Proof: Trivial

#mathematics #books #krantz

Have you ever come across something like: "This course has no prerequisites except a certain level of mathematical maturity." To me this sounds just as awful as: "It is easy to see that..." or "Proof: trivial." What is mathematical maturity anyway? As far as I know, the concept of maturity is only used in relation to mathematics. Doesn't it simply means knowing a LOT about mathematics? Anyway, If I would have to describe my own mathematical development then I would not use the words mature or maturity. I would probably say that "I am learning how little I know and how little I will ever know". It is as though if I set one step towards my goal, my goal takes two steps back. I keep walking and learning but I will clearly never reach that final goal. You are never done in mathematics.

Krantz (left) Lederman (right )

Stephen G. Krantz wrote a book about mathematical maturity called "A Mathematician comes of Age.". Sol Lederman interviewed Krantz in his series 'Wild about Math'. Krantz has a website too and I happened to found that he left a copy of his book on it: here ( PDF ). There may be a zillion reasons why he left it there so let's not speculate about it. Get the book while you still can and read it if you are interested in the concept of mathematical maturity.

Link to A Mathematician Comes of Age on Amazon.

M336 Groups and Geometry

Thirty point Open University courses consist of four blocks, where the sixty pointers have eight. M336 Groups and Geometry is a four block course. Since there are two inter-related but independent tracks it doesn't feel like an ordinary 30 point course, somewhat heavier in fact. The geometry course roughly discusses one topic per block:
Block 1: Frieze groups
Block 2: Tilings
Block 3: 2D-Lattices and wallpaper groups
Block 4: 3D-Lattices.

See this previous post about block1 and frieze groups.

I am almost done with block 2 but I am still struggling with tilings ( TMA02 question 4 ). In the meantime I have coded a nice Mathematica pattern editor, ( which I hope will form the base for a Wallpaper Group editor and generator ).

Click to enlarge.

Programming Mathematica is easy and fast, that is: after you have wrestled yourself through the rather steep learning curve. An advanced topic in Mathematica ( i.e. chapter 15 in the Cookbook ) is the programming with DynamicModules and Manipulate. It turns out that, even as a GUI, Mathematica seems to have no limitations to what is possible. In order to code the pattern editor I had to crash myself through Manipulate for which I received invaluable help from the Mathematica experts community at Mathematica StackExchange. Thank you very much!

What is Sacred Geometry ?

One of my ( many ) reasons to study mathematics is my fascination for the geometry of crop circles. Appreciating the art in the crop circles does not mean that I have an opinion on how, or by who, the circles are made. I found out that the more you get into the subject the harder it is to answer that question. Only people that know very little about them will say without hesitation that all circles are 'hoaxes'. Hoaxes or not, they are absolutely beautiful.

Bert Janssen, a Dutch crop researcher, looked into the geometry of some circles.

(c) Bert Janssen. Click to enlarge.

See also: crop circle geometry, by Bert Janssen

Another crop circle researcher is Lucy Pringle.

The Godfather of crop circle research is Colin Andrews, he started the crop circle community.

To the point: sacred geometry.

It is rather common to use that word sacred geometry in the crop circle community. But it is also used in the so called ancient aliens theory which explains the sudden appearance of high tech architecture like the Pyramids in Giza by visitations of extra-terrestrials in that period.

The word sacred means:
- devoted or dedicated to a deity or to some religious purpose;
- pertaining to or connected with religion.
The word geometry means:
- a branch of mathematics concerned with questions of shape, size, position of figures, etc.

In mathematics the field of geometry has many branches, i.e. Eucledian, non-Euclidean, algebraic, analytical and so on, but there is no mathematical branch of geometry called sacred geometry. My conclusion is therefore that:

Sacred geometry is the geometry that has been used in the construction of religious artifacts.

By coincidence or not, it turns out that these geometries are often related to the golden ratio which is considered to have special aesthetic qualities.

Get Mathematica Cookbook from O'Reilly for $4.95

If you are into Mathematica this might be helpful. SWIM owns a paper copy of the Mathematica Cookbook, but he did not own the e-book version yet. The e-book version of the Cookbook is particularly interesting because it is entirely written in Mathematica and is delivered as such: as a set of Mathematica Notebooks. Anyway, SWIM went to the O'Reilly website to register his Cookbook. This required a valid e-mail address but no proof of ownership of the hard copy Cookbook. After the registration was completed he got the offer to update to the e-book version for only USD4.95. I don't know how long this option will work though, publishers publish to make money and unfortunately not to disseminate Mathematica. Anyway, USD4,95 for the e-book is value for money, you get 15 chapters or 800 pages or literally hundreds of workable recipes.

