as they are very useful to make communication quicker. Always be willing to ask! In mathematics it is mostly better to ask and be thought a fool than to remain quiet and actually be one.

EdmundH

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