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Sunday, October 27, 2013

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.

Saturday, October 26, 2013

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.

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Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)