Happy Easter

Dear Readers,

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Dancing in Euclidean Space

I invite you to watch an excerpt of Seventh Sense by the Anarchy Dance Theatre from Taiwan.



There is a somewhat longer excerpt here:


See also previous posts on mathematics and dance:
- Maths and Ballet (2)
- Maths and Ballet (1)
- Music inspired by mathematics ( has item on choreagrapher Crystal Pite )

If you need a personal Super Computer...

#computing

They say that the smartphones of today are more powerful than the mainframes of the 1970s. That's only partially true of course, in fact it is ridiculous, i don't see anyone managing a corporate database on a phone, not even with 70s software. Phones use all that power they have to facilitate the silly games they have on them.

To the point. Super computing.

The Met Office in the UK still has a warehouse sized computer room packed with computer power. ( See BBC Horizon, about Global Weirding ). They need all that power to run their weather forecasting models which are continuously fed with massive amounts of real-time data from data collection points all over the world.

IBM supercomputer at the UK Met Office

Personal computers, from phones to your home rack may have some power but it could still take -weeks-, if not -months- to do a calculation, or the rendering of an image. As from a certain point you need that upgrade that goes beyond your budget, or simply beyond the available space you have at a certain temperature. And as you might know cooling is still an issue.

If you are into rendering computing intensive 3D images, then vSwarm might be a solution for you. vSwarm is a community based free render farm for your 3D animations and images.

Link:
- vSwarm

What is the difference between digital image processing and computer graphics?

#mathematica

Although digital image processing and computer graphics are similar in the sense that they rely heavily on ( advanced ) mathematics and that they work with computer imagery they are in essence quite different technologies. So, what is the difference between digital image processing and computer graphics?

Computer graphics creates new digital images from geometrical descriptions, such as 3D-object models or (scene-)graphs, developers of computer games, special effect programmers in the movie industry all use ( primarily ) computer graphics.

Example: Computer Graphics - Click to enlarge

Digital image processing strips an image down to its core: an array of numbers and can thus be manipulated in any way possible, i.e. rotating, resizing, morphing, etc.

 Example: Digital Image Processing - Click to enlarge 

In certain applications the two fields meet, for example when 3D models are (re-)created from image data. An interesting project is for example the creation of a 3D model showing the distribution of dark matter in the universe.

Both Digital Image Processing and Computer Graphics are playgrounds to show off your math skills. Most, if not all mathematics software have extensive support for computer graphics as well as digital image processing.

Donald Duck in Mathmagic land

#disney# #donaldduck#

Although I am a math geek and a great fan of Donald Duck I had never seen the movie Donald Duck in Mathmagic land.

I knew vaguely of its existence but somehow I never went out to get it. I think I have seen all other movies with a strong link to mathematics though. See my other posts about that. Sol Lederman's Wild About Math blog had a link to it on YouTube. Here it is.



Enjoy the movie!

Playing mathematics

#openuniversity# #m336#

" If it isn't fun it isn't Mathematics. " If any part of mathematics is causing problems make it so that you can play with it. It is easier said than done but it -is- true: mathematics is something you should =DO=. You can't read a mathematics book as if it was just any book in any field, let alone that you can read a maths book as if it was a novel. The mathematics reading protocol has to be applied and that means: verify everything that the author tries to tell you. Mathematics books are notoriously full of errors so you might even find one. If it isn't the best method than it certainly is the method with the most fun involved: study by playing with mathematics. If our brain is the computer then mathematics is a computer game. Really.

Built from scratch with Mathematica

I decorated the cube above with tilings I had to study for M336. Studying tilings is dry, to say the least, so I became 'actively involved' and created tilings myself using Graphics in Mathematica, I put the code I made for Frieze patterns to work on Tilings.

As far as I can remember I have always loved mathematics, except for the darkest two years of my life. I remember myself sitting in the second class of secondary school looking at something I had never seen before: geometric proofs in Euclidean style, i.e.: no algebra, no numbers, no vectors, nothing. The teacher had this gigantic protractor for making drawings on the blackboard. The government dropped this style of math teaching years before but that teacher insisted on Euclid. Looking back I think he just wasn't flexible enough. My grades for math were terrible and so were those of many others. He destroyed the dreams of many children. Although I catched on later, I don't think that I will ever be ready for Euclid.