Archimedean (4,6,12) tiling - Color patterns

Once you programmed the printing of a tiling pattern it is very easy to add colors to the tiles. Some examples.

Archimedean (4,6,12) tiling

This is ( part of ) the Archimedean (4,6,12) tiling.

OU exams more difficult than ever ( ... ) ?

I read a rumor on facebook that the Open University exams were harder than ever. No numbers were shown to substantiate the claim however. You may have been aware that the OU rates have been increased dramatically to align them with the rates of the "Brick Unis" ( = how normal universities are called in OU jargon ). Now one of the commenters said that they are doing the same thing with the exams. Suggesting that until now OU exams were much easier than Brick Uni exams. - To be honest I think it's the other way around. Often homework assignments ( for maths at least ) are part of the grade Brick Uni exams while at the OU you get the lowest grade of homework and exam.

Archimedean (3,4,6,4) tiling - In color

Tilings become appealing when they are colored.

Archimedean (3,4,6,4) tiling

This is ( part of ) the Archimedean (3,4,6,4) tiling.

The (3,4,6,4) means that at every vertex you'll find four tiles with 3,4,6 and 4 vertices respectively. The Archimedean tilings are vertex-uniform.

Video Lectures about Lie Groups

"... A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. ..." ( Wolfram Site )

The (self-) study of Lie Group theory is hard. I found two aids that helped me going somewhat in the subject. A book called Naive Lie Theory by John Stillwell and a series of weblectures by Erik van den Ban of the University of Utrecht. On van den Ban's homepage under Lecture Notes you'll find a Lie Group's prerequisites pdf with explanations of manifolds, tangent maps etc. which appear frequently in texts about Lie Theory.

E8 structure visualized

Visual Mathematics

Years ago when I started studying mathematics besides my job in IT I never considered that I would be able to apply mathematics in my day-to-day job. But I do now, because I work on the development of a drawing program ( a specialized drawing program on Android ). Does that make me happy? Just a bit. Because  applied mathematics can get as dirty as computer programming. Once applied, mathematics has lost most if not all of its beauty ( although nothing of its power ). I love pure mathematics, and even more so visual mathematics. The three pictures below are the result of applying a function to respectively the integers 1, 2 and three. The construction ( Mathematica programming, if you like ) of that function however required understanding of calculus, geometry, linear algebra, group theory and tling theory ( they are all parts of the Archimedean tiling of the whole plane with vertex type 4,8,8 ). My point being: this is the mathematics I like so much.

When I really like a picture I made I add it to 'The Gallery'. I am far from being able to create art with mathematics but there is a point where mathematics becomes art or where art becomes ( laying the groundwork for future ) mathematics. M.C. Escher explored mathematics decades before general theories about the subject were formulated.

If you are interested in Mathematics and Art then I can recommend this book: Connections: The geometric bridge between art and science.

Reading Challenge: The Road to Reality

The Road to Reality. A complete guide to the laws of the universe.

Prior to my more serious interests in Mathematics I challenged myself to read Goedel, Escher, Bach. Not an easy task if you aren't familiar with musical theory at all. It took me about six months to complete the book. Of course I was in doubt if I really understood it all ( I didn't, not sure if I do know, even after M381 Mathematical Logic, but math grows on you ). Anyway, I have been told that The Road to Reality by Roger Penrose is a book of similar importance as GEB. With it's 1100+ pages it's a challenge alright.

I intend to read it while commuting. Who says commute time can't be made productive? Clearly this is a self-motivating post ;-), but an announcement of a log as well. Sort of a summary of the book in parts. More later. I completed one chapter sofar or 25 pages which is less than 3%.

The Big LaTeX discussion

Although it is perfectly acceptable to handwrite your TMAs at the beginning of every course someone starts the "Big Latex Discussion" again.

It's usually a nerd that starts off by listing the irrelevant but impressive specs of his ( not often her ) hardware, or bloats how much computing know how he has ( part-time math students often work in IT ). He then announces that he 'is going to make his TMAs in LaTeX.

Wow. Jaws dropping. Not.

Not often he also provides us with a list of software artefacts required, version numbers included. Nerdy but deep in the autistic spectrum.

I haven't decided how I'll produce my TMAs this time. I see three options. Hand-written, LaTeX typeset or (new!) handwritten but digitally stored in vector graphics format.

LaTeX became a problem ( pita ) for me in the past when I had to add drawings and stuff. Making and including mathematical drawings in LaTeX is not trivial. Besides it is a major distraction, while you should be thinking about math you are figuring out how some latex drawing package works. Serious waste of time.

Since you can draw on a tablet, store it in SVG and thus manipulate it any way you want, doing the formulas in LaTeX and the drawings 'by hand' is probably the route I take.

Since I do Android work Eclipse became more or less my IDE of (only) choice so I'll give TeXlipse with PDF4Eclipse a try.

Tilings resources

I am on a OU course again, ( more about that in posts to follow ). The course site opened today, that really kicks off the course for me.

For now some resources you might find interesting.

A tiling is a covering of the whole plane with non-overlapping tiles, each of which is a topological disc. The classic work on tilings is Tilings and Patterns by Grunbaum, Shephard. A list with other books on the subject can be found here.

M.C. Escher used tilings in his graphics work in an ingenious way. This book contains a nice collection of the work of Escher.

It's interesting to note that in Escher's time (1901-1972) there was hardly any mathematical theory about tilings. The foundational work on tilings was published five years after Escher died. Yet from Escher's work it is clear that he understood tilings, and the related line symmetries ( Frieze Patterns ), lattices and plane symmetries ( Wallpaper Patterns ) as no other.

If you are interested in puzzles at all it's likely that you came across Jaap's Puzzle Page, a vast resource of information regarding puzzles. The website is maintained by Jaap Scherphuis. You'll find his YouTube site here with many puzzle demonstrations.

To my astonishment Scherphuizen also maintains an impressive collection of tilings on a Tilings Page. His tilings demonstrations Java Applet is as impressive which you'll find on the same page.

'Pentagon Flower' ( background tiling is the [4,8,8] Laves tiling )
(c) nilo de roock 2012

The mathematics of beauty.

A mathematics of beauty and art. There is no such thing one would expect. Yet, Owen Jones (1809 - 1874), London architect, wrote the 'General Principles in the Arrangement of Form and Colour, in Architecture and the Decorative Arts ... ' in his book The Grammar of Ornament.

Proposition 4.
True beauty results from that repose which the mind feels when the eye, the intellect and the affections are satisfied from the absence of any want.

from Page PL. I

You'll find the other propositions at the link above in the Digital Copy of the 'The Grammar of Ornament'.

Mathematics: created or evolved?

Some ( if not most ) mathematicians think about mathematics as being 'created'. Paul Erdos said that there must be a book, The Book, containing the most important and most beautiful proofs. Who other than God could have written The Book? Einstein once said that 'God does not play dice' and Kronecker said that "God made the integers; all else is the work of man." There is still some creationism left among mathematicians. If there are several ways to attack a problem there is always one best, and that's the one described in The Book.

Where do our mathematical insights come from? Some say they are a form of Divine Intervention. Or is the entirety of mathematics somewhere hidden in our DNA? Evolution Theory versus Intelligent Design. We, the human species, are that part of the physical universe where it becomes self-aware. We ( the physical universe ) describe the behavior of the universe with the help of mathematics. In that sense mathematics evolved over the centuries. In the concept of a created mathematics, mathematics is there, for us humans, to be discovered. Is mathematics finite, created, timeless, deterministic and waiting for intelligent beings to be discovered, understood and applied? Or is mathematics human, a property of our species, a way to handle the complex reality around us. Unlikely as it seems there may be alternative bodies of mathematics possible.

Clearly, compared to the other sciences, mathematics evolves ( is discovered ) at a slow pace. But that's because mathematical truths are eternal ( they say ). The fundamental theorems are, but is the same true for the tools used by engineers and scientists in their daily work? Determinants have lost some of their appeal, quaternions may face the same destiny. Mathematics does change. But the idea that there is one ( created ) mathematics goes deep. You can't patent a mathematical idea for example.

Mathematicians lack a sense of urgency, except for their own careers or immortality maybe. If mathematics in its entirety could be owned by a company, would that ( have ) speed up the development of mathematics? And thus bring the solution of major planetary problems forward? What if there were competing bodies of mathematics, like operating systems, say Windows versus Linux? I think that that would have been possible in some alternate version of history. There -are- different ways of doing things in mathematics. Take geometry for example. In computer graphics you can choose between ( standard ) linear algebra, add quaternions if you like, or a portion of projective geometry so that you can compose translations and linear transformations with matrices, or: you can use geometric algebra which until not so long ago was either abondoned or exclusively used in quantum physics. My point being we may polish the mathematics until its ready for a place in The Book.